The purpose of tuning is to provide useful intervals; useful intervals are those corresponding to frequency ratios that are close to simple rational numbers. So the first step in understanding a tuning system is to see which simple ratios it approximates well. The building blocks of rational numbers are the primes. So, a good start is to look at the primes.
Dividing octaves into 70 equal microsteps: this division is fine enough that most any interval will be approximated tolerably well. The table above shows that primes 5 and 7 fall right in the middle beween the microsteps of 70edo, and yet the resulting error is only about 8 cents. In conventional 12edo, the prime 5 is about 14 cents off; despite that, conventional tuning works quite well enough. If we don't want to worry about precision, 70edo provides an adequate tuning palette to approximate any interval we might want. But precision needn't be discarded so casually. In the right harmonic context, tuning intervals to within a cent or two really helps music sound exquisite. Precise tuning also helps the listener to discern the structure of the music. If the structure is already familiar to the listener, precision is not really necessary. But if the structure is unusual, then precision will help guide the listener.
The primes that 70edo approximates with high precision are 3 and 13. Could I make something musical with 3 and 13? It's definitely an unusual palette!
The next step with a new tuning system is to look at the way intervals combine. A Tonnetz diagram is a useful tool for this. In the diagram above, a shift to the right is movement by a perfect fifth, a frequency ratio of 3:2. A shift up is movement by a ratio 13:8, which I see gets called a tridecimal neutral sixth. The tonnetz diagram shows the commas that are tempered out by the tuning. For example, moving by ten neutral sixths bring one back around to the starting point (ignoring octaves). This reflects the fact that 13^10 is very close to 2^37. Another tempered out comma is traversed by moving seven perfect fifths and then three neutral thirds. This corresponds to the comma 2197:2187, which I see has been called the threedie comma.
I looked for a way to make a scale that would support traversing the threedie comma. The 33 microstep interval, corresponding to a frequency ratio of 18:13, works well for this. A scale with 17 notes per octave, generated by this interval, has steps that are almost all equal: fifteen of the scale steps are 4 microsteps of 70edo, the other two are 5 microsteps. This scale is very close to 17edo. The precision of this tuning also constrains composition. For example, in 17edo, every note has the possibility of moving by a neutral third. In the more precise 70edo scale, only some notes have this possibility.
I used my algorithmic composition software to construct a piece in this scale. I tried to coax it to produce a traversal of this threedie comma, but looking at the scores from my various attempts, I don't think I ever succeeded. But here is a piece that at least sounded musical to my ears!
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