Juhani wrote:But you are now talking about a notation system for 224edo aren't you? Of course such a system, and indeed a much smaller one, is fine for playback. Some EDO will be used in a playback engine, anyway - 1200EDO is common in hardware synths.
But by JI notations I always refer to a one-symbol-per-prime system such as Multi-Sagittal and Sabat-v.Schweinitz, with 3-limit nominals, as well as Johnston's system.
Can you explain how the library of combined symbols will not quickly grow when using such a system? The coverage of the 15-limit diamond is minuscule.
No, I wasn't talking about a notation for 224-edo. What I mean is that it would be OK if every combination of symbols that is actually used, from some JI notation such as Multi-Sagittal or Sabat-v.Schweinitz, was mapped to the appropriate number of degrees of 224-EDO, assuming the nominals are in a chain of its best fifths. 224-EDO is better than 1200-EDO in many ways.
But I take your point that, when using one symbol per prime, the number of combinations does not depend on which EDO is used to approximate it. It does however depend on your choice of nominals. I suspect that Chrysalid Requiem (to take an extreme example) would not require so many combinations, and the combinations that are required would not contain so many symbols, if it was notated using Pythagorean nominals.
I suspect that after about 50 JI accidental combinations have been defined, the rate at which new ones would need to be added would be very slow.
I mentioned the 15-limit diamond because its notes are represented uniquely in 224-EDO whereas in other EDOs some of them map to the same note, such as 15/14 and 16/15 in 72-EDO.
Since the primes 2 and 3 are handled by the nominals and sharps and flats, then the accidental combinations only need to correspond to combinations of primes 5 and above (which will then also need to be combined with sharp and flat).
I pointed out that the frequency of use of such combinations in the Scala scale archive falls off faster than Zipf's law
I show the first 42 here:
You can see that by the 42nd combination, we are down to accidental combinations that occur only once in 600 scale pitches. When it comes to notating musical pieces rather than scales, the fall-off in usage frequency will be even more rapid, since the most commonly used scales will tend to be those with the more common pitches, and the more common pitches will tend to be repeated more often in a piece.
Here's a list of such combinations generated by a simple algorithm, in an order that approximates the frequency order from the Scala archive: