Interval dividing method
- Start with 1/1 and 2/1
- Find the largest interval between two adjacent notes
- Use a method to create an interval between those notes
- Insert the new interval into your list
- 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.
! fliJI1.scl ! Created using Scale Workshop 1.2 ! ! https://sevish.com/scaleworkshop/index.htm?name=fliJI1&data=21%2F20%0A9%2F8%0A7%2F6%0A5%2F4%0A4%2F3%0A7%2F5%0A3%2F2%0A14%2F9%0A5%2F3%0A7%2F4%0A15%2F8%0A2%2F1&freq=440&midi=69&vert=1&horiz=3&colors=white%20black%20white%20white%20black%20white%20black%20white%20white%20black%20white%20black&waveform=triangle&env=organ ! fliJI1 12 ! 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8 2/1