Synchronisation and rhythm

Especially for generative music


In the deserts of Sudan
And the gardens of Japan
From Milan to Yucatan
Every woman, every man

Hit me with your rhythm stick
Hit me, hit me
Je t'adore, ich liebe dich
Hit me, hit me, hit me
Hit me with your rhythm stick
Hit me slowly, hit me quick
Hit me, hit me, hit me

— Ian Dury & The Blockheads - Hit Me With Your Rhythm Stick

Note on rhythm, the understanding and making, amnd detecting of it.

To mention/taxonomise/disambiguate:

  • correlograms
  • “vector strength” and event-based doohickeys versus continuous-signal systems
  • Kuramoto oscillators
  • Circle map
  • phase locked loops
  • “Entrainment” (Is that phase locking from the analysis rather than design perspective?)
  • The attraction of Pythagorean rhythms
  • machine listening for it
  • Lucky People Center International
  • Autocorrelation structures and simulating point processes from them

Making rhythms

  • recurrent neural networks

  • hidden markov models

  • sync oscillator system

    • continuous Kuramoto, possibly on a graph with an interesting topology
    • or Strogatz/Perkel/Haken-style pulse-sync, possibly also on a graph
    • (Pikovsky and Rosenblum 2007)
    • Pulse coupled oscillators (Canavier and Achuthan 2007)
    • Deville’s Brownian calculus treatment suggests N!=4 and nearest neighbour networks
    • left-field idea: work using decaying harmonics through feedback - macroscopic karplus-strong.
    • any of these would be interesting with driving noise and arbitrary topologies. See also MIMO allpass.

Neurological/Psychological basis

Doelling:

“We’ve isolated the rhythms in the brain that match rhythms in music,” explains Keith Doelling, an NYU Ph.D. student and the study’s lead author. “Specifically, our findings show that the presence of these rhythms enhances our perception of music and of pitch changes.” […] Brain rhythms, they add, therefore appear to play a role in parsing and grouping sound streams into ‘chunks’ that are then analyzed as speech or music.

Or how about a dynamical systems approach? Famously, (Haken, Kelso, and Bunz 1985) does this: I found this paper fascinating, although the fact they spun several books out of it afterwards seemed to me to be gilding the lily.

To read

Breakbeat cuts

Slicing up your percussion line into mad junglist syncopations is a whole world of its own. Asides from selling a lot of vinyl, it has attracted significant academic interest.

Think of group theory angle, like a Rubik’s cube. Is it a pure group theoretic problem? Or are there additional constraints on a breakbeat cut such that it is still considered rhythmic?

Periodicity Analysis

I’m coming at this from a musical angle; The correlation at different scales can be weird and wonderful and I wonder if some algorithm from the world of Nonlinear Time Series Wizardry could help.

This overlaps with a lot of things, but my core question is best summarised:

Can I use machine learning to identify what about breakbeat cuts makes them rhythmically interesting? What range of repetition between infinite sameness and total chaos is musically attractive?

More generally, identifying cycles and periodicity is itself interesting, so I’ll collect some notes to that purely abstract end here too.

Absil, P.-A, R Mahony, and R Sepulchre. 2008. Optimization Algorithms on Matrix Manifolds. Princeton, N.J.; Woodstock: Princeton University Press.

Acebrón, Juan A., L. L. Bonilla, Conrad J. Pérez Vicente, Félix Ritort, and Renato Spigler. 2005. “The Kuramoto Model: A Simple Paradigm for Synchronization Phenomena.” Reviews of Modern Physics 77 (1): 137–85. https://doi.org/10.1103/RevModPhys.77.137.

Adamo, Mike. 2010. The Breakbeat Bible: The Fundamentals of Breakbeat Drumming. Pap/Com edition. S.l.: Hudson Music.

Anderson, Christopher, and Arne Eigenfeldt. 2011. “A New Analytical Method for the Musical Study of Electronica.” In Proceedings of the Electroacoustic Music Studies Conference, Sforzando.

Boumal, Nicolas. 2013. “On Intrinsic Cramér-Rao Bounds for Riemannian Submanifolds and Quotient Manifolds.” IEEE Transactions on Signal Processing 61 (7): 1809–21. https://doi.org/10.1109/TSP.2013.2242068.

Boumal, Nicolas, Amit Singer, and P.-A Absil. 2013. “Robust Estimation of Rotations from Relative Measurements by Maximum Likelihood.” In 52nd IEEE Conference on Decision and Control, 1156–61. Firenze: IEEE. https://doi.org/10.1109/CDC.2013.6760038.

Boumal, Nicolas, Amit Singer, P.-A. Absil, and Vincent D. Blondel. 2014. “Cramér-Rao Bounds for Synchronization of Rotations.” Information and Inference 3 (1): 1–39. https://doi.org/10.1093/imaiai/iat006.

Buhusi, Catalin V., and Warren H. Meck. 2005. “What Makes Us Tick? Functional and Neural Mechanisms of Interval Timing.” Nature Reviews Neuroscience 6 (10): 755–65. https://doi.org/10.1038/nrn1764.

Canavier, Carmen, and Srisairam Achuthan. 2007. “Pulse Coupled Oscillators.” Scholarpedia 2 (4): 1331. https://doi.org/10.4249/scholarpedia.1331.

Carter, G.Clifford. 1987. “Coherence and Time Delay Estimation.” Proceedings of the IEEE 75 (2): 236–55. https://doi.org/10.1109/PROC.1987.13723.

Clayton, Martin. 1997. “Metre and Tal in North Indian Music.”

———. 2001. Time in Indian Music: Rhythm, Metre, and Form in North Indian Rag Performance. Oxford Monographs on Music. Oxford University Press, USA.

Collins, J. J., and I. N. Stewart. 1992. “Symmetry-Breaking Bifurcation: A Possible Mechanism for 2:1 Frequency-Locking in Animal Locomotion.” Journal of Mathematical Biology 30 (8): 827–38. https://doi.org/10.1007/BF00176458.

Collins, Nick. 2002. “Interactive Evolution of Breakbeat Cut Sequences.” In Proceedings of Cybersonica. London.

———. 2006. “BBCut2: Integrating Beat Tracking and on-the-Fly Event Analysis.” Journal of New Music Research 35 (1): 63–70. https://doi.org/10.1080/09298210600696600.

Demaine, Erik D., Francisco Gomez-Martin, Henk Meijer, David Rappaport, Perouz Taslakian, Godfried T. Toussaint, Terry Winograd, and David R. Wood. 2005. “The Distance Geometry of Deep Rhythms and Scales.” In CCCG, 163–66. http://erikdemaine.org/papers/DeepRhythms_CCCG2005/.

———. 2009. “The Distance Geometry of Music.” In Computational Geometry, 42:429–54. https://doi.org/10.1016/j.comgeo.2008.04.005.

Doelling, Keith B., and David Poeppel. 2015. “Cortical Entrainment to Music and Its Modulation by Expertise.” Proceedings of the National Academy of Sciences 112 (45): E6233–E6242. https://doi.org/10.1073/pnas.1508431112.

Ellis, D. P. W., C. V. Cotton, and M. I. Mandel. 2008. “Cross-Correlation of Beat-Synchronous Representations for Music Similarity.” In IEEE International Conference on Acoustics, Speech and Signal Processing, 2008. ICASSP 2008, 57–60. https://doi.org/10.1109/ICASSP.2008.4517545.

Elman, Jeffrey L. 1990. “Finding Structure in Time.” Cognitive Science 14: 179–211. https://doi.org/10.1016/0364-0213(90)90002-E.

Feldman, David P, and James P Crutchfield. 2004. “Synchronizing to Periodicity: The Transient Information and Synchronization Time of Periodic Sequences.” Advances in Complex Systems 7 (03): 329–55. https://doi.org/10.1142/S0219525904000196.

Fischer, Ingo, Raúl Vicente, Javier M Buldú, Michael Peil, Claudio R Mirasso, M C Torrent, and Jordi García-Ojalvo. 2006. “Zero-Lag Long-Range Synchronization via Dynamical Relaying.”

Fokker, A D. 1968. Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7) Harmonic Lattice of Notes. Koninkl. Nederl. Akademie van Wetenschappen.

Foote, Jonathan. 1999. “Visualizing Music and Audio Using Self-Similarity.” In Proceedings of the Seventh ACM International Conference on Multimedia (Part 1), 77–80. MULTIMEDIA ’99. New York, NY, USA: ACM. https://doi.org/10.1145/319463.319472.

Freidlin, Mark I., and Alexander D. Wentzell. 1993. “Diffusion Processes on Graphs and the Averaging Principle.” The Annals of Probability 21 (4): 2215–45. https://doi.org/10.1214/aop/1176989018.

Freidlin, Mark, and Matthias Weber. 1998. “Random Perturbations of Nonlinear Oscillators.” The Annals of Probability 26 (3): 925–67. https://doi.org/10.1214/aop/1022855739.

Glass, Leon. 1991. “Cardiac Arrhythmias and Circle Maps−A Classical Problem.” Chaos: An Interdisciplinary Journal of Nonlinear Science 1 (1): 13–19. https://doi.org/10.1063/1.165810.

———. 2001. “Synchronization and Rhythmic Processes in Physiology.” Nature 410 (6825): 277–84. https://doi.org/10.1038/35065745.

Guevara, Michael R., and Leon Glass. 1982. “Phase Locking, Period Doubling Bifurcations and Chaos in a Mathematical Model of a Periodically Driven Oscillator: A Theory for the Entrainment of Biological Oscillators and the Generation of Cardiac Dysrhythmias.” Journal of Mathematical Biology 14 (1): 1–23. https://doi.org/10.1007/BF02154750.

Haken, Hermann, J A Scott Kelso, and Bunz. 1985. “A Theoretical Model of Phase Transitions in Human Hand Movements.” Biological Cybernetics 51 (5): 347–56. https://doi.org/10.1007/BF00336922.

Hennig, Holger, Ragnar Fleischmann, Anneke Fredebohm, York Hagmayer, Jan Nagler, Annette Witt, Fabian J Theis, and Theo Geisel. 2011. “The Nature and Perception of Fluctuations in Human Musical Rhythms.” PLoS ONE 6 (10): –26457. https://doi.org/10.1371/journal.pone.0026457.

Hennig, Holger, Ragnar Fleischmann, and Theo Geisel. 2012. “Musical Rhythms: The Science of Being Slightly Off.” Physics Today 65 (7): 64–65. https://doi.org/10.1063/PT.3.1650.

Hockman, Jason. 2014. “An Ethnographic and Technological Study of Breakbeats in Hardcore, Jungle and Drum & Bass.” PhD Thesis, McGill University Libraries. http://oatd.org/oatd/record?record=oai%5C:digitool.library.mcgill.ca%5C:121313.

Ichinomiya, Takashi. 2004. “Frequency Synchronization in a Random Oscillator Network.” Physical Review E 70 (2): 026116. https://doi.org/10.1103/PhysRevE.70.026116.

Keith, W. L., and R. H. Rand. 1984. “1∶1 and 2∶1 Phase Entrainment in a System of Two Coupled Limit Cycle Oscillators.” Journal of Mathematical Biology 20 (2): 133–52. https://doi.org/10.1007/BF00285342.

Lattner, Stefan, and Maarten Grachten. 2019. “High-Level Control of Drum Track Generation Using Learned Patterns of Rhythmic Interaction.” In IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA 2019). http://arxiv.org/abs/1908.00948.

Mateo, David, Nikolaj Horsevad, Vahid Hassani, Mohammadreza Chamanbaz, and Roland Bouffanais. 2019. “Optimal Network Topology for Responsive Collective Behavior.” Science Advances 5 (4). https://doi.org/10.1126/sciadv.aau0999.

Mirollo, R., and S. Strogatz. 1990. “Synchronization of Pulse-Coupled Biological Oscillators.” SIAM Journal on Applied Mathematics 50 (6): 1645–62. https://doi.org/10.1137/0150098.

Mirollo, R., and S. H. Strogatz. 2007. “The Spectrum of the Partially Locked State for the Kuramoto Model.” Journal of Nonlinear Science 17 (4): 309–47. https://doi.org/10.1007/s00332-006-0806-x.

Moreno, Y., and A. F. Pacheco. 2004. “Synchronization of Kuramoto Oscillators in Scale-Free Networks.” EPL (Europhysics Letters) 68 (4): 603. https://doi.org/10.1209/epl/i2004-10238-x.

Parncutt, Richard. 1994. “A Perceptual Model of Pulse Salience and Metrical Accent in Musical Rhythms.” Music Perception: An Interdisciplinary Journal 11 (4): 409–64. https://doi.org/10.2307/40285633.

Pikovsky, Arkady, and Michael Rosenblum. 2007. “Synchronization.” Scholarpedia 2 (12): 1459. https://doi.org/10.4249/scholarpedia.1459.

Pikovsky, Arkady, Michael Rosenblum, and Jürgen Kurths. 2003. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press.

Robertson, Andrew N. 2011. “A Bayesian Approach to Drum Tracking.” In. http://smc.afim-asso.org/smc11/papers/smc2011_185.pdf.

Robertson, Andrew, and Mark Plumbley. 2007. “B-Keeper: A Beat-Tracker for Live Performance.” In Proceedings of the 7th International Conference on New Interfaces for Musical Expression, 234–37. NIME ’07. New York, NY, USA: ACM. https://doi.org/10.1145/1279740.1279787.

Robertson, Andrew, and Mark D. Plumbley. 2013. “Synchronizing Sequencing Software to a Live Drummer.” Computer Music Journal 37 (2): 46–60. https://doi.org/10.1162/COMJ_a_00178.

Robertson, Andrew, Adam Stark, and Matthew EP Davies. 2013. “Percussive Beat Tracking Using Real-Time Median Filtering.” In Proceedings of European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases. http://www.ecmlpkdd2013.org/wp-content/uploads/2013/09/MLMU_Robertson.pdf.

Robertson, Andrew, Adam M. Stark, and Mark D. Plumbley. 2011. “Real-Time Visual Beat Tracking Using a Comb Filter Matrix.” In Proceedings of the International Computer Music Conference 2011. https://www.eecs.qmul.ac.uk/~markp/2011/RobertsonStarkPlumbleyICMC2011_accepted.pdf.

Robertson, A. N., and M. D. Plumbley. 2006. “Real-Time Interactive Musical Systems: An Overview.” Proc. Of the Digital Music Research Network, Goldsmiths University, London, 65–68. http://www.eecs.qmul.ac.uk/~markp/2006/RobertsonPlumbley06-dmrn.pdf.

Schöner, Gregor. 2002. “Timing, Clocks, and Dynamical Systems.” Brain and Cognition 48 (1): 31–51. https://doi.org/10.1006/brcg.2001.1302.

Semjen, Andras, Dirk Vorberg, and Hans-Henning Schulze. 1998. “Getting Synchronized with the Metronome: Comparisons Between Phase and Period Correction.” Psychological Research 61 (1): 44–55. https://doi.org/10.1007/s004260050012.

Strogatz, Steven H. 2000. “From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators.” Physica D: Nonlinear Phenomena 143: 1–20. https://doi.org/10.1016/S0167-2789(00)00094-4.

———. 2004. Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Hyperion.

Temperley, David. 2000. “Meter and Grouping in African Music: A View from Music Theory.” Ethnomusicology 44 (1): 65–96. https://doi.org/10.2307/852655.

Toussaint, Godfried. 2005. “Mathematical Features for Recognizing Preference in Sub-Saharan African Traditional Rhythm Timelines.” In Pattern Recognition and Data Mining, edited by Sameer Singh, Maneesha Singh, Chid Apte, and Petra Perner, 3686:18–27. Berlin, Heidelberg: Springer Berlin Heidelberg. http://cgm.cs.mcgill.ca/~godfried/publications/off-beatness.pdf.

Toussaint, Godfried T. 2004. “A Comparison of Rhythmic Similarity Measures.” In ISMIR. http://cgm.cs.mcgill.ca/~godfried/publications/similarity.pdf.

———. 2005. “The Euclidean Algorithm Generates Traditional Musical Rhythms.” In Proceedings of BRIDGES: Mathematical Connections in Art, Music and Science, 47–56. http://cgm.cs.mcgill.ca/~godfried/publications/banff.pdf.

———. 2010. “Generating ‘Good’ Musical Rhythms Algorithmically.” In Proceedings of the 8th International Conference on Arts and Humanities, Honolulu, Hawaii, 774–91. http://cgm.cs.mcgill.ca/~godfried/publications/Hawaii-Paper-Rhythm-Generation.pdf.

———. 2011. “The Rhythm That Conquered the World: What Makes a ‘Good’ Rhythm Good?” Percussive Notes 2: 52. http://cgm.cs.mcgill.ca/~godfried/publications/Percussive-Notes-Web.pdf.

———. 2013. The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? 1 edition. Boca Raton, FL: Chapman and Hall/CRC.