Melody Harmony Rhythm

See Human hearing.

Equal temperment and just intonation:

https://www.youtube.com/watch?v=VRlp-OH0OEA

See also xenharmonic music..

https://musiclab.chromeexperiments.com/Technology

What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch)

Twelve Tones

http://www.lightnote.co/

Venn Piano

https://books.google.co.uk/books?id=2gYVpfr9C8QC&pg=PA13&lpg=PA13&dq=algebraic+topology+self-assembly&source=bl&ots=HkZONnJC7Y&sig=78G5cZc-mNKmKcnFUqTPqq4Z_PA&hl=en&sa=X&ved=0ahUKEwjm4bS3oeTWAhXPalAKHRYFC8sQ6AEIPzAD#v=onepage&q=algebraic%20topology%20self-assembly&f=false

Why are there twelve notes?

video

In Pythagorean tuning, we build the notes via the two ratios 2:1 and 3:2 (or equivalently 2:1 and 3:1)

That is, we take ratios of the form $\left(\frac{3}{2}\right)^m 2^n$, for integers $m$ and $n$. How many notes fall inside an Octave as we do this? The power of $n$ just moves us between one octave and the next, so we can safely take all powers of $n$, and just focus on what happens as we increase powers of $m$. If we take the $\log_2$ of the ratio, then the first octave lies between $\log_2(1)=0$ and $\log_2(2)=1$. Ratios in the first octave will lie somewhere in between. We can find these by doing $\log_2{\left(\left(\frac{3}{2}\right)^m 2^n\right)} \text{ mod } 1$ $=m\log_2{\left(\frac{3}{2}\right)}\text{ mod } 1$. It turns out that if we take $m=0,...,11$, we get $12$ well-spaced notes, which is what led Pythagoras and all of western music to use these $12$ notes as our chromatic scale! The fundamental reason is that $\log_2{\left(\frac{3}{2}\right)}\approx \frac{7}{12}$, so that after adding $\log_2{\left(\frac{3}{2}\right)}$ $12$ times we are close to an integer, and thus to a beggining of an octave, that is the base note. This is the approximate period. If it was 7/12 exactly, then we would have $m\frac{7}{12}\text{ mod } 1$, which we can recast as $7m \text{ mod }12$, which will visit all points $0,...,11$ as $7$ and $12$ are Coprime, so that $12$ iterations are needed to come back; period of periodic orbit is $12$. Therefore it can't visit the same point twice in these $12$ iterates, so that it visits $12$ points. But there are only $12$ integers here, so that it visits them all! This is fundamentally why it is so well spread out after twelve iterations.

See Modular arithmetic