The theoretical structure of musical space can be found in the circle of fifths. Unfortunately, the following is rather complicated. My version of the circle of fifths is based on Coltrane's tone circle (Coltrane, 2017) and was part of my 2019 dissertation project for the BA Applied Music, UHI.
The circle of fifths
Imagine sixty notes on the piano keyboard wrapped up in a circle, starting and ending on C. There are five octaves (5x12 semitones). 5 octaves of C form a pentagram in a circle. Call C the tonic. Now determine the perfect intervals (the fourths and the fifths of C: F and G). Observe that they are positioned as prime numbers in the circle of sixty keys. Now turn the tone circle upside down and call F# the countertonic. Again determine the perfect intervals and again find them positioned as prime numbers (fourths and fifths of F#: B and C#).
Now you have two opposing keys that embrace each other with their perfect intervals or prime numbers. In the chart below, tonic and countertonic appear as two pentagrams. Notice that prime numbers pair up as harmonic couples. This phenomenon is called the 'Twin Prime Conjecture' in mathematical number theory (Rezgui, 2017)
Finally, apply to this tone circle the circle of fifths, albeit in reverse (compared to the common circle of fifths). Observe that all 12 positions of the circle of fifths have a number divisible by 5. Where one of the perfect intervals of tonic and countertonic coincides with a position in the circle of fifths, it cannot be a prime number, because a prime number is only divisible by itself.
So far, we are still caught flat in a chart of the second dimension. Music, however, happens in three dimensions plus the dimension of time.
Similar to the Coastline Paradox, musical time also is paradoxical because it it works with fractals. These are fractions of a whole that you can zoom into, in ever more detail. If you zoom into a rugged coastline, measuring the crevices of every rock, it will appear longer, than if you zoom out and don't observe the detail.
Slowing a frequency lowers the pitch. Reversely, if you speed up a rhythm, it will become pitch. Poly-rhythms thus describe intervals of pitch. A poly-rhythm of 3 over 2 creates the interval of a fifth for example (Neely, 2018). Intervals therefore have proportionately timed relationships. The relationship between two notes of a tune is triadic; both relate to the fundamental, or tonic. Adam Neely managed to produce fractal tunes; he squeezed midi-sequences in time, and multiplied them in rhythmic proportions according to the interval. The result is a fractal midi tune that is contained in itself. Best to watch his tutorial (Neely, 2017).
You find many paradoxical symmetries in the structure of musical space-time. Like with all natural patterns, you can apply the Mandelbrot equation with its infinite iterations of self-similarity also to music (Hsü & Hsü, 1990). The slow is symmetric to the fast, as well as the big to the small: the proportions within one octave are symmetrically self-similar to that of five octaves within the circle of fifths. Also the difference between high and low pitch seems to dissolve because a circle has neither beginning nor ending. Most mind boggling however, is the symmetry between inside and outside.
The circle of 'fifths' is actually made of fourths. Ascending five semitones, you arrive at the fourth, descending five semitones at the fifth. The two perfect intervals are perfectly symmetric. The tonic (C) is therefore enclosed in its perfect intervals (G,F) by virtue of symmetry. Beyond this enclosure happens to be the countertonic (F#) octave at six semitones in both directions. This is why I started to see the countertonic as a skin, a surface or horizon to the tonic and vice versa. The result is a structure that you can turn inside out in the middle of the tonic octave. Given that F# is dissonant to C, 'outside' would mean 'outside of tonic consonance'.
A structure in musical space
The concept of tonic and countertonic creates a tension through polarity, where harmony could be perceived as the sum of movements that binds them together. Beauty is derived from the paradox of uniting contradictions. Nothing in this structure is discrete or certain, except for the proportions of perfect intervals, or the three-fold relationship between the fundamental, the fourth and the (descending) fifth. Trying to see the circle of fifths in a three-dimensional way, I would like to suggest that it looks like the 'harmonic function on a ring' (Wikipedia). This illustration of a mathematical equation shows an object with a ring, two spatially opposed tonic and countertonic pentagon-octaves and a chromatic scale-line weaving along the vertices. There is also a sense of surface turning inside out. I think it looks exactly like the circle or sphere or cylinder or thingymabob of fifths and fourths.
Coltrane, J. (2017). Open Culture. Abgerufen am 16. 1 2019 von http://www.openculture.com/2017/04/the-tone-circle-john-coltrane-drew-to-illustrate-the-theory-behind-his-most-famous-compositions-1967.html
Hsü, K., & Hsü, A. (1990). Fractal Geometry of Music (Physics of Melody). PNAS, 87(3), pp.938-941.
Mandelbrot, B. (2004). Fractals and Chaos: the Mandelbrot Set and Beyond. New York: Springer.
Neely, A. (2018). Polyrhythms are Polypitch - live @ Ableton Loop, Berlin. Abgerufen am 11. 12 2019 von You Tube: https://www.youtube.com/watch?v=-tRAkWaeepg
Neely, A. (2017). The Coltrane Fractal. Retrieved 1 22, 2019, from https://www.youtube.com/watch?v=J98jwtm5U4E
Rezgui, H. (2017). Conjecture of twin primes (still unsoved problem in Number Theory). Academic Journal/Surveys in Mathematics and its Applications, 12, 229-252.
wikipedia. (2020). By KDS4444 - Own work, CC BY-SA 4.0. Retrieved from Wikimedia.org: https://commons.wikimedia.org/w/index.php?curid=49035989