Exploring the space of perfectly balanced rhythms and scales

Journal of Mathematics and Music

Volumn 11, Issue 2-3, 2017, Pages 101-133

  Milne, Andrew J ^a   Bulger, David ^b   Herff, Steffen A ^a  

^a WESTERN SYDNEY UNIVERSITY   (Australia)

^b MACQUARIE UNIVERSITY   (Australia)

Author keywords

consonance and dissonance; evenness; perfect balance; rhythm and meter; scales

Indexed keywords

EID: 85049933097     PISSN: 17459737     EISSN: 17459745     Source Type: Journal    
DOI: 10.1080/17459737.2017.1395915     Document Type: Article

References (50)

1. Amiot, Emmanuel., 2007. “David Lewin and Maximally Even Sets.” Journal of Mathematics and Music 1 (3): 157–172. doi: 10.1080/17459730701654990
2. Amiot, Emmanuel., 2009. “Discrete Fourier Transform and Bach's Good Temperament.” Music Theory Online 15 (2).
3. Amiot, Emmanuel., 2010. “Sommes Nulles de Racines de l'Unité.” Bulletin de l'Union des Professeurs de Spéciales 230: 30–34.
4. Amiot, Emmanuel., 2016. Music Through Fourier Space: Discrete Fourier Transform in Music Theory. In the series Computational Music Science. Cham, Switzerland: Springer International. doi: 10.1007/978-3-319-45581-5.
5. Amiot, Emmanuel., 2017a. “Decompositions of Nil Sums of Roots of Unity. An Adaptation of ‘Sommes Nulles de Racines de l'Unité’.” Journal of Mathematics and Music 11 (2). https://doi.org/10.1080/17459737.2018.1453951.
6. Amiot, Emmanuel., 2017b. “The Discrete Fourier Transform of a Distribution.” Journal of Mathematics and Music 11 (2). https://doi.org/10.1080/17459737.2018.1453952
7. Amiot, Emmanuel, and William A., Sethares. 2011. “An Algebra for Periodic Rhythms and Scales.” Journal of Mathematics and Music 5 (3): 149–169. doi: 10.1080/17459737.2011.640469
8. Arom, Simha., 1991. African Polyphony and Polyrhythm: Musical Structure and Methodology. Cambridge, UK: Cambridge University Press.
9. Bruns, Winfried, Bogdan, Ichim, Tim, Römer, Richard, Sieg, and Christof, Söger. 2016. “Normaliz. Algorithms for rational cones and affine monoids.” Accessed 30 October 2017. https://www.normaliz.uni-osnabrueck.de.
10. Callender, Clifton., 2007. “Continuous Harmonic Spaces.” Journal of Music Theory 51 (2): 277–332. doi: 10.1215/00222909-2009-004
11. Carey, Norman, and David, Clampitt. 1989. “Aspects of Well-Formed Scales.” Music Theory Spectrum 11 (2): 187–206. doi: 10.2307/745935
12. Clampitt, David., 2007. “Mathematical and Musical Properties of Pairwise Well-Formed Scales.” In Proceedings of the First Conference on Mathematics and Computation in Music (MCM), Berlin, 18–20 May 2007, edited by Timour Klouche and Thomas Noll, 464–468. Vol. 37 of the series Communications in Computer and Information Science. Berlin: Springer-Verlag.
13. Clough, John, and Jack, Douthett. 1991. “Maximally Even Sets.” Journal of Music Theory 35 (1/2): 93–173. doi: 10.2307/843811
14. Cohn, Richard., 1991. “Properties and Generability of Transpositionally Invariant Sets.” Journal of Music Theory 35 (1/2): 1–32. doi: 10.2307/843808
15. Cycling '74. 2017. “Max Software Tools for Media.” Accessed 23 May 2017. https://cycling74.com/products/max/.
16. Davies, Matthew, Guy, Madison, Pedro, Silva, and Fabien, Gouyon. 2013. “The Effect of Microtiming Deviations on the Perception of Groove in Short Rhythms.” Music Perception 30 (5): 498–511. doi: 10.1525/mp.2013.30.5.497
17. Domínguez, Manuel, David, Clampitt, and Thomas, Noll. 2009. “WF Scales, ME Sets, and Christoffel Words.” In Proceedings of the First Conference on Mathematics and Computation in Music (MCM), Berlin, 18–20 May 2007, edited by Timour Klouche and Thomas Noll, 477–488. Vol. 37 of the series Communications in Computer and Information Science. Berlin: Springer-Verlag.
18. Erlich, Paul., 2006. “A Middle Path Between Just Intonation and the Equal Temperaments, Part 1.” Xenharmonikôn 18: 159–199.
19. Fisher, Nicholas I., 1993. Statistical Analysis of Circular Data. Cambridge, UK: Cambridge University Press.
20. Frühauf, Jan, Reinhard, Kopiez, and Friedrich, Platz. 2013. “Music on the Timing Grid: The Influence of Microtiming on the Perceived Groove Quality of a Simple Drum Pattern Performance.” Musicae Scientiae 17 (2): 246–260. doi: 10.1177/1029864913486793
21. Hinrichsen, Haye., 2012. “Entropy-Based Tuning of Musical Instruments.” Revista Brasileira de Ensino de Física 34 (2): 2301-1–2301-8.
22. Hofstadter, Douglas R., 1985. Metamagical Themas: Questing for the Essence of Mind and Pattern. New York: Basic Books.
23. Kassel, Christian, and Vladimir, Turaev. 2008. Braid Groups. New York: Springer.
24. Lam, Tsit Y., and Ka H., Leung. 2000. “On Vanishing Sums of Roots of Unity.” Journal of Algebra 224 (1): 91–109. doi: 10.1006/jabr.1999.8089
25. Lewin, David., 1959. “Re: Intervallic Relations between Two Collections of Notes.” Journal of Music Theory 3 (2): 298–301. doi: 10.2307/842856
26. Lewin, David., 1987. Generalized Musical Intervals and Transformations. New Haven, CT: Yale University Press.
27. London, Justin., 2004. Hearing in Time: Psychological Aspects of Musical Meter. Oxford, UK: Oxford University Press.
28. Mardia, Kanti V., 1972. Statistics of Directional Data. London, UK: Academic Press.
29. Messiaen, Olivier., 1944. Technique de mon Langage Musical. Paris: Leduc.
30. Milne, Andrew J., 2013. “A Computational Model of the Cognition of Tonality.” Ph.D. thesis, The Open University, Milton Keynes, UK.
31. Milne, Andrew J., David, Bulger, Steffen A., Herff, and William A., Sethares. 2015. “Perfect Balance: A Novel Principle for the Construction of Musical Scales and Meters.” In Proceedings of the 5th International Conference on Mathematics and Computation in Music (MCM), London, UK, 22–25 June 2015, edited by Tom Collins, David Meredith, and Anja Volk, 97–108. Vol. 9110 of the series Lecture Notes in Artificial Intelligence. Cham, Switzerland: Springer International Publishing.
32. Milne, Andrew J., and Roger T., Dean. 2016. “Computational Creation and Morphing of Multilevel Rhythms by Control of Evenness.” Computer Music Journal 40 (1): 35–53. doi: 10.1162/COMJ_a_00343
33. Milne, Andrew J., Steffen A., Herff, David, Bulger, William A., Sethares, and Roger T., Dean. 2016. “XronoMorph: Algorithmic Generation of Perfectly Balanced and Well-Formed Rhythms.” In Proceedings of the 16th International Conference on New Interfaces for Musical Expression (NIME 2016), Brisbane, Australia, 11–15 July 2016. Brisbane: Queensland Conservatorium Griffith University. http://www.nime.org/proceedings/2016/nime2016_paper0077.pdf.
34. Milne, Andrew J., and Simon, Holland. 2016. “Empirically Testing Voice-Leading, Tonnetz, and Spectral Models of Perceived Triadic Distance.” Journal of Mathematics and Music 10 (1): 59–85. doi: 10.1080/17459737.2016.1152517
35. Milne, Andrew J., Robin, Laney, and David B., Sharp. 2015. “A Spectral Pitch Class Model of the Probe Tone Data and Scalic Tonality.” Music Perception 32 (4): 364–393. doi: 10.1525/mp.2015.32.4.364
36. Milne, Andrew J., Robin, Laney, and David B., Sharp. 2016. “Testing a Spectral Model of Tonal Affinity with Microtonal Melodies and Inharmonic Spectra.” Musicae Scientiae 20 (4): 465–494. doi: 10.1177/1029864915622682
37. Milne, Andrew J., William A., Sethares, Robin, Laney, and David B., Sharp. 2011. “Modelling the Similarity of Pitch Collections with Expectation Tensors.” Journal of Mathematics and Music 5 (1): 1–20. doi: 10.1080/17459737.2011.573678
38. Milne, Andrew J., William A., Sethares, and James, Plamondon. 2008. “Tuning Continua and Keyboard Layouts.” Journal of Mathematics and Music 2 (1): 1–19. doi: 10.1080/17459730701828677
39. Quinn, Ian., 2004. “A Unified Theory of Chord Quality in Equal Temperaments.” Ph.D. thesis, University of Rochester.
40. Sethares, William A., 1993. “Local Consonance and the Relationship Between Timbre and Scale.” The Journal of the Acoustical Society of America 94 (3): 1218–1228. doi: 10.1121/1.408175
41. Sethares, William A., 2005. Tuning, Timbre, Spectrum, Scale. 2nd ed. London, UK: Springer-Verlag.
42. Sethares, William A., Andrew J., Milne, Stefan, Tiedje, Anthony, Prechtl, and James, Plamondon. 2009. “Spectral Tools for Dynamic Tonality and Audio Morphing.” Computer Music Journal 33 (2): 71–84. doi: 10.1162/comj.2009.33.2.71
43. Shannon, Claude E., 1948. “A Mathematical Theory of Communication.” The Bell System Technical Journal 27 (3): 379–423. doi: 10.1002/j.1538-7305.1948.tb01338.x
44. Sloane, Neil J. A., 2016. “The On-Line Encyclopedia of Integer Sequences.” Accessed 30 October 2017. https://oeis.org.
45. Steinberger, John P., 2008. “Minimal Vanishing Sums of Roots of Unity with Large Coefficients.” Proceedings of the London Mathematical Society 97 (3): 689–717. doi: 10.1112/plms/pdn006
46. Toussaint, Godfried T., 2013. The Geometry of Musical Rhythm: What Makes a “Good” Rhythm Good? Baton Rouge, FL: CRC Press.
47. Tu, Loring W., 2010. An Introduction to Manifolds. New York: Springer-Verlag.
48. Tymoczko, Dmitri., 2006. “The Geometry of Musical Chords.” Science 313 (5783): 72–74. doi: 10.1126/science.1126287
49. Wilson, Erv., 1975. “Letter to Chalmers Pertaining to Moments-of-Symmetry/Tanabe Cycle.” Accessed 31 October 2017. http://www.anaphoria.com/mos.pdf.
50. Zabinsky, Zelda B., Robert L., Smith, J. Fred, McDonald, H. Edwin, Romeijn, and David E., Kaufman. 1993. “Improving Hit-and-Run for Global Optimization.” Journal of Global Optimization 3 (2): 171–192. doi: 10.1007/BF01096737