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Axiomatic scale theory. (English) Zbl 1466.00016

Summary: Scales are a fundamental concept of musical practice around the world. They commonly exhibit symmetry properties that are formally studied using cyclic groups in the field of mathematical scale theory. This paper proposes an axiomatic framework for mathematical scale theory, embeds previous research, and presents the theory of maximally even scales and well-formed scales in a uniform and compact manner. All theorems and lemmata are completely proven in a modern and consistent notation. In particular, new simplified proofs of existing theorems such as the equivalence of non-degenerate well-formedness and Myhill’s property are presented. This model of musical scales explicitly formalizes and utilizes the cyclic order relation of pitch classes.

MSC:

00A65 Mathematics and music
08C10 Axiomatic model classes
06F99 Ordered structures
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[1] Amiot, Emmanuel. 2007. “David Lewin and Maximally Even Sets.” Journal of Mathematics and Music 1 (3): 157-172. https://doi.org/10.1080/17459730701654990. · Zbl 1163.42311
[2] Balzano, G. J. 1980. “The Group-Theoretic Description of 12-Fold and Microtonal Pitch Systems.” Computer Music Journal 4 (4): 66-84. doi: 10.2307/3679467 · doi:10.2307/3679467
[3] Brlek, Srečko, Marc Chemillier, and Christophe Reutenauer. 2018. “Music and Combinatorics on Words: A Historical Survey.” Journal of Mathematics and Music 12 (3): 125-133. doi: 10.1080/17459737.2018.1542055 · Zbl 1427.00031
[4] Burns, Edward M., and W. Dixon Ward. 1978. “Categorical Perception-phenomenon or Epiphenomenon: Evidence from Experiments in the Perception of Melodic Musical Intervals.” The Journal of the Acoustical Society of America 63 (2): 456-468. http://asa.scitation.org/doi/10.1121/1.381737. · doi:10.1121/1.381737
[5] Carey, Norman. 1998. “Distribution Modulo 1 and Musical Scales.” Ph.D. thesis. http://www.researchgate.net/profile/Norman_Carey/publication/235712212_Distribution_modulo_1_and_musical_scales/links/004635153159b6e710000000.pdf.
[6] Carey, Norman, and David Clampitt. 1989. “Aspects of Well-Formed Scales.” Music Theory Spectrum 11 (2): 187-206. doi: 10.2307/745935 · doi:10.2307/745935
[7] Carey, N., and D. Clampitt. 1996. “Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues.” Perspectives of New Music 34 (2): 62-87.
[8] Castrillon Lopez, Marco, and Manuel Domìnguez Romero. 2016. “LR Property of Non-Well-Formed Scales.” Journal of Mathematics and Music 10 (1): 18-35. doi: 10.1080/17459737.2016.1164907 · Zbl 1346.00045
[9] Clampitt, David. 2009. “Mathematical and Musical Properties of Pairwise Well-Formed Scales.” Communications in Computer and Information Science CCIS 37: 464-468. · Zbl 1202.00044 · doi:10.1007/978-3-642-04579-0_46
[10] Clampitt, David, Manuel Domìnguez, and Thomas Noll. 2007. “Well-Formed Scales, Maximally Even Sets and Christoffel Words.” Proceedings of the MCM. · Zbl 1202.00046
[11] Clough, John, and Jack Douthett. 1991. “Maximally Even Sets.” Journal of Music Theory 35 (1/2): 93-173. doi: 10.2307/843811 · doi:10.2307/843811
[12] Clough, John, Jack Douthett, and Lewis Rowell N. Ramanathan. 1993. “Early Indian Heptatonic Scales and Recent Diatonic Theory.” Music Theory Spectrum 15 (1): 36-58. doi: 10.2307/745908 · doi:10.2307/745908
[13] Clough, John, Nora Engebretsen, and Jonathan Kochavi. 1999. “Scales, Sets and Interval Cycles: A Taxonomy.” Music Theory Spectrum 21 (1): 74-104. https://doi.org/10.2307/745921 · doi:10.2307/745921
[14] Clough, John, and Gerald Myerson. 1985. “Variety and Multiplicity in Diatonic Systems.” Journal of Music Theory 29 (2): 249-270. doi: 10.2307/843615 · doi:10.2307/843615
[15] Clough, John, and Gerald Myerson. 1986. “Musical Scales and the Generalized Circle of Fifths.” The American Mathematical Monthly 93 (9): 695-701. doi: 10.1080/00029890.1986.11971924 · Zbl 0659.00023
[16] Cohn, Richard. 1996. “Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions.” Music Analysis 15 (1): 9-40. doi: 10.2307/854168 · doi:10.2307/854168
[17] Cohn, Richard. 1997. “Neo-Riemannian Operations, Parsimonious Trichords, and Their Tonnetz Representations.” Journal of Music Theory 41 (1): 1-66. doi: 10.2307/843761 · doi:10.2307/843761
[18] Conklin, Darrell, and Ian H. Witten. 1995. “Multiple Viewpoint Systems for Music Prediction.” Journal of New Music Research 24 (1): 51-73. https://doi.org/10.1080/09298219508570672.
[19] Domínguez, Manuel, David Clampitt, and Thomas Noll. 2009. “WF Scales, ME Sets, and Christoffel Words.” Communications in Computer and Information Science 37 CCIS: 477-488. · Zbl 1202.00046 · doi:10.1007/978-3-642-04579-0_48
[20] Douthett, Jack. 2008. “Filtered Point-Symmetry and Dynamical Voice-Leading.” In: Music Theory and Mathematics: Chords, Collections, and Transformations, edited by Jack Douthett, Martha M. Hyde, and Charles J. Smith, 72-106.Rochester: University Rochester Press.
[21] Douthett, Jack, and Richard Krantz. 2007. “Maximally Even Sets and Configurations: Common Threads in Mathematics, Physics, and Music.” Journal of Combinatorial Optimization 14 (4): 385-410. doi: 10.1007/s10878-006-9041-5 · Zbl 1149.90163 · doi:10.1007/s10878-006-9041-5
[22] Galil, Zvi, and Nimrod Megiddo. 1977. “Cyclic Ordering Is NP-Complete.” Theoretical Computer Science 5 (2): 179-182. doi: 10.1016/0304-3975(77)90005-6 · Zbl 0383.68045 · doi:10.1016/0304-3975(77)90005-6
[23] Harasim, Daniel, Thomas Noll, and Martin Rohrmeier. 2019. “Distant Neighbors and Interscalar Contiguities.” In Mathematics and Computation in Music, Proceedings of the 7th International Conference (MCM 2019), Madrid, Spain, June 18-21, 2019 Vol. 11502 of Lecture Notes in Computer Science, edited by Mariana Montiel, Francisco Gomez-Martin, and Octavio A. Agustín-Aquino, 172-184. Cham: Springer International. · Zbl 1456.00072 · doi:10.1007/978-3-030-21392-3_14
[24] Harasim, Daniel, Martin Rohrmeier, and Timothy J. O’Donnell. 2018. “A Generalized Parsing Framework for Generative Models of Harmonic Syntax.” 19th International Society for Music Information Retrieval Conference.
[25] Harasim, Daniel, Stefan E. Schmidt, and Martin Rohrmeier. 2016. “Bridging Scale Theory and Geometrical Approaches to Harmony: The Voice-leading Duality between Complementary Chords.” Journal of Mathematics and Music 10 (3): 193-209. https://doi.org/10.1080/17459737.2016.1216186. · Zbl 1354.00057
[26] Huntington, E. V. 1916. “A Set of Independent Postulates for Cyclic Order.” Proceedings of the National Academy of Sciences 2 (11): 630-631. http://www.pnas.org/cgi/doi/10.1073/pnas.2.11.630. · JFM 46.0308.06 · doi:10.1073/pnas.2.11.630
[27] Levine, Mark. 1995. The Jazz Theory Book. Petaluma, CA: Sher Music.
[28] Lewin, David. 1987. Generalized Musical lntervals and Transformations. New Haven: Yale University Press.
[29] Megiddo, Nimrod. 1976. “Partial and Complete Cyclic Orders.” Bulletin of the American Mathematical Society 82 (2): 274-277. doi: 10.1090/S0002-9904-1976-14020-7 · Zbl 0361.06001 · doi:10.1090/S0002-9904-1976-14020-7
[30] Miller, G. A. 1956. “The Magical Number Seven, Plus or Minus Two: Some Limits on Processing Information.” The Psychological Review 63 (2): 81-97. doi: 10.1037/h0043158 · doi:10.1037/h0043158
[31] Moss, Fabian C., Markus Neuwirth, Daniel Harasim, and Martin Rohrmeier. 2019. “Statistical Characteristics of Tonal Harmony: A Corpus Study of Beethoven”s String Quartets.” PLOS ONE 14 (6): e0217242. http://dx.plos.org/10.1371/journal.pone.0217242. · doi:10.1371/journal.pone.0217242
[32] Neumaier, Wilfried, and Rudolf Wille. 1990. “Extensionale Standardsprache der Musiktheorie: eine Schnittstelle zwischen Musik und Informatik”.
[33] Noll, Thomas. 2007. “Musical Intervals and Special Linear Transformations.” Journal of Mathematics and Music 1 (2): 121-137. https://doi.org/10.1080/17459730701375026. · Zbl 1181.00022
[34] Noll, Thomas. 2015. “Triads as Modes within Scales as Modes.” In: Collins T., Meredith D., Volk A. (eds) Mathematics and Computation in Music. 5th International Conference (MCM 2015), London, 22-25 June 2015, vol. 9110 of Lecture Notes in Computer Science, edited by Tom Collins, David Meredith, and Anya Volk, 373-384. Cham: Springer. · Zbl 1321.00085 · doi:10.1007/978-3-319-20603-5_37
[35] Patel, Aniruddh D. 2008. Music, Language, and the Brain. New York: Oxford University Press.
[36] Pressing, Jeff. 1983. “Cognitive Isomorphisms Between Pitch and Rhythm in World Musics: West Africa, the Balkans and Western Tonality.” Studies in Music 17, 38-61.
[37] Quinn, Ian. 2004. “A Unified Theory of Chord Quality in Equal Temperaments.” Ph.D. Thesis, Eastman School of Music.
[38] Rohrmeier, Martin. 2011. “Towards a Generative Syntax of Tonal Harmony.” Journal of Mathematics and Music 5 (1): 35-53. https://doi.org/10.1080/17459737.2011.573676.
[39] Savage, Patrick E., Steven Brown, Emi Sakai, and Thomas E. Currie. 2015. “Statistical Universals Reveal the Structures and Functions of Human Music.” Proceedings of the National Academy of Sciences 112 (29): 8987-8992. doi: 10.1073/pnas.1414495112 · doi:10.1073/pnas.1414495112
[40] Sears, David R. W., Marcus T. Pearce, William E. Caplin, and Stephen McAdams. 2018. “Simulating Melodic and Harmonic Expectations for Tonal Cadences Using Probabilistic Models.” Journal of New Music Research 47 (1): 29-52. https://doi.org/10.1080/09298215.2017.1367010.
[41] Trehub, Sandra E., E. Glenn Schellenberg, and Stuart B. Kamenetsky. 1999. “Infants” and Adult’ Perception of Scale Structure.” Journal of Experimental Psychology 25 (4): 965-975.
[42] Tymoczko, Dmitri. 2006. “The Geometry of Musical Chords.” Science 313 (5783): 72-74. doi: 10.1126/science.1126287 · Zbl 1226.00026 · doi:10.1126/science.1126287
[43] Tymoczko, Dmitri. 2008. “Scale Theory, Serial Theory and Voice Leading.” Music Analysis 27 (1): 1-49. doi: 10.1111/j.1468-2249.2008.00257.x · doi:10.1111/j.1468-2249.2008.00257.x
[44] Wooldridge, Marc Charles. 1992. “Rhythmic Implications of Diatonic Theory: A Study of Scott Joplin”s Ragtime Piano Works”. Ph. D. diss., State University of New York at Buffalo.
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