There is still no mathematics of a simple melody never mind more complex musical structures. Music as we know it has two main dimensions: tonal space and arrow of time (tonal chronotope).
The tonal space is perceived as the phenomenal tonal gravity. However, the gradient of tonal attraction is non-linear (unlike the linear distribution of frequencies in a musical scale) and the tonal space is cyclical. There are some other interesting details of the tonal space but the non-linearity and cyclicality are the most important features of the tonal space.
The perception of tonal distances in music is mildly synesthetic. It would be awesome to find an algorithm for visualization of tonal topography.
Perhaps you have tried the Society for Mathematics and Computation in Music (SMCM) http://www.smcm-net.info/
Thank you for the recommendation. Unfortunately, I do not see that the SMCM addresses the question of topography of music.
There is a paper in the proceedings of the Third International SMCM Conference that refers to the topography of music.
Thank you so much for the link. Very interesting.
The most difficult problem of creating music topology is incorporation of psychophysics of tonal relationships into a robust mathematical model. The tonal relationships are the foundation of the music that we can sing and dance with (Bach, Schubert, Ravel) as compared to aleatoric methods (Berg, Stockhausen, Xenakis). Neither of the papers touches on this very difficult subject of integrating the acousmatics and mathematics.
Perhaps, if you have not already done so, you could attend a SMCM conference and present on that problem of incorporation of psychophysics of tonal relationships into a robust mathematical model. That way, you might find some people who might have ideas on ways to approach the problem.
The problem needs a cooperation of top-noch topologists, musicologists, and music psychologists. Tonal space is fundamentally unlinear. While notes in a scale are arranged lineary frequency-wise, the tonal relationships among the notes is a different story.
Dear M.N. Korsakova-Kreyn,
I just want to present my material as a mapping of musical notes into geometric images Appassionata.pdf using Beethoven's "Appassionata" as an example.
Is this mapping relevant to your topic?
Dear Dr. Fedulov, thank you for the response.
My question was about visualization of time-tonal relationships, which is true to the psychophysics of music perception.
Obtaining mathematical representation of a complex structures or processes when using top-down approach can be challenging. Take the planetary motion fro example; if you try to derive the mathematical representation of it, without knowing/considering the Newtonian gravitational equation can be tricky; knowing the gravitation equation helps to take a down-top approach, where you can model not just the planetary motion, but the motions of even the bigger and more complex structures in the space.
Now, I would speculate that the same pattern applies to the musical structures. Until recently we did not have any theory that explains (at least partially) music at the lowest detailed level, so we were not able to apply down-top modeling.
Luckily, recently in 2017, we have published a paper that attempts to explain consonance, where by using computational neuroscientific model, we were able to reproduce precisely and completely the consonance pattern (Helmholtz’s order of intervals by consonance) for the first time ever. More precisely, we show that ANY system featuring Hebbian neuroplasticity will be able to reproduce some form of a consonance pattern. We go even further, by citing evidence provided by other authors, offering a hypothesis that suggest how the consonance pattern translates to pleasantness (or non-pleasantness for the dissonant sounds).
Therefore, (if) the consonance pattern is produced as a result of a very simple neural network calculations, when feeding with simple input stream, that should be enough for a math expert to derive a purely mathematical model (assuming such is possible).
Here is a link of the paper, and feel free to contact me for more details: https://doi.org/10.1016/j.bica.2017.09.001 ( or the free preprint: https://auditory.solutions/publications/Consonance1preprint.pdf ).
Thank you, Toso.
There are also papers by Gerald Langner, which resonate with your approach.
Krumhansl and Kessler (1982) have actually produced a graph of perceptual differences for all tones (diatonic and chromatic) of a diatonic scale.
There are several problems that make it hard to do the geometrical presentation of music: 1. the accumulation of influences among tones even in a simple melody. 2. the invariance of affective quality for melodic intervals going up and the same intervals going down (this is easy to demonstrate with singing). 2. the lack of complete mathematical model of the gravitation filed, which could be useful for visualization of melodic topography.
Conceptually, Gibsonian model for visual perception can be helpful for explaining tonal patterns (as tonal flax) in music.
Thank you again for your paper,
Marina
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