Squib

Every culture's music is influenced by the tools available to them. The primary tool available to me is a Launchpad with a 9x9 grid of LED buttons (minus the top right corner). So, the most immediately obvious set of pitches to use is a lattice where moving one button to the right is by 2/1 and one button up is 3/1. This theoretically allows these intervals to be combined in any combination, as shown in this image.

There's a problem. 9/8, the whole tone, is as easy to reach as 108/1! Not only does this make useful intervals like 9/8 more difficult to use, but since keyboard space is limited, this adds extremely high and low notes that render much of the space unusable. So instead, notes can be represented as any combination of fourths (4/3) and fifths (3/2), the two simplest non-harmonic intervals. This leaves 2/1 and 3/1 easily accessible while moving very large intervals further away, as well as moving more complex smaller intervals closer together. It's also much more natural to think of 9/8 as the difference between a fourth and a fifth than it is to think of it as the difference between three 2/1s and two 3/1s, for example.

This is the most basic tuning and layout of this music theory, and it works very well. Except for absolute pitch, every place on the keyboard is the same, a property called isomorphism. This allows any interval or chord to be placed anywhere on the keyboard.

One way is to use the notes we already have. Two 9/8s make a rough approximation of a major third, but two 4/3s minus three 9/8s is a much closer one and is nearly indistinguishable from 5/4. It's awkward to use, but we can adjust the layout to change that.

Just like when the layout was changed to fourths and fifths, bringing more complex intervals closer pushes larger intervals further away, but this time the range is reduced to only a little more than an octave and the intervals far to the side are very complex and not very useful. There are other layouts where this 5/4 is in a different direction, which solves some of these problems, but having it so far away from the other notes makes it awkward to play.

The other way to approximate 5/4 is to indeed add more notes between the existing ones. The tuning we already have is called Pythagorean tuning because it uses only ratios of 2 and 3. We can take two sets of Pythagorean and put them together. There are an infinite number of ways to do this, but only 3 preserve isomorphism.