Schisma accents in regular temperament notation

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herman.miller
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Schisma accents in regular temperament notation

Post by herman.miller »

I was looking at 5-limit hanson temperament and I noticed that the schisma tempers to 0.6 cents, and thus the schisma accent might actually be useful if you have a large enough scale to notate. It turns out that the schisma maps to +53 generators of hanson, so that's not likely to be useful, but I wondered if there might be other 5-limit temperaments that would need schisma accents. So I got a big list of 5-limit temperaments in a spreadsheet and looked at what they did with the schisma. One that clearly does need a notation for schisma is atomic temperament, 12p&576p, for the trivial reason that the generator is a schisma. It's essentially indistinguishable from JI, though, and you might as well notate it as JI. Are there any others that might actually be useful?

One where a schisma could be useful to notate is 12p&51p misty [<3 5 6] <0 -1 4]>. The schisma maps to (+1, -4), which is exactly half of a 5-comma. (+1, +4) can be notated as 128/81 A:.::||\: , while 405/256 A:||\: is (+2, +0). But this 12 cent interval is sizable enough that it might be better to notate it with a flag. If you look at temperaments that leave the schisma in the range from 0 to 5 cents, most of them map it to 53 or more generators. Ennealimmal has it at (+7, -19), but it also has 9 periods to the octave.

So as a general rule, it seems that schisma accents won't be required for 5-limit rank 2 temperaments. I haven't looked into rank 3 or higher limit temperaments, but I think we could get away with a general rule that schisma accents can be omitted in rank 2 temperaments at least.
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Dave Keenan
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Re: Schisma accents in regular temperament notation

Post by Dave Keenan »

That's a great result! Thanks for doing that survey, Herman.
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