270edo

A tuning system is a set of pitches that enable the making of music. A tuning system can be used in designing a musical instrument, or as the foundation of a system of notation, or as a theoretical construct for composition and analysis. There are countless possible tuning systems, and even the actual systems in use are very many. One simple class of tuning systems consists of those that divide octaves into some number of equal steps. The conventional tuning system that has spread from Europe around the world, 12edo, divides octaves into twelve equal steps. This system is so dominant, governing the design of instruments, tuners, digital audio workstations, etc., that exploring alternatives is not so easy! But there is a vast world available to explore, with just a bit of effort.

What makes a tuning system good? The availability of consonant intervals is a major criterion. What should count as a consonant interval, that's a challenging issue. A classical starting point is that a consonant interval is one that is very close to a simple frequency ratio. How close? How simple? But at least we are getting close to a concrete idea.

The chart above shows, for a quite wide spectrum of simple ratios, how closely they can be approximated by the available intervals of 12edo. The numbers on the top and left margins define the numerator and denominator of a simple ratio. The green square containing 0.02 in the upper left shows the error when 12edo is approximating the simple ratio 3:1. That ratio would be 19.02 steps of 12edo, but of course the closest actual interval in the tuning system is 19 steps, so the error is 0.02. The error for 5:1 is 0.14, which is not so terribly accurate. I have used bright green to flag the very accurate intervals of 12edo and pale green to flag the moderately accurate intervals. The squares in grey are redundant, because the numerator and denominator are not mutually prime.

The simplest intervals are the ones toward the upper left. The more and brighter green cells that a tuning has, the better the tuning, especially when the green cells are toward the upper left.

One of the simplest alternative tunings is 24edo, where each (half) step of 12edo is divided in half, to form quarter tones. These charts measure errors in terms of the size of the step of the tuning. Since the step size of 24edo is half the size of the steps of 12edo, many of the errors double in size. But some are much smaller. The error for 11:1 is only 0.03. 11:1 falls just about half way between steps of 12edo, so when a new note is added in the middle, it comes very close to 11:1. But overall, comparing 24edo to 12edo, it is not clear which is the winner. 24edo adds a lot of notes for not so much gain.

270edo divides octaves into 270 equal steps instead of the conventional 12, so obviously it is adding an awful lot of extra notes! But its chart shows, it really hits a lot of simple ratios quite accurately. Of course its steps are so small that it can't be too far off! But these errors are in terms of the tiny steps of 270edo. For example, 3:1 is 427.94 steps of 270edo. The available interval of 428 steps is just 0.06 of a step sharp.

Here are three snapshots of the evolution of a composition from my thermodynamic composition system:

• temp=865.154772702373; cost=17487362.4275858;
• temp=447.765805311866; cost=8883781.06482988;
• temp=10; cost=3690000; (oops, this is from memory... I didn't record the exact numbers!)

Even at the final stage, the piece has not settled into monotony:

For this piece I counted as consonant a wide range of simple intervals, such as 14:13, 13:12, 12:11, etc. These ratios can be combined to form commas like 676:675, 1001:1000, and 2401:2400 that 270edo tempers out. Traversals of these commas can get stuck, as topological defects, in the piece. Exactly what pattern of such traversals remains in this final stage... that would be a bit complicated to figure out!