—— qegls x" ‘ , .. r 4|. 'D': '33 UL ~ V v ' ~ t. ‘ 1 1;. ' ‘ l I" "-'-2' u 5_ ,A 8"?!" \u «.1 8 «fianc- .0 fish "f I .. J 'm‘ ! U‘vt .euo area motor: 1; Thiii§t°63fldaéfi3 C ' y. ‘3 "“ '"“ ”’ War—1:: ‘- t l . t I! THE COMPUTER ANALfS WM. HUS-LC: AN APPLICATION PROGRAM USING S‘ET “THEORY - —.—‘h—~ -. r.“— (a, .' 3.. presented by “-~ ‘-____.. W‘- s... -. has been accepted towards fulfillment ' of the requirements for Ph.D. _ Music degree 1n .cewalz) Major professor Date 10/28/83 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 5 lllllllllllfilllllllll/ll”WillL 3 1293 01059 0119 from your record. PLACE N RETURN 30)!» remove figment TO AVOID FINES mum on or baton UE DATE DUE DATE DUE DATE D THE COMPUTER ANALYSIS OF ATONAL MUSIC: AN APPLICATION PROGRAM USING SET THEORY BY Mary Hope Simoni A DISSERTATION Submitted to Michigan State University in partial fufillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Music 1983 Copyright by Mary Hope Simoni 1983 ABSTRACT THE COMPUTER ANALYSIS OF ATONAL MUSIC: AN APPLICATION PROGRAM USING SET THEORY BY Mary Hope Simoni The Purpose The purpose of the study is to provide music theo- rists with software which will reduce the amount of time required by set theory analysis and decrease the probability of human error. The software is capable of executing many facets of set theory analysis as codified by Allen Forte. The Procedure The software is designed for a Digital Equipment Corporation VAX 11/780 minicomputer in the programming language BASIC—Plus 2. Eadh component CHE set theory analysis is modularized in the program BOETHIUS. The pro- gram BOETHIUS is supported by two databases: 1) FORTE -— a database of all prime forms accompanied by their vector and set complement of cardinalities from two to ten, and 2) ALPHONCE -- a database of all subsets and supersets of each set listed in FORTE. BOETHIUS reduces sets to prime form and accesses the databases for set complex relations,* similarity relations, and invariance. BOETHIUS creates two output files: 1) ANALYSIS -- a list of the prime form sets within a composition accompanied by their vector, set complement and Basic Interval Succession pattern and 2) EVALUATE -- set complex relations, similarity relations, and invariance for all pairings of sets listed in ANALYSIS. The software is applied to Five Pieces for Piano by George Crumb for a thorough set—theoretical analysis. Conclusions The computer is an indispensable tool in the set- theoretical analysis of music. Computer-assisted analysis revealed many set—theoretical relationships in Five Pieces for Piano by George Crumb. * Set complex relations and other terms pertinent to computer-assisted set-theoretical analysis are defined in the Glossary of Technical Terms included ix: the disserta— tion. AC I‘QIOW LE DGMENT S Special thanks to Dr. Russell Friedewald, Guidance Committee Chairperson, for his insightful advice, enthusiasm and support. I hOpe I am able to model his insatiable curiousity, acceptance of divergent opinions and profession- alism throughout my career. Thanks to the members of my Guidance Committee: Dr. Theodore Johnson, Dr. James Niblock and Ms. Deborah Moriarty. The dissertation woubd not have been possible without the resources of the Lansing Community College Science Department in the Division of Arts and Sciences. Very special gratitude ‘Uo mgr family, my friends in music, and especially, Kevin Dowd. -i— TABLE OF CONTENTS LIST OF TABLES O I O OOOOOOOOOOOOOO O O O O O O O O O O O O O .......... LIST OF FIGURES 000.........OOOOOOOOOOOOOOO.....OOOOOOO Chapter 1 — Introduction 1.1 FJHOH Atom) 0! UI-waHtu mpwwl—J'U U1 C C a wcooiwtps’whoniwroz‘w WW \IO‘ HrdhaHhaan .waI—‘O wwwwwww Identification of the Problem ................... Statement of Purpose ............................ Statement of Objectives ... ...... ................ Review of the Related Literature: The Application of Computer Technology to the Discipline of Music Methodology ..................................... ter 2 - Description of Databases and External Files Terminology of Database Design .................. Terminology of File Types ....................... Databases Utilized in the Program BOETHIUS ...... Reports Generated by BOETHIUS ................... Program Structure Summary ....................... ter 3 — Description of Modules in Program BOETHIUS Internal Matrices and Dimension Declarations .... Set Entry . ....... ............................... Transfer B.I.S. Patterns to MATRIX BIS .......... Normal Order .................................... Assign Normal Order Arrangements to MATRIX B for Best Normal Order Operations .................... Best Normal Order Operations .................... Best Normal Order Operations: Determine if the Set is in Transposition or Inversion ............ The Set is in Transposition ..................... The Set is in Inversion ......................... Assign Set in Prime Form to MATRIX PRIME$ ....... Retrieve the Set in Prime Form from FORTE ....... Print Set Data to ANALYSIS ...................... Conditional Branching ........................... Compare Sets for Set Complex Relations, Similarity Relations or Invariance .............. Chapter 4 - The Alpha Test: A Computer-Assisted Set- Theoretic Analysis of Five Pieces for Piano by George Crumb 4.1 b-h-b-bob 0“!!wa Summary of Analysis of the First Piece .......... Summary of Analysis of the Second Piece ......... Summary of Analysis of the Third Piece .......... Summary of Analysis of the Fourth Piece ......... Summary of Analysis of the Fifth Piece .......... Nexus Set ...................................... -11. 10 12 13 21 21 23 24 25 27 31 32 33 34 36 38 38 4O 4O 41 46 53 59 66 73 81 Chapter 5 - Summary 5.1 5.2 5.3 5.4 Conclusions ..................................... Strengths of BOETHIUS and Accompanying External Files ........................................... Limitations of BOETHIUS and Accompanying External Files 000......0.00.0...O.....OOOOOOOOOOOIIOOOOOO Futurisms ......OOOOO...I.......OOOOOOOOOOOIOOIOO APPENDICES Appendix A Appendix B Appendix C Database FORTE ........................... Computer Program BOETHIUS ................ Output File EVALUATE for the Third Piece from Five Pieces for Piano by George Crumb Appendix D - A User's Guide to BOETHIUS ............... Glossary of Technical Terms ........................... Bibliography .....OOOOOOOOOO0.0.00000000000000000000000 -iii— 85 85 86 87 88 93 101 106 109 113 Table THE FIRST Table Table Table Table Table 1. 2. 3. 4. 5. 6. LIST OF TABLES A Relational Data Model ....... PIECE ANALYSIS for Frequency of First Piece Set Complex Relations in the First Piece . Similarity Relations in the First Piece .. Inclusion Relations in the First Piece ... the First Piece Occurrence of PC Sets in the THE SECOND PIECE Table Table Table Table Table Table THE THIRD Table Table Table Table Table 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. ANALYSIS for the Second Piece ............ Frequency of Occurrence of PC Sets in the Second Piece Set Complex Relations in the Second Piece Similarity Relations in the Second Piece . Subset Invariance from Sets of Cardinality Six in the Second Piece Inclusion Relations in the Second Piece .. PIECE ANALYSIS for the Third Piece Frequency of Occurrence of PC Sets in the Third Piece Set Complex Relations in the Third Piece . Similarity Relations in the Third Piece .. Inclusion Relations in the Third Piece ... THE FOURTH PIECE Table Table Table Table Table Table THE FIFTH Table Table Table Table Table 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. ANALYSIS for the Fourth Piece Frequency of Occurrence of PC Sets in the Fourth Piece Set Complex Relations in the Fourth Piece Similarity Relations in the Fourth Piece . Subset Invariance from Sets of Cardinality Six in the Fourth Piece .................. Inclusion Relations in the Fourth Piece .. PIECE ANALYSIS for the Fifth Piece ............. Frequency of Occurrence of PC Sets in the Fifth Piece .............................. Set Complex Relations in the Fifth Piece . Similarity Relations in the Fifth Piece .. Inclusion Relations in the Fifth Piece -iv— Page 11 48 49 50 51 52 55 56 57 58 59 59 62 63 64 64 66 69 70 71 71 73 73 77 78 79 80 80 Figure 1. A Hierarchical Data Model Figure 2. Input/Output Design ................. FIVE PIECES FOR PIANO by George Crumb Figure 3. Segmentation of First Piece ......... Figure 4. Segmentation of Second Piece ........ Figure 5. Segmentation of Third Piece ......... Figure 6. Segmentation of Fourth Piece ........ Figure 7. Segmentation of Fifth Piece ......... Figure 8. Set Complex Relations for all PC Sets Figure 9. The Appearance of a Nexus Set .............. LIST OF FIGURES anv— Page 12 22 47 54 61 68 74 82 83 CHAPTER 1 INTRODUCTION 1.1 Identification of the Problem For over 20 years, the mathematical concept of set theory has been applied to atonal music. The method of analysis described by Allen Forte in The Structure of Atonal Music1 yields insightful information into many contemporary compositional processes. But the application of set theory has two substantial disadvantages: 1) analysis is tedious and time consuming and may result in an array of trifling data which is not indicative of the compositional process and 2) analysis requires a mathematical aptitude and data processing skills which increase the probability of human error . 1.2 Statement of Purpose The purpose of this study is to provide music theorists with software which will reduce the amount of time required by set theory analysis and decrease the probability of human error without compromising the multi—faceted analytical approach as codified by Allen Forte. 1 Allen Forte. The Structure of Atonal Music. New Haven: Yale University Press, 1973. _l—. 1.3 Statement of Objectives The purpose of the study will be achieved through the realization of the following objectives: 1) To design software which will perform the following components of set theory analysis. normal order best normal order prime form transposition of PC sets inversion of PC sets equivalence of PC sets under transposition equivalence of PC sets under inversion set complement similarity relations pitch similarity vector similarity set complex relations K relation Kc relation Kh relation invariant subsets between PC sets of the same cardinality and set number 2) To use the aforementioned software to perform a complete set—theoretic analysis of Five Pieces for Piano by George Crumb. 1.4 Review of Related Literature: The Application of Computer Technology to the Discipline of Music Since the introduction of the first commercial computer over 20 years ago, the world has experienced a technological revolution rivaled by none in history. In recent years, the calibrated crescendo of computers into the market has sparked numerous computer applications in the arts. The marriage of music and the computer has benefited the disci- pline of music. If music is defined as the combinations of sounds into logical structures, then the logical objectivity of computer architecture is in harmony with the discipline of music. However, music ‘by «definition is integrally 'bound by principles of aesthetics which resist the regimentation of logic. Such a diametric contrariety has created a polarity between the subjectivity of music and the objectivity of science. How can music succumb to the trenchant objectivity of science? Reconciliation of the paradox between objectiv— ity and subjectivity' may ‘be achieved by’ delegating the logical constructs of music to the computer while reserving aesthetic judgment to humans. Several branches of music have benefited from the contrived wedding of the computer and music: most notably, musicology, composition, pedagogy and theory. Musicology Computer—assisted research in musicology has primarily been devoted to the task of transcription. A report from the 1982 International Computer Music Conference describes a method for the transcription. of German lute tablatures. Formerly, only the pitches and not the rhythm could defini- tively be derived from lute tablatures. Musicologists assert that the graphic position of the pitches in relation to the bass and a search for conjunct melodic lines can determine not only the pitch ‘but also the voicing and rhythm. Researchers are in the process of developing an algorithmic model (See Glossary of Technical Terms) which will accommodate the subtleties of lute transcription. Another research project reported at the International -4- Conference of Musical Grammars and Computer Analysis de— scribes a computer search for musical patterns in the Cantigas d_e Santa Maria —- a lBth-Century secular vocal genre. Linguistic pattern-matching softwane has been adapted to assist in the search for musical patterns. When automated, both methodologies will make a significant con- tribution to the field of musicology. Composition Many composers have excitedly embraced the computer as an indispensable tool in the field of composition. As early as the 19403, engineers at Princeton became interested in a mechanical composing machine. .A statistical study of the melodic characteristics of 12 songs of Stephen Foster pro- duced an algorithmic model which would generate melodies in the style of Stephen Foster. Since the algorithm was de— signed to control the pitch parameter, the resultant melodies predictably lacked every element of music except pitch. Such a consequence resulted in an accelerated interest in sonology, psychoacoustics, and algorithmic com- position as integral aspects of computer music. Sonology is the study of sound and electronic sound synthesis techniques. Current research in synthesis tech- niques is being developed at the Center for Research in Music and Acoustics at Stanford University. One project has successfully synthesized the phonetics of vowel sounds using frequency modulation with four oscillators. Another project has imitated the timbre of the violin, viola, Violoncello, -5- and string bass by employing the acoustical discoveries of Helmholz as a mathematical model. The body of a string instrument is; simulated In! a static filter. Additional filters and variable decay rates are utilized to simulate string tension in relation to a particular frequency. The University of Illinois is pioneering research devoted to developing Central Processing Unit (C.P.U.) and storage- efficient algorithms for simulating traditional orchestral timbres. The University of New Hampshire has researched the digital simulation of the piano and created a mathematical model derived from the pdano's action. 'Various synthesis techniques have been implemented based upon the mathematical model including exercising control over the attack and decay rate of significant partials. The CHANT project at the Institut de Recherche et Coordination de la Musique uses an object-oriented approach in which objects represent in— formation models about sound production cn: perception. .A hierarchy of relationships between the objects has an effect on sound production. The CHANT system is interactive with graphics capability, sound analysis display, and is accom— panied by a database of objects and rules. The forerunners in the realm of psychoacoustics are Stanford University and the Massachusetts Institute of Technology. One research project at Stanford attempts to answer the question, "How do we physically measure the attaCk time of an arbitrary soumd to yiehd a perceptually meaningful result?" During the attack phase, acoustic —6- instruments produce many complex amplitude—envelope func— tions. Researchers have determined that attack time is perceived When a sound exceeds a percentage of its maximum envelope. Another research project in psychoacoustics led to the development of a system which can convert a musical performance iJnx> staff notation. In addition, the system can execute analysis based upon the user's request. Further research is being done in the areas of timbral perception, reverberation modeling, and sound spectrum analysis. Algorithmic composition is the art. of’ combining the fruits of sonological and psychoacoustic research in the program mode of the computer. The compositional algorithm is the series of steps which when executed by a computer will result in a composition. Styles of computer music range from the interpretation of output generated by a computer on traditional acoustic instruments to a computer- synthesized tape. Although algorithmic composition does not presuppose a: particular style, iJ: is noteworthy that tone color generally plays an important role in the formal struc- ture of algorithmically-composed computer music. Pedagogy Computer-Assisted Instruction has benefited the educa- tional pursuits of individuals of all ages in many different subject areas. In the discipline of music, the computer is ideally suited for the drill and practice routines necessary in written theory and ear-training. When used interac- tively, the computer has the potential of being the perfect -7- instructor with seemingly limitless examples coupled with the ability to give immediate, objective feedback. The information processing capabilities of the computer allow for complete record keeping for each individual student. Multi-level program branching in the instructional design can maximize individualized instruction. Computer-Assisted Instruction in music is firmly rooted in the curricula of many institutions of higher education throughout the country. Among them are the University of Delaware, Stan— ford University, Ohio State University, the University of Iowa, the University of Georgia, the State University College at Potsdam, and North Texas State University. Recently, Queen's College of the City University of New York has developed computer-assisted instructional materials in music composition. The instructional design is intended to assist young children in learning the technique of compo- sition without first mastering an instrument. Theory Computers have been utilized in a wide variety of appli- cations in the field of theory. Theorists are exploring systems of equally-tempered scales of other than 12 notes. By expanding the range of available pitches and altering the timbres of pitches within a scale, theorists are hoping to establish a new system of tonality. Perhaps the best-known application of the computer in the field of theory is the -8- 2 The work of Bo Harry Alphonce and "The Invariance Matrix." matrix consists of a twelve-tone set table which is analyzed in. group theoretic terms. The dissertation. includes a computer—generated listing of all of the subsets and super- sets of ‘prime form sets with cardinalities from two to twelve. David L. Jackson of the University of Cincinnati author- ed a computer program. written in FORTRAN that executes analysis of horizontal and vertical sonorities by employing the intervallic relationships of Howard Hanson, the root extraction of Paul Hindemith, and prime form PC sets of Allen Forte.3 A similar project by Timothy Kolosick employs an Apple II to sample "time—slices" and analyze vertical sonorities using set theory.4 Although these research projects are similar in purpose, the aforementioned studies utilize the computer to execute selective components of set theory analysis. The present study surpasses the afore- mentioned studies by encompassing all aspects of set theory analysis. 2 Bo Harry Alphonce, "The Invariance Matrix" (Ph.D. disser- tation, Yale University, 1974). David L. Jackson, "Horizontal and Vertical Analysis Data Extraction Using a Computer Program," (Ph.D. dissertation, University of Cincinnati Conservatory of Music, 1981). 4 Timothy J. Kolosick, "A Computer-Assisted, Set—Theoretic Investigation of vertical Simultaneities of Selected Piano Compositions by Charles E. Ives," (Ph.D. dissertation, University of Wisconsin at Madison, 1981). -9- The use of computers for theoretical analysis as well as composition raises questions of artificial intelligence. Is the computer capable of making the evaluative decisions required of insightful musical analysis? Can the computer compose a "work of art?" The range of thought processes required vary from one musical task to another. Likewise, the sophistication of the program design to complete a musi- cal task will vary based on the complexity of the thought process. Therefore, the computer‘s capacity to simulate human thought is contingent upon a matching of the complex— ity of the thought process and the programmer's ability to quantify it. 1.5 Methodology The objectives will be achieved through the following methodology: 1) Determine the desired output. 2) Develop algorithms, flowcharts and write codes for each component of set theory analysis. 3) Define the content of the databases. 4) Develop a means of accessing the databases. 5) Design the structure of the databases. 6) Build the databases. 7) Execute a computer-assisted analysis of Five Pieces for Piano by George Crumb. 8) Interpret the data. The program employs a Digital Equipment Corporation minicomputer VAX 11/780 using the BASIC-Plus 2 programming language. CHAPTER 2 DESCRIPTION OF DATABASES AND EXTERNAL FILES 2.1 Terminology of Database Design The following terminology and definitions are relevant to the application and are intended to provide a common basis for understanding. Definition 9; Database A database may be defined as a system for collecting, storing and retrieving large quantities of information so that a program can utilize the information and thus fulfill a purpose. Database Structures The relationShip between data items II! a database is termed database structure. There are essentially' three database structures: the relational data model, the hierar- chical data model and the network data model. Structures employed in this application are the relational data model and a hybrid form of the hierarchical data model. The Relational Data Model The relational data model is an application, of set theory to the relationship between data items. Table l is a tabular representation of data pertinent to the analysis of atonal music. -10- -11- Table l. A Relational Data Model 0 l 02- 01 1 100000 10-01 0 l 2 03- 01 7 210000 09—01 0 l 2 3 4 5 6 7 8 10 10- 02 216 898884 02-02 0 l 3 4 6 7 O6-Z13 98 324222 0 2 4 6 8 10 06- 35 120 060603 The data in its entirety is called the relation. Each horizontal row of data within the relation is called a record. Each vertical column of data is called a field. A domain is the entirety of data items which may appear in a field. The unique combination of fields drawn from a domain in the formation of records results in the properties of the relational database model.5 1) Each column is attribute homogeneous; that is, in any column all items are of the same kind. 2) Each column is assigned a distinct name. 3) All records are distinct; duplicate records are not allowed. 4) Both the records and the fields may be viewed in any sequence at any time without affecting the information content. Due to the influence of set theory on the relational data structure, the model is ideally suited as a structure for data pertaining to a domain of unique sets. The Hierarchical Data Model The hierarchical data model is characterized by a single data item having a relationship with many other data items. Abridged from James Martin, Principles of Data-Base Management, (Englewood Cliffs, New Jersey: PFEntice-Hall, Inc., 1976), p. 96. -12- The structure is analogous to a family tree with one set of parents having children and those children having children. The terminology reflects the similitude. Level one is the root of the tree. At level two, the children of the root become the parents of the children at level three. A funda— mental principle of hierarchical structure is that a child may only have one set of parents. The one—to-many corre- spondence :hi the hierarchical data model suggests the relationship between a set and the subsets and supersets of the set. ROOT LEVEL 1 ‘E’ m 2 a a m3 0 o o o a Figure l. A Hierarchical Data Model 2.2 Terminology of File Types The program BOETHIUS was written on a Digital Equipment Corporathmu VAX 11/780. The VAX supports three types of files which may be employed as databases: sequential file, virtual array file, and record I/O (input/output) file. The three types of data files exhibit a similar type of organi— zation. Each horizontal row of data stores a record and each vertical column of data stores a field. -13... Sequential File A. sequential file is a: list of interrelated records stored in a: sequential order. Operations (”1 sequential files must be done serially, i.e., one record after another. Virtual Array File A virtual array file is a one-dimensional or two- dimensional array. The subscript of the array determines a specific location in the array and is the access path to data stored in the array. Record I/O File A record I/O file is a collection of interrelated rec- ords which may be accessed, modified, or deleted on a random basis. The fields in each record are defined by a MAP statement which establishes the boundary of each field in the record. This is in contrast to the cellular structure of a virtual array file. Records may be accessed by the computer on a random basis by defining a field as the pri- mary key and other fields in the record as alternate keys. 2.3 Databases Utilized in the Program BOETHIUS The program BOETHIUS is the controlling program in the analysis of atonal music using set theory. BOETHIUS is supported by two databases entitled FORTE and ALPHONCE. The program BOETHIUS accesses data from FORTE and ALPHONCE to assist in the analysis of music. -14- Database No. 1 -- FORTE FORTE is derived from Appendix One of The Structure gf Atonal Music by Allen Forte with the addition of sets with cardinalities of two and ten. FORTE consists of the prime form sets, their set name, vector and set complement name. Each record in FORTE is assigned a unique index number that is used to identify that particular set. FORTE is a record I/O file exhibiting the relational data structure. Each field is defined in the MAP statement at line 205 of BOETHIUS which specifies the variable name of the field and the maximum number of characters allowable in a field. The number of allowable characters is followed by a percent sign (%) indicating that the number is an integer. Example 1: MAP Statement from BOETHIUS 205 MAP(IDX) PRIME$=23%,CARD$=2%,SN$=6%,IND.NO$=5%,VT$= 8%,N$=8% The variable names of the fields correspond to the following data items: PRIME$ - Pitch class integers of a set in prime form. CARDS - Cardinality of a set. The cardinalities range from two to ten. SN$ - Set name. The set name consists of a hypen followed by the set number. A "Z" is inserted in the second character position of Z-related sets. IND.NO$ - The index number of the set. The index number is utilized as an access path to database no. 2, ALPHONCE. VT$ — The vector of the set. N$ - Set complement name. The set complement name does not separate the cardinality from the set -15- number. In the case of Z-related hexachords, the set complement name is the hexachord's Z- related pair. In the case of hexachords which do not have a z-related pair, the set complement name is blank. The database FORTE is built by an interactive program entitled FBUILD. Thus, the program FBUILD builds the data- base FORTE which is accessed by the program BOETHIUS. The MAP statement in FBUILD is identical to the MAP statement in BOETHIUS assisting in the connection between the three files FBUILD, FORTE, and BOETHIUS. Beside FBUILD and BOETHIUS having identical MAP statements, the two programs must also have identical OPEN statements. The OPEN statement serves to open a channel between a program and secondary storage and in the case of a record I/O data file, define primary and alternate keys. The primary key in FORTE is the field name PRIME$. The primary key is the preferred key in acces- sing the data file. A sample section of FORTE with corresponding labels would appear as follows: Example 2: A Representative Portion of FORTE PRIME$ CARDS SN$ IND.NO$ VT$ NS 0 l 02- 01 1 100000 10-01 0 l 2 03- 01 7 210000 09—01 0 1 2 3 4 5 6 7 8 10 10- 02 216 898884 02-02 0 l 3 4 6 7 06- 213 98 324222 PRIMARY KEY ALTERNATE KEYS The access path between FORTE and BOETHIUS is through the primary key PRIME$. In the program BOETHIUS, the user -16— inputs integers from zero to eleven which represent a seg- ment of pitches in a composition. The integers are manipu- lated by BOETHIUS to produce the set in prime form and store the prime from arrangement in MATRIX PRIME$. BOETHIUS then concatenates the set in prime form into the variable PC$ (lines 2750-2775). The variable PC$ has the same format as PRIME$ in FORTE. The variable PRIME$ is set equal to PC$ at line 2985. The computer is instructed to GET PRIME$ from FORTE at line 2990. The computer accesses the record which has the same primary key. At that time, the entire record may be retrieved. Following acquisition of the record from secondary storage, each field of the record is assigned to a primary storage array called MATRIX SET.NAM$. As an example of data retrieval from FORTE by BOETHIUS, assume the user inputs the pitch class integers 7,5 and 6. BOETHIUS will reduce the set to its prime form [0 1 2] and store the arrangement in MATRIX PRIME$. Each element of MATRIX PRIME$ is concatenated into PC$ so it has the same format as PRIME$ in FORTE. Example 3: Data Retrieval from FORTE by BOETHIUS MATRIX PRIME$ ******************* pc$ = 0 1 2 * * * * 1: 0 t 1 * 2 * * * * * ******************* -17- PRIME$ is set equal to PC$, and the computer scans FORTE searching for the set which is equal to PRIME$. Upon locat- ing the set, the fields of the record may be retrieved and assigned to MATRIX SET.NAM$. Example 4: A Record in FORTE: 0 1 2 03 - 01 7 210000 09- 01 A Portion of a Row in MATRIX SET.NAM$ in BOETHIUS: Col 0 Col 1 Col 2 Col 3 ****1:************************************* 'k * 'k t i * 7 * O3 * — 01 * 09-01 * * * * * 'k *****************************************i: IND.NO$ CARD$ SN$ N$ Database No. 2 -- ALPHONCE ALPHONCE contains the subsets and supersets of all the sets listed in FORTE as obtained from Appendix Nine of "The Invariance Matrix."6 ALPHONCE is a two-dimensional virtual array file. A portion of ALPHONCE may be envisioned as follows: 6 Bo Harry Alphonce, "The Invariance Matrix," (Ph.D. disser- tation, Yale University, 1974). Reproduced by permission of the author. Exampl Row 1 Row 2 Row 3 ~18- e 5: A Portion of ALPHONCE Col 0 Col 1 Col 2 Col 3 Col 4 Col 5 Col 6 ****************************************************** * * * * * 'k 'k 'k * 1 * 195 * lO-Ol * 10-02 * 10-03 * 10-04 * 10-05 * * * 'k * * * 'k * 'k***************************************************** 'k 'k 'k * 'k 'k * * * 2 * 197 * lO-Ol * 10—02 * 10—03 * 10-04 * 10-05 * 'k * 'k 'k * 'k * 'k *****************************************************i * * * * * t * * * 3 * 194 * 10—01 * 10-02 * 10-03 * 10-04 * lO-OS * * * * * * 'k 'k * ~k***~k************************************************'k 7‘ T \ INDEX NO. OF SUBSETS AND SUPERSETS OF SET NO. ENTRIES WHICH CORRESPOND WITH INDEX NO. The purpose of ALPHONCE is to provide data for evaluat- ing set complex relations between any two sets of different cardinalities. For example: GIVEN: The cardinality of SET A does not equal the cardin- ality of SET B. See if SET B is in a K, Kc, or Kh relation with SET A. SEARCH ALGORITHM: 1) 2) 3) 4) Locate SET A and its list of subsets and supersets in ALPHONCE. Search for the list of subsets or supersets of SET A for an occurrence of SET B. If SET B is found, then SET A and SET B are in K relation. Search the list of subsets and supersets of SET A for an occurrence of the complement of SET B. If the complement of SET B is found, then SET A and SET B are in Kc relation. If SET A and SET B are in K relation and Kc rela- tion, then they are in Kh relation. The index number which is stored in column zero corre- sponds with the index number in FORTE and is used to -19- identify SET A. Column one contains the number of subsets and supersets that will map into SET A. Each column in column two through the value stored in column one contains the set name of a subset or superset of SET A. If SET B and/or the complement of SET B is found in columns two through the value stored in column one, then SET A and SET B are in a set complex relation. The index number is the link between the two databases. The index number retrieved from FORTE becomes the access path to a row of data in ALPHONCE through the array sub- script. Consider for example a record in FORTE that is assigned to locations in MATRIX SET.NAM$. Example 6: A Record in FORTE: 0 l 2 03 - 01 7 210000 09- 01 A Portion of a Row in MATRIX SET.NAM$ in BOETHIUS: Col 0 Col 1 Col 2 Col 3 ****************************************** * * * * * * 7 * O3 * - 01 * 09-01 * * * * * * ****************************************** The index number stored in column zero of MATRIX SET.NAM$ is used to identify the row in ALPHONCE where the subset/superset of SET A is stored. Once SET A is located in ALPHONCE through,the index number, the row of data can be -20- searched for the occurrence of SET B (n: the complement of SET B. The relationship 'between data items in each rOW' of ALPHONCE exhibits a quasi-hierarchical structure. The index number may be considered the root of the tree. The number of subsets and supersets is an intermediate data item be- tween the index number and the names of the subsets and supersets. Beginning with column two, a sequential search tests for an occurrence of SET B and/or the complement of SET B. Example 7: A Portion of a Row from ALPHONCE Exhibits Hierarchical Structure. ****************************************************** * * t a * * * * * l * 195 * 10-01 * 10-02 * 10-03 * 10-04 * 10-05 * * * * * * * * * ****************************************************** ROOT LEVEL 1 LEVEL 2 No. of Entries m3 eeeee ALPHONCE is built by the program BOBUILD which refer- ences the sequential file KSET. Thus BOBUILD inputs information from KSET and builds the virtual array file ALPHONCE which is accessed by the program BOETHIUS. -21- 2.4 Reports Generated by BOETHIUS Two reports are generated by BOETHIUS: ANALYSIS and EVALUATE. ANALYSIS The output file ANALYSIS is a sequential file that contains analytical information about each set tflua user inputs into BOETHIUS. Information contained in the report ANALYSIS includes the set in prime form, set name, transpo- sition or inversion Operator, vector, set complement name, and B.I.S. pattern. EVALUATE EVALUATE is a sequential file Which is created upon the user's request to evaluate pairs of sets listed in ANALYSIS. Comparisons include set complex relations, similarity rela- tions, and invariance. 2.5 Program Structure Summary The figure below represents the relationship between the databases, the controlling program BOETHIUS, and the reports generated by BOETHIUS. Input \lz Output Figure 2. -2 2.. KSET FBUILD BOBUILD INDEX NO. FORTE -------------------------- A ALPHONCE ‘\\\\“\N\\T‘-;. BOETHIUS / ANALYSIS Input/Output Design / ‘TTTTTTTTT‘TT‘-_ EVALUATE CHAPTER 3 DESCRIPTION OF MODULES IN PROGRAM BOETHIUS 3.1 Internal Matrices and Dimension Declarations Internal matrices store the data required to reduce sets to prime form and reference databases. The purpose of the internal matrices falls into two categories: 1) Variable Matrix - The information in the matrix changes during the program run. Variable :matrices are either one-dimensional or two-dimensional. An example of a one-dimensional matrix is MATRIX DUP. The DUP matrix col- lects members of a set during entry and tests for duplicate pitch-class integers. An example of a two-dimensional matrix is MATRIX A. The purpose of MATRIX A is to provide a storage location for the circular permutations required to find a set in normal order. Variable matrices are initial- ized to zero upon the user's request to enter a new set. 2) Cumulative Matrix — The cumulative matrices serve to accumulate data about the analysis of sets so set compari- sons may be made. Set comparisons include K Relationships, R Relationships, and Invariance. The cumulative matrices are two-dimensional thus providing greater accessibility to components of the analytical data. -23- -24- 3.2 Set Entry The entry of sets to be reduced to prime form is con- trolled by a conditional loop which is executed upon the user's request. When entering the first set, the pitch- class integer notation which substitutes for letter notation is displayed on the screen. The user is first asked to input the cardinality of the set to be reduced to prime form. The cardinality is tested to make certain it is from two txn ten inclusive. The cardinality input loop (lines 1700—1710) will execute infinitely until a valid cardinality is entered. Upon entry of a valid cardinality, the pitch-class integers are loaded into a one-dimensional matrix to check for duplicate pitch—class integers. ILf a duplicate is de~ tected, the user has the opportunity to adjust the cardinal- ity of the set or re-enter a different pitch-class integer. The phase of data entry which mandates a cardinality from two to ten and does not allow for duplicate pitch-class integers will continue until both conditions are met. Careful testing of data during entry is important for the following reasons: 1) If the cardinality is not from two to ten inclusive, external references to databases cannot be made and a thor- ough analysis is not possible. 2) The cardinality of a set as well as the members of the set must be integers. If a fraction is entered rather than an integer, the digits to the right of the decimal -25- point are truncated. ILf alpha—numeric information (string data) is entered, control is transferred to the error trap (line 19000). The error is suppressed from terminating the program, an error message is displayed, and the user is given another opportunity to enter the data. 3) A duplicate pitch-class integer in set analysis would result in a set that will not correspond to the data- base reference of the sets in prime form. For this reason, pitch-class integers are first loaded into time MATRIX DUP and tested against previously entered pitch-class integers before being assigned to MATRIX A for normal order opera- tions. The aforementioned data entry tests are mandatory to insure that analytical data generated by the program are valid. 3.3 Transfer B.I.S. Pattern to MATRIX BIS The Basic Interval Succession. (B.I.S.) pattern <3ften times lends insight into the ordering of pitches in the compositional process. A compositional process may produce sets which slightly differ from a pmevious appearance of a similar musical event. Two sets may be aurally similar yet set theory analysis may indicate that the sets are dis- similar. For this reason, the analysis of B.I.S. patterns is included in the analysis of atonal music. The B.I.S. pattern is discovered by comparing two ad- jacent pitch-class integers. Two relationships can exist between two pitch-class integers: l) the first may be -26— greater than the second or 2) the first may be less than the second. Irregardless of the two relationships, the smaller pitch-class integer should be subtracted from the larger pitch-class integer. If the difference is greater than six, it is subtracted from the modulus. The result is a B.I.S. pattern which may range from zero to six. The algorithm is as follows: 1) Begin comparisons upon entry of the second pitch- class integer. 2) If the pitch-class integer on the right is greater than the pitch-class integer on the left, then subtract the one on the left from the one on the right. Otherwise, subtract the one on the right from the one on the left. 3) If the difference is greater than six, then it is subtracted from the modulus. Example Eh The first pitch—class integer is greater than the second. A13 0 [IUD 49 d Example 9: The first pitch-class integer is less than the second. .7 ° ‘3 ‘1 .;li 3 Example 10: The difference is greater than six. 0 9 1 1 £19- 4 = 12 — 8 BIS MODULUS - DIFFERENCE -27- The B.I.S. patterns are accumulated ixnxn MATRIX B.I.S. and subsequently' printed to .ANALYSIS for B.I.S. pattern comparisons by the user. 3.4 Normal Order After the B.I.S. pattern is determined and assigned to MATRIX BIS from MATRIX A, normal order operations begin. Normal order is defined as the arrangement(s) of pitches which has the least difference between the first and last member of the set after circular permutations. The process of placing a: wet in normal order occurs over a series of three steps: 1) A bubble sort places the pitch-class integers in ascending numerical order. 2) Produce arrangements of time set through circular permutations. 3) Find the arrangement(s) with the least differnece between the first and last members of the permu- tations. Step 1: The Bubble Sort The computer evaluates two adjacent pitch-class inte- gers. If the one on the right is greater than the one on the left, then the two pitch-class integers are exchanged. Several passes through the matrix eventually result in the set being placed in ascending numerical order. -28— Example 11: Pitch—class Integers Input During Set Entry ************************* * * * * * MATRIX A, * 6 * 3 * 11 * 10 * ************************* Example 12: Arrangement of pitch-class integers as a result of the bubble sort ************************* * * * * * MATRIX A, * 3 * 6 * 10 * ll * ************************* The set in normal order is assigned to the cumulative MATRIX NORM.ORD so strongly represented Rp similarity relation comparisons may be made. The cardinality of the set is stored in column zero. Example 13: MATRIX NORM.ORD Col 0 Col 1 Col 2 Col 3 Col 4 ************************************ * t * * * * * 4 * 3 * 6 * 10 * 11 * * * * * * * ************************************ Step 2: Circular Permutations Once the set is in ascending numerical order, circular permutations will reveal the ordered set with the smallest interval between the first and last member of the permuta- tion. The number of permutations will equal the cardinality minus one. Since the bubble sort yields one arrangement, further permutations will begin with the second arrangement -29- and continue to the cardinality. Each circular permutation will move set members to the left by one column in MATRIX A and take the first set member and add the modulus and place it in the last column of MATRIX A. Example 14: MATRIX A After Circular Permutations Col 1 Col 2 Col 3 Col 4 ************************* t * * i * Row 1 * 3 * 6 * 10 * ll * * * * e * ************************* * * * * * Row 2 * 6 * 10 * ll * 15 * * * * * * ************************* * * * * * Row 3 * 10 * 11 * 15 * 18 * * t * e * ************************* * * * * * Row 4 * 11 * 15 * 18 * 22 * * * * t * ************************* Each row is one permutation of the set and is derived from the previous row. Step 3: Find the Arrangement(s) with the Smallest Interval between the First and Last Member of the Permuta- tion Once the set has undergone circular permutations, it is necessary to find the smallest interval between the first and last member of the set. The computer assigns the vari- able LOW the value of the modulus. The computer evaluates each circular permutation by subtracting the value of the first column from the value of the last column. If the -30- difference is less than the lowest number, the variable LOW assumes the value of the difference. Example 15: Evaluating Permutations in MATRIX A LOW = MODULUS If DIFFERENCE < LOW then LOW = DIFFERENCE Col 1 Col 2 Col 3 Col 4 ************************* * * * * * Row 1 * 3 * 6 * 10 * 11 * 8 < LOW then LOW = 8 * * * * * ************************* * * * * * Row 2 * 6 * 10 * 11 * 15 * 9 <> LOW then LOW = 8 * * * * * ************************* * * * * * Row 3 * 10 * 11 * 15 * 18 * 8 = LOW then LOW = 8 * * * * * ************************* * * * * * Row 4 * 11 * 15 * 18 * 22 * ll <> LOW then LOW = 8 * * * * * ************************* Permutation No. 1 (Row 1) - When three is subtracted from eleven, the difference is eight. Eight is less than the lowest number so the lowest number is assigned the value eight. Permutation No. 2 (Row 2) - When six is subtracted from fifteen, the difference is nine. Since nine is not less than the lowest number (eight), the lowest number remains eight. Permutation No. 3 (Row 3) - when ten is subtracted from eighteen, the difference is eight. Eight is not less than the lowest number (eight), so the value of the lowest number remains eight. -31- Permutation No. 4 (Row 4) - When eleven is subtracted from twenty-two, the difference is eleven. The difference is not less than the lowest number (eight), so the lowest number maintains the value of eight. 3.5 Assign Normal Order Arrangments to MATRIX B for Best Normal Order Operations. The computer re-evaluates the *3*6*10* 11* * * * * * * * * * * ************************* ************************* * * * * * * * * * * *6*10*ll*15* *10*11*15*18* * t * * * * * * * * ************************* ************************* * * * * * *10*ll*15*18* * * * * * ************************* * * * * * * 11 * 15 * 18 * 22 * * * * * * ************************* The sets are counted as they are entered into MATRIX B. The purpose of counting the arrangements with the least difference between the first and last members of the set is -32- so that the computer may later be instructed to evaluate the number of sets counted during best normal order Operations. 3.6 Best Normal Order Operations Best normal order operations begin after the arrange- ment(s) with the smallest interval between the first and last member of the set is assigned to MATRIX B from MATRIX A. One achieves best normal order by evaluating the permu- tation(s) by three criteria: 1) The permutation is in best normal order when the ordering has the least difference between the first and second member of the set and the difference between the last two members of the set is not less than the difference between the first two members of the set. If the above conditions are true than the set is in transposition. 2) In the event the differece between the last two members of the set is less than the difference between the first two members of the set, the set is in inversion. 3) If the difference between the first two members of the set is equal to the difference of the last two members <5f the set, then comparisons of adjacent members continue until a smaller interval is discovered. If the smaller interval is detected on the left half of the set, then the set is in transposition. If the smaller interval is de- tected on the right half of the set, then the set is in inversion. -33- 3.7 Best Normal Order Operations: Determine if the Set is in Transposition or Inversion The algorithm for determining if a set is in transposi- tion or inversion is as follows: 1) Find the difference between the first two members of the permutation and store the difference in the variable STOREl. 2) Find the difference between the last two members of the permutation and store the difference in the variable STORE2. 3) If STOREl does not equal STORE2 or the cardinality is less than or equal to four, then go to step 5. 4) STOREl equals STORE2 and the cardinality is greater than four so continue searching for a smaller interval between adjacent members. If the smaller interval is en— countered on the right side of the permutation, then the permutation is in inversion - otherwise, the permutation is in transposition. If the set is composed of all the same interval, then the program defaults to transposition. 5) If STOREl is greater than STORE2 then the permuta- tion is in inversion. If STOREl is less than STORE2 then the permutation is in transposition. 6) Continue steps 1-5 for each best normal order ar- rangement. -34- Example 17: Best Normal Order Arrangements ************************** * * * * * MATRIX B * 3 * 6 * 10 * 11 * * * * * * ************************** * i * * * * 10 * 11 * 15 * 18 * * * * * * ************************** Col 1 Col 2 Col 3 Col 4 (N-l) ( N ) ************************** * * * * * MATRIX B * 3 * 6 * IO * ll * ************************** STOREl = 3 STORE2 = l STOREl > STORE2 Therefore the Permutation is in Inversion Col 1 Col 2 Col 3 Col 4 (N-l) ( N ) ************************** * * * * * MATRIX B * 10 * 11 * 15 * 18 * ROW 2 * * * * * ************************** STOREl = 1 STORE2 = 3 STOREl < STORE2 Therefore the Permutation is in Transpo- sition 3.8 The Set Is in Transposition After it is determined that the set is in transposition, two Operations must occur to reduce the permutation to prime form: 1) Find the transposition operator. 2) Reduce the set to prime form. -35- Step 1: Find the Transposition Operator The algorithm for determining the transposition operator is as follows: 1) The transposition number is the first member Of the set. 2) If the value of the first member is greater than the modulus, then the modulus should be subtracted from the value. 3) The transposition Operator is the inverse of the transposition number. Example 18: Finding the Transposition Operator Col 1 Col 2 Col 3 COl 4 ************************** * * * * * MATRIX B * 10 * ll * l5 * 18 * Row 2 * * * * * ************************** Transposition Number = 10 Transposition Operator = Modulus - Transposition Number 2 = 12 - 10 Transposition Operator = 2 Step 2: Reduce the Set to Prime Form The set may be reduced to prime form by adding the transposition operator to each nmmber Of the set (modulus twelve). Example 19: MATRIX B After Reduction to Prime Before Reduction Form Col 1 Col 2 Col 3 Col 4 Col 1 Col 2 Col 3 Col 4 ************************* ************************* * * * * * * * * * * Row* 10 * ll * 15 * 18 * * O * 1 * 5 * 8 * 1 * * * * * * i t * * ************************* ************************* -36- 3.9 The Set Is in Inversion Several operations are required to reduce a set that is in inversion to prime form. The series of Operations take place in a subroutine which produces the retrograde inver- sion Of the permutation and reduces it to prime form. 3.9.1 Invert the Members Of the Set Inversion takes place over two steps: 1) If the set member is greater than the modulus, then the modulus should be subtracted from the value. 2) The inverse Of the set member equals the set member subtracted from the modulus. Example 20: MATRIX B Before Inversion After Inversion ************************* ************************* * * * * * * * * * * Row* 3 * 6 * 10 * ll * * 9 * 6 * 2 * l * l * * * * * * * i * * ************************* ************************* 3.9.2 Reverse the Order Of the Set by Storing Values in MATRIX C and Returning the Retrograde Inversion to MATRIX B MATRIX C serves as a storage matrix for the retrograde arrangement of the permutation in MATRIX B. After the set members are reversed, they are reassigned to MATRIX B. Example 21: MATRIX B Before Assigning to MATRIX C ************************* * * * * * Row 1 * 9 * 6 * 2 * l * * * * * * ************************* -37- MATRIX C: Storage Matrix for Retrograde Arrangement ************************* * * * * * Row 1 * 1 * 2 * 6 * 9 * * * * * * ************************* MATRIX B: After Set Members are Reversed ************************* * * * * * Row 1 * 1 * 2 * 6 * 9 * * * * * * ************************* 3.9.3 Find the Inversion Operator The algorithm for determining the inversion Operator is as follows: 1) 2) 3) 4) The inversion number is the first member Of the set in MATRIX B. If other members Of the set are less than the inversion number, then the modulus should be added to the other set members. If the inversion number is greater than the modu- lus, then the modulus should be subtracted from the inversion number. The inversion Operator is the inverse Of the inver- sion number. Example 22: MATRIX B Row 1 ************************* * * * * * * 1 * 2 * 6 * 9 * * * * * * ************************* Inversion Number = 1 INVERSION OPERATOR = MODULUS - INVERSION NUMBER 11 = 12 - l Inversion Operator = 11 -38— 3.9.4 Reduce the Set to Prime Form The set may be reduced to prime form by adding the in- version operator to each member Of the set (modulus twelve). Example 23: MATRIX B Before Reduction to Prime Form Prime Form ************************* ************************* t * t * * * * * * * Row* 1 * 2 * 6 * 9 * * 0 * 1 * 5 * 8 * l * * * * * * * * * * ************************* ************************* 3.10 Assign Set in Prime Form to MATRIX PRIME$ The set in prime form is transformed to string data and placed in the cumulative MATRIX PRIME$. Each member of the set is followed by a space to coincide with the format field of FORTE. The cardinality Of the set is stored in column zero. The set is concatenated into the variable PC$. Example 24: MATRIX PRIME$ ******************************* * * * * * * Row 1 * 4 * 0 * 1 * 5 * 8 * * * * * * * ******************************* 3.11 Retrieve the Set in Prime Form from FORTE FORTE is a Record I/O file which allows random access to data records. The set in prime form as stored in the vari- able PC$ is assigned to the variable PRIME$. The data pointer locates the set in prime form allowing access to the remainder of the data fields in the record. -39- Example 25: Retrieve the Set in Prime Form from FORTE PC$ = 0 1 5 8 PRIME$ = PC$ PRIME$ = 0 1 5 8 Set in Prime Cardin- Set Index Form ality Number Number Vector Set (PRIME$ (CARDS (SN$ (IND.NO$ (VT$ Complement Field) Field) Field) Field) Field) (NS Field) 0 l 5 8 04- 20 38 101220 08— 20 3.11.1 Establish MATRIX SET.NAM$ The data fields for the set are assigned to MATRIX SET.NAM$ for the purpose of accumulating analytical data for set comparisons. Column 0 = The index number of the set (IND.NO$) Column 1 = The cardinality Of the set (CARDS) Column 2 = The set number (SN$) Column 3 = The set complement (NS) Column 4 = Transposition/Inversion Operator (TRANS.INV$) Note: In the event a set can be reduced to prime form by -transposition. or inversion, the TRANS.INVS defaults to transposition. Example 26: MATRIX SET.NAM$ *********************************** * * * * * * Row NSET * 38 * 04 * - 20 * 08- 20 * T2 * * * * t i * *********************************** -40- 3.11.2 Establish MATRIX VECTOR The vector field (VT$) is assigned to the cumulative MATRIX VECTOR so vector similarity relations may be made. Each. interval class is ill a. corresponding column. The vector is preceded by the index number. Example 27: MATRIX VECTOR Col 0 COl 1 Col 2 Col 3 Col 4 Col 5 Col 6 ******************************************* * * * * * * * * Row NSET * 38 * 1 * O * 1 * 2 * 2 * O * * * * * * * * * ******************************************* 3.12 Print Set Data to ANALYSIS The following data are printed to the output file ANALYSIS for the user's reference. 1) The set number being analyzed (NSET) 2) The cardinality Of the set followed by the set number 3) The set in prime form 4) The transposition or inversion Operator 5) The vector 6) The set complement name 7) The B.I.S. pattern 3.13 Conditional Branching After the analytical data has been retrieved, stored, and printed, the user is pmesented with the fOllowing two conditions: Condition NO. 1 — The user is presented with the ques- tion "WOULD YOU LIKE TO FIND THE PRIME FORM OF A SET? Y/N?" _41- If the user responds "Y" or "YES," then the computer branches back to the initialization process in SET ENTRY (3.2) and re-executes until the data is printed to ANALYSIS (3.12). The user should respond "N" or "NO" to invoke execution Of set comparisons. Condition NO. 2 — The user is asked if (s)he would like to make comparisons of previously analyzed sets. Set com- parisons include similarity relations (pitch and vector), set complex relations and invariance. If the user responds with anything other than "Y" or "YES," the program termin- ates. If the user responds affirmatively, then a new output file entitled EVALUATE is created. Set comparisons are made ix: a one-to—One correspondence between all pmeviously analyzed sets. 3.14 Compare Sets for Set Complex Relations, Similarity Relations or Invariance Three conditions are tested to determine the method of analysis between any two sets A and B. Upon meeting a con- dition, the computer branches to a subroutine that executes the analysis. The three conditions Of analysis are as follows: Condition No. l — Set Complex Relations The cardinality of set A does not equal the cardinality of set B. Condition NO. 2 — Similarity Relations The cardinality of set A equals the cardinality of set B but the set number of set A does not equal the set number of set B. -42- Condition NO. 3 — Invariance The cardinality and set number of set A equals the cardinality and set number of set E. The transposition or inversion Operator of set A does not equal the transposition or inversion operator Of set B. Set Complex Subroutine The virtual array SUB.SUP$ allows random access to the database of subsets and supersets. Set A is identified through the index number retrieved from FORTE and stored in MATRIX SET.NNMS. Set B is identified by concatenating the cardinality and the set number into the variable BSETS. The computer accesses the virtual array SUB.SUP$ through the index number and searches for an occurrence Of the variable BSETS. If BSETS is found, then set A and set B are in K relation and the logical variable REL.K becomes true (-1). The computer simultaneously searches for an occurrence Of the set complement. If the set complement is found then set A and set B are in Kc relation and the logical variable REL.KC is true. If set A and set B are in K relation and Kc relation then they are in Kh relation. If both the logical variable REL.K and REL.KC are true, the REL.KH becomes true. The set complex relation is printed to EVALUATE. Similarity Relations Subroutine Similarity relations are divided into strongly repre- sented pitch similarity and vector similarity. Similarity relations are recorded in EVALUATE. -43- Part I: Strongly Represented Pitch Similarity Relations A strongly represented pitch similarity relation is defined as two different sets of the same cardinality in normal order that have a common subset of all but one ele- ment. Strongly represented pitch similarity is indicated by "Rp." Pitch similarity relations are discovered by the computer sequentially searching each member Of set B against each member Of set A. Each time a common element is found, the variable PITCH is incremented by one. If the variable PITCH is equal to the cardinality less one, then the Rp relation exists between sets in normal order. If the Rp relation exists, the logical variable RP becomes true (-1). Example 28: MATRIX NORM.ORD (Sets are in Normal Order) Col 0 Col 1 Col 2 Col 3 Col 4 ******************************* * * * t * * Set A * 4 * 3 * 6 * 10 * 11 * * * * * i * ******************************* / Cardinality Col 0 Col 1 Col 2 Col 3 Col 4 ******************************* * * * * * * Set B * 4 * 3 * 6 * 8 * 10 * * * * * * * ******************************* PITCH = CARDINALITY - 1 3 = 4 - 1 THEREFORE, THE Rp RELATION EXISTS Part II: Vector Similarities Vector similarity is defined as an equal number Of occurrences of the same interval class (IC) between two sets -44- Of the same cardinality. Each IC in the vector Of set A is compared against the corresponding IC in the vector Of set B. If the number of occurrences of an IC in set A is the same as that Of set B, then the variable VEC.SIM is incre- mented by one. Example 29: MATRIX VECTOR IC 1 IC 2 IC 3 IC 4 IC 5 IC 6 Col 0 Col 1 Col 2 Col 3 Col 4 Col 5 Col 6 ******************************************* * * * * * * * * Set A * 38 * 1 * 0 * l * 2 * 2 * 0 * * * * * * a * * ******************************************* 5 1 i 1 \N IC 1 IC 2 IC 3 IC 4 IC 5 IC 6 Col 0 Col 1 Col 2 Col 3 Col 4 Col 5 Col 6 ******************************************* * * * * * * * * Set B * 4O * 0 * 2 * 1 * 1 * 2 * 0 * * * * * * * * * ******************************************* VEC.SIM = 3 THEREFORE VECTOR SIMILARITY = R3 Invariance Subroutine Invariance is defined as a common subset between two sets of the same cardinality and set number but of a dif— ferent transposition or inversion. Invariance is discovered by a sequential comparison of set B against set A. Each time an invariant pitch is discovered, the variable INVARI- ANT is incremented by one and the pitch-class integer is assigned to MATRIX INVAR. Any invariant pitches between set A and set B in normal order are written to EVALUATE. -45- Example 30: MATRIX NORM.ORD Col 0 Col 1 Col 2 Col 3 Col 4 ******************************* * * * * * * Set A * 4 * 1 * 2 * 3 * 4 * 4-1 @ * * * * * * T 11 ******************************* Cardinality Col 0 Col 1 Col 2 Col 3 Col 4 ******************************* * * * * * * Set B * 4 * 2 * 3 * 4 * 5 * 4-1 @ * * * * * * T 10 ******************************* MATRIX INVAR Col 1 COl 2 Col 3 ****************** * * * * * 2 * 3 * 4 * * * * * ****************** CHAPTER 4 THE ALPHA TEST: A COMPUTER-ASSISTED SET-THEORETIC ANALYSIS OF FIVE PIECES FOR PIANO BY GEORGE CRUMB The computer-assisted analysis of Five Pieces for Piano by George Crumb was performed in parallel with a hand analy- sis in order to verify the output. The text Of the present chapter is a conspectus of the compilation Of data. ANALY- SIS is included in the text of this chapter. The output file entitled EVALUATE for the third piece is located in Appendix C. 4.1 Summary of Analysis Of the First Piece The PC sets in Five Pieces for Piano by George Crumb are segmented 'based Luxni rhythmic* organization, articulation, dynamics and registration. The segmentation Of the first piece yields 10 unique PC sets (see Figure 3). Of the 37 presentations Of the sets, 13.5 percent are cardinality two and 70.3 percent are of cardinality three. The majority (56 percent) Of the PC sets are set 03- 01 [0 l 2]. Two of the ten PC sets are comple- mentary sets: 02— 02 / 10- 02. Table 2 is the computer- generated output file entitled ANALYSIS. All Of the sets which are larger than cardinality three contain set 03—01 as a subset. The SET NO. in ANALYSIS refers to the ordering of -46- -u7_ momma umteu ecu Cd cowumucosmmm .m weaned .LVVC w22uee \ .-. - L. a x. - .-. -- - N L. N. a. o .r v .r o .r m .r N .r A. .r c .r c - .r _ .r v .r N. .-. c _ .r v .r N .r v .r - - .r _ _ .r m .r m L. m .r N .-. m. .r c .-. c _ m N. o - L. _ - - - ~ c ._. - L. - — L. N. .-. v .r -C.—.<-.r.-;O >2 - .- "0.21:2. 1‘ I! w: m>dwk LC.- c.-c: -r.U-..->-.:< enema c:u LOO ~NM¢LOOFCOO .Oz sum -48- -49- the sets as they appear in the music. Table 3 is a list Of the 10 PC set names and their frequency of occurrence. Table 3. Frequency of Occurrence of PC Sets in the First Piece Number Of Set Name Occurrences Percentage 02- 01 2 5.4% 02- 02 2 5.4% 03- 01 21 56.8% 03- 05 3 8.1% 03- 08 2 5.4% 05— 01 2 5.4% 08- 05 l 2.7% 10- 02 _1 2.7% 37 100 % 4.1.1 B.I.S. Patterns of Melodic Sets in the First Piece The majority of sets are presented harmonically lending B.I.S. pattern analysis inapplicable. Set numbers 20, 25, 28, 31, 33, 34, 35 and 36 are presented melodicalLy. Set number 36 is the union of sets 33, 34 and 35. Of the melo- dically presented sets, the majority Of basic interval successions are of interval class one. The second most prominent interval class is six. An analysis of the B.I.S. patterns indicates that the m2, M7 and tritone are melo- dically significant. -50_ 4.1.2 Set Complex Relations in the First Piece The data for the set complex relations is contained in the output file EVALUATE. Set complex relation is defined as two sets of different cardinalities that will map into each other and/or the complement of each other. Within the first piece, there are 329 possible set complex relations. It is significant that every pairing Of sets that may form a set complex relation is in a set complex relation. Nearly 80 percent of the set complex relations are Kh relation indicating a compositional preference for strong set complex relationships. Table 4 is a portion Of EVALUATE which demonstrates the set complex relationships. Table 4. Set Complex Relations in the First Piece Number Of Relation Occurrences Percentage K Relation 12 3.6% Kc Relation 55 16.8% Kh Relation 262 79.6% No Relation O 0.0% 329 100 % 4.1.3 Similarity Relations in the First Piece The data for similarity relations is contained in the output file EVALUATE. An Rp relation is defined as two different sets Of the same cardinality that have the cardin- ality minus one (n — 1) pitches in common. Rp relations between sets occur only 14.3 percent of the time. The sets in an Rp relation are separated by intervening sets indicat- ing little regard 'for hmmediate ;pitch. similarity ‘between sets Of the same cardinality. -51- Vector similarity is measured by the number Of vector entries in common for each interval class. The pairing Of sets demonstrates that vectors are in either an R 2, R 3, or R 4 relation. It is important to note that there are no R O or R 1 relations indicating a preference for sets with vec— tor similarity. However, there are not any vector pairings with all six interval class entries in common or Z-related pairs. Table 5 is a portion Of EVALUATE which demonstrates the similarity relations. Table 5. Similarity Relations in the First Piece Number of Relation Occurrences Percentage Rp Relation 17 14.3% 119 100 % R 0 Relation 0 0.0% R 1 Relation 0 0.0% R 2 Relation 69 58.0% R 3 Relation 42 35.2% R 4 Relation 8 6.8% Z—Related Pair 0 0.0% 119 100 % 4.1.4 Inclusion Relations in the First Piece The data for inclusion relations is contained in the output file EVALUATE. The inclusion relation is defined as the degree Of invariance between sets Of the same cardin- ality and set name. Three relationships may exist between two equivalent sets: 1) the sets are identical and demon- strate complete invariance; 2) the sets have no elements in common and demonstrate no invariance; 3) the sets have some elements in common and demonstrate an invariant subset. -52- Of the 218 pairs of equivalent sets, 24 are completely invariant. It is noteworthy that the union of sets 1, 2, 3 and 4 forms the universal set. Sets 18, 20, 21 and 22 likewise form the universal set. The succession Of sets 18, 20, 21 and 22 are completely invariant with sets 4, 3, 2 and 1. Also, the union of sets 21, 22, 23 and 24 form the universal set and are completely invariant with sets 2, l, 4 and 3. Since set 03- 01 is present the majority Of the time, set 03- 01 also demonstrates the highest degree of complete invariance and no invariance. The sets which have no elements in common (no invariance) and appear in succes— sion are contributory in locating appearances of the universal set. The sets Of cardinality three (03— 01, 03- 05 and 03- 08) generate 53 invariant subsets. Of these 53 subsets, 21 have 2 members in common (n.-Il) and 32 have 1 nmmber in common (n - 2). All of the subsets with two members reduce to the set 02- 01 [0 1]. Of the 21 subsets which demon— strate r1- 1 invariance, 6 pairs Of sets share the PCs [0 11]. All of these six subsets are produced by pairings of set 03- 01. Table 6 is a segment Of EVALUATE which displays the inclusion relations in the First Piece. Table 6. Inclusion Relations in the First Piece Number Of Relation Occurrences Percentage Complete Invariance 24 11.0% NO Invariance 141 65.0% Subset Invariance 53 24.0% 218 100 % -53- 4.2 Summary of Analysis Of the Second Piece The segmentation of the second piece indicates there are 13 unique PC sets (See Figure 4). As in the previous piece, set 03- 01 is reiterated a majority of the time. Many of the larger sets contain set 03-01 as a subset reinforcing the importance of the set. Two pairs Of the 13 sets are complementary sets: 03- 01 / 09- 01 and 04- 01 / 08- 01. Table 7 is the computer-generated output file .ANALYSIS. m 3 '5U_ :- ill” I! III" lflfii "I ‘ II 3 u I a al. I ' “"Iii lllllII _ N -C ICC N - m _ N N d -C :00 - N _C ICC _ - n N _ N - v _ N N - v N ~ ~ N u -C ICC a N _C ICC N u do ICC m o N v o N d _C ICC C g u m _ v N CC ICC N A .C ICC o _ _ m u _ N v N _ N - _C ICC v N CC IOC _ N o N 0C INC d N ~C IoC u N —C ICC ~ _ ~C ICC N a _C ICC _ d ~C ICC ~ _ ~C ICC _ ~ N a N v _ A _C InC - d ~C ICC _ N _C ICC _ - ~C ICC N - o - N N - -C ICC N - ~C ICC M mc IC— _ d do ICC — N N - 0 N - -C IQC N _ do ICC N _ C N - N - -0 Ice N - m - N N - -C In: - N -c Ioo - N m N - - N -C ICC N — -C ICC mzmmfifida .m._.m FZMXMJCZCU £52LC QULCtU >2 OLt-L CLO mn— CPI-.....r- n-CCLLV. r:- .. :22: _n c-vam CCCC-N :55 -N O_Nnvm -NmN-v NVN-Nv CCO-Nn cccc-N coco-N NNv-VN Ca::-N mVNch -C-C-C occc - N meNch NVNCNv occc-N oc-CNO -N-N occc-N Ccsc-N occc-N occc-N Cccc-N Cece-N acocNm coco-N occc-N cccc-N nvnch occc-N AC-.--_ N AC-- -pca- 3::c-N Nccch CCCC _ N New-«v nC-_-- _ N c-Nmem ----- N nC---- N NvN-Nv CCC-LN Occc-N moeumw ~— — — ~vvv~v~cocvovo—ovnO-movnvomomOhmoooooo-NmmOnm FFFFPFFPFFFFFPFFPFFSFFFPPHFFFPFFFFNPEFFPFFFFF # IwOe2-.u2<=e Lot m_c>;ML EC» CACC _ C _C ImC ~ C _C ICC — C _C Ino _ C _C InC ~ C _C ICC — C QCNIOC ~ C OCNIOC _ C —C Ivo — 0 do Ino _ C ~C Ino - C NN ICC m C no IMO u o No 100 N C QC InC a C —0 Inc a C no ICC — C so ICC ~ C do Ino N C co Inc — C QC ImC _ C "C Inc — C ~C InC — C ~C InC _ C —C Ino _ C —0 Inc _ C —C INC - C do ICC _ C _C InC _ C _C ImC _ C —C InC _ C NC ICC _ C go InC - C _C 1.5 a C mC INC _ C -C Inc - C .C ICC _ C _C InC - C CCNICC _ C _C InC _ C _C ICC _ C _C Inc - C .C Ino _ C CCNICC a C _C Ino _ C _C Ino mu; mr-cc< —N~w¢m\0t~mc‘ uwz Em -55- -56- Table 8 is the frequency of occurrence Of each PC set. Table 8. Frequency of Occurrence Of PC Sets in the Second Piece Number of Set Name Occurrences Percentage 02— 03 l 2.2% 03- 01 26 57.8% 03- 08 l 2.2% 04— 01 l 2.2% 05- O9 1 2.2% 06- 01 3 6.7% 06-Z04 1 2.2% 06-206 3 6.7% 06- 22 1 2.2% 08- 01 l 2.2% 45 100 % 4.2.1 B.I.S. Patterns Of Melodic Sets in the Second Piece As is true Of the first piece, the majority Of sets in the second piece are presented harmonically. Set numbers 12, 16, 17, 20, 21, 22, 23, 24, 35 and 36 are presented melodically. Of the melodically presented sets, the major- ity of basic interval successions are Of interval class one as is true in the first piece. The second most common interval succession is interval class two. This is in contrast to the first piece where interval class six played a secondary role. In the second piece, all interval classes are represented melodically. Once again, this is different -57- from the first piece where there is an absence Of the melo— dic presentation of interval classes two and three. 4.2.2 Set Complex Relations in the Second Piece Within the second piece, there are 546 possible set complex relations. In contrast to the first piece, the second has 36 pairs Of sets which are not in a set complex relation. Also, the second piece differs from the first in that the K relation is the most prominent set complex rela- tion. Table 9 displays the set complex relations of the second piece. Table 9. Set Complex Relations in the Second Piece Number Of Relation Occurrences Percentage K Relation 256 46.9% Kc Relation 33 6.0% Kh Relation 221 40.5% No Relation 36 6.6% 546 100 % 4.2.3 Similarity Relations in the Second Piece As is the case in the first piece, Rp relations are not prevalent in the second piece. vector comparisons in the second piece show a greater incidence of an.Ii 3 relation. Also, the R 0, R.lq I! 2 and R 4 relation are represented. This is in contrast to the first piece where there are no occurrences Of R 0 or R 1. The second piece does not con- tain any Z-related pairs as is true Of the first piece. ~58- Table 10. Similarity Relations in the Second Piece Number of Relation Occurrences Percentage Rp Relation 6 5.6% 107 100 % R 0 Relation 7 6.6% R 1 Relation 32 29.9% R 2 Relation 3 2.8% R 3 Relation 52 48.6% R 4 Relation 13 12.1% Z-Related Pair 0 0.0% 107 100 % 4.2.4 Inclusion Relations in the Second Piece The percentage Of complete invariance, no invariance, and subset invariance between pairs Of sets is very similar to the inclusion relations in the first piece. The second piece demonstrates that of the 337 pairings of equivalent sets, 208 exhibit no invariance. The union Of sets 20, 22, 23 and 24 form the universal set. The union of sets 12, 13, 14 and 15 also form the universal set but with two d-sharps. It is noteworthy that sets 16, 17 and 18 are completely invariant with sets 22, 23 and 24. Sets of cardinality three produce 76 invariant subsets. Of these subsets, 42 subsets have 2 members in common (n - l) and 34 have 1 member in common (n — 2). All Of the 42 subsets in an n— 1 relationship reduce to the set 02- 01 [0 1]. The reiteration of the invariant subset 02- 01 is consistent with the compositional craftsmanship of the first piece. The recurrence of interval class one reinforces its importance both melodically and harmonically. -59- Unlike the first piece, the second includes sets of cardinality six. The sets of cardinality six also demon- strate subset invariance. Table 11 displays the initial set and the subset which is invariant. Set 02- 01 [0 1] is in a set complex relationship with the initial set as well as the invariant subsets. The set complex relationship reiterates the structural importance Of interval class one. Table 11. Subset Invariance from Sets of Cardinality Six in the Second Piece Initial Set * n — 1 * n - 2 * n - 4 06— 01 * 05- 01 * 04- 01 * --- 06-ZO6 * --- * 04- 08 * --- 06— 07 * -—- * 04- 09 * 02— 01 Table 12. Inclusion Relations in the Second Piece Number of Relation Occurrences Percentage Complete Invariance 42 12.5% NO Invariance 208 61.7% Subset Invariance 87 25.8% 337 100 % 4.3 Summary of Analysis of the Third Piece The third piece Inarks the center Of the five-piece structure. George Crumb cast the central work in a palin- dromic design. Measures 6-13 form the palindrome. An introduction to the palindrome (mm. 1-5) recurs as a codetta (mm. 13-17) but slightly altered. The introduction and —60- codetta reinforce the symmetrical structure. The compo— sitional inventiveness Of the third piece creates many set- theoretic relationships. The third piece is composed Of six unique PC sets (see Figure 5). Consistent with the set selection of the first two pieces, set 03- 01 is reiterated a majority Of the time. In contrast with the previous two pieces, complementary sets are absent. Table 13 is the computer-generated output file ANALYSIS. Once again set 03- 01 is a subset of many of the sets listed in ANALYSIS. tion of the Third Piece Figure 5. Seqmenta -61- E1: '1" .. Hill I|§| 3 :::.........a Ilg|ll|1|||l. g,- ||la||||| |||i|""" 1 Hi | . / 51555:. . |E| | é||Eé|s|| E ‘2 '6. tall“ 3 IE|C|| ||iHFI .F.Peters Corporation. Reprinted by permission. Cbpyrieht (C) 1973 by C m H o H 0 ¢ N 00 N H Ho 0 m H m N H no N H o H N H N Ho N H HO N H HO N H Ho m H N H N H N Ho H N Ho N H HO H N Ho N H o H N H N m H m 0 50 N H HO N o H o H o m mo 0 H o H o m no Imo Imo Imo loo Imo I00 I00 I00 I00 100 I00 Imo Imo Iwo Imo mzmm99 m09ZHtmz¢MB $5 (I) [‘0 INC") "OcmHm MOM mwome m>Hm How womHm GHHSB wfiu MOM mHm>H NHH..H.<~HEQO >2H‘mz2 OOQHC :Hu:CL IIIiI womwa :.t:cu ozI tot mHm>uHm uCC .CH OHSCF use h\DHmc< a-INMQU‘QY‘QO‘ .oz sum .69. -70- Table 19. Frequency of Occurrence of PC Sets in the Fourth Piece Number of Set Name Occurrences Percentage 03- 01 17 47.2% 04— 05 2 5.5% 05- 02 1 2.8% 05- 03 l 2.8% 06— 01 5 13.9% 06-Z04 2 5.5% 06—Z06 3 8.4% 06- 07 2 5.5% 07- 19 1 2.8% 08- 01 l 2.8% 36 100 % 4.4.1 B.I.S. Patterns of Melodic Sets in the Fourth Piece Due to the similarities between the second and fourth piece, there is a recurrence of similar B.I.S. patterns. The majority of the melodic movement is by interval class one with interval class two playing a secondary role. As is true of the second piece, all interval classes are present. 4.4.2 Set Complex Relations Present in the Fourth Piece The set complex relationships are proportionally similar to the set complex relationships in the second piece. Table 20 summarizes the set complex relationships in the fourth piece. -71- Table 20. Set Complex Relations in the Fourth Piece Number of Relation Occurrences Percentage K Relation 189 44.4% Kc Relation 6 1.4% Kb Relation 193 45.3% No Relation 38 8.9% 426 100 % 4.4.3 Similarity Relations Present in the Fourth Piece Once again, the Rp relation does not play an especially significant role in the selection of sets. However, on one occasion, two contiguous sets are in an Rp relation. Sets 24 and 25 share the PCs 3, 4, 5, 8, 9 and 10. This partic- ular Rp relationship accentuates the importance of set 03- 01. The most frequent vector similarity relation is fl 1 followed by R 0 relation and the R 4 relation. There is an absence of similarity relations with two or three vector entries in common. As is the case in the previous pieces, there are not any Z-related pairs. Table 21 demonstrates the similarity relations for the fourth piece. Table 21. Similarity Relations for the Fourth Piece Number of Relation Occurrences Percentage Rp Relation _1_ 13.5% 52 100 % R 0 Relation 10 19.2% R 1 Relation 35 67.3% R 2 Relation 0 0.0% R 3 Relation 0 0.0% R 4 Relation 7 13.5% Z-Related Pair _0 0.0% 52 100 % .72. 4.4.4 Inclusion Relations Present in the Fourth Piece The fourth piece consists of relationships which are similar to those of the second piece. The union of sets 10, 11, 12 and 13 are invariant tri-chords which form the universal set” This presentation. of the universal set corresponds with sets 20, 22, 23 and 24 in the second piece. In the fourth piece, set 9 and the union of sets 11, 12 and 1&3 are completely invariant. This invariance corresponds with the relationship between sets 16, 17 and 18 and sets 22, 23 and 24 in the second piece. Pairings of set 03- 01 generates 55 invariant subsets. Of these 55 subsets, 22 of the subsets have 2 members (n - 1) and reduce to the set 02- 01 [0 le. In one instance, a pairing of set 04- 05 produces the invariant subset 02- 06 U3 6]. Sets of cardinality six produce several invariant subsets. These subsets are similar to the invariant subsets which are present in pairings of equivalent sets in the sec- ond piece. Table 22 shows the initial set and the invariant subsets which are formed. Table 23 displays the inclusion relations present in the fourth piece. -73- Table 22. Subset Invariance from Sets of Cardinality Six in the Fourth Piece Initial Set * n - 1 * n _ 2 * n _ 4 06— 01 * 05- 01 * ——- * HH. * * * 06-204 * -—- * -—— * 02- 05 ‘k * * 06-ZO6 * —-- * 04— 06 * ___ * * * O6-ZO6 * —-— * 04. 08 * --- * 'k * O6-ZO6 * --- * 04. 09 * --- Table 23. Inclusion Relations in the Fourth Piece Number of Relation Occurrences Percentage Complete Invariance 11 7.2% No Invariance 84 55.3% Subset Invariance 57 37.5% 152 100 % 4.5 Summary of Analysis of Fifth Piece Segmentation of the fifth piece yields 19 unique PC sets (see Figure 7). Of these sets, set 03- 01 is sounded one- half of_the time. Set 03- 01 is frequently a subset of the larger sets lending additional importance to the signifi- cance of the set. Three pairs of the 19 sets are comple— mentary: 03- 01 / 09- Ol, 04- 01 / 08— 01, and 05- 13 / 07— 13. Table 24 is ANALYSIS for the Fifth Piece. Table 25 lists the frequency of occurrence of each PC set. .conmHELco Nb occhLQom .coHmechoo mpmpom.h.o an mNOH Hov uanLNooo ...“... .... ... {cure ...": a. Cad-53 *IIInIOIIOIOOIIOIIo g. ' El.- HN Qt Edi-«v. “...... N a... 526 01.3.... .8... TI , $ T! .HH 11:. .L :mewrnw.. Hu .\ a. ........ mums: mm. I.r N I H H/I I .. . . o .1 ) H . 1 n E “ II III \ . n o 9% II II I‘ n‘ \ ANSI-u. Aim. .2 8.1"! 013 i is .5.“ .95.an datum Aid-v a1!» mumHa eptHu ass to coHumucmEamm .N mgsoHa __—_ II... "VI I‘Tl 3. HIIEL: ~: “II "II— V IIIII f|$ .III 11."! ;; _||l| .. :: to _ .. I. . --‘ 2' 'l 11 l :' . 2.: I'm-r ‘ -. fl 2. " :zo IIB """""""" @lfllflfi H... |" 4mgl . 5 ' |'--— 5: |.|| "LIN-{j palm! I ”J's—— “I'“ -75- Dachau of "In 030:1 9‘; Iain. SET NO. \Dmuo‘mbwuw Analytical Data SET 93225. 03- 01 08- 01 03- 01 03- 01 03- 01 03- 01 03- 01 03- 01 03- 01 06- 01 03- 01 03- 01 03- 01 02- 02 04- 03 03- 01 03- 01 05- 15 03- 01 03- 06 04- 01 03- 04 07-217 05- 01 08- 05 05- 13 07- 13 03- 01 05- 02 09- 01 09- 01 03- 01 06- 01 03- 01 03- 08 03- 01 03- 01 04- 02 04- 06 03- 01 “U (U (n OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO HHPHHNHb—‘HHHHHHHHHHHHMHWHHHNt—‘b—lO—lt—‘b—‘HHHHHWHH NNNNNNNMNNNNM NNNMNO‘NNMNNNMNNNNNWNbNMNNw a-btthA u (gnaw # asbcn mIDJben U \1 Table 24. for Five Pieces for Piano: ‘1“ cam ANALYSIS for the Fifth Piece Fifth Piece by George Crumb TRANS‘INV OPERATOR VECTOR COHPLEMENT B.I.S. PATTERNS T 6 210000 09- 01 2 1 T 1 765442 04- 01 1 1 1 5 1 1 1 T 0 210000 09- 01 2 1 T 3 210000 09- 01 l 2 T 9 210000 09- 01 1 2 T 7 210000 09- 01 1 2 T 1 210000 09- 01 2 1 T 10 210000 09- 01 2 1 T 4 210000 09- 01 2 1 T 10 543210 2 1 2 1 1 T 1 210000 09- 01 1 1 T 5 210000 09- 01 1 1 T 8 210000 09- 01 1 2 T 0 010000 10- 02 2 T 5 212100 08- 03 1 3 1 T 6 210000 09- 01 1 1 T 9 210000 09- 01 1 1 T 0 220222 07- 15 1 1 6 2 T 3 210000 09- 01 1 1 T 7 020100 09- 06 2 2 T 11 321000 08- 01 1 1 3 T 5 100110 09- 04 5 1 T 3 434541 05-217 1 4 4 1 2 4 T 8 432100 07- 01 1 2 1 2 T 8 654553 04- 05 6 1 5 1 2 1 4 T 1 221311 07- 13 3 1 2 6 I 10 443532 05- 13 1 2 4 2 2 1 T 10 210000 09- 01 1 1 T 4 332110 07- 02 3 2 3 1 T 0 876663 03- 01 1 2 4 3 4 3 4 3 T 4 876663 03- 01 4 1 2 2 1 3 S 4 T 7 210000 09- 01 1 2 T 4 543210 2 1 3 1 2 T 10 210000 09- 01 1 2 I 1 010101 09- 08 2 6 T 1 210000 09- 01 1 2 T 10 210000 09- 01 2 1 I 7 221100 08- 02 2 1 3 T 4 210021 08- 06 2 1 6 T 4 210000 09- 01 2 1 -77- -78- Table 25. Frequency of Occurrence of PC Sets in the Fifth Piece Number of Set Name Occurrences Percentage 02- 02 1 2.5% 03- 01 20 50.0% 04- 01 l 2.5% 04- 02 l 2.5% 04- 03 1 2.5% 04- 06 1 2.5% 05- 01 l 2.5% 05- 02 l 2.5% 05- 13 1 2.5% 05- 15 1 2.5% 06- 01 2 5.0% 08- 05 l 2.5% 09- 01 _2 5.0% 40 100 % 4.5.1 B.I.S. Patterns in Melodic Sets in the Fifth Piece As is true of the previous four pieces, the interval classes one, two and six occur with the greatest frequency. Set number 2 assumes an important role because of its in- sistent repetition. The basic interval succession of set 2 is all interval class one with the exception of one occur- rence of interval class five which separates two four-note groups. -79- 4.5.2 Set Complex Relations Present in the Fifth Piece The majority of the set complex comparisons form the Kh relation. The high degree of set complex membership is significant given the increased number of PC sets in com- parison to previous pieces. Table 26 summarizes the set complex relations. Table 26. Set Complex Relations in the Fifth Piece Number of Relation Occurrences Percentage K Relation 121 23.7% Kc Relation 44 8.6% Kh Relation 313 61.3% No Relation 33 6.4% 511 100 % 4.5.3 Similarity Relations in the Fifth Piece As is true in the previous piece, there is one occur- rence of two contiguous sets in an Rp relation. Sets 35 and 36 share the PCs l and 11. Vectors with two and three entries in common are most prevalent in the fifth piece. As is the case in the previous pieces, there are not any Z- related pairs. Table 27 demonstrates the frequency of the similarity relations. .80. Table 27. Similarity Relations in the Fifth Piece Number of Relation Occurrences Percentage Rp Relation _9 13.0% 77 100 % R 0 Relation 4 5.2% R 1 Relation 3 3.9% R 2 Relation 25 32.5% R 3 Relation 41 53.2% R 4 Relation 4 5.2% Z-Related Pair _0 0.0% 77 100 % 4.5.4 Inclusion Relations in the Fifth Piece Sets 6, 7, 8 and 9 are invariant tri-chords which when united form the universal set. There is another presenta— tion of the universal set during the introduction prior to measure one. Sets of cardinality three join together to form invariant subsets. All of the subsets with two members reduce to the set 02- 01 [0 1]. In one instance, successive statments of 09- 01 at m. 14 produce the invariant hexachord 06-Z37 [O 1.22 3 4 8]. Table 28 lists the inclusion rela- tions in the fifth piece. Table 28. Inclusion Relations in the Fifth Piece Number of Relation Occurrences Percentage Complete Invariance 14 7.3% No Invariance 122 63.5% subset Invariance 56 29.2% 192 100 % -81- 4.6 Nexus Set Set complex relations of all 33 unique PC sets through- out the entire composition do not readily expose a nexus set. Analysis of set complex relations does not even un- cover a primary and secondary nexus set which satisfies most of the PC sets. Figure 8 is a chart of set complex rela- tions of all pairs of PC sets throughout the composition. FL; "I .9 a ' 1 1 'q I 5 l c . . . I > t 11“ 9199 31 01 N Z ‘N “M ”N 1N 1N 1M UM ”X ‘N ‘N 101 ‘ 'N N ‘N UM 1” 1N ‘M DI UN to. q“ l [N ‘N ‘N ‘N l ‘01 N 1 0' 01 ‘ ‘ ‘0' b1 10! ‘01 [N ‘N M 'N ‘N 1N ‘N 1M ‘N ”1 'M ‘N ‘N ‘l 61 W ‘N 1N ‘N 1N 1M ‘N 'N ‘M ‘N ‘N '”—_"Z':Z ~L ’ II7-1 II I (II “N LIZ-L ‘Dl fT‘L ‘N ‘M ‘N NM ”M H! ‘M ‘M ‘N ‘U ‘N ‘N ‘N ‘N UM L *l [u ‘N ‘N ”H 7 [f N ZZ-Q H.HH:3 P7'9 I q z.- » ~9 9 7 9 zz—O L LI.— X L X 92-9 X X X 92-9 2 x “N 91* [N {1’9 x ‘N a“ “N N! a” N” ‘N N ‘N X as um ‘N 'N NM U! “N “X ‘0! “I “M “I UN on 1N ‘N W3)! 2: Q! “X on 'N ‘N *9 “N “X 6 “N ‘N :51 a" -s E us an fix an “M an “a 3x 1» ux 3» q” q” 1 on us an 3,; 5’)! :51 *S Di “M Z I “M I '9 -V bl “N 9 I ‘M “M a“ S -b 3H u» “N “N 3H q” a“ a” 3M “i “N 3% q” q” a" “M lbl a“ a“ an an ___..— —-—-—-—- ......— __H— _——— ‘N on 3M ‘N -V -V I V f —? Z .99.. 9 -v an an c an ax an an on an an 1x H: -b H! “I Z an an 3M “X on I “V 1D! “N -[ DI 3N 8 I “M an 3N UN on an an —_— ...-— ...—H..— “‘- ....— -c I v 9 -t s -r —r R my ‘N T 9 -f -f S “H a“ -f “N v 3” 3N ‘N 3M 'N -33- However, two successive statements of set 09- 01 merge during the fifth piece to form. the invariant hexachord 06-237 [0 1 2 3 4 8]. This unearthed invariant subset serves as one of two nexus sets. Figure 9. The Appearance of a Nexus Set mo|‘|’o Jramma‘h'co, Con Fuoco mode ...—“'3 14 Set O6-Z37 is accompanied by its Z-related pair, set 06-Z04, as the other nexus set. There are 19 mappings into set O6-Z37 and 20 mappings into set O6-ZO4. Figure 8 includes set 06-Z37 in the analysis of set complex relations. There are four sets (05- 02, 05- 13, 07- 07 and 07- 19) that will not map into 06-Z37 or 06-ZO4. These four sets do not occur frequently. -84- Frequency Set Name Location of Occurrence 05- 02 Fourth piece Set 6 Fifth piece Set 29 05- 13 Fifth piece Set 26 07- 07 Third piece Sets 1, 14, 16 07- 19 Fourth piece Set 20 Sets O6-Z37/06-ZO4 as the nexus sets creates a masterful link between the set complex about the nexus sets and the set selection throughout the composition. It has been noted in the summary analysis of each piece that set 03- 01 occurs with the greatest frequency. The two successive statements of set 09- 01 (the complement of 03- 01) are marked "molto drammatico, con fuoco" contributing to the climactic essence of the statement. It seems stratagemic that one of the nexus sets is buried within the complement of the most persistently employed set. The conspicuous absence of Z- related pairs throughout the composition lends a: touoh of jocular irony to Z-related hexachords as the nexus sets. CHAPTER 5 S UMMARY 5.1 Conclusions The primary conclusion is that the computer is an indi- spensable tool in the analysis of music. The computer is a supple instrument whose purpose is shaped by the conscien- tious logic of the programmer. Chances of human error during analysis are obliterated by the exactitude of the machine. Theorists are relieved of the tedious numerical manipulation of set theory without compromising the benefits of a thorough analysis. A secondary conclusion is that the Five Pieces for Piano by George Crumb are rich in set-theoretic relationships which are contributory to the delicate balance of unity and form within the composition. Set complex relations, simi— larity relations and invariance play an integral part in Crumb's selection of sets. 5.2 Strengths of Boethius and Accompanying External Files 1) BOETHIUS provides the user with a complete analysis of the relationships between PC sets. .85. -86- 2) The flexibility of BOETHIUS allows for melodic and/or harmonic set-theoretic analysis. 3) BOETHIUS is a well-structured program.9 a) The program exhibits top-down, modular design. b) the program is documented throughout the source code. c) The program is user-friendly and allows for several levels of twanching based upon the user's request. d) Input data is tested before being committed to analysis. e) The program is supported by an error-handling routine which permits error recovery' without aborting the program. f) The flexible design of the databases allows for future expansion of BOETHIUS. 5.3 Limitations of BOETHIUS and Accompanying External Files 1) BOETHIUS was written specifically for the DEC VAX 11/780. 2) BOETHIUS has been thoroughly tested in all facets of set theory analysis. However, bugs have a precarious way of creeping into even the most well-structured programs. 3) BOETHIUS is capable of a thorough analysis of the parameter of pitch. The program analyzes rhythm only as it relates to segmentation criteria. The program does not 9 Verified by Claude Watson, Director of Arts and Sciences Division Computer Laboratory; Lansing Community College; Lansing, Michigan. -37- analyze many pertinent aspects of a musical composition; for example, form, amplitude, timbre, tessitura and articula- tion. 4) BOETHIUS does not recommend a nexus set or a primary and secondary nexus set. 5.4 Futurisms 1) BOETHIUS will be molded to be microcomputer- compatible thus making the software more accessible. 12) Once BOETHIUS is ndcrocomputer-compatible, several universities throughout the country will be contacted to participate in a BETA test of the software. 3) Further research will be done in the hOpes of creating software Which will analyze all facets of a musical composition. 4) Artificial Intelligence techniques will be utilized to discover the subliminal direction of contemporary compo- sition. APPENDICES APPENDIX A \OKDCDGDOD KO \OCDCDCDCDQVNQQQ 03m~dxlH~q~1q(moxm<36\m¢>b4>bnb¢sbwb¢sp.hcshubcshubcswLuOJNIDOIwcuuiwIpuawcnuawcuu>wcouiwInuiwcpunbcow Hummmmmmmmmmmmmmmmmmmmmmmmmbbb O‘C‘O‘O‘O‘ONO‘UIU‘U‘IU‘UIU‘UILDWWU’IUIUIUIUIU‘ meQQQQOmONONO‘mO‘O‘O‘O‘ (D\l\l \OCDQQ \1 CD (DODH-JNIH~4~Jm<30~m¢ub¢b¢4>¢bb4>¢bb4>¢5b4>bn§¢>h ~40»mcnoxmtnUIanuwmtnuvm4>¢>pd>¢.bis quqmmmmmmmmmmmtflmmm (DQxlxlONOOO‘O‘ CDNQVQG‘OW (1)\1\l mmqqqmo‘mmm \DCD \DCOCD \Dmmfl 00 \OQCDCDQQQ mflxlflONO‘m \DCD mmmu \OCD CDCDxl \DCI) 10 \0 O .90. 07- 30 06-Z46 05-236 06-Zl7 07- 20 06-Z47 05- 13 04- 04 05- 06 06-206 06-Z43 07- 22 06-Z44 05- 14 06- 18 06-Z48 05-Z38 04- 05 05- 07 06- 07 05- 15 04- 06 03- 02 03- 03 08- 12 07- 11 08- 17 06-Z37 07-Z37 07- 26 06- 14 05- 10 06-213 07- 31 08- 28 06-224 07- 34 07- 32 06- 27 05- 16 06-Zl9 06-Z49 05-Zl7 04- 11 05-Z12 07- 28 06-Z25 07- 35 06-228 05- 24 06-226 06- 34 05- 27 06- 31 165 131 83 102 155 132 60 22 53 91 128 157 129 61 103 133 85 23 54 92 62 24 21 185 146 190 95 172 161 99 57 98 166 201 109 169 167 112 63 104 134 64 29 59 163 110 170 113 71 111 119 74 116 343542 233331 222121 322332 433452 233241 221311 211110 311221 421242 322332 424542 313431 221131 322242 232341 212221 210111 310132 420243 220222 210021 111000 212100 556543 444441 546652 333321 434541 344532 323430 223111 324222 336333 448444 233331 254442 335442 225222 213211 313431 224322 212320 121110 222121 344433 233241 254361 224322 131221 232341 142422 122230 223431 05- 30 06-224 07-236 06-Z43 05- 20 06-Z25 07- 13 08- 04 07- 06 06-Z38 06-Zl7 05- 22 06-Zl9 07- 14 06-Z26 07-Z38 08— 05 07- 07 07- 15 08- 06 09- 02 08- 03 04- 12 05- 11 04- 17 06-239 05-Z37 05- 26 07- 10 06-Z42 05- 31 04- 28 06-Z46 05- 34 05- 32 07- 16 06-Z44 06-228 07-Zl7 08- 11 07-212 05- 28 06-Z47 05- 35 06-249 07- 24 06-Z48 07- 27 OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO NMNNNNNNNNNNNNNNNNNNNNNNNNNHHI—‘HHI—‘I—‘I—‘HHI—‘l—‘l—JHHHI—‘l—‘Hl—‘I—‘F—‘l—‘HI—‘l—‘I—J oxmunmmmbbbbbbbbbbpbpwwwwwwww b.b4>b-b¢-¢ubuow(Duiwcpoaw(HuiwcgquIH0302w H~JONmChoxmcn m «Dmcnxiq \l 03\JO\ (I)\|\IO\O\O\O\U1U1U1UIUI \Immmmmmmbbbbbbbbbh qmmmmmm mmqox (I) \OCDx) \Imammmmmmm (1341410 03 @0000 CD“ \Omfl (I)\l\1030'\ (I) 10 mu \0 -91- 04- 13 05- 19 06- 30 05- 29 06-Z29 05- 31 04-229 05- 20 03- 03 04- 07 06- 16 05-218 05- 21 06- 20 04-Z15 06-Z50 05- 30 05- 32 04- 18 05- 22 04- 19 03- 04 04- 08 04- 16 04- 20 03- 05 04- 09 02- 02 08- 10 07- 08 06- 08 07- 23 06-Z39 05- 08 07- 25 06- 21 06-Z45 05- 11 04- 10 06-Z23 05- 23 06- 33 05- 25 04- 12 05- 28 04- 14 03- 06 06- 32 05- 26 04- 21 05- 33 06- 35 05- 34 04- 22 31 66 115 76 114 78 47 67 25 101 65 68 105 33 135 77 79 36 69 37 10 26 34 38 11 27 183 143 93 158 124 55 160 106 130 58 28 108 70 118 72 30 75 32 12 117 39 80 120 81 40 112011 212122 224223 122131 224232 114112 111111 211231 101100 201210 322431 212221 202420 303630 111111 224232 121321 113221 102111 202321 101310 100110 200121 110121 101220 100011 200022 010000 566452 454422 343230 354351 333321 232201 345342 242412 234222 222220 122010 234222 132130 143241 123121 112101 122212 111120 020100 143250 122311 030201 040402 060603 032221 021120 08- 13 07- 19 07- 29 06-250 07- 31 08-229 07- 20 09- 03 08- 07 07-Zl8 07- 21 08-215 06-Z29 07- 30 07- 32 08- 18 07- 22 08- 19 09- 04 08- 08 08- 16 08- 20 09- 05 08- 09 10- 02 04- 10 05- 08 05- 23 06-Z10 07- 08 05- 25 06-Z23 07- 11 08- 10 06-Z45 07- 23 07- 25 08- 12 07- 28 08- 14 09- 06 07- 26 08— 21 07- 33 07- 34 08- 22 OOOOOOOOOOOOOOOOOOO mmbbwwwwwwwwwwmwmwm \lmO‘UIUlUlub-b \Immmpp (D \D mxlUl -92- 05- 35 04- 24 03- 07 04- 23 04- 27 03- 08 04- 25 03- 09 02- 03 05-237 04- 17 04- 26 03- 10 04- 38 03- 11 02- 04 03- 12 02- 05 02- 06 82 42 13 41 45 14 43 15 84 35 44 16 46 17 18 032140 020301 011010 021030 012111 010101 020202 010020 001000 212320 102210 012120 002001 004002 001110 000100 000300 000010 000001 07- 35 08- 24 09- 07 08- 23 08- 27 09- 08 08- 25 09- 09 10- 03 07-Z37 08- 17 08- 26 09- 10 08- 28 09- 11 10- 04 09- 12 10- 05 10- 06 APPENDI X B COMPUTER PROGRA“ BOETHILC IO REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA GET-THEORY ANALYSIS OF ATONAL MUSIC AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AUTHOR: DATE: MARY SEPTEMDER l, 1983 9AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 20 I ALL RIGHTS RESERVED. COPYRIGHT, SHALL AT LANSING, TITLE ALL TIMES DELONG COPYRIGHT NOTICE MARY SIMON], MICHIGAN TO AND OUNERSHTP TO THE AUTHOR. 1983 OF THIS SOFTUARE REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI IAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 30 I THIS PROGRAM NORMAL ORDER, POSITION OR INVERSION PRIME FORMS, PROGRAM ADSTRACT ANALYZES PITCH-CLASS SETS REST NORMAL ORDER. OPERATOR. SET NAMES, AND PLACES THE SETS IN AND PRIME FORM UXTH THE TRANS- B.I.S. AND VECTORS ARE ACCUMULATED ALLOUING PATTERNS, NORMAL ORDER. ANALYSIS OF SET COMPLEX RELATIONS AND SIMILARITY RELATIONS. REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAE IAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 40 o VERSION NORMAL.DAS SET.DAT rs:ns.sas FORM.DAS ANA.DAT DTSS.DAS APEX.DAS KEV.DAS DOHARR.DAS MA!N.DAS DOHARRY.DAS ALPHONCE.DAT ALLEN.SAS FORTE.DAT ANALYSIS.OUT EVALUATE.OUT DOETHIUS.DAS 100 ON ERROR 200 CHANNEL ---------‘-~---- MODIFICATION HISTORY LOG DATE 1-15-83 SIS-83 3-25-83 3-3-83 3-14-83 3-15-83 SIRS-GS 4-23-83 6-21-83 6-21-83 7-16-83 S-Sl-SS G-Sl-SS 8-31-83 8-31-83 8-31-83 S-Sl-GS 0-31-83 GO TO 19000 REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA8 DESCR! PROGRAM TO FIND THE NORMAL ORDER DATA F OP ATONAL MUSTC', PTION ILE OF FORTE, ALLEN. OF A SET 'THE STRUCTURE APPENDIX 1 PROGRAM TO FIND THE PRIME FORM OF A SET UTTH THE TRANSPOSITTON/INVERSION OPERATOR AND RE TRIEVE THE SET FROM 'SET.DAT' PROGRAM SIMILAR IN DESIGN TO PR1ME.DAS NITHOUT THE USE OF INPUT FILE 'SET.DAT' SAMPLE INPUT/OUTPUT FILE FOR SIMILARITY RELATT PROGRAM SIMILAR TN DESIGN TO ADDXT! 'RELAT! PROGRA ON OF ’SET.DAT' ON MODULE M MODULE AND TO PORM.DAS UTTH SIMILARITY TEST THE LOGIC OF SIMILARITY RELATION ANALYSIS 'SET.D 'SUPSU R RELA PROGRA 'SURSU FINAL VERSION OF PROGRAM USING AND ° AT' P.DAT' TTONS M TO DUILD THE P.DAT' SUDSUP.DAT‘ AS DATA AS TNDEXED RECORD 1/0 AS VIRTUAL ARRAY FOR VIRTUAL ARRAY FILE 'SET.DAT° DASES REVISED PROGRAM TO DUTLD VIRTUAL ARRAY FILE ' 'ALPHO 'ALLEN ‘FORTE 'ANALY ’EVALU REVIST ALPHONCE.DAT' NCE.DAT' - .DAS' - 'REC.DAS' .DAT' I 'SET.DAT' SIS.OUT' I 'ANA.DAT' ATR.OUT' I ON OF 'MATNolAS' CHANNEL ASSIGNMENTS ASSIGNMENT 'FORTE.DAT',RECORD T/O FILE OF FORTE. ALLEN. “STRUCTURE APPENDIX 1 'ANALYSIS.OUT‘. ANALYTICAL DATA PIECE OF MUSIC. B.I.S. PATTERNS, PRXME FORM, INVERSION OPERATOR, MENT NAME. OF SET NAME, 'SUDSUP.DAT' ‘COMPARS.SET' OF ATONAL MUSIC' SEOUENTIAL FILE OF A PARTICULAR DATA INCLUDES NORMAL ORDER. TRANSPOSITION AND SET COMPLE’ REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA8 D...IUIIDII.“MINI-..flflflflflflflflflflflflflflpfl I I I 03 'RVALUATI.OUT'. SEOUENTIAL EILE I I NNICN LIsrs RELATIONSHIPS DETHEEN I I PAIRS or BEIS As LIBIED IN I I 'ANALYSIS.OUT' I I COMPARISONS INCLUDE SIMILARITY I I RELATIONS: SET COMPLEX RELATIONS I I (K,KC,KH); INTERSECTION; AND I I INVARIANCE. ! I I I II ‘ALPHONCE.DAT' VIRTUAL ARRAY EILE I I or SUDSETS AND SUPERSETS. I I ALPHONCE, no HARRY. I I 'THE INVARIANCE MATRIX‘. I I APPENDIX 9 I I I I Is 'KBET.DAT' SEQUENTIAL EILE to I I to IUILD 'ALPHONCE.DAT' I I A 205 MAP(IDX) rtINEI-aaz.CAnoI-ax.ENI-Ix.IND:N0I-SI.UII-EI.NI-az 210 OPEN 'EORTE.DAI' AS EILEIII. ORGANIZATION INnExEo FIXED I .PRIMARY KEY PRIME$ 8 .ALTERNAYE KEY CARD. DUPLICATE. CHANGES I ,ALTERNATE KEY 8N0 DUPLICATBS CHANGES I .ALTERNATE KEY IND.NOI CHANGES I .ALTERNATB KEY 910 DUPLIEAIEI ENANaEE I ,ALTERNATE KEY NI DUPLICAYEB CHANGES I 320 230 335 300 SIG 320 330 340 370 380 400 AIO I30 440 ISO 300 600 605 SIO SIS 620 630 640 650 660 670 600 SSS 690 700 710 720 730 740 750 ,MAP Inx. ACCEss MODIFY, ALLou MODIFY OPEN 'ANALYSIS.OUT’ ROI ourrut AS III! .2: OPEN 'EVALUATE.OUT' Eon OUTPU! AS EILE .3: OPEN 'ALPHONCI.DAT' A8 [III III. ORGANIZATION VIRTUAL REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI I DIMENSION DICLARATIONB I DIM AIIRE,IREI DIM RCIRT.ISEI DIM DISISOOE.ISE) h'” LTIRZ.I32) DIM DUPIIRI) DIM INVARIIREI DIM NORM.ORD(ROOE,I22) DIM PRIMEI(2001.122) DIM SET.NAMI(2002,43) DIM .4.SUD.SUPI(2201,3301I-GE DIM VECTORIROOZ.SI) RRMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI I I I SUDROUTINES I I I I LINE NO. DESCRIPTION I I ------------------- I I IOOOO SUDROUTINE TO RETURN THE RETROGRADE I I INVERSION OE A SET IN REST NORMAL I I ORDER I I I I IIOOO SET COMPLEX I I I I IROOO SIMILARITY RELATIONS I I PART I: STRONGLY REPRESENTED I I PITCH SIMILARITIES I I PART II: VECTOR SIMILARITIES I I UITHOUT INTERCHANGE I I I I 13000 INVARIANT PITCHES DETUEEN THO I I SETS IN NORMAL ORDER I I REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI I PRINT HEADINGS TO OUTPUT EILES I IAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA INPUT 'COMPOSER':COMPOSERI INPUT 'TITLE OE COMPOSITION': TITLES MARGIN .3. I332 \ PRINTIR ‘ PRINT IZ.‘ANALYTICAL DATA EOR' PRINT .2."’:TITLEI;"':' DY “SCOMPOSERI PRINT .2, ‘ ' PRINT .2,'SET':TADIS):’SET'3TAD(IS):'PCS':TADIQI):‘TRANSAINV':TAD(52)3 I 'VECTOR'STADIIOI:'COMPLEMENT':TAD(7RI:'D.I.S. PATTERNS' PRINT 02,'NO.':TAR(S):'NAME':TAD(IS):'AAA‘:TAD(4II:'OPERATOR'3TADCSRI: I 'AAAAAA'BTADISOI:'AAAAAAAAAA‘:TADI73):‘AAAAAAAAAAAAAAA' PRINT .2.‘AAA':TAD(S):'AAAA':TAD(II):'AAAAAAA' PRINT IE.’ ’ MARGIN .3. I321 \ PRINTOG PRINT .3,'SET COMPARISONS POR' PRINT .3."':TITLEI:"‘:' DY ’3COMPOSERI PRINT .3,“ ’ PRINT IS.TAD(32I:'SET COMPLEX’3TADII7):‘SIMILARITY':TAD(72):‘INCLUSION' PRINT 03,'SET SET NO. A SET SET NO. '3TADI32):‘RELATIONS’:TAD(QS): I ‘RELATIONS‘:TADI72):‘RELATIONS' PRINT .3,‘AAA AAAAAA A AAA AAAAAA °3TAR(32):‘AAAAAAAAAAA':TAR(QS):I 'AAAAAAAAA'3TADI7R):‘AAAAAAAAA' PRINT .3,TAD(IQI:‘A':TAD(30):'A' .00 REMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI I I I CUMULATIVE DICTIONARY AND INITIALIZATION I I 003 304 805 S06 007 S08 009 SIO S20 S30 COMP.INV I 0 INV.REL I 0 INV.SUD I 0 N.COUNT I 0 NC.COUNT I 0 NH.COUNT I 0 KX.COUNT I 0 MAT DIS I ZER MAT COMP I ZER MAT NORM.ORD I EER COMP.INV I COUNTER EOR INV.SUS I COUNTER EOR INVARIANT SUDSETS K.COUNT I COUNTER EOR SETS IN K RELATION NC.COUNT I COUNTER EOR SETS IN KC RELATION KH.COUNT I COUNTER EOR SETS IN KH RELATION NX.COUNT I COUNTER EOR SETS NOT IN N RELATION MAT DIS I MATRIX OE DIS PATTERNS MAT COMP I MATRIX OE SET COMPLEMENTS MAT NORM.ORD I MATRIX OE SETS IN NORMAL ORDER COMPLETELY INVARIANT SETS INV.REL I COUNTER EOR TOTAL INVARIANT RELATIONS S63 MAT VECTOR I ZER MAT VECTOR I MATRIX OE VECTORS S64 MODULUS! I I21 MODULUS I NUMDER OE AVAILABLE PITCHES IN AN OCTAVE S65 NSETX I 0 NSET I SET NO. I RON DESIGNATION EOR CUMULATIVE MATRICES S6S MAT PRIME. I NULI MATRIX PRIME. I MATRIX OE SETS IN PRIME EORM SSO MAT SET.NAMI I NULI SET.NAMI I MATRIX OE SET NAMES AND S90 TRANSPOSITION/INVERSION OPERATORS I COMPLEMENT NAMES 900 NO.INV I 0 NO.INV I COUNTER EOR SETS UITH NO INVARIANCE 910 NO.IP - o NO.IP - NUNIEE or SITE IN Rp RELATION 920 NO.VECO - o NO.VRC - COUNTIR EOR NUNEEI or IC ENTRIES IN cannon 930 NO.VECI - o 940 No.vec2 - o 950 NO.VEC3 - o 950 NO.VEC4 - o 970 NO.VEC6 - o 990 sErcon.IEL - o SETCOH.RBL - TOTAL SET COMPLEX RELATION COUNTER 990 SIH.REL - o SIM.R£L - TOTAL SIMILARITY RELATION COUNTER 995 TOT.REL - o TOT.R£L - CARTESIAN raooucr or IE: COMPARISONS 1000 REM IEoIN ANALYSIS 1005 PRINT ono INPUT ‘UOULD you LIKE to EIND INE PRIME EOIN or A 82!? r/N-;ANsuEas 1020 It AusuEaI - ’NO' on ANSNEII - 'N‘ THEN 32740 I030 PRINT 'EIRST YOU NILL DB ASKED TO ENTER THE CARDINALITY or THE SET.’ 1040 PRINT ‘THE CARDINALITY or THE IE! INOULD NOT II LIBS THAN run 03- I050 PRINT 'GIEATER THEN TEN' ‘ Ioeo PRINT -vou NILL BE PROMPTED IACN TIMI you AIE TO ENTER A' I070 PRINT °rxrcu CLASS INTEGER.‘ 1090 PRINT 'USE THE EOLLouINo INTEGER NOTATION:' 1090 PRINT - a c - o A' IIoo PRINT . A CO.Db - I I- 1110 PRINT - A o - a A' 1120 PRINT - A no.Eo - a A' Ixao PRINT - A E - 4 A' 1140 PRINT - A E - 5 a- IIso PRINT - A ro.ob - 6 I- 1160 PRINT - A o - 7 A‘ 1170 PRINT - A GO,Ab - A' 1190 PRINT - a A - 9 A' 1190 PRINT - A A0,Db - 10 A' I200 PRINT - A I - II A' Ian I IAI ° - I220 PRINT '00 NOT DUPLICATE ANY PITCH ELAEI INIEOEI- 1230 PRINT 'AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA‘ I240 REMAAAAANAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAI VARIADLE DICTIONARY AND INITIALIEATION I IAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA ANSUERI I" I ANSHERI I ANSUER TO PRIME EORM SORT PROMPT DIS.COLE I 0 DIS.COLE I COLUMNS IN DIS MATRIX I250 I260 I265 DSETII“ DSETI I CONCATENATION OE CARDINALITY I SET NAME IN ' SET COMPARISONS I270 CE I 0 CE I COLUMNS IN MATRIX A I275 CARDI I ’ ' CARD. I CARDINALITY EROM 'SET.DAT' I2S0 I290 I300 I320 I330 I335 CHECK! I 0 COLE I 0 COLUMN! I 0 COUNTERX I 0 DIFFERENCE! I 0 ENTRIESE I 0E CHECKE I LOCATIONS IN DUP ARRAY COLE I COLUMNS IN SET/MATRIX COMPARISONS COLUMN! I COLUMNS IN MATRIX A I MATRIX D COUNTERI I ROSS IN NORMAL ORDER/COLUMNS IN RETRO INV. DIEEERENCEE I DIEEERENCE DETNEEN IST I LAST INTEGERS ENTRIESX I NO. OE ENTRIES OE SUDSETS I SUPERSETS IN SUD.SUPO I340 GRADES I 0 GRADES I COLUMNS IN MATRIX C I350 I! I 0 IX I INVERSION OPERATOR I360 ICE I 0 ICX I COLUMNS IN VECTOR MATRIX I365 IND.NOI I " IND.NOI I INDEX NO. OE RCORDS IN 'SET.DAT° I370 INVE I 0 INVT I INVERSION OPERATION 1300 INVARIANT! I 0 INVARIANT! I COLUMNS IN MATRIX INVAR I390 ID! I OE IOX I INVERSION OPERATOR I450 LOUX I 0 LOU! I ”BEST NO. IN SORT I460 MAT A I ZER MATRIX A I NORMAL ORDER PERMUTATIONS II70 MAT D I ZER MATRIX D I DEST NORMAL ORDER PERMUTATIONS I4S0 MAT C I ZER I490 MAT DUP I ZER I500 MAT INVAR I ZER MATRIX C I RETROGRADE OE MATRIX D MATRIX DUP I ARRAY TO CHECK EOR DUPLICATE ENTRIES MATRIX INVAR I ARRAY TO STORE INVARIANT PCS I5I0 NX I 0 NZ I CARDINALITY OE THE SET I512 NO I " NI I SET COMPLEMENT NAME EROM ’SET.DAT' I515 NO! I 0 NO! I EOR/NEXT COUNTER I520 NSETX I NSETZ O I NSETX I NUMDER OE SETS ANALYEED I530 PASSX I O PASSX I NO. OE PASSES IN NORMAL ORDER SORT I540 PCX I 0 PC! I PITCH CLASS INTEGER -96- I545 PC. I " I550 PITCH! I 0 I560 PITCH]! I 0 I570 R! I 0 I575 REC! I 0 I580 REL.K! I 0 1590 REL.KC! I 0 I600 REL.KH! I 0 I6I0 RETRO! I 0 I620 ROBy I 0 I62I ROH.! - 0 I622 RP! I 0 I623 SLIST! I 0 I625 SN. I ’ ’ I630 STORE! I 0 I640 STOREI! I 0 I650 STORE2! I 0 O 0 PC. I SET IN PRIME EORM (PRIME.) PITCH! I COUNTER EOR SIMILARITY RELATIONS PITCHI! I COUNTER EOR SIMILARITY RELATIONS R! I ROHS IN MATRIX D REC! I INDEX NO. EOR ACCESSING SUD.SUP. REL.K! I LOGICAL VARIADLE EOR RELATION K REL.KC! I LOGICAL VARIADLE EOR RELATION KC REL.KH! I LOGICAL VARIADLE EOR RELATION KH RETRO! I EOR/NEXT RETROGRADE DO”! I EOR/NEXT ROH ROHI! I EOR/NEXT ROH IN MATRIX D RP! I LOGICAL VARIADLE FOR R9 RELATION SLIST! I EOR/NEXT COUNTER TO SEARCH SUD.SUP. SN. I SET NAME EROM 'SET.DAT' STORE! I STORAGE SPACES IN SORT I660 STORE3! I I670 STORE4! I I680 T! I 0 I690 TO! I 0 I695 TRANS.INV. I " T! I TRANSPOSITION TO! I TRANSPOSITION OPERATOR ' TRANS.INV. I CHARACTER 'T' OR 'I‘ HITH TRANSPOSITION OPERATOR VT. I VECTOR EROM 'SET.DAT' I696 VT. I " I697 PRINT\PRINT I698 PRINT 'SET NUMDER‘SNSET! I699 PRINT ’ENTER -I TO RE-INITIATE SET ENTRY' I700 INPUT 'HHAT IS THE CARDINALITY OF THE SET'IN! I7I0 IE N! I -I THEN I697 I7I5 IE N!<2! OR N!>I0! THEN PRINT 'THE CARDINALITY MUST DE EROM 2 TO I0'\I GO TO I700 I720 REM ASSIGN THE PITCH CLASS INTEGERS TO MATRIX A I730 EOR COLUMN! I I! TO N! I740 ON COLUMN! GO TO I750. I760. I770, I780. I790. I300. ISIO. I820, I830, I835 I750 COLUMN. I ’EIRST'\GO TO I840 I760 COLUMN. I 'SECOND'\GO TO IS40 I770 COLUMN. I 'THIRD'\GO TO I840 I780 COLUMN. I ’EOURTH'\GO TO I840 I790 COLUMN. I 'EIETH'\GO TO I840 I800 COLUMN. I 'SIXTH'\GO TO I840 ISIO COLUMN. I 'SEVENTH’\GO TO I840 I820 COLUMN. I ‘EIGHTH'\GO TO I840 IS30 COLUMN. I 'NINTH’\GO TO I840 I835 COLUMN. I ’TENTH' IS40 PRINT 'ENTER THE '3COLUMN." PITCH CLASS INTEGER': I850 INPUT PC! I855 IE PC! I -I THEN I697 I860 REM CHECK IF PITCH CLASS INTEGER IS VALID I870 IE PC!<0 OR PC!>II THEN I PRINT 'PITCH CLASS INTEGER MUST DE FROM 0 TO II'\GO TO I740 I880 REM CHECK TO SEE IE DUPLICATE PITCH CLASS HAS ENTERED I890 DUPICOLUMN!) I PC! I900 IE COLUMN!(2 THEN AII.IIIPC! \ GO TO 2030 I9I0 EOR CHECK! I I! TO COLUMN! . I! 'I920 IE DUP(COLUMN!I<> DUP(CHECK!I THEN I960 I930 PRINT 'DUPLICATE PITCH CLASS INTEGER HAS ENTERED“ I940 INPUT 'DO YOU HANT TO CHANGE THE CARDINALITY OF THE SET? Y/N';ANSHER. I950 IE ANSHER. I ‘NO‘ OR ANSHER. I 'N’ THEN I730 ELSE I700 I960 NEXT CHECK! I970 AII,COLUMN!) I PC! I980 REM TRANSFER DIS PATTERN TO DIS MATRIX I990 IE COLUMN!<2! THEN 2030 2000 DIS.COL! I DIS.COL! O I! 20I0 IE AII!,COLUMN!)>A(I!.(COLUMN!-I!)) THEN I DISINSET!,DIS.COL!)IA(I!.COLUMN!)-A(I!,(COLUMN!-I!)I ELSE I DISINSET!.DIS.COL!)IA(1!,(COLUMN!-I!)I'A(I!.COLUMN!) 2020 IF DISINSET!.DIS.COL!) > 6! I THEN DIS(NSET!,DIS.COL!) I MODULUS! - DISINSET!.DIS.COLE) 2030 NEXT COLUMN! 2040 REM SORT ROUTINE TO FIND NORMAL ORDER: PART I I DUDDLE SORT 2050 FOR PASS! I I! TO N! - I! 2060 FOR COLUMN! I I! TO N! - PASS! 2070 IE A(I.(COLUMN!OI!))LOH! THEN 2370 23I0 COUNTER! I COUNTER! 9 I! 2315 IF AIROH!,I!) I 0! THEN MAT D I A\GO TO 2380 2330 FOR COLUMN! I I! TO N! 2340 D‘COUNTER!,COLUMN!)IAIROH!.COLUMN!) 2350 NEXT COLUMN! 2370 NEXT ROH! 2380 REM ROUTINE TO FIND THE REST NORMAL ORDER: I DETERMINE IE SET IS IN TRANSPOSITION OR INVERSION 2390 FOR ROH! I I! TO COUNTER! 2400 STOREI! I DIROH!,2) - DIROH!.I) 24I0 STORE2! I DIROH!,N!‘ - D(ROH!.(NE-I!II 2420 IF STOREI!<>STORE2! OR N! (I 4! THEN 2490 2430 FOR COLUMN! I I! TO INTIIN!°I!)/2!) 2440 S?ORE3! - Itlouz,(c0Lunnzozx)) - DIROH!,¢COLUMN!OI)) 2450 3:02:42 - D(ROH!,(N!-COLUMN!)) - S(ROH!,(N!-(COLULN!OI!)I) 2460 IE STORES! - STORE4! THEN 2480 2470 It EIOIEax > 8TORE4! THEN oosua Ioooo ILSR 2500 2480 NEXT COLUMN! 2490 IF STOREI! ) STORE2! THEN 2590 2500 REM THE SET IS IN TRANSPOSITION I PART I I FIND TRANSPOSITION OPERATOR 2510 T! I DIROH!.1I 2520 IF T!>MODULUS! THEN TEITE-MODULUS! 2530 TO! I MODULUS!-T! 2540 REM TRANSPOSITION OPERATION: I PART 2 I REDUCE THE SET TO PRIME FORM 2550 FOR COLUMN! I I TO N! - 2560 DCROH!.COLUMN!)IDIROH!,COLUMN!I-T! 2570 NEXT COLUMN! 2580 IF COLUMN!>IN! THEN 2600 2590 GOSUD I0000 ISUDROUTINE FOR RETROGRADE INVERSION 2600 NEXT ROH! 26I0 IF COUNTER! I 1! THEN ROHI! I I! \ GO TO 2720 2620 FOR COLUMN! I I! TO N! 2630 LOH! I MODULUS! 2640 FOR ROH! I 1! TO COUNTER! - I! 2650 FOR R! I (ROH! O 1!) TO COUNTER! 2660 IF D(ROH!,COLUMN!) I D(R!,COLUMN!I THEN 2680 2670 IF DIROH!,COLUMN!I < DCR!,COLUMN!I AND DIROH!,COLUMN!) < LOH! I THEN LOH! I DIROH!.COLUMN!) I ELSE LOH! I D(R!,COLUMN!I 2680 NEXT R! 2690 NEXT ROH! 2700 703 ROH!! - I! to counrlnx 2705 It I(ROHI!,COLUMN!I - LOH! THIN 2730 27I0 NEXT ROHI! 27I5 NEXT COLUMN! 2720 men TRANSFER SET IN ruxna roan to rnxnzo MATRIX 2730 PRIME.(NSET!.0!I I NUMI.(N!) 2750 FOR COLUMN! I I! TO N! - I! 2755 PRIME.(NSET!,COLUMN!I I NUMISID‘ROHI!.COLUMN!II 2760 PC. - rcs o PIIMB.(NBIT!.COLUMN!) . SPACS.(I!) 2755 next coLunuz 2770 PRIME.(NSBT!.N!) - NUMI.(D(ROHI!,N!)) 2775 PC. I PC. 9 PRIME.‘NSET!.N!I 2975 REM RETRIEVE DATA FROM 'EORTE.DAT’ (FILEIII 2900 nesroaeox 2985 PRIME. I PC. 2990 GETSI KEYIC ES PRIME. 2991 Ian ESTABLISH MATRIX III.NAn0 2995 SET.NAM.(NSET!,0I I IND.NO. 3000 SET.NAH.(NSET!.I!I - CARD. 30Io SET.NAM.(NSET!,2!I - SN. 3013 IET.NAM.(NSET!.3!I - N. 3020 Ir :0: > 02 AND :0: < MODULUS! THEN TRAN..INV. - -I - 9 NUMI.(I0!) 3030 II :0: - nonuLus: THEN TRANS.INV. - 'I 0- 3040 I! to: > 02 AND to: < MODULUS! INEN TRANI.INV. - 'T - o NUMI.(TO!) 3030 I! to: - noouLusx THEN TRANB.INV. - '2 0' 3070 SET.NAMSINSETE.42I I TEANS.INVS 3071 VECTOR¢N83T!,0!I - VALIIND.N0.) 3072 IEn ESTABLISH MATRIX uncro- 3073 to: IE: - I! I0 I: 3074 VECTORINSET!,IC!I - vALINIncvro.IcI.IIII 3075 NEXT IC! 3000 IEn PRINT DATA :0 'ANALYSIS.OUT' (FILBOZI 3090 PEINT02.NSET!3TAD(5I:SET.NAM.(NSET!,I!I38ET.NAMS(NSET!,2EI: I TADIISISPRIME.3TAD(43ISSET.NAMS(NSET!..EI3TADC52IIVT03TADI63):I IEr.NAnocnsEII.3zIgrAII72I; 3092 FOI c0Lx-1x to u: - 12 3094 791w: 42,319<~sarz.c0131; 3090 mix: COL! ' 3100 rlxuro2 3103 PRINT 3110 rnxur 'NOULD you LIKE to FIND run 721": roan or 4 our? 11~°z 3120 xnrur ANSUERO 3130 17 4~su244 . °129° on 4~3usno - °v° tHBN 1240 3140 PRINT °90010 you Lxxl to 44x: conr4l1son2 FOR rnuvxouva 4u4valn sure? vzn-; 3130 INPUT 4usu244 3100 17 4usue99 - °130° on 4usuzno - °1° tulu 3170 ELIE 32740 3170 nun 3:014 337 conr42130us or 3333 onrnn IN 'ANALYIIB.OUT' (FILE 421 4 AND rlxut to ‘IVALUATI.OUT' (IILI 431 3173 nan BE! conr49190~s INCLUDE 01n1142177 RELATIONS¢PITCH I vucroa»: an: coansx RELATIONS, AND 1~04314uca 3100 709 4: 4 1x to user: - 1: 3190 to: 3: - 14x 4 1x1 to ”our: 3200 791u143.4x;t49¢0139:1.u4n414x.12130.9.u4n014x.2z)3:43114z1s°4°g 3205 731u143.31:t43¢2213332.44n4¢ox.121:92!.u4n4¢33.2x)3343130213° °; 3210 1! SET.NAHO(AZ,IZ) <> s:r.u4n4¢3x.1x1 1 rne~ sercon.naL - IETCOH.R£L 4 1x \ 1 coauo 11000 . .I In: COMPLEX RELATIONS IUIROUTINE 3220 1! 3:3.94n4142.2x1 <> 33:.u4n0493.2x1 1 AND 33r.~4u4¢42.1x1 . 3:3.u4n443:.1x1 2 THEN BIM.REL - IIM.REL 4 1x \ 4 oosus 12000 1 SIMILARITY RELATIONS 203-ou71u: 3225 17 ser.~4n4¢4z.2z1 - 9:2.u4n449z.2z1 AND sar.n4n4(43.1x1 - 3:3.u4n413z.121 AND 9:1.u4n414x,4x1 <> szr.u4n4(lz.4z> tneu INU.R£L - INU.RBL 4 1x \ 4 oosun 13000 1 1uv4-14uce auunourxua 3227 17 0:1.u4n414x.1x) - aar.u4n4(3z.1x) AND ser.u4n4(4z,2x> - aer.u4n4¢0z.2x1 AND 8ET.NAMO(A!.4!) I BET.NAMI(EI,4!) rueu INV.IEL - INV.REL 4 1x \ 1 conr.1uv - cour.1uv 4 1x \ 1 anur 43.343¢721;°conrnzrc INVARIANCR' 3230 NEXT 92 3240 NEXT 4x 324: 791~r 43x rnxur 43 3247 101.4c1 - sarcon.3zL 4 8!M.REL 4 INU.REL 3240 1: TOT.IEL - 02 tnsu 32740 3250 REM rn1ur out at: con9491so~ aunn4av 3260 PRINT 43.°sunn491 or 5:1 con749xsous roa- 3270 rnxur 43.°°°;:11Le4;°°°;° av °;conrosnno 3200 721»: 43\ ru1ur 43 3290 poxut 43.7431401:°nun333 DF':TAD(35):' ranceur4oe°3343(701:°raucuur4oa° 3295 731a: 43.343139);°occunneuc25°:3491351;°os THE NHOLE';TAR(70):'RY CATEGORY' 3300 PRINT 03.343439):°44444444444°;349¢331;°444444444444°334917013°44444444444° 3310 7|1~r 43 \ 73141 43 3320 rnxu: 43.°uuna£a or 5319-374314013~32123249tss1gussrx/usarx 4 1002:: 340(701gusexx/usarx 4 100: 3330 Plxuz 43 3340 PRINT 03.°0zr coannx RILATION8'32AI(40):BETCOM.RRL:TAI(53)g BETCOM.RlL/TOT.RRL 4 1002::43<701: 3343 17 RITCOM.IZL - oz tutu PRINT 03.° 0° ' :19: 721»: 43, IETCOM.RlL/IRTCDM.RIL 4 100: 3330 PRINT 03.343¢31:°n RRLATION':2AI(40):X.CDUNT;TAD(55)a K.COUNt/!OT.REL 4 100233431701: 33:: 1! IETCOM.R£L - oz ruau ra1~3 43.° 0° ELSE 731w: 43. x.counr/sarcou.321 4 1002 3300 721a: 03.149¢s>;°xc RELATION';!Al(40):KC.COUNT:TAI(55); xc.cou~3zxor.9:1 4 1002:749c70); 3365 1! szrcon.921 - ox ruzu ruxut 43.° 0° ELSE PRINT 43. xc.couu:/sarcon.llL 4 1002 3370 731a: 43.749131:°xu RELATIDN':TAIt40):KN.COUNT:TAR(53)3 un.cou~r/tor.lzL 4 10023343170); 3373 17 EETCOM.REL - ox THEN rnxut 43.° 0° ELSE 791": 43. xn.cou~r/sarcon.laL 4 100: 3300 PRINT 43.149151:°u0 22L4110~°31431401:xx.counr:743¢ss1s KX.COUN!/TOT.RIL 4 10023349470): 3303 1! EETCDM.REL - 0; tuna rnxnt 03.° 0° ELSE 72141 03. KX.COUN!/IETCOM.IEL 4 1002 3390 PRINT 03 \ 7319! 03 3400 raxnr 03,'SIMILARITY 35L411o~8°13401401:sxn.nzL3740MODULUS! THEN RIROU!,COLUMN!)IDCROU!.COLUMN!)-MODULUS! INV! I MODULUS! R(ROU!.COLUMN!) R(ROU!.COLUMN!) I INV! NEXT COLUMN! REM RETROGRADE INVERSION OPERATIONS: PART 2 I REVERSE ORDER I STORE IN MATRIX C GRADE! I 0! FOR RETRO! I N! TO I! STEP GRADE! I GRADE! 0 I! C(ROU!.GRADE!I I BCRON!,RETRO!) NEXT RETRO! FOR COLUMN! I I! TO N! R‘ROU!,COLUMN!) I C(RON!,COLUMN!) NEXT COLUMN! REM RETROGRADE INVERSION OPERATIONS: I! I R(ROU!.I!) FOR COLUMN! I 2! TO N! If R(ROU!.COLUMN!) NORM.ORDCR!,COLUMN!) THEN I3090 I3070 INVARIANT! I INVARIANT! O I! I3080 INVAR(INVARIANT!I I NORM.ORD(A!.COL!) 13090 NEXT COLUMN! 13100 NEXT COL! - 1310: PRINT 43.T43¢711: 13106 IE INVARIANT! - 02 TNEN PRINT 43.° NO INVARIANCE'\ I NO.INV - NO.INV 4 12 \ 00 To 13112 13107 FOR NO! 4 12 To INVARIANT! 13100 PR'774J,INUAR(N021; 13109 NEXT NO! 13110 INV.SUB - INV.SUR 4 12 13111 PRINT43 13112 RETURN 19000 REM ERROR TR4P 19005 It ERR - 50 AND ERL - 1700 TNEN RESUME 1700 19006 IE ERR - 50 AND ERL - 1930 THEN RESUME I840 19010 PRINT °ERROR°;ERR;°4T LINE°;ERL;ERT4¢ERR) 19030 RESUME 32740 CLOSE 41 32750 CLosE 42 32760 CLOSE 43 32763 CLOSE 44 32767 END APPENDIX C U) ['1 H) wwaMNMNNNMMNMNNN MNMMF—‘HP—‘P—lHHHHP‘HP—‘t—IF—‘HHb—ov—JH I 08- 08- 08- O3- 03- Z O (D (’_)OC)O()(') () C' ‘\-I ‘J \J \’ \I \J \J \J ‘J \J O 07 (,1 C) r) 00 U‘m\1\l"\1 OOCJOOOOOOOOOO(DOOO F’UIU'UIU‘U‘U‘UIU‘UTU‘U‘U1U1U‘U' .7 I \r & Ol COMPARISONS ° 55. 5:? ° 2 05- ° 3 03- ° 4 07- * 5 06- ° 6 03- ° 7 03- ° 8 03- ° 9 c3- * 1: 06- ° 11 03- * 12 03- ° 13 03- * 14 03- * 15 06- ° 16 07- ° 17 03- ° 15 03- ° 19 06- ° 3 03- ° 4 07— ° 5 06- ° 6 o:- 4 7 03- ° 8 03— ° 9 c3- ° 10 06- ° 11 03- ° 12 03- ° 13 03- ° 14 03- ° 1s 06- ° 16 07- ° 17 03- ° 19 03- ° 19 06— ° 4 07- ° 5 06— ° 6 03- OUTPUT FILE EVALUATE FOR FIVE PIECES FOR PIANO: THIRD MOVEMENT BY GEORGE CRUMB SET COMPLEX SIMILARITY INCLUSION NC. * RELATIONS RELATIONS RELATIONS CE * K5 3; * KS 07 * Complete Invarian O7 * K 01 * KB 01 ’ KH Cl * K5 Cl * 131 01 * No Relat1on C- ’ RH 01 * KB 01 * KB 01 * RH 07 * x C7 * Complete Invarian O; * RH 06 * No Relation O7 * K 01 * KS 07 * KB 07 * K 01 * KH Cl * KS 01 * KH Cl * RH 01 * No Relation l * RH 01 * RH 31 * KH Cl * KH OI ‘ K 07 * KH C. * KH Cd * K 07 * K C7 * KH OT * K Cl * No Invariance -101- :e 0) Im a O‘O‘O‘O‘O’IO‘OO‘O‘UIUIUIUILDUIUIUIUIUIUIUIUIUI-b-bbbbhbbbbbbbbbmwwmmwwwwwmww ~102- SET COMPLEX SIMILARIT ' INCLUSION SET NO. * S_E:1‘_ SET NO "' RELATIONS RELATIONS RELATIONS 3- 01 * 7 03- O- * No Invariance 03- Ol * 8 03- Cl ‘ No Invariance 03- Cl ’ 9 O:- 31 * O 11 03- Ol * 10 06- cl * K 03— Ol * 11 03- Ol * O 11 03- Ol * 12 03- Ol * No Invariance 03- 01 * 13 03- 01 * No Invariance CE- 01 * 14 CE- 31 * No Invariance 03- Cl * 15 06- O7 * K 03- Ol * 16 07- O7 * RH 03- Ol * 17 03- 01 * Complete Invariance CE- 01 * 19 CE- 96 * R o R 3 03- C; * :? Cé- :7 * K 07- O7 * 5 06- O7 * K 07- O7 * 6 03- Ol * KS 07- O7 * 7 03- Ol * KB 07- C7 * 8 03- Ol * KS 07- 07 * 9 03- 01 * KS 07- C7 * 10 06- Ol * No Relation 07- 07 * 11 Ol- 01 * KB 07- 07 * 12 03- O: * RH 07- O7 * 13 CE- 01 * KS 07- O7 * 14 03- 01 * RH 07- O7 * 15 06- O7 * K 07- 07 * 16 07- O7 * Complete Invariance 07- O7 * 17 03- Ol * KB 07- O7 * 18 03- O6 ’ No Relation 07— O7 * 19 06- C7 * K 06- C7 * 6 03- 31 * KB 06- C17 * 7 03- Ol * KB 06- C7 * 8 03- C; * KB 06- O7 * 9 03- 01 * KB 06- O7 * 10 06- Ol * R p R 1 06— O7 * ll 03- 1 * KB 06- O7 * 12 03- l * KB 06- O7 * 13 03- C'- * KB 06- O7 * 14 03- 01 * RH 06- O7 * 15 06- 7 * Complete Invariance 06- O7 * 16 07— O7 * KB 06- 07 ' 17 03- 01 * RH 06- O7 * 18 03- O6 * No Relation 06- O7 * 19 06- O7 * 1 7 03- 01 * 7 03- Ol * No Invariance 03- 01 * 8 03- 01 * No Invariance 03- 01 * 9 03— 01 * 10 03- 01 * 13 06- 31 * K 03- 01 * 11 O3- 31 * 10 3— 01 * 12 03- 01 * No Invariance 03- ‘1 * 13 Cl— 31 * No Invariance C3— 01 * 14 03- 31 * Complete Invariance 03- 01 * 15 06- O7 * K (D a \O\O\O\O\D\OO®\O\Ommmmmmmmmmm\l\1\l\1\l\lxl~J\J\I\J\JO\O\O\O‘ -103- SET COMPLEX SIMILARITY INCLUSIDN SET N3. * S_E'_.'_ SET 21’). * RELATIONS RELATIONS RELATIONS 03- 01 * 16 07- O7 * KH 03- 01 * 17 03- 01 * No Invariance 03- Cl * 18 03- O6 ‘ R p R 3 CE- 01 * 19 06- C7 * K 33- 01 * 8 03- 01 * No Invariance 03- 01 * 9 03- 01 * No Invariance 03- 01 * 10 06- 01 * K 03- 01 * 11 03- 1 * No Invariance 03- 01 * 12 03- O- * No Invariance 03- 01 ‘ 13 03- ul * Complete Invariance 03- 01 ‘ 14 03- 01 * No Invariance 03- 01 * 15 06- O7 * K 03- Cl * 1 C7— Oz * RH 03- C- * 17 CE- 2. * No Invariance 03- 31 * 18 03- 06 * R p R 3 03- 01 * 19 06- :7 * K 03- 01 * 9 03- 01 * No Invariance 03- 01 * 1O 06- 01 * K 03- 01 * 11 03- 01 * No Invariance 03- 01 * 12 3- 1 * Complete Invariance 03- 01 * 13 03- 1 * No Invariance 03- 01 * 14 03- 01 * No Invariance 03- 01 * 15 06- O7 * K 03- 01 * 16 07- C7 * RH 03- 01 * 17 03- C1 * No Invariance 03- 01 * 18 03- O6 * R p R 3 03- 01 * 19 06- O7 * K 03- 01 * 10 06— 01 * K 93- 31 * 11 03— C- * Complete Invariance 03- 01 * 1 03- 31 * No Invariance 03- 01 * 13 03- 01 * No Invariance 03- 01 * 14 03- 01 * 10 03- 01 * 15 06- O7 * K 03- 01 * 16 07- O7 * RH 03- 01 * 17 03- 01 * O 11 03- 01 * 18 03- 06 * R p R 3 03- 01 * 19 06— C7 * K 06- 01 * ll 03— 01 * RH 06- 01 * 12 03- 01 * RH 06- 01 * 13 03- 01 * KH 06- 01 * 14 03- 01 * RH 06- 01 * 15 06- O7 * R p R 1 06- 01 * 16 07- O7 * No Relation 06- 01 * 17 03- 01 * RH 06- 01 * 18 03- O6 * KB 06- 01 * 19 06- O7 * R p R 1 03- 01 * 12 03- 01 * No Invariance 03- 01 * 13 03- 01 * No Invariance 03- 01 * 14 03- 31 * C 03- 01 * 15 06- O7 * K 03- 01 * 16 07- O7 * KR U) Hr—I [11 Hb—J lba H ... ..I \J I PHD-4H MNNN ...! ...n—a )WM I—Jl—‘F‘y-‘v- (”Inbounb HHHH bbbh -104— SET COMPLEX SIMILARITY INCLUSI ; SET NO. * §§:_ SET NO * RELATIONS RELATIONS RELATIONS 03— 1 * 17 C3- 01 * O 11 03- Cl * 18 03- 06 * R p R 3 03— 01 ‘ 19 06- C7 * K CE- :1 ‘ 13 C: 31 ‘ No Invariance 03- 01 * 14 03- O1 ‘ No Invariance 03- 01 * 15 06- O7 * K 03- 01 * 16 07- O7 * KB 03- O1 * 17 CE- 01 * No Invariance 03- 01 * 18 C3- 06 * R p R 3 03- 01 * 19 06— O7 * K 03- 01 * 14 03- 01 * No Invariance *3- C1 * 15 06- C7 * K 03- 01 * 16 C’- 87 * KR 03- 01 * 17 O’— C1 * No Invarianc; 3- C1 * 18 CE- 06 * R p R 3 03— Cl * 19 06- O7 * K 03- Cl * 15 06- O7 * K 03- 01 * 16 07- O7 * KB 03- Cl * 17 03- Cl ‘ No Invariance 03- 1 * 18 03- OS * R p R 3 C3— 01 * 19 06— C7 * K 06- O7 * 16 07- CS * KB 06- O7 * 17 3- O1 * K8 06- O7 * 18 03- O6 * No Relation 06- O7 * 19 06- O7 * l 7 07- O7 * 17 '3- 01 * KB 07— O7 * 18 03- O * No Relation 07- 07 ‘ 19 76~ C7 * F C3— 01 * 18 C3- :6 ‘ R p R 3 C8- 31 * 19 Qc- C7 * K 03- O6 * 19 06- C7 * No Relation ~105- SUMMARY OF SET COMPARISONS FOR FIVE PIECES FOR PIANO: THIRD MOVEMENT BY GEORGE CRUMB Number of Percentage Percentage Occurrences of the Whole by Category Number of Sets 19 100 100 Set Complex Relations 107 62.5731 100 K Relation 36 21.0526 33.6449 Kc Relation 0 O 0 Kb Relation 61 35.6725 57.0093 No Relation 10 5.84795 9.34579 Similarity Relations 13 7.60234 100 Rp Relation 13 7.60234 100 R 0 Relation 0 0 0 R 1 Relation 3 1.75439 23.0769 R 2 Relation 0 0 0 R 3 Relation 10 5.84795 76.9231 R 4 Relation 0 0 0 Z-Related Pair 0 0 0 Inclusion Relation 51 29.8246 100 Complete Invariance 9 5.26316 17.6471 No Invariance 32 18.7135 62.7451 Subset Invariance 10 5.84795 19.6078 APPENDIX D A USER'S GUIDE TO BOETHIUS The following steps are intended to assist the user in performing an effective analysis of a musical composition with the assistance of BOETHIUS. STEP 1: Segmentation of the Composition The user should segment the piece into sets of pitch- class integers based (n1 segmentation criteria relevant to the composition, i.e., rhythmic organization, notation, timbre, registration, articulation and dynamics. IEach set should be numbered consecutively. The consecutive numbering will correspond with SET NO. in ANALYSIS. The user may wish to assign each pitch its pitch-class integer. STEP 2: User Interface The user begins execution of the program by entering the command RUN BOETHIUS. A series of instructions are dis- played on the screen. First you will be asked to enter the cardinality of the set. The cardinality of the set should not be less than two or greater than ten. You will be prompted each time you are to enter a pitch-class integer. Use the following integer notation: -lO6- -lO7- * C=O * * C#,Db = 1 * * D=2 * * D#,Eb = 3 * * E=4 * * F=5 * * F#,Gb = 6 * * G=7 * * G#,Ab = 8 * * A=9 * * A#,Bb = 10 * * B=11 * DQ_NOT DUPLICATE ANY PITCH-CLASS INTEGER The user is presented the question "Would you like to find the prime form of a set? y/n?" If the user responds affirmatively, (s)he is asked to enter the cardinality of the set. BOETHIUS prompts the user for entry of each pitch- class integer. If the user makes an error, set entry can be reinitiated by entering a —1. BOETHIUS reduces each set to prime form, calculates the transposition or inversion Operator, names the set— complement and vector, calculates the B.I.S. pattern, and writes the data to ANALYSIS. Execution of the aforemen- tioned process is controlled by an affirmative response to the question, "Would you like to find the prime form of a set? y/n?" Should the user respond negatively, BOETHIUS presents the question "WOuld you like to make comparisons for previously analyzed set? y/n?" If the user responds affirmatively, BOETHIUS quantifies the relationship between all previously analyzed sets, writes the data to EVALUATE, and terminates the program. A negative response terminates the program. -108- STEP 3: Examination of Output Files ANALYSIS and EVALUATE The user should critically examine the data presented in ANALYSIS and EVALUATE in order to assess the compositional process. GLOSSARY GLOSSARY OF TECHNICAL TERMS ALGORITHM. A series of steps which are utilized to complete a particular task. ANALYSIS. The computer-generated output file that lists the set in prime form, the transposition or inversion opera- tor, the vector, the set-complement name and the B.I.S. pattern. Inn the case of Z-related hexachords, the Z- related pair is listed as the set-complement. Hexa- chords without a Z-related pair are not accompanied by a set-complement. ARRAY. A collection of storage cells with the same variable name. Arrays may contain numerical or string (alpha- numeric) data. BASIC-PLUS 2. A dialect of the programming language BASIC developed by the Digital Equipment Corporation. BOETHIUS. The program name assigned by the author for the series of instructions to the computer that execute set theory analysis. CATHODE RAY TUBE (C.R.T.). A video screen display. CENTRAL PROCESSING UNIT (C.P.U.). A component of a computer system that executes program statements. CONCATENATION. The process of linking string (alpha- numeric) data. DOMAIN. The entirety of data which may appear in a field. ENVELOPE. The attack, decay, sustain and release time of a sound. EVALUATE. The computer-generated output file that lists set comparisons for all sets listed in ANALYSIS. Set com- parisons include similarity relations, set complex relations and inclusion relations of sets of the same cardinality and set name. All set relationships are counted and the data is represented in a tabular sta- tistical analysis. FIELD. A vertical collection of data. -lO9- -llO- FLOWCHART. A symbolic representation of the functions that a computer performs. Example A: Flowchart with Corresponding Program Code T———— 10 REM PROGRAM PURPOSE @ PaoenAM 20 NSET = NSET + 1 ————— -1 puaposg 3O INPUT ANSWERs 40 IF ANSWERS -"'- "Y" THEN 20 NSET=N$ET+1, L_____. 50 END 1 z//