SOME ACOUS’FICAL FROPFRFES 0F TREANGLES AND CYMBAIj AND THEIR WHON T0 PERFORMANCE PRACTICES THESIS FDR THE EfiEGREE OF PH. ‘5. 2‘5 MUSIC MICE-HOAN STATE. UNNERSITY JOHN BALDWIN E970 LIBRARY Michigan State University paw-“3 ( This is to certify that the thesis entitled Some Acoustical Properties of Triangles and Cymbals and Their Relation to Perfonmance Practices presented by John Bardo Baldwin has been accepted towards fulfillment of the requirements for ii— degree mills—1°— Major professor £25 / gar/175%? ‘ Date July 21L1970 0-169 —v-—-— 1" v-.- v —" -va‘UII‘UII‘v v-0, ABSTRACT SOME.ACOUSTICAL PROPERTIES OF TRIANGLES AND CYMBALS AND THEIR RELATION TO PERFORMANCE PRACTICES By John Baldwin very little specific information is currently available concerning the vibrational aspects of triangles and cymbals. Further, much of the information that is available is based on subjective personal opinions which are often confusing and contradictory. The data from this study should provide a basis for predicting the sounds of various triangles and cymbals (when instrument size, implement size and.material, striking point and angle, and dynamic level are known), thus eliminating excessive experimentation. The data was also used to substantiate (and sometimes invalidate) certain typical perfonmance practices. The study investigated, measured, and compared the overtone structures, produced by six triangles (6" Abel, 6" Ludwig, 6" Sonor, 6" Zildjian, 6" Pigstail, 10" Pigstail), and five cymbals (16" Avedis Zildjian, 16” New K. Zildjian, 16” Old K. Zildfiian, l7" Paiste, 20" Paiste). The triangles were struck with three implements, each 9" in length: 7/32" drill rod; 5/32” drill rod; 7/32" cold-rolled steel. The cymbals were struck with three implements: .Mhsser yellow yarn; Musser red yarn; Deagan brown cord with red stitching. The triangles were struck at three points: near the t0p of the closed side; the middle of the bottom side; near the closed corner of the John Baldwin bottom side. A 90° angle of incidence was used in all instances. However, at impact, the implement was perpendicular to the plane of the triangle at the top and bottom striking points (90° striking angle), and parallel to the plane of the triangle at the corner striking point (0° striking angle). The cymbals were struck at two points: near the edge; near the cup. .A 90° angle of incidence and a 0° striking angle were used in all instances. Three dynamic levels (ff, mf, and pp) were used on all triangles and cymbals. The sounds produced by variations in instrument size, implement size and.materia1, striking point and angle, and dynamic level were recorded with a Magnecord recorder, played back through an Ampex Recorder / Reproducer, and analyzed with a Bruel and Kjaer Frequency Analyzer. The resulting graphs were printed out with a Bruél and Kjaer Level Recorder. A.mathematical investigation of the triangles was made using the finite element method in conjunction with the Structural Analysis and Matrix Interpretive System (SAMIS), a computer program developed by the Philco Corporation. The program produced a set of predicted frequencies for each triangle plus the information necessary to diagram.each frequency's mode of vibration. These recommendations were made for high sounds on most triangles: use a relatively short and/or thick steel triangle; use an implement of either drill rod or cold-rolled steel; strike the triangle near one of the closed corners with a 90° striking angle. These recommendations were made for low sounds on most triangles: use a relatively large and/or thin John Baldwin steel triangle; use an implement of either drill rod or Cold-rolled steel; strike the triangle on either open side with a 0° striking angle. These recommendations were made for high sounds on.most cymbals: use a relatively small and/or thick cymbal; use a hard yarn implement if playing at a mf or pp level; strike the cymbal near the cup. These recommendations were made for low sounds on most cymbals: use a relatively large and/or thin cymbal; use a soft yarn implement if playing at a mf or pp level; strike the cymbal near the edge. Recommendations for further research included the determination of the influence of these variables: instrument material; lengths of the sides of a triangle; angles formed by the sides of a triangle; shape, size, and height of a cymbal's bow and cup; other implement materials and sizes; and aging of the instrument. SOME ACOUSTICAL PROPERTIES OF TRIANGLES AND CYMBALS AND THEIR RELATION TO PERFORMANCE PRACTICES BY ,. Baird's John’Baldwin .A THESIS Submitted to Michigan State University in Partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of.Mhsic 1970 f a /. v . l i/ O 9. nc / I'll l l l I I III. I I ll l \s G. .. (‘9; __J i' ,4,» / _. / (I v ”j! 3/ Copyright by JOHN BARDO BALDWIN 1971 ACKNOWLEDGMENTS For their invaluable aid and cooperation in the preparation of this dissertation, the author wishes to express his appreciation and gratitude to the following pe0p1e: his former advisor, Dr. George Duerksen, for his genuine concern and painstaking guidance: Dr. Robert Little for the mathematical background for the SAMIS computer program; oMr. Robert Buell and the Ford Motor Co. for their assistance with the SAMIS computer program and the contribution of the necessary computer time;er. Leonard Ott for his assistance with the recording equipment; and the many pe0p1e (especially Donald Andrus, Cloyd Duff, H.R. Spencer, and the late Harold Thompson) who took the time and thought to answer the author's questionnaire. The author also wishes to acknowledge his wife, Alison, for her constant encouragement and willing assistance in the preparation of this dissertation. ii U3TOFTABLES . LIST OF FIGIRES I. DHRODUCTIOX Statemen' Backgrou: Related Inst Impl Stri Stri Pitc Gene SUhmI‘)’ “- 51mm Ax Emdmmn Inst SL151: Impl Stri Stri Stri Recc Play pTOCeduI Ekpe Math IH' REPORT Am r Triangle EXpE SUmH Mat}. AETE CYTHbal s EXpE Skim TABLE OF CONTENTS LIST OF TABLES ............................ V LIST OF FIGURES .......................... Vi I. INTRODUCTION . . . . ....................... 1 Statement of Purpose Background of Problem Related Literature Instrument Mhterial Implement Size and Material Striking Point Striking Angle Pitch Levels General Sound or Timbre Summary II. EQUIPMENT AND PROCEDURE .................... 20 Equipment Instruments Suspension Setups Implements Striking Mbchanism Striking Points and.Angles Striking Force or Loudness Recording Equipment and Studio Playback and.Analyzing Equipment Procedure Experimental Mhthematical III. REPORT.AND DISCUSSION OF RESULTS ................ 51 Triangles Experimental Results Summary of Experimental Results and Related Research .Mhthematical Results Agreement of Mathematical and Experimental Results Cymbals Experimental Results Summary of Experimental Results and Related Researdh iii IL (DNCLUS IONS A Conclusic that l That That 18th Wha \Vna "Ah: IV} CONCLUSIONS AND RECOMMENDATIONS ................. 105 Conclusions What are the comparative overtone structures produced by triangles when played with large and small implements of the same material? What are the comparative overtone structures produced by cymbals when played with hard and soft implements of the same material? What is the effect, if any, of the material of the implement on the overtone structures produced by triangles and cymbals? What is the precise relationship between the striking angle and/or point and the predominantly high and low pitch areas within one triangle or cymbal? What are the comparative overtone structures produced by triangles and cymbals when played at various dynamic levels? What are the relative strengths or intensities of the overtones produced by triangles and cymbals? What are the similarities, if any, among the overtone structures produced by different types of triangles (e.g., spindle and pigstail) and different brands of cymbals (e.g., Avedis Zildjian and Paiste)? What are the modes of vibration of a triangle suspended at one corner? Instrument Size and.Material Pitch Level Recommendations for Performance Instrument Size and Material Implement Size or Hardness and Material Striking Point and Angle Recommendations for Further Research Triangles Cymbals BIBLIOGRAPHY .......................... . . 117 APPENDIX.A : LETTER.OF INQUIRY ................... 129 APPENDIX B : BRUEL.AND KJAER LEVEL RECORDER GRAPHS ......... 131 APPENDIX C : NDDES OF VIBRATION- -MA'I'RIX AND DIAGRAMS ........ 134 iv rx) \1 o 10. 11. 12. 13. 14. 1S. . Decibel Le” Frequencie MlCl‘Op. Fundamenta Triang Fundamenta Triang Fundamenta Triang I"Ilndamenta Triang Fundailienta Triang Fundamenta Triang Predicted Agl‘eement Furldamenta Zildji Fundamen t a Zildji FundameUta Zildji Fundamenta Cyrba; Fundamen ta Cmb 31 \lb ratiOna 10. ll. 12. 13. 14. 15. Al. LIST OF TABLES Decibel Levels.At Impact .................... 40 Frequencies Having Nodal or Antinodal Points.At or Near the ZMicrophone Location .................... 52 FUndamentals, Upper Limits, and Energy Peaks of the Abel Triangle .......................... 54 Fundamentals, Upper Limits, and Energy Peaks of the Ludwig Triangle .......................... 56 Fundamentals, Upper Limits, and Energy Peaks of the Sonor Triangle .......................... 59 Fundamentals, Upper Limits, and Energy Peaks of the Zildjian Triangle .......................... 62 Fundamentals, Upper Limits, and Energy Peaks of the 6" Pigstail Triangle .......................... 65 Fundamentals, Upper Limits, and Energy Peaks of the 10" Pigstail Triangle .......................... 68 Predicted Triangle Frequencies ................. 77 Agreement Between Experimental andeathematical Results . . . . 84 Fundamentals, Upper Limits, and Energy Peaks of the.Avedis Zildjian Cymbal ...................... 87 Fundamentals, Upper Limits, and Energy Peaks of the New K. Zildjian Cymbal ...................... 89 Fundamentals, Upper Limits, and Energy Peaks of the Old K. Zildjian Cymbal ...................... 91 Fundamentals, Upper Limits, and Energy Peaks of the 17" Paiste Cymbal ........................... 94 Fundamentals, Upper Limits, and Energy Peaks of the 20" Paiste Cymbal ........................... 96 Vibrational Mode Matrix for Zildjian Triangle ........ 135 V (\J . 11. 12. Generalized Abel Triang' Ludwig Tria Senor Trim: Zildjian T1 5" Pigstai‘ 10" Pigsta General 1 :e - AVGdiS :i] 10. Old K, [‘J i Striking ' Striking Striking RECOrd'n . Playb 3C] 10. ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. LIST OF FIGURES Generalized Triangle ...................... 22 Abel Triangle ......................... 23 Ludwig Triangle ........................ 24 Sonor Triangle ......................... 25 Zildjian Triangle ....................... 26 6" Pigstail Triangle ...................... 27 10" Pigstail Triangle ..................... 28 Generalized Cymbal ....................... 29 Avedis Zildjian Cymbal ..................... 30 New K. Zildjian Cymbal ..................... 31 Old K. Zildjian Cymbal ..................... 32 17" Paiste Cymbal ....................... 33 20" Paiste Cymbal ....................... 34 Striking Mechanism ....................... 37 Striking Points Used on Triangles ............... 38 Striking Points Used on Cymbals ................ 39 Recording Studio ........................ 42 Playback and.Analyzing Equipment ................ 43 Settings on Frequency.Analyzer #2107 .............. 44 Settings on Level Recorder #2305 ................ 45 Numbering of Elements and Nodal Points ............. 47 Analyzed Overtone Structures for Triangles ........... 75 vi AZ. Predicted O . In-plane \‘i‘ Out-of-plan . Analyzed 0\ . Bmt‘l and 1 Modes of \’ 23. 24. 25. 26. A1. A2. Predicted Overtone Structures for Triangles ........... 79 In-plane Vibration--Partia1 4 .................. 81 Out-of-plane Vibration-~Partial ll ............... 82 Analyzed Overtone Structures for Cymbals ............ 103 Bruél and Kjaer Level Recorder Graphs .............. 132 Modes of Vibration for Zildjian Triangle ............ 144 vii I. INTRODUCTION Statement of Purpose The tone quality [of percussion instruments] depends on such factors as the dimensions, character of the metal or other material, point of striking, material and shape or the striking point, and the manner of reinforcement. In addition, various types or brands of instruments produce different sounds (e.g., spindle and pigstail triangles; Avedis Zildjian and Paiste cymbals). But as yet, there is no body of organized knowledge available to provide the scientific basis for these different sounds and tone qualities. As Cloyd Duff of the Cleveland Orchestra says, I do not know the technical points about cymbals and triangles . I only know them from a performance point. If they are good instruments and possess the qualities that I require in these instruments for my purpose, I use thep, and if not they are poor instruments and not for my need. Thus, the performing percussionist is forced to grope rather blindly and gradually learn, through trial and error or oral tradition, how to utilize the various tonal capabilities of his instruments to produce the most appropriate sound at the proper musical moment. This study investigated and measured the overtone structures produced by triangles and suspended cymbals. In addition, the mathe- matical predictions of a triangle's vibrational behavior (based on 1Wilmer Bartholomew, Acoustics of Music, p. 130. 2Cloyd Duff, personal letter, October 17, 1967. material, shape, and dimensions) were compared.with the acoustical measurements of the actual sounds produced. The following questions served as a guide to this investigation and measurement: 1. What are the comparative overtone structures produced by triangles when played with large and small implements of the same material? 2. What are the comparative overtone structures produced by cymbals when played with hard and soft implements of the same material? 3. What is the effect, if any, of the material of the implement on the overtone structures produced by triangles and cymbals? 4. What is the precise relationship between the striking angle and/or point and the predominantly high and low pitch areas within one triangle or cymbal? 5. What are the comparative overtone structures produced by triangles and cymbals when played at various dynamic levels? 6. What are the relative strengths or intensities of the overtones produced by triangles and cymbals? 7. What are the similarities, if any, among the overtone structures produced by different types of triangles (e.g., spindle and pigstail) and different brands of cymbals (e.g., Avedis Zildjian and Paiste)? 8. What are the modes of vibration of a triangle suspended at one corner? The data which resulted from answering the above questions were then used to substantiate (and sometimes invalidate) certain typical perfomance prac also provided a 1 cymbals produce 1 eliminating exce to point the way and cymbals. General tub-es, plates, (‘1 general discussi body: uniform C tension-require instruments. In actual percussic and explanation. other UllSlClalls, not attempted or and properties o Since t‘ ' ~ | thEHts, HOE indefinite nolSr performance practices used on triangles and suspended cymbals. The data also provided a basis for predicting the sounds that various triangles and cymbals produce (when material, shape, and dimensions are known), thus eliminating excessive experimentation. .Another use of the data would be to point the way for further research into the actual design of triangles and cymbals. Background of Problem General discussions of the acoustics of membranes, rods, bars, tubes, plates, and bells are readily available. Unfortunately, these general discussions seem.always to be in terms of the ideal vibrating body: uniform cross-section or thickness; homogeneous material; uniform tension--requirements which are rarely, if ever, met in actual percussion instruments. In other words, the specific acoustical properties of actual percussion instruments have been quite neglected, both in research and explanation. This is partially due to the fact that acousticians, other musicians, and "particularly percussionists as a whole have largely not attempted or had the means to explore more fully the acoustical bases and properties of these instruments."1 Since the fundamentals and overtone structures of many percussion instruments, notably triangles and cymbals, form.what has been called "an "2 indefinite noise mixture (rather than a regular harmonic series with integer ratios), some musical acousticians apparently justify their neglect of these instruments by saying that "it is easy to understand why ¥ 1JameslMoore, "Percussion.Acoustics: .An Introductory Evaluation,” Percussionis ,‘V (October, 1967), 218. 2Harry Olson, MUsic, Physics and Engineering, p. 177. percussion inst] Buck seems to SI by saying that ' etc., are of int . . ,2 nisimans.‘ However, percussion instr their sound is "I to the author's graphical materi Of MUSIC “1601”). also have bc In general. Can about t; on thelr act KnOlSIEd‘ That Percussion norm, tO be mrber of musica 1t 13 necessary percussion instruments have only a limited use musically."l Indeed, Buck seems to sum up the apathy or disinterest of musical acousticians by saying that "the sounds that can be produCed from rods, plates, bells, etc., are of interest and importance to physicists rather than to musicians."2 However, physicists are apparently not overly interested in percussion instruments either (Wood dismisses cymbals by saying that their sound is "called by no accident 'kitchen music'"3). In response to the author's inquiry concerning the availability of relevant biblio- graphical materials of either a musical or physical nature, an Instructor of Music Theory at the University of Illinois wrote: I have not only been trying to find information on tamtams but also have been checking out metal-plate percussion instruments in general. I have now exhausted most of the possibilities and can say the same thing about cymbals and triangles as I can about tamtams: I have found prafitically no information on the1r actual harmonic structures. Knowledgeable percussionists will take issue with Buck's inference that percussion sounds are not musically important. For them, it is very important to be able to accurately and consistently produce any of a number of musical sounds. However, with the present lack of information, it is necessary for the percussionists to experiment with various combinations of instruments, implements, and striking angles and/or points in order to be reasonably confident of producing a particular sound at any particular moment. 1Bartholomew,.Acoustics of Mhsic, p. 130. 2Percy Buck, Acoustics for Mhsicians, p. 84. SAlexander Weed,.Acoustics, p. 448. 4DonaldAndrus, personal letter, October 28, 1967. The present acoustical investigation of the sounds of triangles and cymbals, coupled with the mathematical investigation of the triangles, should remove some of the uncertainty of percussion performance by making it possible to accurately predict the tonal results when the major performance variables are known (i.e., the dimensions of the instrument, the material of the instrument and implement, the striking angle and/or point, and the striking force). Related Literature The following information concerning triangles and cymbals has been compiled from pedagogical materials, magazine articles, books on percussion, musical and physical acoustics texts, and personal corre- spondence. Much of the information is subjective in nature or is based on tradition or personal experience. It should be noted that the questions presented in this study are not answered with any degree of scientific satisfaction. Instrument Material Tri les Triangles made of aluminum are presently available, but they are almost always "avoided.because they produce an inferior tonal response."1 Although Hart vaguely reports that the "triangle is a piece of metal,"2 most instruments in use today are made of steel. But there are only 1James Ross, "The Triangle: Don't Underestimate It,” The Instrumentalist, XIX (April, 1965), 84. 2WilliamSebastianHart, ”Percussion Clinic," The Instrumentalist, XIII (February, 1959), 69. general comm high tempera The I Symphony, cor nickel or chr not as yet fo plated one."3 Ihe a fly aluminum . 01365, or whv “L 1 Al erCussion, Ea) he, 3a ar01d 4 Charle 1'0 - J-M. F i We 6 general comments as to the specific kind of steel used: "very hard 1 and "plated steel."2 high tempered carbonized steel" The late Harold Thompson, former percussionist with the Boston Symphony, confirms that the presence of some kind of plating (usually nickel or chrome) is essential for the desired triangle sound: "I have not as yet found an unplated triangle that sounds as good as a highly plated one."3 The author was unable to find any definitive information as to why aluminum is inferior to steel, why some steel is better than other types, or yhy the plating is essential. Cymbals Regardless of the brand, all cymbals seem to consist of a “metal alloy, made from copper, tin, lead, and iron."4 Quoting Zildjian, Flagler states that their cymbals are "'roughly eighty per cent copper and twenty per cent tin'” with a "'small amount of silver.'"5 Peters reports the use of a specific combination of 78.55% copper, 20.28% tin, .54% lead, and .18% iron.6 Two cymbal manufacturers state that the 1A1 Payson and Jack MCKenzie, Music Educators' Guide to Percussion, p. 58. ZHarry Bartlett, Guide to Teaching Percussion, p. 110. 3Harold Thompson, personal letter, October, 1967. 4Charles Spohn, The Percussion, p. 47. 5J.IM. Flagler, "Onward and Upward With the Arts," The New Yorker, December 6, 1958, p. 156. 6Gordon Peters, "Treatise on Percussion," p. 114. metal in thei: highly temper: One pe determine if d more prominent changing the 5 Thomps. WUBIMIOVGTtO] to use is drili the"1<'=lrgest n: Spencer I Chose sta hard temper Plating. . p1ayed than Steel has a rec{Uil‘es p1 \ lAvedis 2Phil 5 3H.R, S 4 . Thomps SROSS’ f)Hart , 7 Spence metal in their cymbals (no matter what the exact content) is very highly tempered.1’2 Implement Size and.Material Triangles One percussion innovator "used a Stroboconn in an attempt to determine if different sizes of beaters caused different overtones to be more prominent. . . . [He] found no change in the overtones due to changing the size of the beater."3 Thompson feels that a "soft and dense stick will generate maximum.overtones and long sound."4 Ross states that the ”best material to use is drill rod cut up in 9 inch lengths."5 And Hart suggests that the "largest nails available"6 should be used. Spencer prefers stainless steel for the following reasons: I chose stainless steel because it combines a steel of medium hard temper with an attractive metal which doesn't require plating. . . . Cold rolled steel . . . dents more when played than does the stainless steel. Highly tempered steel has an excellent sound but is hard to cut and requires plating. lAvedis Zildjian Co., Cymbal Notes. ZPhil Grant, Tested Tips for School Mhsic Supervisors. 3H.R. Spencer, personal letter, December 2, 1968. 4Thompson, personal letter. 5Ross, "The Triangle," p. 84. 6Hart, "Percussion Clinic,” p. 69. 7Spencer, personal letter. als .Although the variety of usable implements is limited only by the percussionist's imagination, only two sources yielded any specific information as to the effect of the size and material of the implement on the sound produced. Firth states that ”a larger, heavier stick will bring out the fundamental and its overtones much quicker and clearer.”1 And Bartlett reports that the tips of wooden snare drum implements "will cause the higher frequencies in the cymbal to predominate, producing a more tinkly sound."2 Striking Point Triangles Bartlett simply states that the ”usual playing spot is on the outside,"3 and Wildman rather vaguely says the triangle is "hit on the opposite side from the open corner."4 Leidig is more specific when he recommends that the triangle be struck "on the upper third of [the] right side (corner with the Opening to the left).”5 When Tilles says to strike the "closed end of the triangle and have this side facing your right hand,” his stated reason is "because the sound travels 1Vic Firth, Percussion Symposium, p. 26. 2Harry Bartlett, Percussion Ensemble Method, p. 65. 3 Bartlett, Teachinngercussion, p. 110. 4Louis Wildman, Practical Understanding of the Percussion Section, p. 71. 5Vernon Leidig, Contemporary Percussion Technique and.Method, p. 12. 1 However, he includes no further toward the ends of the instrument.” explanation or supporting evidence for this recommendation. Leach reports that "physical laws tell us it [the triangle] will sound better if struck in the middle"2 (but then neglects to say BREE physical laws, hgy_or yhy_it will sound better, or BREE defines "the middle”). Spencer seems to agree with Leach, reporting that "generally . . the middle of the closed sides produced the cleaner clearer tone."3 Blades, however, feels that ”the best tone is produced by striking the triangle on the outer [open] side near the top corner."4 Price emphatically disagrees with the above Opinions and asserts that "the triangle is never struck on the outside, but always on the base."5 Collins and Green corrOborate this view, stating that the "beater should fall upon the center of the bottom angle Of the triangle when single notes are played."6 .Although Gardner concurs with the use Of the base, he includes a picture showing a striking point very near the Open end.7 1Bob Tilles, "The Bob Tilles Column," The Ludwig Drummer, VIII (Spring, 1968), 31. 2Joel Leach, Percussion Manual for Music Educators, p. 78. 3Spencer, personal letter. 4James Blades,.Orchestral Percussion Techniques, p. 26. 5Paul Price, Techniques and Exercises for Playing Triangle, Tambourine, and Castanets, p. 7. GMyron Collins and John Green, Playing and Teaching Percussion Instruments, p. 123. 7Carl Gardner, The Gardner MOdern Method, p. 80. Hart a correspond wit toward the mid. directly in tln It is : among percussic occasions. Var upper third Of outer side near However, none c accurate measm that, as of non mentation which While m obtained, depen 10 Hart and others1 recommend a change Of striking point to correspond with dynamic changes: ”near the top for pianissimo work, toward the middle of the right hand [side] for louder work, and directly in the middle for loudest effect."2 It is readily apparent that there is no universal agreement among percussionists as to a single best striking point for all occasions. 'Various authorities have recommended "the outside," "the upper third of the right side,” "the closed end," "the middle," "the outer side near the top corner," and "the center of the bottom angle. However, none Of these striking pointS'were defined with any type of accurate measurement, nor related to any specific sound. This means that, as of now, the percussionist still "must determine by experi- mentation which playing spot produces the best sound for that particular triangle and musical passage."3 C als While most authorities agree that "a number Of sounds can be Obtained, depending upon which part of the surface is struck,"4 they also seem to agree that normally the best striking point is fairly close to the edge: ”the cymbal, generally, should be struck close to the 1Thomas Brown and Willard Musser, Percussion Studies I, p. 5. 2Hart, "Percussion Clinic," p. 69. 3Mitchell Peters, "Triangle Technique," The Instrumentalist, XXII (February, 1968), 79. 4Bartlett, Teaching Percussion, p. 71. edgef'l "313011 edge of the Cj played upon n1 near the edge on the extreme Denov a rapid rhyth: "so each strO} play near to, 11 edge;"l "about 2 1/2 inches from [the] edge;"2 u stick striking near the edge of the cymbal;"3 "for almost all strokes and rolls, the cymbal is played upon near the edge;"4 and "the cymbal should be struck on top near the edge."5 However, Thompson feels that if a cymbal "is activated on the extreme edges, the full spectrum of sound does not emerge."6 Denov agrees with the close-to-the-edge striking point, "unless a rapid rhythmic figure is to be played."7 Leach goes on to say that "so each stroke can be easily distinguished, it is Often necessary to play near to, or on the bell of the cymbal."8 Striking Angle Almost without exception, the typical angle of incidence for playing both triangles and cymbals is 90°; that is, the plane of the implement's stroke is perpendicular to the playing surface. However, the term "striking angle" used in this study refers ng£_to the angle of incidence, but rather to the angle of the implement to the plane of the 1Sam Denov, The.Art of Playing Cymbals, p. 7. 2 Leidig, Contemporary Percussion Technique, p. 12. 3Collins and Green, Playing and Teaching, p. 121. 4Morris Goldenberg, MOdern School for Snare Drum, p. 92. SBartlett, Percussion Ensemble Method, p. 65. 6Thompson, personal letter. 7Sam Denov, "Techniques of Cymbal Playing," The Instrumentalist, XIX (September, 1964), 58. 8 Leach, Percussion.Manual, p. 38. 12 instrument at the moment Of impact. In other words, a striking angle of 90° indicates that the implement is perpendicular to the plane of the instrument at impact; and a striking angle Of 0° indicates that the implement is parallel to the plane of the instrument at impact. Triangles Although.most Of the literature concerning the triangle shows the implement perpendicular to the plane of the triangle at impact (90° striking angle), Ross states that the triangle should be struck on the horizontal leg with the implement ”at about a 45 degree angle."1 Peters states that "striking the triangle with the beater perpendicular [i.e., parallel to the plane of the triangle--0° striking angle] will produce a more diffuse sound with.more overtones."2 Thompson states that he generally strikes the triangle with "the stick vertical-—not against gravity."3 C als Very little information is available concerning the striking angle used on cymbals, but Denov and Bartlett include illustrations which indicate that the implement is parallel to the plane of the cymbal at impact--0° striking angle.4’5 1Ross, "The Triangle," p. 84. GWitchell Peters, "Triangle Technique," p. 82. 3Thompson, personal letter. 4Denov, Playing Cymbals, pp. 18-19. 5Bartlett, Teaching Percussion, pp. 71-72. 13 Pitch Levels Triangles Although different triangles will produce varied pitch levels, the word pi:gh_should not be used "in a sense Of definite pitch, for triangles are not tuned."1 Peters very explicitly states that "it is out of the question to talk Of pitch."2 WOod agrees, saying that the triangle "gives numerous strong partials and no definite pitch."3 .Although Briggs has found that some triangles exhibit a "trace Of fundamental at about six per milisecond, say round about [gig] 6,000 c/s,”4 Stauder reports a fundamental of about 700 cps, with the most intense partials occurring between 7000 and 9500 cps.5 Contrary to the apparent majority Opinion, SpOhn emphasizes the point that "because Of the variety of triangles which are available and the variation in pitch of different triangles of the same size, the teacher and student should select triangles by pitch for specific compositions."6 However, the author is unable to find any information telling what pitches are to be found in what sizes Of triangles, nor any evidence of music written for triangles tuned to definite pitches. lMitchell Peters, "Triangle Technique," p. 79. 2Gordon Peters, ”Treatise on Percussion," p. 280. °Alexander Wood, The Physics of Music, p. 149. 4G..A. Briggs, Musical Instruments and.AudiO, p. 96. SWilhelmStauder, "Schlaginstrumente-eAkustik," Die Musik in Qeschichte und Gegenwart, XI (1963), 1747. 6Spohn, Percussion, p. 51. 14 als It seems to be commonly accepted that good cymbals ”'are the one kind Of instrument that doesn't have positive pitch. Instead, they have a rough dominant pitch,'”1 Often called the bell tone. Burns agrees, saying that a "good cymbal will sound all pitches or their harmonics simultaneously, even though it cannot be tuned to a specific note."2 In fact, one company states that ”'if any single note does dominate a cymbal's tone, it's Obviously an inferior instrument.'"3 Sewrey found that he could raise the overall pitch level of the cymbal "by playing closer to the cup, in order to bring out the higher overtones."4 One of the few published acoustical studies on.musical instruments includes this statement relative to the pitch level of a cymbal: "The spectrum is particularly rich in high frequencies, the higher peaks lying 5 above 8,000 cps." And although Briggs reports finding several cymbal sounds up to 25000 cps, he states that "most Of the energy is located in the range above about 5 kc/s [5000 cps]."6 When questioned, one professional percussionist commented on the influence of the shape Of the bow on the pitch level of a cymbal: 1Flagler, ”Onward and Upward,” p. 136. zRoy Burns, The Selection, Use and Care of Cymbals in the Stage and Dance Band, p. 3. 3Flagler, "Onward and Upward," p. 138. 4James Sewrey, "Percussion Clinic," The School MUsician, XXXIII (January, 1962), 14. SL. Sivian, H. Dunn, and S. White, "Absolute Amplitudes and Spectra of Certaianusical Instruments," Journal of the Acoustical Society of America, II (January, 1931), 353. 6Briggs,M’usical Instruments and.Audio, p. 73. "The shape of t 105. . . cymba lower more bodi to an extent th Triangles The £01 to describe the "delicate, extr quality which 1 triangle sound goes fEilrther, : can be frivolor Vote and arti states that th Firth on the general \ l ThOmp ZBI‘OMT] 3 priCe 4Bart1 S Char] a... 7Wood 15 ”The shape Of the arc or bow is the most determining factor of high or low . . . cymbal sound. [A] flat and relatively straight bow produces a lower more bodied sound as a rule. The noticeably rounded arc suppresses to an extent the lower vibrations, and therefore allows a higher sound."l General Sound or Timbre Tri les The following subjective terms have all been used in attempts to describe the desired triangle sound: "tinkling, shrill sound;"2 3 "delicate, extremely high pitched 'tinkle;'" "shimmering metallic quality which has considerable brilliance."4 White compares a good 5 Hart triangle sound to the "shimmering sparkle of exquisite jewels." goes farther, saying that "the sound Of the triangle is a Eiflhlfi: This can be frivolous, thunderous, exciting, melancholy, according to the taste and artistry of the person playing it."6 One physicist simply states that the triangle "produces a jangle of partial tones."7 Firth recognizes that the size of the triangle has some influence on the general sound when he reports that the sound of a smaller one is 1Thompson, personal letter. 2Brown andeusser, Percussion Studies, p. 5. 3 Price, Triangle, Tambourine, and Castanets, p. 7. 4Bartlett, Teaching Percussion, p. 110. 5Charles White, Drums Through the Ages, p. 57. 6Hart, "Percussion Clinic,” p. 69. 7Wood, Acoustics, p. 425. 16 "thinner in texture."1 Leach agrees, saying that the ”larger the triangle, the more it tends to sound a 'bong' rather than a 'ting.'"2 Ross feels that there is a definite relation between the striking point and the general sound: "the tonal response on the bottom of the triangle amplifies the lower overtones, the upper side, the higher overtones."3 Spencer reports finding the same relationships in his experimentation with various triangles.4 But neither Ross nor Spencer provide any explanations for these relationships. Although discriminating percussionists eventually come to associate the above subjective terms with various triangle sounds, to date there has been no systematic attempt to accurately describe these sounds in terms of their partial or overtone structures. als In general the use of subjective terms also characterizes most attempts to describe a good cymbal sound: "unique shimmering sound;”5 "brilliant, crashy tone;"6 ”a thick quality."7 Spohn feels that the "sound of a good cymbal will tend to rise after it is struc ."8 1Firth, Percussion Symposium, p. 30. 2Leach, Percussion Manual, p. 78. 3Ross, "The Triangle," p. 84. 4Spencer, personal letter. 5Leach, Percussioananual, p. 38. 6Bartlett, Teaching Percussion, p. 12. 7Denov, Playing Cymbals, p. 7. 8Spohn, Percussion, p. 47. 17 Perhaps this is what Lang refers to when he says "it is especially important that its highs 'come out.'"1 Payson and MCKenzie report that ”there are several brands of cymbals and each brand has its own particular sound."2 Sewrey apparently agrees, saying that there is a "wide difference between a [new] 'K,' an 91_<_1_ 'K,‘ and an 'A' Zildjian cymbal."3 (Avedis Zildjian-- made in America since 1929; New K. Zildjianr-made in Turkey since 1929; and Old K. Zildjianr-made in Turkey before 1929.) Lang finds that an Old ”'K' is somewhat thicker and with a wider range of overtones."4 Thompson finds that when New K.'s are compared with Avedis Zildjians, "the K's sound more 'lows"with the tendency to become kind Of 'brashy' the louder played."5 One rather prevalent Opinion is that the general sound depends upon the striking point. For example, "if struck on the edge, it has a 'splashy' sound. If struck on the cup, it has a hollow and 'clanging' sound."6 Thompson writes that he finds "the tremelo when played on [the] edges of cymbal . . . a different color than if the roll is played more lMOrris Lang, "Percussion Clinic,” The School Musician, XXXIII (December, 1961), 16. zPayson and MCKenzie, MUsic Educators' Guide, p. 52. 3James Sewrey, "Percussion Clinic," The School Musician, XXXIII (February, 1962), 54. 4Lang, "Percussion Clinic,” p. 16. 5Thompson, personal letter. 6Firth, Percussion Symposium, p. 26. 18 'amidships."'1 He then attempts to explain this Observation by saying that "probably the outside is the more easy to vibrate as this surface suggests that the action flows in toward the cup--when struck at a halfway'point the force must flow both directions."2 Denov recognizes the influence of the cymbal's size on the overall sound when he relates that the cymbal tone "is enhanced in direct prOportion to the amount of metal contained in the cymbal. Plurality of overtones and sustaining quality increases as the guantigy . 3 Of metal increases." Summary It has been pointed out that very little concrete information is known concerning the vibrational aspects of triangles and cymbals. Further, many of the Opinions and much Of the material pertaining to these instruments is confusing, misleading, and contradictory. This means that, at the present time, percussionists have no way of knowing exactly what sounds any given instrument will produce without actually experimenting with different implements, striking points and angles, and striking forces. This acoustical study Of triangles and cymbals, coupled with the mathematical investigation of the triangles, should remove some of the uncertainty of percussion performance by enabling the pUblic school music lThompson, personal letter. 2Thompson, personal letter. 3 . . . . Sam.Denov, "Equ1pp1ng the Cymballst," The Instrumentalist, XVIII (June, 1964), 60. director and when the maj' material of ' strikim poi] instrument a1 the proper m. 19 director and the performing percussionist to I) predict the tonal results when the major performance variables are known (i.e., the dimensions and material of the instrument, the size and material of the implement, the striking point and angle, and the striking force), and 2) select an instrument and implement to produce the appropriate musical sound at the proper musical moment. Six t] by ludwig Inch professional 6 triangles (one Tool Company. Five c Zildjian lepa one 16" mOdel : [one 17" mOdel AS the II. EQUIPMENT AND PROCEDURE Eguipment Instruments Six triangles were used: three commercial 6" models marketed by Ludwig Industries, Sonor Company, and Avedis Zildjian Company; one professional 6” model manufactured by Alan Abel; and two Pigstail triangles (one 6” model and one 10" model) manufactured by the MOund Tool Company. Five cymbals were used: two 16" models marketed by the Avedis Zildj ian Company (A.) and Fred Gretsch Manufacturing Company (New K.); one 16" model manufactured by Zildjian (Old K.); and two Paiste cymbals (one 17" model and one 20” model) marketed by Ludwig Industries. As the majority of these instruments were selected from large stocks by experienced professional percussionists (including Alan Abel, Maurie Lishon, Dick Schory, James Sewrey, and the late Harold Thompsonl), IAlan.Abel: presently percussionist with the Philadelphia Orchestra, and percussion innovator and inventor’. Maurie Lishon: presently owner of Franks Drum Shop, Inc. , formerly professional percussionist with name bands and staff percussionist‘with.CBS4WBBM Radio in Chicago. Dick Schory: presently Senior Vice-President of Ludwig Industries, director Of the Percussion Pops Orchestra, percussion clinician, formerly percussionist with the Chicago Symphony Orchestra and free-lance arranger and studio percussionist. James Sewrey: presently Product Manager and Educational Director of Ludwig Industries and percussion clinician, formerly public school music director and university percussion instructor, percussionist with the Wichita (Kansas) Symphony Orchestra. Harold Thompson: formerly percussionist with the Boston Symphony Orchestra and consultant for the Avedis Zildjian Company. 20 21 it may be assumed that these instruments are Of above-average quality according to present standards. Although the two Pigstail triangles are not commercially available today, the above instruments are typical in shape and size Of the instruments used by both amateur and professional percussionists. The generalized triangle in Figure 1 illustrates the various measurements made for this study. The lengths of the segments were measured from the Open ends to the dotted lines. These numbers are placed inside the outline of the triangle. The diameters of the ends of each segment are located outside the outline. The angles formed by the legs of the triangle are indicated within the outline, as is the total length. .All length measurements were made with a ruler graduated in tenths of inches and all diameters were measured with a.micrometer graduated in thousandths of inches. The dimensions and distinguishing characteristics of each of the triangles used for this investigation are shown in Figures 2 through 7. The generalized cymbal in Figure 8 illustrates the various measurements made for this study. The diameters of the hole, the cup, and the entire cymbal are given in the upper portion. In the lower portion, the height of the cup, the height Of the bow, and the thickness are indicated. Except for the thickness (measured with a micrometer graduated in thousandths of inches), all measurements were made with a ruler graduated in tenths Of inches. Dimensions and distinguishing characteristics of the cymbals used in this investigation are shown in Figures 9 through 13. 22 .621" E .565” 60° 6.4" 6.0" .621" Total Length = 18.7" .565" 60° 60° ’ 6.3" j R453" .453", Figure 1.--Generalized Triangle . 56S" . 56S" 23 50° . 565" 3.4" .385" .4" Total Length = 17.3" 4.2" .385" 65° 4 F l I :.1:l .9" AA A \ .565" Figure 2.--Abel Triangle .565"/ .312" .312" Distinguishing characteristics: isocoles shape; smaller segments of unequal length and diameter; relatively large overall diameter. 24 ~‘-.442" 6.0" .442" Total Length = 18.0” .442" 60° 60° 8 i 6.0" \ .442" .442" Figure 3.--Ludwig Triangle Distinguishing characteristics: equality of all measurements. 25 .1" .291" g .291" 60° 6.2' ~.0" .291' Total Length = 18.4" 291" 60° 60° ,. i t 6.0" 2) / \5291" .291" Figure 4.--Sonor Triangle Distinguishing characteristics: equal angles; equal, and relatively small, diameter at all points; small hole in apex. 26 .343" u .343" 55° 6.4” 6.2" .453" Total Length = 18.55 .220" 1 65° 60° ’1" , x r \ 5.7 /2? .346" .222"/r .05" .15" Figure 5.--Zildjian Triangle Distinguishing characteristics: unequal angles; tapered segments of unequal lengths. 27 .375" ' .375" 60° 6.5' 6.2" .375" Total Length = 19.4" .375" 60° 60° ,, t l _____ I 5.6" : 1_1 \.375" .375"/ av.2575" .140" Figure 6.--6” Pigstail Triangle Distinguishing characteristics: tapered segment on lower leg; equal angles; equality of diameter (with exception of tapered segment); downward angle of tapered segment. 28 .500" g .500" 60° 10.7' 0.0' .500" Total Length = 31.8" .500" 60° jp° 6 5.. r 2 0,, . " ““““ . /{\\ ° ll 2'6" / I \ I .500" .500" .490" .490" av.315" .140” Figure 7.--10" Pigstail Triangle Distinguishing characteristics: tapered segment on lower leg; equal angles; equality of diameter (with exception of tapered segment); downward angle Of tapered segment. 29 \ .050" Figure 8.--Generalized Cymbal Figure 9 . -. or. the other < 30 ' ‘ .055" Figure 9.--Avedis Zildjian Cymbal Distinguishing characteristics: no unique features compared to the other cymbals . 31 .4" 16.1" _._1r.... .9" i .049" Figure lO.--New K. Zildjian Cymbal Distinguishing characteristics: relatively small cup diameter; relatively high cup. 32 .052" Figure ll.--Old K. Zildjian Cymbal Distinguishing characteristics: relatively large cup diameter; flat bow; small hole. 33 1.4" A ‘ .043" Figure 12.--l7" Paiste Cymbal Distinguishing characteristics: largest hole of all cymbals studied; thinnest of all cymbals studied. Figur913,-- Dist largESt b 0w 34 Figure l3.--20" Paiste Cymbal Distinguishing characteristics: cup height same as New K. Zildjian; largest bow height; largest overall diameter of all cymbals studied. 35 Suspension Setups The triangles were suspended by a heavyewire "Podemski" holder, insulated with rubber tUbing, with a supporting cord of dental floss. All triangles were held so that the top of each.was five feet from the floor. The cymbals were supported by a regular floor stand, consisting of a Premier flush-base with rubber feet, a metal shaft, and a Slingerland cymbal tilter. The tilter post was insulated with rubber tUbing and the cymbal rested on a felt washer. The stand was adjusted so the edges of all the cymbals were three feet from the floor. These suspension setups are typical of those used by both amateur and professional percussionists. Implements Three triangle implements were used: 7/32” (.218") x 9" in ' drill rod and cold-rolled steel, and 5/32" (.156") x 9" in drill rod. Cold-rolled steel is relatively soft with a carbon content of .15 to .25 per cent, while drill rod is much harder with a carbon content Of at least .85 per cent.1 Three implements were used on the cymbals: iMusser yellow'yarn 0W8); Musser red yarn 0M6); and Deagan brown cord with red stitching (#2014-C). The Musser series of yarn.implements is color and number coded, with the yellow 0M8) being larger and softer than the red 0M6). The Deagan series of brown cord implements is also color and number lErnest Edgar Thum and Richard Edward Grace, "Iron and Steel-- Classification and Uses of Plain Carbon Steels,” Encyclopaedia Britannica, 1963, XII, 666. 36 coded, with the #2014-C being the hardest of a set of four. Each implement head was mounted on a 3/8" x 13” birch shaft. These implements are typical of those used by both amateur and professional percussionists. Striking.Mechanism The basis of the striking mechanism (Figure 14) was an "Eaton's Vibration Demonstrator" (#3325), a flat-spring apparatus built by the Welch Scientific Company. This apparatus was mounted on a heavy metal stand, adjustable for the proper striking angles, and insulated with rubber tubing. Wooden blocks fastened to the springs were drilled out to accomodate the various implement shafts. .An arbitrary scale was also fastened to the apparatus in order to help maintain continuity of the striking forces. Striking Points and Angles The three striking points used on the triangles are shown in Figure 15. The first point (Egp)'was located 1" down on the closed side of the small triangles, and 1 1/2" down on the closed side of the 10" Pigstail triangle. The second point (£93223) was located on the bottom side 1" from the closed corner on the small triangles, and 1 1/2" from the closed corner on the 10" Pigstail triangle. The third point (buggg Q was located at the midpoint of the bottom side on all of the triangles. Impler Imple Figure 14' 37 Arbitrary Scale Flat Spring Implement Head A / Birch Bloc / Implement Shaft//// iii:L ‘ Clamp with rubber / insulation Welch Spring Apparatus 1" Black Pipe / Heavy Metal Base >— Figure l4.--Striking Mechanism 38 Top (1" down on small triangles; 1 1/2" down on lO"Pigstail triangle) Corner (1” in on small triangles; 1 1/2" in on 10" Pigstail triangle) / Bottom (midpoint on all triangles) Figure 15.--Striking Points Used on Triangles 39 At the top and bottom striking points, the implement was perpendicular to the plane Of the triangle at impact (90° striking angle). At the corner striking point, the implement was parallel to the plane of the triangle at impact (0° striking angle). The two striking points used on the cymbals are shown in Figure 16. The first point (edge) was located 1" from the edge on the small cymbals, and 1 1/2" from the edge on the 20" Paiste cymbal. The second point (SEE) was located near the cup, but was still measured from the edge: 5” from the edge on the Avedis Zildjian, the New K. Zildjian, and the Old K. Zildjian cymbals; 5 1/2" from the edge on the 17" Paiste cymbal; and 6 1/2" from the edge on the 20" Paiste cymbal. Edge (1" in on small cymbals; 1 1/2" in on 20" Paiste ‘. cymbal) Cup (5" in on.Avedis, New K., and Old K. Zildjian cymbals; 5 l/2" in on 17” Paiste cymbal; 6 1/2” in on 20" Paiste cymbal) Figure 16.--Striking Points Used on Cymbals At plane of ti. The both amateu The nf, and pp) Company (#1 microphone-- height of 6 set on "C" I only by the The for both tri Table l.-- 40 At both striking points, the implement was parallel to the plane Of the cymbal at impact (0° striking angle). These striking points and angles are typical of those used by both amateur and professional percussionists. Striking Force or Loudness The force or loudness Of the various dynamic levels used (ff, mf, and pp) was measured by a sound level meter built by General Radio Company (#1551-C). The meter was held at the same position as the microphone--at a distance of 5 l/4' from the instruments and at a height Of 6 1/2'. The meter was set on "fast” and the weighting was set on "C" (thus insuring equal influence from 20 to 20000 cps, limited only by the capabilities of the microphone). The dynamic levels and their decibel equivalents at impact for both triangles and suspended cymbals are shown in Table 1. Table l.--Decibel Levels at Impact = Dynamic Level Decibel Level Triangles Cymbals ff 89 95 mf 80 85 PP 74 78 41 Consistency Of these levels was maintained throughout the recording sessions by a combined use of the arbitrary scale on the striking mechanism and the VU meter on the recorder. Recording Equipment and Studio The microphone used was a Neuman condensor microphone (#C-47/64). For both the triangles and cymbals, the microphone was positioned 5 1/4' from the instruments at a height of 6 1/2'. The recorder used was a.Magnecord (#1028). The recording was done on "Channel 1,” with the record level set at "4" and with a tape speed of 15" per second. The tape used was Scotch #210, cut into 31" loops spliced with Scotch splicing tape. The overall dimensions of the recording studio were: length = 21', width = 17 1/4', and height = 8 1/4'. The locations of the microphone stand, the instrument stands, other equipment, and miscellaneous furniture, as well as the presence of various wall materials, are indicated in Figure 17. The numbers inside the outlines of the filing cabinets and bookshelves refer to the heights of these items. Playback and Analyzing Equipment The playback machine was an.Ampex console-mounted Recorder/ Reproducer (#AG-350). The playback levels used were ”8” for the triangles and "7" for the cymbals. The sounds of the instruments were analyzed by a Brubl and Kjaer Frequency Analyzer (#2107) and the resulting graphs were printed out by a Bruél and Kjaer Level Recorder (#2305)- Figures 18, 19, and 20 show the playback and analyzing setup, as well as the settings used on the analyzing equipment. 42 eeeeem meeeeouem--.AH menace any .o.HN 1.“ . Hoummam wouofimm xuou _ - Ill . | II goon maoom , r .o .e .e 4 eeemeoe seem .e eHeee we on woucfimm .Ooo figpm moaflh uamnom< :m / \\aooe:»umoH Beam, ‘ .mellv .ma.m , .mN.AH Hm .m~.m e. no omqu xx mum \ 20 am. gamuhflu\ \ .mm m 1. mafia oumam cameo oomum moonoonuwz Mmom cameo Homemwsom mofieuooom mo>Honm xoom .m OHHm .N illL IE... mcwmpnso moan oomam ~.—_—_—————- Flexible D Figure 18.- 43 B 8 K Frequency Analyzer Flexible Drive Shaft / B 8 K Level Recorder Earphones Figure l8.--P1ayback and Analyzing Equipment 1 Ampex Recorder / Reproducer 44 Input Potentiometer H5" ”Direct” Weighting Network "Linear 20 - 40,000” Frequency Range - c/s ”200 - 630 630 - 2000 2000 - 6300 6300 - 20,000" .Meter Range "80 dB SL -40 dB 100 mv" Meter Switch "Fast - RMS" Range.Multiplier "-20 dB x 0.1" Frequency Rejection "Balance” Frequency.Analysis Octave Selectivity "40 dB" Function Selector "Auto" Figure l9.--Settings on Frequency Analyzer #2107 Potent: Inpur Figure 20 4S Potentiometer "SO" "4" Range - dB Rectifier Response HM" Paper Speed - sec. ”101" Input Potentiometer Lower Limiting Frequency - c/s "20" Writing Speed - mm/sec. 'IlOOI' Drive Shaft Speed - rpm "0.36" Input.Attenuator "30H Figure 20.--Settings on Level Recorder #2305 46 Procedure Experimental Following the proper placement and adjustment of all necessary equipment, the various sounds that were produced on the triangles and cymbals were recorded. To avoid the influence of initial or impact transient sounds, the recorder was activated slightly less than one second after impact.1 Within the limits of the Operator's reflexes, there was little or no overlap of recorded sound on the tape loops. This meant that a full two seconds of analyzable sound was obtained. The tape loops were then played back through a reproducer and the sounds analyzed by a frequency analyzer. The resulting graphs of the sounding partials (from 20 to 20000 cps) and their relative strengths or intensities in decibels were printed out by a level recorder (see Appendix B for examples of the graphs). Mathematical The mathematical investigation of the triangles' vibrational behavior was accomplished through the use of the finite element method. The principle of the finite element method is: "a structure may be satisfactorily represented by an assembly of discrete elements having "2 and which are connected with each other at a "finite number of nodal points."3 In this investigation, each simplified elastic properties, hAlthough the author recognizes that initial transients are an important aspect of instrumental timbre, their influence was intentionally avoided in this study. ZRobert W. Little, "Finite Element Method," p. l. 3O.C. Zienkiewicz, The Finite Element Method in Structural and Continuum Mechanics, p. 1. 47 triangle was represented by fifteen elements having sixteen nodal points, with point sixteen fixed to eliminate rigid body motions (Figure 21). l 1011 Nodal Points 10 12 12 9 9 l3 l3 8 l4 8 l4 7 15 7 Elemenf:) 15 6 5 4 3 2 1 1° . Cr 1g.1, Ct tor ;CD 5 4 3 2 1 Figure 21.--Numbering of Elements and Nodal Points On the basis of each element's physical characteristics and geometry, the nodal forces and displacements (axial, bending, and torsional) may be calculated for each individual element. The displace- ments, forces, stiffness characteristics, and mass relevant to each element are expressed in matrix form, When these individual matrices are properly assembled into large matrices for the entire triangle, the resulting matrix equation can then be solved for the resonant frequencies. 48 These resonant frequencies (expressed in cycles per second--cps) are functions of the expression. VEEA1I2 (the constants which can.be factored out of the matrices mentioned above). Two of these constants are material properties: Young's modulus of elasticity--E; and.mass density--p (steel = ;Z§§). The other constants are geometric properties: 386 the moments of inertia resisting twist about the y and z axes--I (—%%: where d = diameter); the cross-sectional area-rA; and the length-~1. Therefore, with a minimum of calculation, it is possible to ascertain the relative influence on the frequency of a change in material, diameter, or length. Another material property which should be noted here-~damping--has been described by Wood: ”Any source of sound if set in vibration and left to itself vibrates in its own natural frequency, producing a note which gradually dies away . . . but remains constant in pitch."1 Thus, triangles made of metals with differing damping properties will produce sounds of varying duration. This particular property, however, is independent of any of the above material or geometric properties and does not affect the frequency in any way. The actual generation and manipulation of the matrices was done by the Structural Analysis and.Matrix Interpretive System.(SAMIS) computer program developed by the Philco Corporation, Western Development Laboratories, under contract to and in association with the Jet Propulsion Laboratory. The objective of this program.is "to automate analysis of structures composed of . . . line elements with unifonm cross-sections. 1Wood, Physics of Music, p. 23. 49 This includes predictions of deflections and stresses . . . and in addition, resonant frequencies can be Obtained.”1 The SNMIS program is a segmented system within the guidelines of the FORTRAN II computer language. The selection or sequencing of the various segments is controlled by the user and is accomplished by writing a set of pseudo instructions ("a pseudo instruction calls for a set of subprograms to perform a matrix operation rather than defining each step of the operationVZ). In addition to the pseudo instructions, the principal input data for this investigation included material tables and element data. The material tables define the mechanical properties of the material(s) used in the various elements, and the element data defines the "local geometry (member thickness, cross-sectional areas, moments of inertia), grid- points (numbers and locations), coordinate systems, temperature, weight, and pressure on each structural element."3 (The works of Lang, and Melosh and Christiansen are recommended for further reference to the SAMIS program.4’5) LMLE. Lakser, User's Guide-~Structural.Analysis and Matrix Interpretive System (SAMIS), p. l-l. zTheodore E. Lang, Summary of the Functions and Capabilities of the Structural Analysis and.Matrix Interpretive System Computer Program #32-1075, p. 2. 3RObert J. Melosh, Philip.A,Diether, and Mary Brennan, Structural Analysis and.Matrix Interpretive System (SAMIS) Program.Report #33-307, Revision #1, p. 107. 4Theodore E. Lang, Structural Analysis and.Matrix Interpretive Systemg(SAMIS)gUSer Report #33-305. 5Robert J. Melosh and Henry N. Christiansen, Structural Analysis and.Matrix Interpretive System (SAMIS) Program: Technical Report #33-311. 50 The principal output data was of two types. The first was a listing of the frequencies (in cps) of the predicted partials. The second was a matrix indicating the relative movements (in three- dimensional space) of each nodal point for each partial. From this data, diagrams were plotted showing the actual vibration patterns which ‘were responsible for producing the predicted frequencies (see Appendix C). Thre might be in} equipment. the presence 3 blank tape from 0 to 2" ground ”noi. Hon-ever, th 0f the into The 64-99 Cps. Points at 0 0f the trier However, as 0f the inhil in Tables \. Tod 0/321 III. REPORT AND DISCUSSION OF RESULTS Three graphs were printed to determine what, if any, sounds might be inherent in the recording studio, and the recording and playback equipment. .A graph printed with no tape on the playback machine indicated the presence of detectable sounds at 60 and 500 cps. .A graph printed from a blank tape indicated the same two frequencies, plus a band of "noise" from 0 to 200 cps. .A third graph printed from a tape of the studio back- ground "noise" indicated a broad band of "noise" from 0 to 2000 cps. However, these sounds and "noise" did not seem to influence the results of the investigation to any noticeable extent. The fundamental resonant frequencies of the studio were 53.33 cps, 64.99 cps, and 125.76 cps. Higher frequencies having nodal or antinodal points at or near the microphone location are shown in Table 2. Several of the triangles and cymbals produced frequencies near those in Table 2. However, as all recording was done in the same studio, the extent (if any) of the inhibiting or reinforcing effects of the standing waves was not isolated. Triangles Experimental Results The experimental results of the six triangles examined are listed in Tables 3 through 8 under these headings: implement (Imp)--7/32" drill rod (7/32 Drill), 7/32" cold-rolled steel (7/32 Cold), 5/32" drill rod 51 52 (5/32 Drill); striking point1 (St Pt)--top (Top), corner (Cor), bottom (Bot); dynamic level (Lev)--ff, mr, pp; fundamental (Fund); upper limit (UL); and Energy Peaks in Decreasing Order of Intensity. Table 2.--Frequencies Having Nodal or Antinodal Points At or Near the MicrOphone Location Frequencies in cps Nodes Antinodes 266.65 259.96 454.93 319.98 533.30 789.88 628.80 879.32 909.86 1013.27 1066.60 2369.64 1131.84 2637.96 1819.72 3039.81 2133.20 7108.92 2389.44 7913.88 3639.44 9117.43 4266.40 4778.88 7278.88 8532.80 9557.76 14557.76 17065.60 19115.52 The fundamentals, upper limits, and energy peaks are given in cycles per second (cps). The energy peaks are further identified by decibel ratings (dB)--e.g., 7150 / 33 indicates a frequency of 7150 cps at an intensity level of 33 dB. It should be noted that the energy peaks, or partials are listed in decreasing order of intensity and 295.1“ order of frequency. 1It should be noted that the tOp and bottom striking points were used with a 90° striking angle, and the corner striking point was used with a 0° striking angle. 53 Each table is accompanied by a discussion of the effects (if any) on each triangle's overtone structures produced by changes in implement size and material, dynamic level, and striking point and angle. General trends in pitch levels and overtone strengths or intensities are also noted for each triangle. Abel Triangle (Table 3) Implement Size and Material There were no consistent differences in the overtone structures produced by the two éiE§§.°f implements on the Abel triangle. There were no consistent changes in the overtone structures produced by the two kinds of implements used on the Abel triangle. Dynamic Level The ff level produced two effects on the.Abel triangle's overtone structures: higher upper limits were produced;-and the amplitudes of the partials were generally increased. In two instances (7/32" drill rod / top, and 7/32" drill rod / corner), more partials were produced at the pp level than at the ff level. And more partials were produced at the ff level than at the pp level in only two instances (5/32" drill rod / top, and 5/32” drill rod / bottom). The remaining five instances exhibited the same number of partials at both dynamic levels. Striking Point and Angle The two larger implements produced more partials at the bottom than at the top striking point. However, both of these implements produced a higher strongest partial at the top than at the bottom striking 54- NH coon co cocN NN coco oN cch mm coco cocco coco co co ccomo mo coco NH coNcN HN coco mN ccHN cm cmoN cm ccem cc coco cocoa coco we poo co cccmo cc cmNco mN coco cN coce NN ccoc cN chN cm cocN cm comm cocoa coco we NH coco no ccoo mo comma co ooeN oN coco cN cmoe NN coon cocoa coco co co comma NN ccoc HN coco mN cmeN oN cho cN come on coon cocNN coco we ecu Hooec NN cocoa No coco no come on coco mm coon mo cmeN mo coNe +ccch como me Nc\m o cccoo o comma co coco cN cch cN coHN cN coco Nm comm No coco cocoa coco co no ccccH mu cocoa HN cmNcH cN coco oN coco on come on coco om ocNN He coon No coco ccch coco us ooh NH coco cN coco cN comma cm coco on cch on ccoc cc ocNm co ccem no ccce +ccch coco co co occco NH cch co occoo mN coHc cN cocoa cN coNN mm comm em cch No coco coch cch co HN cccmo NN coco NN cocoa o cooc NN ccoN oN coco cm comm on che mm cch om ccce cocoa coco we ooc MN ccmcc cN ccmNo NN cmmN oN.cocc on coco Nm coco mm coco cm coco co coco cccoo coco co co coco cc come co mNeN co ocmeo NN oooe NN chc on coco cocoa coco oo co mNmN mN cocoa cN mNoN mm cho mm cmoc om coco No coco cccco coco we coo coco cN comma Nm ccmN on coco mm ccmN co coco No coco co coco eocch coco cc Nm\N mo cccmo mo coco cN cmoN NN coco NN coco NN coco cm comm cocoo coco oo mo cocoa HN ccHN NN cch oN mNoH cm coco om coco No comm ccoNo mNco we coo . co cocoa oN coco cN coco on chN co cch co comm cm coco +ccch coco me No cccmc NH cocoa co coco co mNcH NN chN oN coco NN comm mm mNcN No coco ccomo coco oo oo comma No comma oN cocoa co occo HN cch NN och NN ccmc cN coNN no coco ccoNH coco we ooc Ho cocoa oN occmo eN coco cN coco cN chN oN coco cm coco cc cch No coco +ccch coco co mo cccmo NH cocoa cN cch HN mNmH mN coon NN come cN coco cocoa mNmo co cN cmeN NN occcc cN mNmH cm coco em couo om come cm coNe cccoo mNmo co e8 Hooec c coco co coco eN cocoa om coco Ho ococ +ccch coco oc NN\N co cocoa mo cccmo oN cch AN coco NN coco cm chc om coco co ccem cccco coco cc NH cmomo on occmo oN coco NN coco NN och No coco co coco No comm cocoa coco be: coo No coco cN ccoco on coco No cch on chN co coco cm comm +ccch coco co me moo a: may no moo no moo me one no moo me moo m1 moo me moo no moo woo woo >94 um um ash xpomcouco mo pooeo acommouuom co mxeoo xmuocm a: vase oawomoae Hoo<.onu mo mxmom xmuoom pom .muosaq homo: . moeooofiofio: . m oocec 55 point. The smaller implement produced more partials at the top striking point, and the most intense first partials were the same for both the top and bottom striking points. The 0° striking angle consistently produced a lower fundamental (325 to 375 cps lower) than the 90° striking angle. With only one exception (7/32" cold-rolled steel / tOp) , fewer partials were produced with the 0° striking angle than with the 90° striking angle. Pitch Level and Overtone Intensities The Abel triangle's partials seemed to fall into two sections: the majority of the partials occurred from 1500 to about 9000 cps; and a smaller number occurred from about 10500 to 16000 cps. One 17000 cps partial and two 18000 cps partials were found. The Abel triangle produced two rather definite fundamentals: 1900 to 1950 cps at the tOp and bottom striking points; and 1525 to 1575 cps at the corner striking point with the 0° striking angle. These partials were rated as high as the fourth most intense partial in only seven of the eighteen examples, and were the weakest partial in another six instances. The most intense partial was the third partial in seventeen of the twenty-seven examples and the fourth partial in another eight instances. The two strongest energy peaks most frequently occurred near the frequencies of 4000 and 5500 cps. @ig Triangle (Table 4) Implement Size and Material There was no consistent variation in the Ludwig triangle's OVeI‘tone structures due to a change in implement size. A change in the 5(5 HH cccm HH cccH mH cmHN cN och mm cch occHH occH mm oH cccm NH cmcH mH occHH HN comm HN cmHN cN cch mm comm cccmH cmcH mo” com mH occHH mH cccm cN comm cN cccm cN mNcH mN coco Nm cccm cc cch cccmH mNmH mm m chH cH cccHH NH chm mN comm mN ccmc cm con ccoNH coNH mm m chH cH cccH mH cocN HN occHH NN cmNm mN comm mm chm cm ccmc cccmH chH mc_ moo HHHmo cH cmNH mH cccHH HN cocN NN ccHN mN chm mm ocmc Hc comm mo cccm ccocH cmNH mm NmNm m cccmH NH cchH mH cccHH cH cch NH cccH HN ccmm mm cccm mm cccm ccccH cccH mm HH ccmmH NH occHH NH chm mm cccm mm mNcH Ho cch cc coco cccmH mNcH mac mom HH cmNNH mH occHH cH cchH mH chm Hm coco mm mNcH No cch mc coco cccmH mNcH mm m ccmcH cH comm NH comm mH cmNcH mH coNc HN omHN NN cch NN cccm cN cmcH ccoNH cmcH firm oH occHH mH cccm cN comm mN cmcH mN ccmo mm cch cc comm cccmH cmcH .mc com cH cccmH mH ccmNH mH cccm mN ccmo Hm comm mm cmcH mm chc Hc comm cc ccmN cccmH cmcH mm N cmNH NH cccHH cN ccmm mN ccmo mN cmNm Nm comm mm comm ccoNH cm~H mm m cmNH HH occHH HH cch NH ooHN mN comm Hm comm cm comm mm ccmc cccmH cmNH _mc moo cHoo N chH NH cmNcH mN chm cN ccHN cN cch Nm comm mo ccmm co comm cccmH chH mm NmNN m cccmH cH cmNNH mH chm mH comm cN cch mN coco oN mNcH mN cccm cccmH mNcH mm m occHH cH commH mH comm mN cch mN cccH cm coco cc ccmc cccmH cccH _mc mom cH cccmH mH occHH NN comm Nm cccm mm cccm mN cccH Hm cccm mm cch cccmH occH mm HH cccm NH cccHH mH cccH mN coco mN comm cm cch cocNH cccH mm cH chm HH cmcH cH cmNHH mH chc Hm cch mm comm cccmH cmvH cs pom HH ccmNH mH chm mm cccH Nm cch mm chc mm comm cccmH cccH mm m chH cH cch NH comm cH cccHH mH coco HN comm Hm ommc cm cmNm cccmH chH mm m chH cH cch HH cmcH HH cccmH cH cccHH mH comm mN comm mm comm mm comm Nm cmNm ccccH chH Ame h8 HHHmc HH chH mH ccmH mH coHN oH cch cN occHH cc ccmc Ho cccc No comm cccmH chH mm NmNN NH ccmNH cH ccHN mH cccm mN comm mN cccH mN cch cm cccm cccmH cccH mm mH cccmH mH ccoNH mH occHH mH cccm mN cch mN cccH Hc coco No cccm cccmH cccH me mom cH cccmH oH cchH mN comm cm coHN mm cch mm cccH Hc cccm mc comm cccmH cccH mm mo moo mp ago no moo Bo moo mo moo mo «do So moo Bo moo mo moo mo moo moo moo >3 um um 95 33555 mo .355 933983 5 93mm .335 .5 BE ofimqmmhh mazpzq map mo mxmom ammoqm paw .mumEHq Homo: .mHmucoewwq: --.v oHan 57 implement material had no consistent effect on the overtone structures of the Ludwig triangle. Dynamic Level The louder dynamic level (ff) seemed to have two effects on the Ludwig triangle's overtone structures: the amplitudes of the partials were generally increased; and higher upper limits were produced. In one instance (7/32" cold-rolled steel / bottom), the pp level produced more partials than the ff level. The ff level produced more overtones than the pp level in feur instances (7/32" drill rod / top, 7/32" cold-rolled steel / corner, 5/32” drill rod / corner, and 5/32" drill rod / bottom). The number of partials was the same at the two dynamic levels in the remaining four examples. Striking Point and Angle One noticeable effect of a change in striking point was the higher upper limits produced at the top striking point. The partials produced at the top striking point were also generally of greater intensity than those produced at the bottom striking point. The fUndamental frequencies produced by the 0° striking angle were consistently 150 to 200 cps lower than those produced by the 90° striking angle. In addition, the most intense partials produced by the 0° striking angle were generally higher than those produced by the 90° striking angle. With only two exceptions (5/32" drill rod / corner / pp, and 7/32” drill rod / corner / mf), the 0° striking angle produced a lower top partial (around 11000 cps) than the 90° striking angle (around 15000 cps). 58 Pitch Level and Overtone Intensities TWO main groups of partials were found for the Ludwig triangle. The majority of the partials occurred between 1400 and 7000 cps, with a small number occurring between about 8800 and 15000 cps. One 18000 cps partial, one 17000 cps partial, and two 15500 cps partials were found. Two rather definite fundamental partials were produced on the Ludwig triangle: 1400 to 1450 cps at the t0p and bottom striking points, and 1200 to 1250 cps at the corner striking point with a 0° striking angle. However, these frequencies occurred as the most intense partial only once (7/32" cold-rolled steel / bottom / PP), and were usually the third most intense partial at the top striking point, the weakest at the corner striking point, and the fourth or fifth.most intense partial at the bottom striking point. There was no consistency as to which partial was the most intense on the Ludwig triangle. However, the three strongest energy peaks at the top striking point occurred most frequently around 4500, 6600, and 1400 cps. The three strongest energy peaks at the bottom striking point occurred most frequently around 2400, 5500, and 4700 cps. And the three strongest energy peaks at the corner striking point occurred.most frequently around 5750, 3900, and 4800 cps. Sonor Triangle (Table 5) Implement Size and.Material .A change of implement siz§_produced no consistent changes in the overtone structures of the Sonor triangle. The use of different implement materials did not seem to have any consistent effect on the Sonor triangle's overtone structures. mamgfihh hocom 0:“ MO 93on Ehocm pcm .mquHq hunk: .mHmucoEHwUCJuTIQ cacao SE) c com m ccmH mN ccmm cm cccc cm ccmN mm coco cc cccm ccoNH ccm mm o ccm NH cmNH cH cccoH mH cccH mH chcH mN cccm NN cccm Nm cccc cm coco mm cch cccmH ccm coo pom cH cccmH mH cNm mN cccH mm cccm mm comm mm cch Hm cccm cm coco cccmH cNm mm m ccHH cH cmNN oH chcH mN coco mN cccm cN comN Hm comm cm coco ccoNH ccHH mm . c ocNH N cccmH m ccmcH HH ccmm mH ccoH mH ccmc mH cccm cH comN N coco HN comm cccmH chH mom moo HHHmc cH cNm NH cccmH cH occHH cN cccm N cmNm mNcoccm cN cch cm coco Hm cccH mm ccmc cccmH cNm mm NmNm m cccH HH cNm mH cmNmH 4H cccmH HN chcH mN cmNN N cccm om ccmc Nm cccm mm comm mm coco ccch cNm mm cH chmH cN cNm HN chcH mN ccmm mN cccH mm cccm mm cccm N.”1.Hc mm cccm cccmH cNm moo mom cH cchH mH cNm NN ccccH cm ccccH cm cccm mN comN mm comm cc cccm mm cccm +ccch cNm mm N mmm N chH HH ccccH mH cccH oH ccmcH mH cccm cN ccmN cm cccm Hm cccm cm coco ccoNH mmm mo NH cNm mH ccmmH NH ccccH mN ccmcH NN comm mN cccm N cccH mm ccmN mm coco mm cccm .N cccm +ccch cNm cc_ com NH cNm mH ccmcH cN ccccH cm cccH Hm cch mm comm mm ccmc mm cccm cc cccc No cccm +ccch cNm cm L N cmNN m cmNH cH cmNN NN cccm N coco cm ccmN Hm comm occHH cmNH mm c ccmH mH cchH NH cmNNH cN cmmc cN ccmcH N comm NN cccc mN ccmm cccmH ccmH c=_ too cHoc m cmNH m cccH mH cccmH cN ccmm HN cccm Hm ccmc Nm comm mm ccmN mo coco cccmH cmNH cc NmNN . m omm o cccH m occHH cH ccoNH cN comN mN cccm mN cccm mN coco cm ccmc cccmH omm .mcc m cmm cN mNmH NN ochH mN comN Nm comm mm cmmN mm cccm mm coco mm cccm cccmH omm com com HH cmNmH mH cNm Hm comN mm cccH cm comm cm cccm mm cccm mm cccm Nc cccc cccmH cNm co m cNm cH_cchH cH ccccH mH cccm cN cccm mN comm m- cccm mN comN mm cccc cccmH cNm mm m cNm cH cccmH cH cchH mm cch cm cccm cm comm om cccH No comm ccoNH cNm com com _ HH cNm mH cccmH mH occoH mN ccmcH mm cccH cm comm mm cccc cc cccm Hm ccmN Nc cccc mm cccm .ccch cNm cc . m cmNH HH cccmH mH ccccH NH coco HN cch NN coco cN comm ccccH cmNH mm c ccHH NH cccmH mH ccmcH mN cccm cm cccm Nm comm mm cch cc coco cccmH ccHH cc“ moo cHHc: m occHH m cmNH HH comm NH cmcm cH cmNN mH cccH NN ccmc Nm comm cchH cmNH mH NmNN c ccm m cccH mH ccmcH mN cccm N cccm mN comm Nm cccc mm cccm cccmH ccm mm o ccm cH cmNcH.HN cmmH mN chcH mN cccm mm cccm Nm mNmN co cccm mm cccc ccoNH com mc_ com HH ccccH HN ch NN ccmcH mm comN mm mNmH cc cccm No cccm mo cccm cc cccm cchH ch cc mo. moo mo moo mp moo mo moo rd moo Bo. moo no moo Bo moo M8 moo 33— 3.3 33 moo moo moo H . c. .nu.mcH mm mm maH 33:35 .3 .820 9:83qu 5 mmmom 3.8:.» .5 2:: mHmcmth Hocom ecu mo mxmom mmmocm vow .mumqu Home: .mHmucoEmpaSm--.m manmc 60 Dynamic Level The ff level consistently increased the amplitudes of the partials, but did not alter the basic overtone structures of the Sonor triangle. In all but two instances (7/32" drill rod / corner, and 7/32" cold-rolled steel / top), the ff level produced higher upper limits than did the pp level. The pp level produced more overtones than the ff level in only one instance (5/32” drill rod / top). The two levels produced equal numbers of overtones in two instances (7/32” cold-rolled steel / top, and 7/32” cold-rolled steel / bottom). In the remaining six instances, the ff level produced more overtones than the pp level. Striking Point and.Angle With the two larger implements, the upper limits produced at the bottom striking point were higher than those produced at the top striking point. The 0° striking angle consistently produced a higher fUndamental (about 300 cps higher) than the 90° striking angle. The upper limits produced by the 0° striking angle were generally lower than those with the 90° striking angle. With the exception of the 5/32" drill rod implement, the 0° striking angle produced fewer partials than did the 90° striking angle. The partials produced with the 0° striking angle were generally of lesser intensity than those produced with the 90° striking angle. Pitch Level and Overtone Intensities .Although the Sonor triangle did produce two rather definite fundamentals (900 to 920 cps at the t0p and bottom striking points, and 1200 to 1250 cps at the corner striking point with 3 0° striking angle), 61 these frequencies were consistently one of the two weakest partials (nineteen were the weakest and eight were the next-toeweakest). The two most intense partials at the top and bottom striking points generally occurred around 4400 and 6000 cps, and those at the corner striking point generally occurred around 3500 and 6400 cps. The majority of the Sonor triangle's partials occurred below 10500 cps. The few partials above that level were all weak and only slightly more intense than the fUndamentals noted above. There was no consistency as to which partial was the most intense: the second partial twice; the third partial seven times; the fourth partial four times; the fifth partial seven times; the sixth partial six times; and the eighth partial once. Zildjian Triangle (Table 6) Implement Size andeaterial No consistent changes in the Zildjian triangle's overtone structures could be attributed to a change of implement size, ‘With only three exceptions (7/32" cold-rolled steel / top / mf, 7/32" cold-rolled steel / top / pp, and 7/32" cold-rolled steel / corner / pp), the cold- rolled steel implement produced more partials in each instance than did the drill rod implement. Dynamic Level The ff level produced higher upper limits in all but one instance (5/32” drill rod / bottom). The ff level produced more overtones than did the pp level in seven instances, and an equal number of overtones in the remaining two instances (7/32" drill rod / top, and 5/32" drill (32 cH cNHH NH cmNm oH cmmH mH ccmN mH cmNm NN cch mN coco ccoNH cNHH mm NH cNHH mH cmmH mH cch mN chm mN ccmN mN chm mm cccm cccmH cNHH ms com NH cccm mH ccmH mH cNHH cH cmNm mH ccmH HN cccm NN ccmN mm coco ccoNH cNHH mm c cmm m ocNH HH cccm NH chN mH cccm mH cccm cH cmHm cN ccmm occHH cmm Hem o ocNH m omm cH cch mN coco cN cmNm mN cmNN Hm chm cm comm cccmH com me mac HHHmc mH ccmH NH cmm cN comm HN cccm cN cch mN chN mN chm mm comm ccccH omm mm NmNm m ccHH cH cmmH HH cccHH NH ccmH cN cmmN mN coco mm chm cm cccc cccmH ccHH mm m ccmNH HH cNHH cH chm mN comN mN cmmH Nm coco mm chm cc cch cccmH cNHH cc mom mH comm mH cmNcH cN ccmN HN cox. mN cNHH cm cccm Nm cch mo cmNm mm cch ccccH cNHH mm m chcH cH cHHH mH comN mH cch mH cch cH cccm mH cmNm NN chc cccm cHaH mm mH ccmcH mH cNHH mH ccmH oH chN mH cmNm mH cccm NN cch mN coco mN cch cccmH chH we pom NH ccmH NH cNHH HN ccmcH HN cch NN cccm cm comm mm cccm mm cccm mm cch cccoH cNHH mm o cccm N cmm m cmNH m chm mH cmNm NN comm ccoNH omm mm N cNm m mNNH m cch mH chm mN chN mN cccm oN chm Nm comm mm cccm ccoNH cNm mc_ .mou cHoc HH mNcH NH cccmH mH ccmm mH cNm mH cch cN chN mN cccm cm chm mm cmmc Nm comm cchH cNm mm NmNN m ccHH cH chcH mH cch cN comN cN cmo mN cccm Hm cch ccoNH ccHH mo HH ccHH HH cmmH NH ccmNH NN comN mm cccm mm cch No chm ccccH ccHH com com mH ccmNH cH ccmH mH ccmcH cN ccmH mN ccHH mN cccm mN comm mm cch cc cmNm cccmH ccHH mm HH comm mH cccm cH cmmH mH cNHH NN cccm cN cch mN coco cccHH cNHH mm NH ccmH HN ccmH NN cNHH cN coco Hm cmNm cccHH cNaH me com NN ccmH NN cNHH mN cch mN cccm cm cmmH cm cmNm Hm cccm mm cccm cccmH cNHH mm N cNm m chH mH chm mN chN cN chm mN coco cm comm ccoNH cNm cm N cmNH m ccm cH cccm mH coco mN cmNN Hm comm mm chm ccoNH ccm “cm moo HHHNc m ccmH cH cchH mH cmm NN cccm cN chm NN coco cm chN Nm cmmm mm chc ccccH omm mm NmNN N ccHH m cmNH m cccmH HH cccm mH ccmH mH ccmN mm cccm mm chm mm cch cccmH ccHH mm HH ccmNH cH ccHH NN ccmm oN ccmH mN ccmN om cccm cm cccc Nm cccm No chm cccmH ccHH mm" mom cH cmNNH mH ccmH cN ccccH HN ccHH NN ccmH Hm ccmN Nm coco Hc cmNm Nc chc cccmH ccHH mm mo moo mu moo mp moo m0 moo Bo moo mo moo mp moo m0 moo mo 30 mo moo moo moo >3 um um 95 bmmcoufi mo .830 $333qu 5 8360 3.85 .5 0:3 ll onfimmc 5&3: of ma 98m 3.85 com .323 8mm: .mHficmsmccan--.c Bea. 63 rod / corner). The ff level also generally increased the amplitudes of the various partials without altering the basic overtone structures. Striking Point and Angle With only one exception (5/32" drill rod / top / mf), the overtone structures produced at the top striking point had higher upper limits than those produced at the bottom striking point. While partials with frequencies of 11000 to 13000 cps were consistently present in the overtone structures of the top striking point, the partials of the bottom striking point were consistently lower: none higher than 8000 cps with the large drill rod implement; none higher than 10800 cps With the cold- rolled steel implement; and none higher than 9700 CpS'With the small drill rod implement. With only two exceptions (7/32" cold-rolled steel / top / mf, and 7/32" cold-rolled steel / top / pp), more partials were produced at the top striking point than at the bottom striking point. The partials produced at the top striking point were generally of greater intensity than those produced at the bottom striking point. .A striking angle of 0° produced a consistently lower fundamental partial (about 300 cps lower) than the 90° striking angle. The upper limits produced by the 0° striking angle were always lower than, or the same as, the upper limits produced by the 90° striking angle. The partials produced with the 0° striking angle were generally of lesser intensity than those produced with the 90° striking angle. Pitch Level and Overtone Intensities The Zildjian triangle's partials seemed to occur in three sections: from 900 to 5000 cps; from 6000 to 10800 cps; and from 11000 to 17000 cps. 64 The first group (lower frequencies) contained the greatest number of partials, and the third group (higher frequencies) contained the least (only nine partials in all). The Zildjian triangle produced two rather definite fUndamental partials: 1100 to 1120 cps at the top and bottom striking points; and 900 to 920 cps at the corner striking point with a 0° striking sngle. However, these frequencies were rated higher than the sixth.most intense partial only three times: the third most intense--7/32" drill rod / bottom / mf; the fourth most intense--7/32" drill rod / bottom / pp; and the fifth most intense--5/32" drill rod / top / ff. And these frequencies were the weakest partials in nine instances. The three strongest energy peaks at the top and bottom striking points generally occurred near the frequencies of 4100, 3250, and 6400 cps. The three strongest energy peaks at the corner striking point were generally near the frequencies of 3200, 3900, and 7700 cps. There was no consistency as to which partial was the most intense. However, the fourth partial was the most intense nine times and the fifth partial was the most intense eleven times. 6” Pigstail Triangle (Table 7) Implement Size and Material .A difference in implement size did not have any consistent effect on the overtone structures of the small Pigstail triangle. .A difference in implement material did not seem to have any consistent effect on the small Pigstail triangle's overtone structures. 6S N cmNH o ccoNH HH cmo cH comH NH ccoN mH comm oH come oN coco HN chm «N cmNH cN coco ccccH cmo on mH cNo cH cch cN ccmH NN cmNH NN cmHm oN coco oN cmmm cc ccoc cooHH cNo we pom HH ccmNH NH ccmcH mN omoH cN coHo mN oNo oN ccmv Nm cmHm mm coco cN comm Nm cmNH cccmH cNo mm o ccmH NH cccmH mH ccccH cN comN cN comm NN coco cm comN Nm cccm cccmH ccmH co m ccmH mH ccccH HN comN mN coco cm cccm Nm cccm mm comN ccoNH ccmH oo_ moo HHHHo «H comN mH ccmNH NH coco NH comN mH ccoo oH ccmH cN cmmm mN comm om comm occcH ccmH oo NmNm o ccoNH VH cNo oH ccmmH mH occHH cN coNo NN cmNH NN cmHm mN comm mm comm cm ccoc cccoH cNo co m mNNH cH ccHo cH comN cN cNo mN cmNH NN chm Hm comc Nm ccoc Nm comm ccoHH cNo moo coo cH cccmH mH ccmcH mN cNo on chN om cmHm oN mNNH Ho comm No ococ mc ccmc cccoH cNo cc o coo HH cmNNH NH occHH «N ccoN mN cmNH cN cccm mN ccom mm ccoc mm cmHm ocooH coo on o chcH NH omo cN ccmH mN coco NN comN mN ommv oN cmNH cm con cccmH cmo we Hom mH cccmH cH cNo cH cccmH mN coco Hm cmmN cm cmNH cm ccmc Ne ccmc cc con cccoH cNo mm o ccmH o occHH cH cccmH mH coco mH comN NN coco mN cccm NN ccmN mm cccm ooocH ccmH on «H ccmNH mH occHH NH ccmH cN cccmH mN ccmN mm comm mm cmmN om con Nm coco coch ccmH .oc Hou cHoo VH ccmNH NH ccmH oH occHH cN cccmH Hm comN Nm comN cm coco cm comm Nm cccm ccch ccmH mm NmNN N cmNH o ocovH HH ccmcH HH oNo NH coco mH mNNH mm cmHm Nm comm cm comm Nm come ccoNH oNo no oH cNo VH ccmcH NH cmNH oH ooHo Nu cooN N come mN coov Hm mNHm mm comm cccmH oNo mac ooh VH coo oH cmNH oH cmNcH HN coco NN ocNN N~.mmNH Nm och co ocmc me come no comm cccmH coo om m omm mH cccm 3 cccHH 2 82 S ccmH N chm 3.. cccm mN ccmc mN cmNH cm comm cchH cmm mm NH occH mH cmNmH mN cmmH N comN oN comm cm ccmH mm cccm cc coco cccmH occH mom pom cH cccmH NH chcH NH cmo cN cmmH mN cccm Nm ccmo Nm ccmH mm coom cm con mm cccm cccoH cmo mm N ccmH cH occHH mH cccmH cN comN mN cccm «N comN cN coco oN comm cccmH ccmH on N ccmH o ocNNH NN cmNN m coco mm oon Nm comN Nm cccm coccH ccmH ocH you HHHHu oH ccmH NN oocN Nm comN on come Nm cccm ooomH ccmH mm NM\N N cmo m ocNH NH cmNcH cH cmNH mH coco NN comc mN cch No comm mm comm ccoNH omo on cH coo cH ccmH oH ocNH ac coco cN cmNN cm och Nm comm cm comm Nm ccmv ccoNH cNo we moo mH cccVH cH ccmH cN cNo mN cmcH NN coco oN cooN mm comm He ommm me come Ne cmHm cccmH cNo 00 mo moo mo moo mo moo mo moo mo moo mo ,moo mo moo mo moo mo moo mo mac mo woo woo moo >04 um um QEH mummnmpaH mo Nacho mammmouuoo :H mxmoo Nmmoqm H: ccsm II II onSmHHH. Hmmummmm :o 23 mo monom 3.85 com .323 Homo: .mHmuHHocBHocsm--. N oHomH. 66 Dynamic Level The ff level produced more partials than did the pp level in only two instances (7/32" drill rod / top, and 5/32" drill rod / corner), with the reverse occurring in three instances. The partials were of equal number in the other four examples. The ff level increased the amplitudes of the partials without actually altering the overall overtone structures of the small Pigstail triangle. Striking Point and Angle The partials under 2000 cps were of greater intensity when produced at the bottom striking point, but the partials above 2000 cps were generally of greater intensity when produced by striking at the top striking point. The top striking point produced six partials with frequencies of 11000+ cps, while the bottom striking point produced ten With only one exception (5/32” drill rod / bottom / pp), the bottom striking point produced upper limits which were higher than, or the same as, the upper limits produced by the top striking point. The 0° striking angle produced a fundamental frequency almost double that of the 90° striking angle: 1800 cps as compared to 950 to 980 cps.~ The 0° striking angle also consistently produced fewer partials than the 90° striking angle. The partials produced by the 0° striking angle were generally of lesser intensity than those produced by the 90° striking angle. Pitch Level and Overtone Intensities The partials produced at the top and bottom striking points seemed to appear in three groups: 900 to 1800 cps; 3100 to 5000 cps; 67 and 6300 cps and above. The third group (higher frequencies) usually contained the most partials, with a few occurring as high as 18000 cps. The partials produced at the corner striking point (with a 0° striking angle) also seemed to occur in three groups: 1800 to 3850 cps; 5000 to 7800 cps; and 9000 cps and above. The third group (higher frequencies) contained the fewest partials with only a few partials occurring as high as 17800 cps. . Ill‘l‘ The small Pigstail triangle produced two quite definite funda- mentals: 950 to 980 cps at the top and bottom striking points; and 1800 cps at the corner striking point with a 0° striking angle. However, these frequencies were generally quite weak, being rated higher than the seventh most intense partial only four times. And these frequencies were the weakest partials fourteen times. The three strongest energy peaks at the top and bottom striking points were near the frequencies of 3200, 3800, and 6300 cps, and the three strongest energy peaks at the corner striking point were near the frequencies of 5300, 3600, and 2800 cps. There was no consistency as to which partials were the most intense. 10” Pigstail Triangle (Table 8) Implement Size and.Mmterial With the exception of the ff level, the smaller implement (5/32" drill rod) produced slightly more intense partials than did the larger implement (7/32" drill rod). With only two exceptions (7/32" cold-rolled steel / top / ff, and 7/32" cold-rolled steel / top / mf), the cold-rolled steel implement produced slightly more intense partials than did the drill rod implement. (58 mH ccm «H com mH coo mH cmmH cH comN NH coco cN comN «N chm 0N cmmm cm ch« mm ch« cccmH com on HH cccmH mH ch HN com «N coo oN ccoN mN ccccH «m comN mm chm Nm ch« mm ch« c« cc«m ccccH ch weH pom HH ccccH «H c«« mN ccm mN cccm cN cmcH NN cco mN ccoN Nm comN cm ocNm Nm cc«m H« ch« N« ch« cchH c«« mm o cco cH ccmmH NH mN« mH cmNNH «H cmNcH cH ccoNH NH cccm NN ccoH mNocmmN oN ch« cm comm ccccH mN« on NH occcH mH cmNNH cH ccmmH NH c«« HN ccoc HN ocNH NN cm«H «N cccm mN ccoN oN cmmN Hm ch« mm comm ccoNH c«« mac you HHHHQ «H coo NH ccccH mH ccmNH «N c«« «N ccmcH mN cm«N NN ccoc cm cccm Nm ccoH mm cmmN H« ch« m« comm ccoNH c«« mm Nm\m NH ccccH mH ccmNH «H com mH ccm NN ccmo Hm ccH« Nm cmoN mm cccm cm cmmm ccoNH com on cH coo cH cccmH NH com mH ccmNH cN cc«o mm cmoN cm mNcH c« ccH« H« cmmm N« cccm ccoNH com Lac moo o ccm NH ccmmH mH coN mN ccmo oN cmNcH Hm cc«o cm comm Nm cmcH H« ccH« N« cmoN m« cc«m ccoNH com mm o cco HH ch mH comN mH ccmcH cH cmoN NH chH mm cc«m mm cmN« om ccm« ccoNH ch on cH ccm NH cmNmH NH coo oH cmcH oH com NN cocN cN comN NN comm mN ccmo Nm ch« c« ch« N« cc«m ccoNH com “a: pom NH cmc mH cccmH «H com mN ccm oN comN cm cmoN cm ccmo N« cco« m« ch« «« cc«m cccmH com mm m cco HH cm« mH ccmNH mH cmNcH cH ccmmH cN comm- NN och mN ccoH cN cccm NN comm mm cch ccoNH cm« no HH coo NH c«« mH cmNNH «H ccccH NH cmNcH NN ccoH mN cc«H NN ccoN oN cmNm Nm cmmN mm cmN« om comm cccmH c«« meH H00 cHou NH cNN mH coo oH ccccH oN mN«H oN chcH cm c«« Hm omoH mm comm cm ccoN c« ch« «« chm c« cccm cccmH c«« mm Nm\N o ch «H ccm mH coco cN cmoN NN mNcH Hm cccm cm cmmm mm ccH« cccmH com mm o ccmmH 0H cNm mH ccmcH mH cho NH chm 0H cmoN cm mNcH mm cccm mm cmmm oN ccH« ccc«H cNm .HE moo m ch NH cNm «H occHH NN coco «m mN«H c« 0000 H« cc«m c« cmN« occcH cNm um HH ch NH chH oH ch cN ccoN HN ccccH cm ch« Nm cc«m cccmH on on HH ccccH HH com mH ccoH «H com «N comN NN comN oN ccmo mm cmHm N« ccm« cccmH com mcH pom cH ccmH mH cmNmH mH ch cN cco NN ccoN mN ccm oN coco mm omHm m« ch« m« ccm« cccmH ch ww 0H ch NH 83 2 cm«H mH ccoN «H ccoN NH 0««. oH cmNNH cN cccm NN ccmcH oN comm 0003 c«« an 0H coo NH cmN«H mH ccoN oH cmNNH cm 0«« NN cmNcH «N ccoH «N co«H oN ch« Nm cch mm cccm 083 c«« .HE you HHHHQ mH coo cN cc«cH NN cm«H mN cccmH oN mNcH Nm cccm Nm ccoH «m c«« mm cmoc Nm cmN« H« cmmN m« comm cchH c«« mm NmNN cH mNo «H cccmH cH ch cH cmoN NH chcH mH ccoN oN coco HN cc«o NN ocNH NN cc«m oN cmNm Hm cmN« ccc«H ch on m co« «H ccmNH mN ccmo oN cmoN Nm cmcH mm ch« Nm cmmm c« cccm ccccH co« we ooh HH mNo NH ccm mH com «H cccmH mN cmNNH mN ccoN Hm cc«o mm cmoN N« chH N« chm c« cc«m o« ccH« cccmH com mm mo mao mo moo mo «no mo moo mo moo mo moo mo moo mo moo mo moo mo moo mo moo mo moo moo moo >oH um um meH NuHmcoucH mo Hocuo mchmouooa :H mxmoo Nmmocm .oo wean oncwHHH HmmummHm :cH.ocu mo mHmom Nmmocm cow .mHHEHH Homo: .mHmucoemccsm--.m oHcmN 69 Dynamic Level In eight of the nine examples, the ff level produced a higher upper limit than did the pp level. With only two exceptions (7/32" drill rod / top, and 7/32" cold-rolled steel / top), the ff level produced more partials than did the pp level. In addition, the partials produced at the ff level were generally of greater intensity than the partials produced at the pp level. Striking Point and Angle r The partials under 1000 cps were consistently more intense when ' produced at the bottom striking point; however, those partials between 1000 and 3000 cps were more intense when produced at the t0p striking point. The top striking point produced more partials between 9000 and 15000 cps than the bottom striking point did by a nineteen to eight margin. But the bottom striking point produced more partials above 15000 cps by a six to four margin. The 0° striking angle produced a consistently lower fundamental partial (60 to 70 cps lower) than that produced by the 90° striking angle. The 0° striking angle also produced more partials up to 2000 cps by a thirty-six to thirty margin. Pitch Level and Overtone Intensities The partials produced at the t0p and bottom striking points seemed to fall into feur groups, with decreasing numbers of partials: 490 to 2500 cps; 3100 to 5750 cps; 6400 to 8600 cps; and 9000 to 16000 cps. The partials produced at the corner striking point seemed to fall into three groups: 425 to 2000 cps; 2850 to 7000 cps; and 8500 to 70 16000 cps. The middle group had the most partials, and the last group (higher partials) had the fewest (including only seven partials above 13000 cps). .Although the large Pigstail triangle produced two rather definite fundamental partials (490 to 520 cps at the top and bottom striking points, and 440 cps at the corner striking point with a 0° striking angle), these frequencies were always relatively weak. They were rated higher than the seventh most intense partial only twice (7/32" drill rod / corner / ff, and 7/32" drill rod / corner / pp). The most frequent strong partials at the top and bottom striking points were near the frequencies of 4200, 3400, 4700, and 5600 cps. The most frequent strong partials at the corner striking point were near the frequencies of 3500, 4200, and 2850 cps. The most intense partials were usually either the sixth or seventh partials, but there was no consistent pattern with regard to any of the variations in implements or dynamics. Summary of Experimental Results and Related Research Instrument Size and.Mbterial Recalling that frequency is a function of the expression nggié (see above on p. 48), it is evident that changes in the geometric properties (I, A, and l) affect the resonant frequency of the triangle. With all other properties constant, these relationships are valid: 3 longer length lowers the resonant frequency; and a larger diameter raises the resonant frequency (the A value involves a radius squared, while the I value involves a diameter to the fourth power). ‘5'” 71 .Although the 10" Pigstail triangle was .07” thinner than the Abel triangle, it was felt that the large difference in the average lengths of the sides of the two triangles (the average length of the sides of the 10" Pigstail triangle was ahmost twice that of the Abel triangle) would effectively minimize the influence of this very slight difference in diameter. Thus, the result of the length principle may be seen in the following comparison: 10" Pigstail triangle (.5" thick and sides averaging 10.6" long)--490 to 520 cps at the top and bottom striking points, and 440 cps at the corner striking point; Abel triangle (.57" thick and sides averaging almost 5.8" long)--1900 to 1950 cps at the top and bottom striking points, and 1525 to 1575 cps at the corner striking point. The 10" Pigstail triangle also produced almost twice as many partials under 5000 cps than did the.Abel triangle. Although the average length of the sides of the Abel triangle was .4“ shorter than that of the Sonor triangle, it was felt that the large difference in diameters (the Abel triangle was very nearly twice as thick as the Sonor triangle) would effectively minimize the influence of the slight difference in the average lengths of the sides. Thus, the results of the diameter principle may be seen in the following comparison: Abel triangle (.57" thick and sides averaging almost 5.8" long)--1900 to 1950 cps at the top and bottom striking points, and 1525 to 1575 cps at the corner striking point; Sonor triangle (.29" thick and sides averaging almost 6.2" long)--900 to 920 cps at the t0p and bottom striking,points, and 1200 to 1250 cps at the corner striking point. The Abel triangle also produced fewer partials under 5000 cps than did the Sonor triangle. 72 Referring again to the expression.v—§Kli2, it is evident that a 0 change in the material, and thus the material properties E and p, Will affect the resonant frequency of the triangle. As it is almost impossible to consider the E and p values separately, it is more valid to consider the ratio of the two values (E : p). Therefore, with all geometric properties equal, triangles made from metals with larger ratios than steel (30 x 106 : 'ggg), such as beryllium (42 x 106 : ;%§%., molybdenum (49.3 x 106 : '35 or chromium (34.1 x 106 : 'gg), would have higher 386 ’ 3 resonant frequencies than steel. Conversely, triangles made from.metals with smaller ratios than steel, such as brass (16 x 106 : éggq, silver (11.42 x 106 : 3:2), or aluminum (10.2 x 106 : 36g , would have lower resonant frequencies than steel. .Although.many authorities state that a plated triangle is superior in sound to an unplated one, by applying the expression.vE§KlI2, it can be seen that the overall effect of a plating even several thousandths of an inch thick would be very minimal at best. The damping property of the metal, although not affecting the frequency, should also be considered for its effect on the resonance or duration of the sound. However, damping properties vary considerably with chemical content, frequency, and heat, thus making it very difficult to readily obtain exact comparative figures for various metals. Implement Size and.Material .A change of implement size produced no consistent changes in the overtone structures of any of the five small triangles (Abel, Ludwig, Sonor, Zildjian, and 6" Pigstail). However, the smaller implement (5/32" drill rod) produced partials of slightly greater intensity than the 73 larger implement (7/32" drill rod) on the 10” Pigstail triangle at the mf and pp dynamic levels. .A change of implement material produced no consistent changes in the overtone structures of the Abel, Ludwig, Sonor, and 6" Pigstail triangles. However, the cold-rolled steel implement produced more partials than the drill rod implement in six of the nine instances on the Zildjian triangle. And in seven of the nine instances on the 10” Pigstail triangle, the cold-rolled steel implement produced partials of slightly greater intensity than did the drill rod implement. Striking Point and Angle There was no consistent pattern among the six triangles in their reaction to a change from the top striking point to the bottom striking point. The 0° striking angle produced lower fundamentals in four triangles (Abel, Ludwig, Zildjian, and 10” Pigstail), and higher funda- mentals in the other two (Sonor and 6" Pigstail). Other general effects of the 0° striking angle were lower upper limits, fewer partials, and weaker partials. Dynamic Level Compared to the pp level, the ff level seemed to have three general effects on the overtone structures of the triangles: it produced higher upper limits in.most cases; it consistently produced partials of greater intensity; and it produced more partials in twenty-eight of the fifty-four examples. 74 Pitch Level and Overtone Intensities Each triangle produced two rather definite fundamental partials-- one at the t0p and bottom striking points with a 90° striking angle, and one at the corner striking point with a 0° striking angle. The funda- mentals produced at the top and bottom striking points on the five small triangles ranged from the 900 to 920 cps partials of the Sonor triangle to the 1900 to 1950 cps partials of the Abel triangle. The fundamentals produced at the corner striking point on the five small triangles ranged from the 900 to 990 cps partials of the Zildjian triangle to the 1800 cps partials of the 6” Pigstail triangle. The fundamentals of the 10" Pigstail triangle were considerably lower: 490 to 520 cps at the tOp and bottom striking points, and 440 cps at the corner striking point. No funda- mentals were found that approached the 6000 cps level reported by Briggs (see above on p. 13). With one exception (the corner striking point on the Zildjian triangle), the most frequent strong partials on all the triangles were below the 7000 cps level. .A fundamental partial was the most intense partial only once (Ludwig / 7/32" cold-rolled steel / bottom / pp). And the fundamental partials were actually the weakest partials in sixty-one of the one hundred sixty-two examples. There was no consistency among the triangles as to which partials were the most intense. Similarity of Overtone Structures The partials and their frequencies presented above in Tables 3 throughLS form the basis of Figure 22. By plotting each partial's frequency on a horizontal scale marked in thousands of cycles per second (kCS), the overtone structures of the triangles may be compared. 75 monquHo mom momsuochum ocoumo>0 comNqu0 couoHcoHc--.mN omcmmm mox mH NH 3 mH «H 2 NH HH 3 o m N c m « m N H c H H H H H J H H H H H H H H H H H H H mooH 5 555.0on oqun.mo-uso _ _ _ rH _ _ _~||| oemHa-:H _ _ _ HHHHmme HTcH 05300-50 H _ _ — _ = _ oSmHmHéH _ a _ H H JHmeHo :o oamHo-mo-uso _ — _ __ _H _ 053-5 _ _ _ _ H _ _Sm—HNHHHHN 283N930 H! _ H — _ _ r— oamHNHLNH _ _ _ H {fl _ _ _Hocom 05300-qu _ _ H _ _ : _ 283-5 — _ _ _ _ H _ _ No.63 28300-30 — H _ H P H — occHHHLHH _ _ _ _ H _ H HmmH/u 80 Three of the triangles (Sonor, Zildjian, and 6" Pigstail) exhibited strikingly similar overtone structures. The sixteen predicted partials were divided into three definite groups: six low partials from about 800 to 1600 cps; six intermediate partials from about 2700 to 4400 cps; and four high partials from about 5400 to 6400 cps. The Ludwig triangle's overtone structure was similar, but placed higher on the frequency scale with more space between the groups: six low partials from about 1700 to 2100 cps; six intermediate partials from about 4300 to 5600 cps; and four high partials from about 7100 to 8600 cps. The Abel and 10" Pigstail triangles exhibited overtone structures with less definitely grouped and more evenly distributed partials. gredicted Modes of Vibration All six triangles exhibited similar modes of vibration (see Appendix C for an explanation of the derivation of the vibrational modes). Depending upon the frequency, each side showed from 0 to 3 nodes during ill-plane vibration, and from 0 to 4 nodes during out-of-plane vibration. Although the triangles were not entirely consistent as to which partials were in- or out-of-plane, the vibrational modes for two of the Six partials that were consistent with all of the triangles are shown in Figures 24 and 25. In both Figures 24 and 25, the shapes of the triangles are shown by the solid lines, and the deformities produced by vibration are shown by the dotted lines. As mentioned before (see above on p. 47) , Point l6--the free end of the right leg of each triangle in the Figures-- was fixed to eliminate rigid body motions, thus accounting for the apparent lack of vibration at the free ends of the right legs of the triangles. 81 Figure 24 . - - In-plane Vibration- -Partial 4 82 Zildj ian ’--§ $ ~-—‘ ‘ \ 6" Pigstail ,’ 10" Pigstail \ g \ !—--~ $ *- Figure 25.-~Out-of-plane Vibration--Partial ll 83 In Figure 24, each triangle is shown as if observed from a point perpendicular to the plane of the triangle. In Figure 25, however, the point of observation is just above the plane of the triangle looking from the base to the apex. Agreement of Mathematical and Experimental Results There was no consistency among the triangles either as to which predicted partials did not appear, or as to which analyzed partials were not predicted. However, the agreement between the mathematical (predicted) and experimental (analyzed) results was quite high (at least 85% in all but two examples). Table 10 shows the predicted partials and the corresponding analyzed partials that were produced by both in-plane and out-of-plane vibration. The intensities of the analyzed partials are also included, but there seemed to be no consistent relationship between intensity and the presence or absence of partials. Dashes in the "predicted" column indicate that the corresponding analyzed partials were not predicted; and dashes in the "analyzed" column indicate that the corresponding predicted partials were apparently not produced. The predicted and analyzed overtone structures for all but one (Sonor) of the six triangles were similar. The manner in which the partials of the Ludwig, Zildjian, and 6" Pigstail triangles were grouped was quite similar, although the frequency ranges of the analyzed groupings were consistently greater and somewhat higher 84 Table 10.--Agreement Between Experimental and Mathematical Results In-plane Vibration Out-of-plane Vibration Triangle Predicted Analyzed Analyzed Predicted Analyzed Analyzed partials in partials in intensities partials in partials in intensities cps cps in dB cps cps in dB 1777 1800-2000 15-42 1732 1400-1900 6-35 2778 2550-2700 20-40 2645 2450-2500 18-32 ---- 3200 40 3442 3800-3900 25-42 3472 3800-4000 40-50 5303 4700-4800 22-48 Abel 5202 4600-5500 35—50 ---- 6300 16-40 6821 6300-7200 22—35 7339 , ---- ---- 7649 ---- ---- 7607 8100-8300 13-41 7872 7900—9000 15-31 9863 ---- ---- 10440 10300 - 23 11460 10750-11000 12-28 1324 1400-1450 ll-39 1318 1200-1450 7-11 1930 ---- ---- 1895 1900 15 2132 2100-2450 14-39 2000 2100-2450 10-24 4331 3900-4500 10-43 4380 3900-4000 25-46 Ludwig 4893 4600-4900 19-47 4830 4800-4950 31-40 5589 5400-5500 19-41 5233 5500-5900 30-43 7139 6500-6700 19-41 ---- 6800-6900 21-37 8716 8700-9200 10-29 8477 ---- ---- 8558 8700-8900 12-27 832.2 895-980 5-21 828 920 10 1243 1200-1280 7-12 1214 1100-1250 6-9 1480 1550-1600 6-36 1390 1300-1600 6-31 2800 2950-3000 23-48 2750 2230-2750 7-16 Sonor 3278 ---- ---- 3139 3000 21-23 3987 3600-4500 25-50 3763 3500-3650 11-32 5450 ---- ---- ---- 4300-4400 15-29 6192 6000 28-43 5432 4800-5400 15-33 5851 6400 20-43 922.3 ---- ---- 908 900-920 6-17 1276 1100-1350 8-29 1230 1200-1275 6-8 554 1500 13-32 1411 1475-1500 11-13 ---- 1800-2000 11-30 ---- 1800-2000 9-26 Zildjian 3038 ~--- ---- 304$ ---- ---- 3391 3200—3300 11-46 3184 3150-3250 13-33 3872 4000-4200 19-48 3713 3800-3900 20-37 5450 5000 16-30 ---- 4550-4700 13-38 6000 6400 16-38 5751 ---- ---- 6016 6000-6400 11-35 1009 900-1000 7-25 1006 ---- ---- ---- 1200-127 7-8 1439 ---- ---- 1481 1450-1530 14-23 1551 1800 6-19 6” 1657 1650-1930 16-37 3075 2800-2850 14-36 Pigstail 3223 3100-3209 21-47 3519 3600-3850 20-37 3676 3800-3990 18-46 4190 ---- ---- 4396 4800-4900 19-45 5447 5000-5500 25-37 5631 5000 26-38 6411 6300-6400 22-37 6197 6300-6500 23-43 469.7 440-; 0 8-19 471.2 125-450 12-34 672.4 650-100 8-12 645.7 700 10 861.9 790—900 13-77 ---- 770 12 10" 1299 814.2 900 8-18 Pigstail 1492 1425-1700 10-42 1226 ---- ---- 2034 1900-2030 13-38 1502 1450—1700 12—29 2779 2400-3500 16—41 1942 1900-2000 13—31 3501 3150-3400 26-46 2692 2450-2850 23-41 3277 3100-5600 27-48 85 than the comparable predicted groupings. Only the first group of the Sonor triangle's analyzed partials was similar to the predicted grouping, and it was also placed somewhat higher on the frequency scale. Both the predicted and analyzed partials of the Abel and 10” Pigstail triangles were less definitely grouped and more evenly distributed than those of the other four triangles. C als Experimental Results The experimental results of the five cymbals examined are listed in Tables 11 through 15 under these headings: implement (Imp--yellow yarn (Yel), red yarn (Red), brown cord (Cord); striking pointl (St Pt)--edge (Edge), cup (Cup); dynamic level (Lev)--ff, mi, PP; fundamental (Fund); upper limit (UL); and Energy Peaks in Decreasing Order of Intensity. The fUndamentals, upper limits, and energy peaks are given in cycles per second (cps). The energy peaks are further inentified by decibel ratings (dB)--e.g., 1220 / 46 indicates a frequency of 1220 cps at an intensity level of 46 dB. It should be noted that the energy peaks, or partials, are listed in decreasing order of intensity, and net in order of frequency. Each table is accompanied by a discussion of the effects (if any) on each cymbal's overtone structures produced by changes in implement 1It should be noted that both striking points were used.with a 0° striking angle. 86 hardness and material, dynamic level, and striking point. General trends in pitch levels and overtone strengths or intensities are also noted. Avedis Zildjian Cymbal (Table 11) Implement Hardness and Material A change of implement hardness did not produce any consistent differences in the.Avedis Zildjian cymbal's overtone structures. The cord implement produced more and more intense partials of 6000+ cps than the red yarn implement did by a nine to four margin. Dynamic Level With only one exception (cord / cup / ff), the ff level produced up to six more partials than did the pp level. The majority of these additional partials occurred above 4000 cps. The ff level consistently 'produced mUch higher upper limits (20000+ cps) than did the pp level (2200 to 7000 cps). Without exception, the partials at the ff level were of considerably greater intensity than those at the pp level. Striking Point . With only one exception (yellow / edge / pp), the edge striking point produced upper limits higher than, or as high as, those produced at the cup striking point. The cup striking point consistently produced a fundamental 6O cps higher than the edge striking point. The edge striking point produced more partials below 1000 cps by a thirty-six to twenty-six margin, but only one more partial above 5000 cps. 87 n on ooon HN own «N oove n~ omen om onHN on omnN o~ ooon om oov Hn oNo on on“ on ooNH oono oov on o oonn nN omm om ooee ow oonn on onHN on ooon Hn ono~ Nn oov on oooo an ooNH oono oov oe_ ago on oooo on oono en ovm on oov Ne ooHN Nv ooow nv.oonn av ooov on ooNH no omm +oooo~ oov mm . ouou o oonm on oonv nN ooon n~ oona “N onnN oN onn o~ ooHN on ome nn own on ooNH an omen on omm ooon ovn on NH oono on oovn nn oonv nn onHN an onnu nn onnN mn owe on oona an ooNH nn oen ow ooon Ne och ooooH own we omnm HN oooNH nn oooo on own on one on ovn oe oouN He oooo no onHN No onoN no oomn no oonn no oooo on oon +oooo~ ovn mm o ooon 5H onnN on own HN when ON ooo “N ooe on on“ wn ooNH oonv oov an on oovh an oovv nn oonn on ooon nn oooN on ooHH on onHN on oov an ooNH on oon oooo oov ma_ 9:0 nN oooo on own on ooHH on ooe ov oonH no onHN He ooon He once No ooev no ooen no ooo +ooooN ooe mm . omm o4 ovv on ooon on ooBN NN onHN nu own ow mmv on ooun an ooNH Hn ovn kn oou ooon own on 0N omm an ooev on onHN on oonm ~n oonH nn ooNH nn owe on ooon nn oen on ohm oooo own we omow Nn oonn on omm on one on oooo on onHN ow oen no oomn Ne oonm no oouH no ooov ov ooh +oooo~ oen mm o ooHn on oonN HN ooua on oHHH oN ooon on oov nn oon nn own ooon oov go ON ooon nN oonn om ooon ow oooN on ooeo Nn ooHH an omHN on oov on omn oonn oov ma_ 9:0 .nN ooNoH on ooon on oonn on oov oe omNN no onn Hm onHN Ne omen no ooHH .ne ooon me omm +ooooN ooe mm Hm» n onHN ma oonn HN ooHH nN ovn on one Nn oen nn omm ooNN ovn on nH oooo no oovv oN oonw BN ovm on ooon on owe mn onHN on ooHH 5n oonH on one on own oonn own we omom Nn oooo ov onm ov owe no omm He oooo He oona Ne omen Ne oove Ne onHN no ovn no oonn ow ooNH me one +oooo~ own mm me moo mo moo mo moo mo moo mo moo mo moo mo moo no man mo ago .mo moo mu moo mo moo mo moo moo moo _ >83 pm pm QEH xpnmcoucn mo nacho mcwmmohooo an axeom xmuoqm A: wand chexu :mwmonnw mfioo><.mcu mo mxmmm xmhocm one .muHqu Home: .mneucoEmwcsm--.HH macaw 88 Pitch Level and Overtone Intensities The Avedis Zildjian cymbal produced two definite fundamental partials: 340 cps at the edge striking point, and 400 cps at the cup striking point. There was no consistency as to the intensity of these frequencies--they appeared in positions one to four, six, and eight to nine. The overtone structures produced at the edge striking point consistently exhibited three strong partials between 340 and 540 cps. However, with only one exception (yellow / edge / mf), the most intense partials produced at the edge striking point were immediately above this band (from 760 to 790 cps). The overtone structures produced at the cup striking point generally exhibited two strong partials: 400 and 540 cps. However, with only three exceptions (red / cup / pp, cord / cup / mf, and cord / cup / pp), the most intense partials produced at the cup striking point occurred between 740 and 800 cps. New K. Zildjian Cymbal (Table 12) Implement Hardness and.Material The harder red yarn implement consistently produced upper limits as high as, or higher than, the yellow yarn implement. The cymbal's overtone structures were not affected by a change in implement material. Dynamic Level ‘With only one exception (cord / edge / ff), the ff level produced up to five more partials than the pp level. .Mbst of these extra partials occurred above 5000 cps. The partials of the ff level were consistently of greater intensity than those of the pp level. The ff level produced 89 Hm ooHe eN one eN ooNn em onnN on ooeH nn one on oeo nn oon some one on on oooe nn one en ooHH nn ooHe on oon on onoH en onmm on oee oe oomn ocooH ooe we gnu on one on oHHH on oNoH on ooeo on oneN on ooeH He oee He ooHe Ne ooNn +oooo~ one wm euou on nnm NH oNn on onNe om nme mm ooNn om oonN oN ooon Nn onnH nn one on ooo oooo nnN on nH nnN ow oHn oN one nn ooHH nn ooen on oone on onnN en ooNn on oee He oon ooomn nnN we omom NN nnm om oHn on ooHH oe onnN He onoH He one we oon ne ooNn ee ooHe +oooo~ nnN ow en ooHe on ooNn mm oonN om oooH eN ooeH em oeo on one mn oee en oen oonn one no Nn oooH nn oomn nn one on ooem en ooNe on oneH on ooo oe oomn He oen ooooH owe oec onu NN oon nn one on ooNe on oHoH He ooen Ne oee Ne onnm ne oon ee oomn ne ooHe +oooo~ oon mm mom on nnN en oHn oN oone nm one eN oomn n ooeN em omon on ooeH on oee oe oon oooo nnN do 3 ea nnN nN oHn nn ooo en ooHH nn ooen ON ooNn en onnm en one on oee He oon oooon nnm we doom NH nnN an oHn on one on ooHH on ooeo oe onnm oe ooeH me one we oon Ne ooHe ne oomn +oooom nnN on ma ooon on ommn om oen em one on one oonm oen on nn ooNe em oen nn oeNH Nn ooNe nn oan en onnN en ooon on one we oon ooooH oen ne_ gnu nn oee en ooHH en onon on oooe oe oooH He onnm ne ooo ne onnn ee oNe oe oooe +ooooN oee ww no» o ooHn NH nnN nn onem nH oHn on one on onoa em ooHH en one on oon ooon nnN no on nnN en oHn nn ooNe nn onnN on oNHH en onon nn oee on oomn Ne onn oono nnN ma ewem nN nnN nn oHn nn ooNe on omm on ooHH oe ooeo He ooeH Ne oee ne onn ne ooeN ee ooNn ne ooHe +oooo~ nnm on me moo me moo mo moo me moo me moo me moo me moo me moo me mnu mo moo me new me moo moo moo >3 “E an on: eufimcoucH mo gecko wcnmmouuoo an mxmom emuocm A: nude H253 Salome. .x 3% 23 as 98a asem Ea .mfifin Home: .mflmpemfiefia--.2 03$ 90 consistently higher upper limits (all were 20000+ cps) than the pp level (from 2300 to 6300 cps). Striking Point The cup striking point consistently produced a higher fundamental partial (from 85 to 185 cps higher) than the edge striking point. The cup striking point never produced more partials than the edge striking point in parallel instances. The edge striking point produced.more partials under 1000 cps by a forty-five to thirty-one margin, but the cup striking point produced two more partials above 5000 cps. Pitch Level and Overtone Intensities The edge striking point consistently produced a.fundamental partial of 235 cps, but the cup striking point varied from 340 to 440 cps. These frequencies occurred only twice in a position higher than the seventh.most intense partial (yellow / cup / pp, and red / cup / pp). The New K. Zildjian cymbal's overtone structures included a strong and relatively narrow band of three to six partials between 235 and 980 cps. Except for the six ff instances and one mf instance (cord / cup / mf), the most intense partial of each example occurred in this frequency band (usually between 550 and S70 cps). Old K. Zildjian Qymbal (Table 13) Implement Hardness and Material Except for the ff level, the harder red yarn implement produced higher upper limits than the yellow yarn implement.. With one exception (cord / edge / ff), the cord implement produced.more partials than the red yarn implement did in parallel instances. 91 NH ooen oH oonN HN onoN eN oHn eN onn nN oeoH eN ono eN oooN oN nNo oN nNnH nn oHe en ooe oone onn an oN onn Hn onn Hn oooN Nn oee Nn ooHH Nn nNnH nn oneH nn oooH nn oonN en oeo Ne ooe oone onn me_ use en oee en oNn en onn en oone en oen on oooH He ooe fie oonN Ne oonH ne ooen ee onoN ee oooN +ooooN onn mm euou HH ooen NH nNN oH oonN NN oonN nN neoH eN onoN eN oHn eN neo oN onnH Hn one en oNn en oHe oone nNN an nH nNN oN ooee oN ooen en onnH en neoH en oeo nn one nn oHn on oooH oe oonN He oee Ne oNn ne oHe ooon nNN we omen eH oNN en one en oon on oHn oN oone He nee Ne oonH Ne oHnH ne ooe ee neN ne onon ee ooon on oooN +ooooN oNN mm e onoN eH oooH nN ooHH eN oNn nN onNH eN onn on ooe Nn oHe ooon onn on eN onen on onoN Hn ono nn ooHH nn oee en onn en oee en neHN on onNH Ne ooe oooe onn new oso . nn ooon en oNn en one en onn oe oee oe ono ne ooe ne oonH ee ooen we onoN on oooH +ooooN onn we omm n oonN NH oNN nH oooH oH nNnH oN oooH oN one eN oee eN oHn Hn oNn en oHe ooon oNN an nH nNN nn oHn nn onoH en oooe nn one en onnH on nee oe oooH oe oonN He oNn Ne oHe oooo oNN we omem eH nNN en one en oHn on oee oe onoH He ooNH He ooo Ne nNn Ne ooe ne ooen ee oooN ne oonN ee onoH +ooooN nNN mm HH onoH nH onNH eN ooe eN onn Hn oHe ooeH onn no oH ooon en oonH oN onn oN oooN N oen nn nNn nn onNH en oooN nn oHe en one oonn onn we. use eN oono nn onn en onn on oee on oneH He onNH Ne oHe ee one ne onnN we oooN on onNH +ooooN onn mm Ho» e oooH NH oNN oH oenH eN oHn eN one Nn ooe nn oNn en oee ooHN oNN an eH nNN eN ooen Nn oonN nn onoN en oHn en oeo on onnH on oee on oooH oe ooe ne oNn oooe nNN me «new oN nNN en oHn an one on oone He oNn Ne ooe ne ooe ee ooee ee ooon ee onnH ee ooen oe ooeN +ooooN nNN mm me man me won me man me mmu me man no moo me mmu me mad me moo me moo me moo me moo me woo woo mnu >3 3 an 9: euHmcoucH mo nacho mchmouuoo qH mxmom emuocm H: econ Ease 92.85. .x Bo ago no 98a 686 e5 .323 “and: .mHSamfiefim--.nH 838 92 Dynamic Level The ff level consistently produced much higher upper limits (20000+ cps) than the pp level (from 1400 to 4500 cps). 'With only one exception (cord / cup / ff), the ff level produced up to six more partials than the pp level. The majority of the additional partials occurred above 3000 cps. The partials produced by the ff level were consistently of greater intensity than those produced by the pp level. Striking Point The cup striking point consistently produced a fundamental 105 to 110 cps higher than the fundamental produced at the edge striking point. The edge striking point produced.more partials in all but one instance (cord / edge / pp). The edge striking point also produced partials which were generally of greater intensity than those produced at the cup striking point. The upper limits at the edge striking point were consistently either equal to, or higher than, the upper limits at the cup striking point. Pitch Level and Overtone Intensities The two striking points produced two rather definite fundamental partials: 220 to 225 cps at the edge striking point; and 330 cps at the cup striking point. However, these frequencies were rated higher than the seventh.most intense partial only three times: yellow / cup / pp, red / cup / mf, and red / cup / pp. Each example of the Old K. Zildjian cymbal's overtone structures included a concentrated band of three to seven relatively strong partials in the range from 220 to 1000 cps. Except for the six ff examples and 93 one mf example (red / cup / mf), the two most intense partials of each instance fell in this frequency band.. The most intense partials of the ff examples ranged from 1900 to 2400 cps. There were only eight partials of 4000+ cps, and only two of these were above 7000 cps. l7” Paiste gymbal (Table 14) Implement Hardness and Material Although both the red and yellow yarn implements produced upper limits of 20000+ cps at the ff level, the harder red yarn implement produced higher upper limits at the mf and pp levels. The cord implement consistently produced upper lhmits as high as, or higher than, the red yarn implement. Dynamic Level The ff level consistently produced higher upper limits (20000+ cps) than the pp level (2450 to 6000 cps). Without exception, the ff level produced up to six more partials than the pp level in parallel instances. Mbst of the additional partials occurred above 4000 cps. The partials produced at the ff level were consistently of greater intensity than those produced at the pp level. Striking Point The cup striking point consistently produced a higher fundamental (100 to 115 cps higher) than the edge striking point. ‘With only one exception (cord / edge / pp), the edge striking point produced up to four more partials than the cup striking point in parallel instances. The edge striking point produced upper limits which were higher than, or at least equal to, the upper limits produced at the cup striking point. 94- oN ooon nN ooNN eN oooN oN ooHH on ooe nn oonH nn oNo en ooe on one oooe ooe gg e oooo oH ooon Hn oon Nn ooee nn ooNn en ooNH en onoH en ooon on oneH Ne one ne one oono oon .ae gnu en ooon on oon oe oen oe neeH He onHN Ne ooo ne oonn ee oee ee oooN we ooon on one +ooooN oon we opou nH ooN nN oen eN nne eN oooN nN oen nn ooHH oN oonH en oNo en one oooe ooN gg nH ooN on oen Nn one nn one nn oeHN en oooH en onee en oon nn oonH on ooen oe oooN He oee oooHH ooN we omen eN ooN nn oooHH nn oNe en onn nn oHo on ooHH oe ooNH He oen He onoN ne oneH ee oooe ee oooN oe ooe +ooooN ooN we nH onnn HN onnN nN ooHH oN oon on ooeH nn ooo en oHe oe one oone oon gg nN oonN Nn ono Nn oon nn oNNH n ooon en oeeH nn one ee one oooo oon we gnu nn oone nn oNo on ooNH oe oeoH He oon Ne oonH ne ooee ee.ooNN Ne oooN we oonn on one +ooooN oon we omm nH ooen nH ooN oH onnN HN ooHN eN oonH eN oonH nN ooHH oN oen on oen nn nNe en oNo ooon ooN gg eN nnN Nn nnn en ooNN nn nNe en ooee nn onnN nn onn on oNo on oooH on oonH Ne ooen ee oNe oooHH nnN HE omen eN nnN Hn ooHHH en onn oe oon oe ooHN He oNo Ne oonH ne oone ee oone ne ooHe ee oooN we ooe +ooooN nnN nu nH ooeH oH ooNH NN oNo oN ooe en oNe en one onHN ooe gg Hn onHN nn ooon nn onoN en ooeH en ooNH nn oNo nn oon ne one we one ooon oon ma: gnu nN ooHH nn oonn oe nNHH He oooH HN neeH He nNn Ne ooN Ne ooNN ne oNo we ooen oe ooN on oee +ooooN oon me How oH ooNN NH oooN nH nnN NN oen eN oonH on oNn Hn oNo en ooe ooHn an gg nN nnN oN ooee on nnn Nn ooen nn.onoN en nNe nn nNn en oonH en oooH Ne ooe oono nnN we omen en nnN oe onn He nNe He onoH He ooHN Ne oooe Ne oNo ne oen ne ooon ee oonH ne onnN ee ooen ee oee +ooooN nnN me me mgo me mgu me mgo me mgu me mgu me mgo me ngu me mgu me mgu me mgu me mgu me mgu me mgu mgo mgu >04 pg on men euHmaoucH mo uoeho mchmoHuoo :H mxmog emuocm a: ecsg 4.35 8...qu s: we. no 98a .385. e8 .834 8%: 53858592: 23. 95 The partials produced at the cup striking point were generally of Vgreater intensity than those produced at the edge striking point. The edge striking point produced.more partials under 1000 cps than the cup striking point by a fifty-three to thirty-six margin. Pitch Level and Overtone Intensities Two rather definite fundamental partials were produced on the smaller Paiste cymbal: 285 to 290 cps at the edge, and 390 to 400 cps at the cup. However, there was no consistency as to the strength of these frequencies. They appeared as the third most intense partial twice, the fifth most intense twice, the sixth.most intense three times, the seventh.most intense once, the ninth.most intense twice, the tenth most intense three times, the twelfth most intense three times, and the thirteenth most intense twice. The overtone structures included a band of strong partials from 285 to 1100 cps. Almost half of the total partials produced occurred within this frequency band. ‘Without exception, the most intense partials in each example occurred within this range. Only eleven partials of 5000+ cps were found. 20" Paiste Cymbal (Table 15) Implement Hardness andeaterial The harder red yarn implement produced higher upper limits at the pp and.mf levels (the upper limdts at the ff level were the same). The red yarn.implement also tended to produce partials of slightly greater intensity at the mf and pp levels. The cord implement produced upper limits equal to, or higher than, the upper limits produced by the red 963 oN oonn eN onoN eN ooeN nN nnn nN onNN nN onn oN ooeH en oeNH oN onn ooon own gg eN oonn oN one Nn ooen Nn oonH nn ooHH nn oooN en onnN nn oNn nn ooo Ne onn oooe own .3: gnu nN oone nn one an oen on oon oe oonH oe ooN He oone Ne ooNH ee oonN ee oooN we own +ooooN onn mm ekou nN oooe on oooN Hn oNn Hn oonH Nn onNH Nn onn nn oon en ooNN He one oooe oNn gg Nn oNn nn oooH nn ooeH en onnn en onNH en onn nn ooon oe oonN He oooN Ne one ooooH oNn we «new Nn ooee nn oNn on oee He nNn Ne ooeH Ne onNn ne ooon ne ooNH ee oonn ne oooN ee oonN me one +ooooN oNn we eH onnN oH one nN ooNN nN ooeH eN ooo nN oen on onNH oe onn oooe own gg eN oonn nN oee on one Hn ooen Hn ooon Hn ooeH Nn oon nn oonN en onn nn oeHH He own oooe own .3: gnu nn one nn oon on oen on ooNH oe oon Ne oonH ee oonn ne ooNN ee ooon ee onn +ooooN onn we co« 8 oonn NN 82 a one a. 8: 3 82 on com 3 .32 M. an 9.. Sn on one en can 88 OS 8 ooee Nn onn nn ooon en oon nn ooon nn ooeN nn ooeH en oonN en ooNn en oon en oen oe oeHH ne nNe oooo onn we omen en one nn nNn oe oonH oe oeHH He ooHn He oonH Ne oon ne onNN ee nNe on oonn +ooooN nNn mm e oooH oH oee oH oooH NN ooo N oonH eN onn oe oon oooN own gg H ooHe eN ooHn oN onNN on ooeH Nn ooo en oee en oen p oeHH ne onn oooe onn we" gnu eH onNHH on oon oe ooNn oe onn He oon Ne ooeH Ne ooNH ne oonN we oooN oe oee on own +ooooN onn we Hoe n oooN eH ooNN NN ooeH nN onNH nN oee oN oon nn oNn en oen oe one oonn oNn gg _ nN ooNn on one Hn oooH Hn.ooon nn onn nn oon nn ooeH nn ooNn en ooNH en oonN on onn ne one oone onn we amen . HN oooHH en oonn ee onn en ooo on one oe oon ne ooNH ee oee ee ooeH ne ooon me oonN on ooon +ooooN onn we mgu me mgo me mgu me mgu me mgu me mgo me mgu me mgu me mgu me mgu mo ago me mgu me mgu mgu mgu _ e >04 pg an oen epHmcoucH mo noeuo mchmouuon aH axeog emnonm n: ecng H.825 Bag :8 on no and Queen e8 82.5.: 8&3 .mHSco_eme§e--.nH 238. 97 yarn implement. The cord implement also tended to produce partials of slightly greater intensity at each dynamic level. Dynamic Level With one exception (red / edge / ff), the ff level produced up to four more partials than did the pp level. The majority of the additional partials occurred above 3500 cps. The partials produced at the ff level were consistently of greater intensity than those produced at the pp level. The ff level also consistently produced higher upper limits (20000+ cps) than the pp level (2000 to 6000 cps). Striking Point The fundamental partials produced at the cup striking point on the 20" Paiste cymbal were consistently 50 to 60 cps higher than the fundamental partials produced at the edge striking point. The upper limits produced at the cup striking point were lower at the mf and pp levels than the upper limits produced at the edge striking point. The partials produced at the edge striking point were generally of greater intensity than the partials produced at the cup striking point. The edge striking point produced more partials of 4000+ cps than the cup striking point by a ten to six margin. Pitch Level and Overtone Intensities The larger Paiste cymbal produced two quite definite fundamental frequencies: 300 to 330 cps at the edge striking point; and 380 cps at the cup striking point. The 380 cps partial was the most intense partial in each of the overtone structures produced at the cup striking point. 98 However, the edge fundamental rated only as high as the third most intense partial once and the fourth most intense once. The remaining edge fundamentals appeared as the seventh through the twelfth most intense partials. The majority of the partials produced occurred in the frequency range of 320 to 1300 cps. With only two exceptions (yellow / edge / ff, and red / edge / ff), the most intense partial in each example occurred within this range. Only sixteen partials were found with frequencies of 4000+ cps. Summary of Experimental Results and Related Research Instrument Size andeaterial The basic acoustical principles of circular plates provide a basis for comparing cymbals and their overtone structures. With all other properties constant, these relationships are valid for cymbals: an increased thickness raises the resonant frequency; and a larger diameter lowers the resonant frequency. Although the "other properties" were rarely constant on the cymbals studied, the above principles were confirmed to a certain extent. The Avedis Zildjian cymbal (.055" thick, with a diameter of 15.9”) did produce higher fundamental frequencies at both the edge and cup striking points-~340 and 400 cps--than the Old K. Zildjian cymbal (.052" thick, with a diameter of 16.1”)--225 and 330 cps. However, the New K. Zildjian and the 17" Paiste cymbals (with dimensions of .049" and 16.1", and .043" and 17.0" respectively) also produced higher fbndamental frequencies than the thicker Old K. Zildjian cymbal: New K. Zildjian's edge = 235 cps and cup = 410 cps; and 17" Paiste's edge = 285 to 290 cps and cup = 390 cps. 99 The increased size of the 20" Paiste cymbal seemed to have no consistent effect on the frequency of the edge or cup fundamental partials. In addition to the diameter and thickness, it would seem that the various other dimensions (cup width, cup height, and bow height) also affect the resonant frequencies of cymbals. However, except for the Old K. Zildjian cymbal (which had the flattest bow, the lowest fundamental, and the most partials under 1000 cps), the effect of the shape of the bow was not consistent with Thompson's statement that a "flat . . . bow produced a lower more bodied sound." The Avedis Zildjian cymbal had a flatter bow (.8") than the New K. Zildjian and 17" Paiste cymbals (each .9”), yet it produced a higher fundamental at both the edge (340 cps compared to 235 and 285 to 290 cps) and the cup (400 cps compared to about 400 and 390 cps). None of the various dimensions (diameter, thickness, cup width, cup height, and bow height) seemed to have any consistent effect on either the total number of partials or the number of partials below 1000 cps or above 5000 cps for any of the five cymbals studied. As has been stated before, the leading cymbal manufacturers apparently make their cymbals from the same basic ingredients (copper and tin, with small amounts of iron, lead, or silver). The success or failure of each instrument is therefore attributed to the processing and/or the use of catalytic agents during the processing. These items are closely guarded secrets of each company. 100 Implement Hardness and Material The harder red yarn implement produced higher upper limits at the mf and pp levels than the yellow yarn implement on the Old K. Zildjian, the 17" Paiste, and the 20” Paiste cymbals. There was no effect on the Avedis Zildjian cymbal's overtone structures due to a change in implement hardness. .And on the New K. Zildjian cymbal, the red yarn implement consistently produced upper limits as high as, or higher than, the yellow yarn implement. In addition to the higher upper limits, the red yarn implement also produced partials of slightly greater intensity at the mf and pp levels on the 20" Paiste cymbal. The cord implement produced upper limits equal to, or higher than, the upper limits produced by the red yarn implement on the two Paiste cymbals. On the Avedis Zildjian cymbal, the cord implement produced more and more intense partials of 6000+ cps. The cord implement generally produced more total partials on the Old K. Zildjian cymbal and generally partials of greater intensity on the 20" Paiste cymbal. .A change of implement material had no consistent effect on the overtone structures of the New K. Zildjian cymbal. Striking_Point The edge striking point produced upper limits equal to, or higher than, the upper limits produced by the cup striking point. .All five cymbals consistently produced higher fundamentals when struck at the cup striking point than when struck at the edge striking point. The greatest difference occurred on the New K. Zildjian cymbal (up to 185 cps) and the smallest difference occurred on the Avedis Zildjian and the 20" Paiste cymbals (up to 60 cps). 101 Other noticeable effects of the edge striking point were: more partials below the 1000 cps level on the.Avedis Zildjian, the New K. Zildjian, and the 17" Paiste cymbals; usually more and.more intense partials on all cymbals; and more partials of 4000+ cps on the 20” Paiste cymbal. Dynamic Level The ff level consistently produced upper limits considerably higher (20000+ cps) than did the pp level (never higher than 7000 cps). The partials produced at the ff level were consistently of greater intensity than those produced at the pp level. With very few exceptions, the ff level produced from four to six more partials than did the pp level in parallel instances. .Mbst of these additional partials occurred above the 3000 cps level on the Old K. Zildjian cymbal, the 3500 cps level on the 20" Paiste cymbal, the 4000 cps level on the.Avedis Zildjian and 17” Paiste cymbals, and the 5000 cps level on the New K. Zildjian cymbal. Pitch Level and Overtone Intensities Each cymbal produced rather definite fundamental partials at each striking point. The fundamentals produced at the edge striking point on the smaller cymbals ranged from the 220 to 225 cps partials of the Old K. Zildjian cymbal to the 340 cps partials of the Avedis Zildjian cymbal. The fundamentals produced at the cup striking point on the smaller cymbals ranged from the 330 cps level of the Old K. Zildjian cymbal to the 340 to 440 cps partials of the New K. Zildjian cymbal. 102 The 20” Paiste cymbal's edge fundamental was 300 to 330 cps, and the cup fundamental was 380 cps. The above fundamental partials were generally relatively weak. A fundamental partial occurred as the most intense partial only once on the four smaller cymbals (Avedis Zildjian / edge / yellow / mf). However, the fundamental partials at the cup striking point were consistently the most intense partials with all implements and all dynamic levels on the 20” Paiste cymbal. The fundamental partials were one of the two weakest partials in thirty-five of the fifty-four examples on the New K. Zildjian, the Old K. Zildjian, and the 17" Paiste cymbals. There was no consistency among the cymbals as to which partials were the most intense. Except for the ff examples, the most intense partials on all the cymbals were rarely as high as 1000 cps; and in all examples, the strongest three partials rarely went as high as 4000 cps. Only twenty-one energy peaks were found above the 8000 cps level mentioned by Sivian, Dunn, and White (see above on p. 14). Even the 5000 cps level reported by Briggs (see above on p. 14) seemed too high, for only fifty-eight energy peaks were found above that level. Similarity of Overtone Structures The partials and their frequencies presented above in Tables 11 through 15 form the basis for Figure 26. By plotting each partial's frequency on a horizontal scale marked in thousands of cycles per second (kcs), the overtone structures of the cymbals may be observed and compared. Frequencies produced by striking at the edge of the cymbals are indicated by the vertical lines abgye_the frequency scales, and those produced by striking at the cup of the cymbals are indicated by 103 mHmanRu How mounpunhum magnet/o eo~3m5TfieN ehang 86H 2 S 2 2 S 2 NH 2 S a m e e m e n N H o H H H _ H J H H H _i H H H H j H mHuvH :H nwcongougH .5 H _ _ _ _T H: H: _ :. H :z: a... __ _ 2 H :__ jH __ __ HH H._HH._._.H:._.: .5 _ H: H _: __ if: _ _ _ : E __H _:H..._.__.H_.:_..__E. .5 _r E _ :: :CEHH: .... l _ E i Z. LEHH. as : _E H H _: :C: __ E = _ _: __ _ H E _ __ n __ :21 104 the vertical lines beggw_the frequency scales. Overtone intensities are not a factor in Figure 26. The five cymbals studied exhibited somewhat similar overtone structures, producing five rather definite groups of partials under 5000 cps. The 20" Paiste cymbal's groups were the lowest in frequency (320 to 1275 cps; 1500 to 1900 cps; 2200 to 2600 cps; 2850 to 3400 cps; and 3800 to 4300 cps), and the 17" Paiste cymbal's groups were the highest in frequency (285 to 1640 cps; 1900 to 2200 cps; 2850 to 2950 cps; 3600 to 4100 cps; and 4600 to 5000 cps). A further similarity among the cymbals was the presence of relatively few partials above 5000 cps. The.Avedis Zildjian cymbal had the most evenly distributed partials, and the New K. Zildjian cymbal had the most clearly grouped partials with wide spaces between the groups. IV} CONCLUSIONS AND RECOMMENDATIONS Conclusions The questions presented in Chapter I are restated below accom- panied by their answers as concluded from the results presented in Chapter III. Conclusions are also drawn from related research concerning instru- ment size and material, and pitch level. What are the comparative overtone structures produced by triangles when played with large and small implements of the same material? Based on the results of the study of the five small triangles (Abel, Ludwig, Sonor, Zildjian, and the 6” Pigstail), it must be concluded that a change in implement size from a diameter of 5/32” to one of 7/32” produces no consistent effect on the overtone structures. This confirms the findings of Spencer (no overtone changes due to implement size). However, the smaller implement did produce partials of slightly greater intensity on the 10" Pigstail triangle at the mf and pp dynamic levels. What are the comparative overtone structures produced by cymbals when played with hard and soft implements of the same material? On all but the.Avedis Zildjian cymbal, a harder implement (red yarn) produces upper limits at the mf and pp levels higher than, or at least equal to, those produced by a softer implement (yellow yarn). At the ff level, both implements produce sound above 20000 cps. .According 105 106 to this study, a harder implement also produces partials of slightly greater intensity on the 20” Paiste cymbal than a softer implement. What is the effect, if any, of the material of the implement on the overtone structures produced by triangles and cymbals? Triangles On the four small, untapered triangles (Abel, Ludwig, Sonor, and 6” Pigstail), a change of implement material from drill rod to cold-rolled steel does not affect the overtone structures in any consistent manner. However, a cold-rolled steel implement will generally produce more partials on the tapered Zildjian triangle, and partials of slightly greater intensity on the 10" Pigstail triangle than the drill rod imple- ment . C als Based on the results of this study, it must be concluded that a change of implement material from cord to yarn has no consistent effect on the overtone structures of cymbals. This seems to refute Bartlett's implication that harder implements cause the higher partials to predomi- nate. What is the precise relationship between the striking angle and/or point and the predominantly high and low pitch areas within one triangle or cymbal? Triangles .As the top and bottom striking points were used with the implement perpendicular to the plane of the triangle (90° striking angle), both of 107 these points produced in-plane, or transverse, vibration. Four of the six triangles studied (Abel, Ludwig, Zildjian, and 10" Pigstail) exhibited higher fUndamentals in this type of vibration. The out-of-plane vibration produced by the combination of corner striking point and 0° striking angle induced torsional vibration in at least one side of the triangle (the closed side). Zahm's statement that the frequency produced by torsional vibration is lower than that produced by transverse vibration1 is thus confirmed by the results of the Abel, Ludwig, Zildjian, and 10” Pigstail triangles. However, Peters'statement that a 0° striking angle ”will produce a more diffuse sound with.more overtones" seems to be refuted due to the general effects of this striking angle (lower upper limits; fewer partials; and weaker partials). .Although not confirmed by this study, acoustical principles state that ”if the point of impact coincides with a node for a particular partial, that partial will tend to be absent or at.minimum.intensity, while if it coincides with an antinode, the partial will be present with greater strengthflhz Therefore, the percussionist could expect to elicit the highest sounds by striking the triangle at a point near one of the closed corners with a 90° striking angle. The middle of any side should produce slightly lower frequencies when struck at the same angle. And striking either open leg with a 0° striking angle should produce the lowest sounds. This sUbstantiates the findings of Ross and Spencer that the sounds produced on the open sides seem to be lower than the sounds produced on the closed side. 1J.A. Zahm, Sound and Music, 2nd ed., pp. 189-90. 2Wood,.Acoustics, p. 94. 108 als As the cup striking point consistently produced higher fundamental partials, and upper limits equal to or less than those produced by the edge striking point, it must be concluded that a relatively narrow band of sound is produced by striking near the cup and a relatively full sound is produced by striking near the edge of the cymbal. These findings confirm the statements of the authorities quoted in Chapter I (Denov, Leidig, Collins and Green, Goldenberg, Thompson, and Leach). What are the comparative overtone structures produced by triangles and cymbals when played at various dynamic levels? On both triangles and cymbals, a dynamic level of ff will have three effects on the overtone structures: higher upper limits; all partials will be of greater intensity; and more partials will be produced in the majority of instances. What are the relative strengths or intensities of the overtones produced by triangles and cymbals? Triangles Three definite conclusions may be drawn concerning relative over- tone strengths or intensities on triangles: the fundamental partials are usually rather weak; with few exceptions, the most frequent strong partials are below 7000 cps; and there is no consistency as to which partials are most intense. C als The following conclusions may be drawn concerning relative over- tone strengths or intensities on cymbals: the fUndamental partials are 109 usually relatively weak; the strongest three partials will usually be below 4000 cps; and there is no consistency as to which partials are most intense. What are the similarities, if any, among the overtone structures produced by different types of triangles (e.g., spindle and pigstail) and different brands of cymbals (e.g., Avedis Zildjian and Paiste)? Triangles Based on the predicted and analyzed overtone structures, the four relatively thin triangles (Ludwig, Sonor, Zildjian, and 6" Pigstail) exhibited some degree of similarity in the way their partials were grouped. The relatively thick triangles (Abel and 10" Pigstail) were similar in that they produced partials which were more evenly distributed throughout the frequency range. C als .Although the frequency ranges of each group of partials were different for each cymbal, the five cymbals studied produced overtone structures which were similar in two ways: five groups of partials under 5000 cps, and relatively few partials above 5000 cps. What are the modes of vibration of a triangle suspended at one corner? .A 90° striking angle induces primarily transverse, or in-plane, vibration in all three sides of the triangle; and a 0° striking angle induces primarily out-of-plane (but still transverse) vibration in all sides and torsional vibration in at least one side (the closed side). 110 In the mathematical study, no side exhibited more than three nodes for in-plane vibration, or more than four nodes for out-of-plane vibration. There is no consistency, either for any one triangle or among all the triangles, as to the number of nodes occurring in each side at any given frequency or partial number. Both closed corners and both free ends of each triangle are in.motion, either from in-plane or out-of-plane vibration, at each frequency. Instrument Size anthaterial Triangles The mathematical and experimental results confirmed that a longer length does indeed tend to produce a lower resonant frequency on a triangle: 10" Pigstail = 469.7 cps predicted and about 500 cps analyzed; and Abel = 1732 cps predicted and about 1550 cps analyzed. The results also confirmed that a larger diameter does tend to produce a higher resonant frequency: Abel = 1732 cps predicted and about 1550 cps analyzed; and Sonor = 828 cps predicted and about 900 cps analyzed. As all the triangles studied were plated, no conclusions as to the effect of the plating can be drawn from the experimental results. However, the application of the expreSsion indicates that the m2 influence of a plating even several thousandths of an inch thick would be very minimal at best. Triangles made from.metals with larger E : p ratios than steel, such as beryllium, molybdenum, or chromium, will produce higher resonant frequencies than steel triangles. And triangles made from metals with ratios smaller than steel, such as brass, silver, or aluminum, will produce lower resonant frequencies than steel triangles. However, steel 111 apparently has the combination of E and p values, plus the appropriate damping characteristics, most conducive to the production of a high resonant frequency with a relatively long duration of sound. C als It must be concluded that, although the relationships of diameter to frequency and thickness to frequency are generally applicable, the principles did not provide an infallible basis for comparing the overtone structures of the five cymbals studied. No conclusions could be made concerning the influence of the heights of the cup and bow, and the width of the cup. As the three leading cymbal manufacturers apparently make their instruments from the same basic ingredients (copper and tin, with small amounts of iron, lead, or silver), it must be concluded that the chemical makeup of the cymbal material does not affect the overtone structures of the cymbals. Pitch Level Given the right combination of triangle, implement, striking point and angle, and dynamic level, it is conceivable (though improbable) that some partial(s) could be emphasized to the extent of being noticeably stronger than other partials, thus approaching a comparatively ”definite" pitch. And Bartholomew reports that it is possible, through careful selection of the striking point, stroke, and implement, to isolate certain I of a cymbal's many simultaneously-sounding partials, thus creating a somewhat well-defined pitch. These practices, however, would disregard 1Bartholomew,.Acoustics of Music, p. 134. 112 the common opinion that good triangles and cymbals do not produce definite pitches. And no instrument examined did produce anything like a definite pitch, or even a noticeably strong band of frequencies which could be heard as definite pitches. Thompson's conclusion that a New K. Zildjian cymbal seems to have more lows than an Avedis Zildjian cymbal is substantiated. However, the fact that the upper partials decayed more rapidly than the lower partials on all the cymbals seems to refute the statements of Spohn (the sound ”will tend to rise") and Lang (it is important ”that its highs 'come out'") concerning the projection and duration of the upper partials. Recommendations for Performance It should again be noted that this study is not concerned with the strike tones (or initial transients), but only with the overtone structures present after_impact. The author recognizes that implement size, hardness, and.material, striking point, and dynamic level all undoUbtedly contribute significantly to the overall sound (impact plus sustaining sound) produced by a triangle or cymbal. With this in mind, the following recommendations for performance (including choice of instrument, choice of implement, and striking point and angle) are presented based on the preceding conclusions drawn from the results in Chapter III. Instrument Size and Material Triangles For a high resonant frequency, a triangle with a combination of a relatively short length and a relatively large diameter is recommended. 113 For a lower resonant frequency, either a larger or thinner triangle is recommended. Steel triangles are recommended, for, at the present time, steel apparently has the combination of material properties, plus the appropriate damping characteristics, most conducive to the production of high resonant frequencies with relatively long duration of sound. However, further research may contribute metals with Higher E : p ratios plus superior damping qualities, thus resulting in triangles with even higher resonant frequencies and greater duration of sound. C als Only two general recommendations can be made concerning cymbal size:y given two cymbals of equal diameter, the thicker one should be used if higher sounds are desired; and given two cymbals of equal thick- ness, the larger one should be used if lower sounds are desired. Based on this study, no recommendations can be made for choosing among Avedis Zildjian, New and Old K. Zildjian, or Paiste cymbals on the basis of material. Implement Size or Hardness and.Material Triangles As a change of implement size from 5/32" to 7/32” produced no consistent effects on the overtone structures of the triangles in this study, no recommendations can be made concerning implement size. When playing one of the small, untapered triangles (Abel, Ludwig, Sonor, and 6" Pigstail), implements of either drill rod or cold-rolled steel are recommended (with some percussionists preferring the drill rod 114 as it resists denting). However, a cold-rolled steel implement is recommended to produce a fuller sound (more partials) on a Zildjian triangle; and a drill rod implement is recommended for the 10" Pigstail triangle is partials of greater intensity are desired. C als .A harder yarn implement is recommended if the desired cymbal sound at a mf or pp dynamic level is to be as high as possible. .As a change of implement material from cord to yarn produced no consistent effects on the overtone structures of the cymbals in this study, no recommendations can be made concerning implement material. Striking Point and.Angle Triangles The following recommendations are made for playing Abel, Ludwig, Zildjian, and 10" Pigstail triangles: strike near one of the closed corners with a 90° striking angle for the highest sounds; strike in the middle of any side with a 90° striking angle for slightly lower sounds; and strike on either open side with 3 0° striking angle for the lowest sounds. C als Striking near the cup of a cymbal is recommended if a high, relatively narrow sound is desired. Conversely, striking near the edge of a cymbal is recommended if low, full sounds are desired. Striking at both points simultaneously should help to bring out the total sound inherent in the cymbal. 115 Recommendations for Further Research The following topics are presented to illustrate areas of research which would contribute the information to confirm or refute many of the theories and opinions not covered in this study relative to the selection of, and performance on, triangles and cymbals. It should be noted that similar studies of other percussion instruments (e.g., wood block, tambourine, tamtam, cowbell, temple blocks, claves) would also benefit public school music directors and performing percussionists. Triangles 1. Construct triangles of equal geometric properties, but of different materials, in order to determine the exact influence of these materials on frequency and duration. 2. Determine the influence of the equality or inequality of the lengths of the sides of a triangle. 3. Determine the influence of the equality or inequality of the angles formed by the sides of a triangle. 4. Determine the influence of implements of more extreme sizes and other materials. 5. Determine the exact influence of the striking point and angle by using other points and angles in various combinations. 6. Determine the influence of implement size and material, striking point and angle, and dynamic level on the strike tones (or starting transients) of a triangle. 7. Determine the influence of aging on the acoustical properties of triangles. 116 8. .Make comparative analyses of the acoustical prOperties of the favorite triangles of leading percussionists. Cymbals 1. Determine the exact influence of diameter and thickness by constructing cymbals differing only in these dimensions. 2. Determine the influence of the shape, size, and height of the cup and bow. 3. Determine the influence of implements of more extreme sizes and hardnesses, as well as other materials. 4. Determine the influence of implement hardness, size, and material, striking point, and dynamic level on the strike tones (or starting transients) of a cymbal. 5. Determine the influence of aging on the acoustical properties of cymbals. 6. Make comparative analyses of the acoustical properties of the favorite cymbals of leading percussionists. BIBLIOGRAPHY BIBLIOGRAPHY , .Articles Denov, Sam. "Equipping the Cymbalist." The Instrumentalist, XVIII (June, 1964), 60-61. . "Techniques of Cymbal Playing." The Instrumentalist, XIX (September, 1964), 58-64. Flagler, J.M. ”Onward and Upward with the.Arts." The New Yorker, December 6, 1958, pp. 135-63. Hart, William sebastian. "Percussion Clinic." The Instrumentalist, XIII (February, 1959), 69-71. Lang, Mbrris. "Percussion Clinic.” The School Musician, XXXIII (December, 1961), 16-17. Moore, James. "Percussion Acoustics: .An Introductory Evaluation." Percussionist, V (October, 1967), 218-20. Navin, Thomas. "werld's Leading Cymbal Maker: Avedis Zildjian Company." Reprinted from the Bulletin of the Business Historical Society, December, 1949. Peters, Mitchell. "Triangle Technique." The Instrumentalist, XXII (February, 1968), 79-84. Ross, James. "The Triangle: Don‘t Underestimate It." The Instrumen- talist, XIX (April, 1965), 84-86. Sewrey, James. "Percussion Clinic." The School Musician, XXXIII (February, 1962), 54-56. . "Percussion Clinic." The School.Musician, XXXIII (January, 1962), 14-15. Sivian, L., Dunn, H., and White, S. "Absolute Amplitudes and Spectra of Certain Mosical Instrumenta." Journal of the Acoustical Society of America, II (January, 1931), 330-71. Stauder, Wilhelm. "Schlaginstrumente--Akustik." Die Musik in GeSChichte und Gegenwart, 1963. V01. XI. 117 118 Thum, Ernest Edgar, and Grace, Richard Edward. "Iron and Steel-~C1assi- fication and Uses of Plain Carbon Steels." Encyclopaedia Britannica, 1963. V61. XII. Tilles, Bob. "The Bob Tilles Column." Ludwig DrUmmer, VIII (Spring, 1968), 31. Books Bartholomew, Wilmer. Acoustics of Music. New York: Prentice-Hall, 1942. Bartlett, Harry. Guide to Teaching Percussion. Dubuque, Iowa: wm. C. Brown Co., 1964. . Percussion Ensemble Method. Dubuque, Iowa: Wm. C. Brown Co., 1961. Blades, James. Orchestral Percussion Techniques. London: Oxford University Press, 1961. Briggs, G.A. Musical Instruments and Audio. Yorkshire, England: Wharfedale Wireless Works, Ltd., 1965. Brown, Thomas, and Musser, Willard. Percussion Studies, I. Delevan, New York: Kendor Music, Inc., 1962. Buck, Percy. .Acoustics for Musicians. London: Oxford University Press, 1918. Burns, Roy. The Selection, Use and Care of Cymbals in the Stage and Dance Band. New York: Henry Adler, Inc., 1964. Collins, Myron, and Green, John. Playing and Teaching Percussion Instru- ments. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1962. Cymbal and Accessory Catalog. North Quincy, Massachusetts: Avedis Zildjian Co., 1966. Cymbal Hints. North Quincy,.Massachusetts: Ayedis Zildjian Co., n.d. Denov, Sam. The Art of Playing the Cymbals. New York: Henry Adler, Inc., 1963. Firth, Vic. Percussion Symposium. New York: Carl Fischer, Inc., 1966. Gardner, Carl. The Gardner MOdern Method. New York: Carl Fischer, 1938. 119 Goldenberg, Morris. .Modern School for Snare Drum. New York: Chappell 8 Co., Inc., 1955. Grant, Phil. Tested Tips for School Music Supervisors. Brooklyn, New York: The Fred Gretsch Co., Inc., n.d. Lakser, M.E. User's Guide--Structural Analysis and Matrix Interpretive System (SAMIS). Dearborn, Michigan: Fordeotor Co., 1967. Lang, Theodore E. Structural Analysis and Matrix Interpretive System (SAMIS) User Report #33-305. Pasadena, CalifOrnia: Jet PrOpulsion Laboratory--Ca1ifornia Institute of Technology, 1967. Summary of the Functions and Capabilities of the Structural Analysis and Matrix Interpretive System Computer Program #32-1075. Pasadéna, California: Jet Propulsion Laboratory-- California Institute of Technology, 1966. Leach, Joel. Percussion.Manual for Music Educators. New York: Henry Adler, Inc., 1964. Leidig, Vernon. Contemporary Percussion Technique and Method. Hollywood: Highland Mosic Co., 1960. Melosh, Robert J., and Christiansen, Henry N. Structural Analysis and Matrix Interpretive System iSAMIS) Program: Technical Report #33-311. Pasadena, California: Jet Propulsion’Laboratory-- California Institute of Technology, 1966. Melosh, Robert J., Diether, Philip A., and Brennan, Mary. Structural Analysis and Matrix Interpretive System (SAMIS) Program Report #33-307, Revision #1. Pasadena, Califbrnia: Jet Propulsion Laboratory--California Institute of Technology, 1966. Olson, Harry. Music, Physics and Engineering. 2nd ed. New York: Dover Publications, Inc., 1967. Payson, A1, and McKenzie, Jack. .MUsic Educators' Guide to Percussion. Rockville Centre, New York: Belwin, Inc., 1966. Price, Paul. Techniques and Exercises for Playing Triangle, Tambourine, and Castanets. Revised ed. New York: jMusic for Percussion, 1967. Spohn, Charles. The Percussion. Boston: .Allyn 8 Bacon, Inc., 1967. White, Charles. Drums Through the Ages. Los Angeles: The Sterling Press, 1960. Wildman, Louis. Practical Understanding of the Percussion Section. Boston: Bruce Humphries, 1964} 120 Wood, Alexander, .Acoustics. New York: Interscience Publishers, Inc., 1941. . The Physics of Music. 5th ed. London: lMethuen 8 Co., Ltd., 1950. Zahm, J.A. Sound and Music. 2nd ed. Chicago: A.C. McClurg 8 Co., 1900. Zienkiewicz, O.C. The Finite Elementhethod in Structural and Continuum Mechanics. London: IMcGraweHill Pub. Co., Ltd., 1967. UnpublishedlMaterials Andrus, Donald. Personal letter. OctOber 28, 1967. Duff, Cloyd. Personal letter. October 17, 1967. Little, Robert W. "Finite Element Method." Unpublished notes, Michigan State university, n.d. Peters, Gordon. "Treatise on Percussion." UnpUblished Master's thesis, Eastman School of Music, 1962. Spencer, H.R. Personal letter. December 2, 1968. Thompson, Harold. Personal letter. October, 1967. General References .Articles .Arts, Jan. "Jottings From.My'Experiences'With the Sound of Bells." Journal of the Acoustical Society of America, XVII (January, 1946), 231. . "Sound of Bells--Jottings from My Experiences in the Domain of the Sound of Bells." Journal of the Acoustical Society of America, IX (April, 1938), 344-47. . "Sounds of Bells-~The Secondary Strike Note." Journal of the .Acoustical Society of.America, X (April, 1939), 327-29. Barakat, Richard. "Transverse'Vibration of a Mbving Thin Rod." Journal of the Acoustical Society of.America, XLIII (Marth, 1968), $33-39. Beck, John. ”Membranophones and Idiophones-oH new? Hone Hmnhm--.H< 0.3mm _ x go\goe\HHHoo e\Hoo< _ ... H H . . H H ., M___:n. mo\ooe\HHHno eNHoo< ..H.. r _ a; ee\ooe\HHHoo e\Hoo< 133 decay. Thus, the frequencies represented in section 6 of Figure.Al decayed much.more rapidly within the two seconds of recording time than the frequencies represented in section 5. APPENDIX C iModes of'Vibration-éMatrix and Diagrams Table A1 shows the matrix developed by the SAMIS computer program for the vibrational modes of the Zildjian triangle. The partial numbers are listed in the first column. The next three columns are labeled points and locations. The numbers at the top of each column serve to identify the nodal points (1 through 16) and locations in three-dimensional space (1 = x, 2 = y, and 3 = 2). That is, the numbers 11, 12, and 13 serve to locate nodal point 1 in the x, y, and 2 directions. The partials with numbers in the first two columns only (i.e., in the x and y columns) were produced by in-plane vibration. The partials with a number in the third column only (i.e., in the 2 column) were produced by out-of-plane vibration. For in-plane vibration, ”plus" numbers in the first two columns indicate positive directions for x (to the right) and y (up), while "minus" numbers indicate negative directions for x (to the left) and y (down). For out-of-plane vibration, "plus" numbers in the third column indicate a.positive direction for 2 (above the plane of the triangle), while "minus” numbers indicate a negative direction for 2 (below the plane of the triangle). The last three digits of the numbers indicate the placement of the decimal point: +001 indicates that the decimal point is to be moved to the right one place; and -002 indicates that the decimal point is to be moved to the left two places. 134 135 Table.A1.--Vibrationa1 Mbde Matrix for Zildjian Triangle Partial Points and Locations Numbers 11 12 13 1 0. 0. -0.3237+000 2 -0.8980-001 0.2665+000 0. 3 0. 0. -0.3756+000 4 0.3925-001 -0.2950+000 0. 5 0. 0. -0.3650+000 6 0.4471-001 0.4649+000 0. 7 -0.3715-001 -0.ll73+000 0. 8 0. 0. -0.1983+000 9 0. 0. 0.4142+000 10 -0.2451-001 -0.3739+000 0. 11 0. 0. -0.1410+000 12 0.4351-001 -0.2387+000 0. 13 0.4054-001 -0.1069+000 0. l4 0. 0. -0.1332+000 15 -0.5416-002 0.2745-001 0. 16 0. 0. -0.2563+000 Partial Points and Locations NUmbers 21 22 23 1 0. 0. 0.1373+000 2 -0.l337+000 0.1158+000 0. 3 0. 0. -0.6517-001 4 0.5841-001 -0.4519-001 0. 5 0. 0. -0.3086-001 6 0.6650-001 -0.2702-001 0. 7 -0.5449-001 _ 0.9894-001 0. 8 0. 0. 0.1618+000 9 0. ‘ 0. -0.3755+000 10 -0.3756-001 0.3731+000 0. ll 0. 0. 0.1617+000 12 0.6416-001 0.2867+000 0. 13 0.5919-001 0.19l4+000 0. 14 0. 0. 0.2516+000 15 -0.7870-002 -0.5402-001 0. 16 0. 0. 0.5056+000 136 Table.A1.--(cont'd.) _— , Points and Locations Partial Numbers 31 32 33 l 0. 0. 0.1287+000 2 -0.l46l+000 -0.9703-001 0. 3 0. 0. 0.2693+000 4 0.6373-001 0.2133+000 0. 5 0. 0. 0.2760+000 6 0.7245-001 -0.4388+000 0. 7 -0.5921-001 0.1030+000 0. 8 0. 0. 0.1706+000 9 0. 0. -0.3253+000 10 -0.4027-001 0.2530+000 0. 11 0. 0. 0.7312-001 12 0.6836-001 0.8966-001 0. l3 0.6138-001 -0.1020+000 0. 14 0. 0. -0.l678+000 15 -0.8070-002 0.4155-001 0. 16 0. 0. -0.3933+000 —-_T_——-—#——_ Partial Points and Locations Numbers 41 42 43 1 0. 0. 0.2683+000 2 -0.1589+000 -0.1990+000 0. 3 0. 0. 0.3539+000 4 0.6916-001 0.2599+000 0. 5 0. 0. 0.2671+000 6 0.7846-001 -0.4223+000 0. 7 -0.6294-001 -0.6048-001 0. 8 0. 0. -0.1124+000 9 0. 0. 0.3119+000 10 -0.4254-001 -0.3581+000 0. 11 0. 0. -0.1516+000 12 0.7148-001 -0.3191+000 0. l3 0.6142-001 -0.l328+000 0. l4 0. 0. -0.1356+000 15 -0.7931-002 0.2062-001 0. l6 0. 0. -0.l935+000 Table Al.--(cont'd.) Points and Locations 137 Partial Numbers 51 52 53 1 0. o. 0.2463+000 2 -0.1719+ooo -0.1563+000 0. 3 0. 0. 0.2023+ooo 4 0.7465-001 0.1079+000 o. 5 o. o. -o.2259-001 6 0.8445-001 -o.1411+ooo o. 7 -0.6611-001 -0.9155—001 o. 8 o. 0. -0.l789+000 9 o. o. 0.4418+000 10 -O.4428-001 -O.3967+000 0. 11 o. o. -0.4389-001 12 0.7342-001 -0.2036+000 0. 13 0.5928-001 0.1522+000 o. 14 0. o. 0.2269+000 15 -o.7443-002 -0.5600-001 o. 16 o. o. o.4922+ooo _—_——_————— Points and Locations Partial Numbers 61 62 63 1 0. o. 0.1063+000 2 -0.2145+ooo 0.1685-001 o. 3 o. o. -0.4879-001 4 0.9281-001 -0.1491+ooo 0. 5 0. o. -0.5130+ooo 6 0.1047+ooo 0.3118-001 0. 7 -0.7946-001 0.1473+ooo o. 3 o. o. o.1s7o+ooo 9 o. o. -0.8990-001 10 -0.5263-001 0.5574-001 o. 11 0. o. o.3995+ooo 12 0.8574-001 -0.3893-001 o. 13 0.6355-001 0.5486-001 o. 14 0. o. -o.2439-oo1 15 -0.7625-002 -o.7199-002 0. 16 o. o. -0.1563+000 Table.Al.--(cont'd.) 138 Points and Locations Partial Numbers 71 72 73 1 0. o. -o.1148+000 2 0.4706-001 -O.ll46+000 o. 3 0. o. -o.1712+000 4 -0.1491+ooo -O.4964-001 0. v 5 o. o. -o.3553-001 6 -o.3793-oo1 0.1084+000 o. 7 0.2847+000 -0.9087-002 o. 8 o. o. -o.1621-001 9 o. 0. -0.2584+000 10 0.1817+000 -o.4416-001 o. 11 o. o. -0.3651+000 12 -0.2911+ooo o.1401+ooo o. 13 -0.2527+000 0.2045+ooo o. 14 o. o. -0.9868-001 15 0.3329-001 -o.2536-001 o. 16 o. o. -o.8314-001 __—-————H___———_—-—___-——-——-——_-—_— Points and Locations Partial Numbers 81 82 83 1 0. o. -0.2652+000 2 0.2646+ooo -0.2107+ooo o. 3 o. o. -o.2075+ooo 4 -o.2783+ooo 0.1631-001 o. 5 o. o. o.337o+ooo 6 -o.2414+000 0.1996+000 o. 7 0.294o+ooo -o.2128-oo1 o. 8 o. o. -o.1o71+ooo 9 o. o. -o.1232+ooo 10 0.3680-001 0.2649-001 o. 11 o. o. -o.3557+ooo 12 -0.291o+ooo 0.1430+ooo 0. 13 0.3950-001 0.5142-001 o. 14 o. o. 0.4428-001 15 -0.2371-001 0.4581-002 o. 16 0. o. 0.7628-001 Table.Al.--(cont'd.) 139 Points and Locations Partial Numbers 91 92 93 1 o. o. -o.2823+000 2 0.324o+ooo -o.2326+ooo o. 3 o. o. -o.1092+ooo 4 -0.2505+000 0.9651-002 o. 5 o. o. 0 3726+000 6 -0.2982+000 0.2218+000 o. 7 -o.6084-001 0.1343+ooo o. 8 o. o. -o.1212-oo1 9 o. o. 0.1783+ooo 10 -0.1994+000 0.1388+ooo o. 11 0. o. 0.3158+ooo 12 o.2035+ooo -0 8477-001 0 13 0.3036+000 -o.9204-001 0. 14 0. o. 0.8066-001 15 -o.4080-001 0.1631-001 0. 16 0. o. 0.2824-001 '._?‘,..7 - - - w—fl w Points and Locations Partial Numbers 101 102 103 1 o. 0. -0.1718+000 2 0.1997+ooo -0.1691+000 0. 3 0. 0. 0.1138+ooo 4 -0.1307+ooo -o.3985-001 o. 5 0. 0. 0.6328-001 6 -o.1495+ooo 0.1481+000 o. 7 -0.2786+000 0.2248+ooo 0. 8 o. o. o.1547+ooo 9 0. o. 0.2287+ooo 10 —0.1503+ooo o.1174+000 o. 11 0. o. -0.4527+000 12 0.4132+ooo -o.1799+ooo o. 13 -o.1371+ooo 0.9097-001 0. 14 0. 0. -o.1015+000 15 -o.919o-002 0.5470-002 0. 16 0. 0. -0.7844-001 Table‘Al.--(cont'd.) 140 '——__1 P . Points and Locations art1al me“ 111 112 113 1 0. 0. -0.S828-002 2 -0.5527-001 -O.4389-001 O. 3 0. 0. 0.4062+000 4 -0.9662-001 -O.4853-001 O. 5 0. O. -0.4031+000 6 0.5553-001 0.4714-001 0. 7 0.1687-001 0.7379-001 O. 8 0. 0. 0.2532+000 9 0. 0. -0.3470-001 10 0.2110-001 0.3764-001 0. 11 0. 0. -0.3819+000 12 0.1365+000 -0.4873-001 0. 13 -0.4682+000 0.2216+000 0. 14 O. 0. -0.1586+000 15 -0.2804+000 0.1357+000 O. 16 0. 0. 0.1431+000 Partial P01nts and Locations me“ 121 122 123 1 0. 0. O.3019+000 2 0.1900+000 0.1034+000 0. 3 0. O. -0.1496-001 4 0.1266+000 0.8157-001 0. 5 0. 0. -0.1380+000 6 0.3381-001 0.3147-001 0. 7 -0.3931+000 -0.1805+000 0. 8 0. 0. -0.4469+000 9 0. 0. -0.1932+000 10 0.2038+000 0.1412+000 0. 11 0. 0. 0.1486-001 12 -0.5719-001 -0.1409+OOO O. 13 -0.3590+000 0.2150+000 0. 14 0. O. 0.4860+000 15 0.3247+000 0.4548+000 0. 16 O. 0. —O.2382+000 Table.A1.--(cont'd.) Points and Locations 141 1 .a .At. Int-(J. .tr Partial Numbers 131 132 133 1 o. o. 0.4780+000 2 o.3322+000 0.1897+000 0. 3 0. o. -0.3584+000 4 0.3899+000 0.2357+000 o. 5 0. o. 0.3774-001 6 0.1106+000 0.7407-001 o. 7 -0.2471+000 -0.1063+000 o. 8 0. o. -o.3035+000 9 o. o. -0.1398+000 10 0.1921+ooo 0.1295+000 o. 11 0. 0. 0.1351+000 12 -o.1947+000 -0.1980+000 0. 13 -0.4895-001 0.3106+000 o. 14 o. o. -0.2789+OOO 15 -O.2558+000 0.5999-001 o. 16 0. o. 0 9023-001 Partial Points and Locations Numbers 141 142 143 1 o 0. 0.4174+000 2 0 2978+000 0.1716+000 o. 3 0 o. -o.4271+ooo 4 0 4333+000 o.2597+ooo 0. s o 0. 0.1039+000 6 0 1443+000 0.9112-001 0. 7 0 3146+000 0.2128+000 0. 8 o. 0. o.4403+000 9 0 0. 0.1478+000 10 -0 1623+000 -o.8531-001 0. 11 0 o. -o.8615-001 12 0 1247+000 0.1778-001 0. 13 -0 4508-001 0.2073+ooo 0. 14 0 0. -0.2800+OOO 15 -0 3567+000 -0.6421-001 0. 16 0. o. 0.1665+000 Tab1e7A1.--(cont'd.) 142 Points and Locations Partial Numbers 151 152 153 1 o. o. 0.1707+000 2 o.1234+ooo 0.7075-001 0. 3 o. o. -o.2007+000 4 0 2013+000 0.1208+000 0. 5 o. o. 0.5728-001 6 0.7309-001 0.4614-001 0. 7 0 3614+000 o.2272+ooo o. 8 o. o. O.4984+000 9 o. o. 0.1873+ooo 1o -o.2380+ooo -0.1354+000 0. 11 o. o. -0.1669+ooo 12 0.2567+ooo 0 1235+000 o. 13 -o.2289+ooo -o.1552-001 o. 14 0. o. 0 6118+000 15 o.4921+ooo 0.3666+000 o. 16 0. o. -o.2953+ooo Partial Points and Locations Numbers 161 162 163 1 0. o o. 2 o. o o. 3 o. o 0. 4 o. 0 0. 5 O. 0 0. 6 o. 0 o. 7 0. 0 O. 3 o. o o. 9 0. 0 O. 10 0. o o. 11 0. 0 o. 12 0. o o. 13 0. o o. 14 0. o 0. 15 0. o 0. 16 0. o o. 143 As the unit of measurement was not specified by the computer, the arbitrary scale chosen for the diagrams of the vibrational modes in Figure.A2 was: a 5/16" square equaled a number of 0.2000+000. Referring to Table Al, the location of point 1 in the second mode of vibration (partial 2) was found by moving .0898 to the left (almost half a square) and .2665 up (slightly more than two and one-quarter squares). The same procedure was followed for each point for each partial, resulting in the accurate plotting of the vibrational modes for each partial. In Figure A2, the shape of the triangle is shown by the solid ‘1 lines, and the deformities produced by vibration are shown by the dotted E lines. .As mentioned before, point 16 was fixed to eliminate rigid body motions, thus accounting for the apparent lack of vibration at the free end of the upper right-hand leg. The diagrams for in-plane vibration (partials 2, 4, 6, 7, 10, 12, 13, and 15) are shown as if the triangle were being observed from a point perpendicular to the plane of the triangle. The diagrams for out-of-plane vibration (partials l, 3, 5, 8, 9, ll, 14, and 16) are shown as if the triangle were being observed from a point just above the plane of the triangle looking from the base toward the apex. 144 ,0 / Partial l - 908.0 cps \ t5 0: ..‘~\ o A. ‘$“ “* ' \ ' \ I I \ ‘ \ ’ \ / \ / I .\ / \ P \ / \ / x R ’ \ f \ / \ / \ / Q / \ I Partial 2 - 922.3 cps \\ ’ \ / // ...... ...-0'” Figure AZ.--Modes of Vibration for Zildj ian Triangle Partial 3 - 1230 cps —o——_ ’ ‘ ’ 4 \ 4 \\ ‘~. P \ / \ I / \ / ‘~\ I \ / \ ’ b / \ d \ I \ / \ / \ / b / \ 4. \ I \ I \ I \ I ‘0 + \ Partial 4 - 1276 cps \ ’ ,4—""""“"~~.c \‘~.. Figure1A2.--(cont'd.) 146 Partial 5 - 1411 cps ‘ .1}! Mu'f ’4’ -—“.~.~ ’ i 70” 4 4 ‘fiF / /’ \\ / ’ \\ éx/ ‘. \ / \\ /” \ p / / \\ ’ \ / ‘ I ’K / \\ I \ I \ I , ‘R I \ \ I, ‘ I \ I o \ I Partial 6 - 1554 cps \ \ —. I I, I ",,” UL ./ \\\ // Figure.A2.--(cont'd.) 147 I \\ I I \ . I ‘~ 0 \ \ \ \ \ I Vt \ I \ i \ I \ ’ \ / 9 / l a , \ / Part1a1 7 - 3028 cps \ ’ \ ’ \ / t——- \. ‘ ’0' ‘~.-—-’ Figure A2.--(cont'd.) 148 // Partial 9 - 3184 cps I, 4— - ~ ‘ /; a \ \.~\ ~ "’/ / \ / \ / \ / \ x x \ / / \\ ’ \ / 6 f \ l \\ I ’ 1 ’ 4 \ I \ / \\ ’ x ‘ Partial 10 - 3391 cps \\ l \ // _ \\ ,r’ ”x / / \\ ~\ ,I ‘\ \ \ / \ \.‘ __ _ _’ ,7 Figure A2 . -- (cont'd.) 149 I\ I I I \ ,’ / \ / \ / \ / \\ /'/.( ‘\\ // \ / ‘\ / \ / \ I \ f’ I. \ I Partial 12 - 3872 cps \\ I \ ””--.~\ ‘\ / \ \.~ I Figure.A2.--(cont'd.) 150 ”\ I, \ | I \, ‘, ‘\ \ \ I I “\ I I \ I \ ' / ‘ F / i / i‘ /_ i /// \ . / \ / \ “ \ I .\~ I Partial 13 - 5450 CPS \ I \ I I .-—"""--"‘.~\‘ l”—'-.~‘\ \:k_—”.7 i .,o~ \ \ \ \ Partial l4 - 5751 cpS \‘ f ‘fl Figure A2.--(c0nt'd-) Figure A2. -- (cont'd.)