User:Flirora/JI Scales

Interval dividing method

1. Start with 1/1 and 2/1
2. Find the largest interval between two adjacent notes
3. Use a method to create an interval between those notes
4. Insert the new interval into your list
5. Repeat steps 2 – 4 until you're satisfied

Naïvely taking the mediant results in too many similar superparticular intervals (since the mediant of 1 and a superparticular interval is another superparticular interval). I found better results from taking either the mediant or the mean, whichever one had a lower prime limit (arbitrarily selecting the mean if the prime limits are equal). 12 notes gives

1/1 11/10 10/9 9/8 7/6 5/4 4/3 7/5 3/2 5/3 7/4 15/8 2/1

but since there's only one 11-limit interval, I decided to take 11/10 out and replace it with the next note to be added, 8/5. I additionally decided to take out 10/9 to replace it with 21/20, giving

1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8 2/1

or in Functional Just System,

C Db⁷₅ D Eb⁷ E⁵ F Gb⁷₅ G Ab₅ A⁵ Bb⁷ B⁵ C

Ab₅ is the oddball here; you could plausibly replace it with Ab⁷ (14/9) for an extra 3/2 to make the flat keys more usable.

It might be possible to construct larger scales using this method.