Musical tuning is a very small technical discipline, simple enough yet rich enough to provide a good sandbox for exploring what non-convergent theorizing can look like. The conventional tuning system, 12 equal divisions per octave, or 12edo, is so well established that it can easily seem that the process of convergence is complete and the ultimate theory has been achieved. These are the universally and absolutely true notes or intervals. Any alternative is necessarily a lesser approximation to the truth.

The limits and flaws of the conventional tuning system, 12edo, are sufficiently evident that many people have explored alternate systems. There are two ways to think about this kind of exploration. One can see it as convergent, that somehow a better tuning system than 12edo will be found, and then perhaps an even better system. The other perspective is that the exploration is more about real alternatives. Tuning systems are not particularly better or worse, but simply different. One or another tuning system might be better or worse for some particular purpose, for a particular piece of music or for a particular instrument. But there may be no absolute ranking independent of the details of some particular intended use.

Just intonation is a tuning system that can easily tempt a person to believe in, as an ultimate theory. But looking at the practical use of tuning systems, there is a lot of music that just cannot work with just intonation. With just intonation, intervals will conform to rational frequency ratios involving small primes, e.g. a perfect fifth of 3:2 and a major third of 5:4. But these intervals can easily be combined to form intervals such as 81:80 which are very close to 1:1. These small intervals are known as *commas*. Tempered tuning systems will provide some more limited collection of pitches, treating as equivalant some pairs of intervals that would differ in just intonation by such commas.

This is a chart of just intonation intervals involving the prime factors of 3 and 5. All intervals have been folded into a single octave range. The numbers are in terms of cents: 1200 times the base 2 logarithm of the ratio. Moving from one cell to the cell on its right is multiplying by 3, which, when folded into a single octave, becomes 3/2 or 3/4. The base 2 logarithm of 3/2 is 0.585. Multiplying this by 1200 gives 701.955 cents. Moving up or down a cell in the chart is moving up or down by major thirds, by 5/4 or 8/5.

I have highlighted in the chart the small intervals, the commas, that are less than 50 cents. This chart could of course be expanded indefinitely. It should be clear that there are very many such small intervals. This is an echo of the one of the earliest crises in the project to build an ultimate theory of reality. The Pythagoreans discovered irrational numbers, which ruined their project to understand the world in terms of rational numbers.

Here I have zoomed in to the central part of the chart, and labeled most of the commas by their conventional names. (I used tonalsoft as a source for these names.) I would like to focus here on three commas, all of which are tempered out by 12edo.

- the syntonic comma is a combination of four perfect fifths and a major sixth. This is the ratio 81:80.
- the diesis is a combination of three major thirds, the ratio 125:128.
- the diaschisma is four perfect fifths and two major thirds, the ratio 2025:2048.

The simplest class of tuning systems is those that divide octaves into some number of equal parts. This table shows many of the most useful such systems. It gives the accuracy of the tuning, as the difference from just intonation. It also highlights which of these three commas are tempered out by the tuning system.

Conventional music in the tradition of e.g. Mozart is based on tempering the syntonic comma. This means that dividing octaves into 19 or 31 steps instead of the conventional 12 will still allow one to play most such music just as it is written. Dividing octaves into 53 equal steps provides intervals very close to those of just intonation, but since none of these simple commas are tempered out... one is forced into rather unconventional music.

This leaves 34edo. It has an attractive degree of tuning accuracy, but also tempers out the diaschisma. It is not as accurate as 53edo, but a bit more conventional.

A few years ago I presented a 12 note per octave subset of 34edo. Here is a new algorithmic composition using this 12 note subset.

With 12 notes per octave, this composition can be mapped straightforwardly into a 12edo version. This provides a good demonstration of what tuning accuracy is about, what difference it makes.