Dipentatonic scales

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A dipentatonic scale is a 10-note scale where every other note gives an MOS pentatonic scale generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Most useful are di-meantone pentatonic and di-superpyth pentatonic. Equivalently, they are scales that have a maximaum of two sizes for intervals that are an even number of steps. Some dipentatonic scales are MOSes, such as the Erlich decatonic in 22edo. Dipentatonic scales are useful because modulating by fifths is easy in them.

Dipentatonic scales can be classified by their fifth size and the separation between the two chains of fifths. This article surveys dipentatonic scales you can get in various edos with relatively good fifths. Dipentatonic scales tend to work better with sharper fifths, because it's easier to interleave a minor third of one pentatonic inside a major third of the second pentatonic.

Sharp-fifth edos

10edo

The trivial case.

15edo

Up to modal rotation, blackwood[10] is the only dipentatonic scale in 15edo.

17edo

The following are, up to modal rotation, a complete list of 17edo dipentatonic scales.

  • 0-2-3-6-7-9-10-13-14-16-17: separation by 16\17
  • 0-2-3-6-7-9-10-12-14-16-17: separation by 9\17 (~13/9)
  • 0-2-3-5-7-9-10-12-14-16-17: separation by 2\17 (~13/12) (neutral thirds MODMOS)
  • 0-2-3-5-7-9-10-12-14-15-17: separation by 12\17 (~13/8) (neutral thirds MOS)

22edo

The following are, up to modal rotation, a complete list of 22edo dipentatonic scales. The first three or so are similar to corresponding 12edo pajara scales.

  • 0-2-4-7-9-11-13-15-18-20-22: separation by 11\22 (pajara MOS)
  • 0-2-4-7-9-11-13-16-18-20-22: separation by 20\22 (~15/8) (pajara MODMOS)

(above is complete list with no 1\22)

  • 0-3-4-7-9-11-13-16-18-20-22: separation by 7\22 (~5/4) (pajara MODMOS)
  • 0-3-4-7-9-12-13-16-18-20-22: separation by 16\22
  • 0-3-4-7-9-12-13-16-18-21-22: separation by 3\22
  • 0-3-4-8-9-12-13-16-18-21-22: separation by 12\22
  • 0-3-4-8-9-12-13-17-18-21-22: separation by 21\22 (superpyth subset)

27edo

The following are, up to modal rotation, a complete list of 27edo dipentatonic scales with no step of size 1\27. (I'm excluding 1\27 because it's near maximum dissonance.)

  • 0-3-5-9-11-14-16-19-22-25-27: separation by 14\27 (~13/9) (more 5-limit harmony, with one 4:5:6:9:13; no wolf fifths)
  • 0-3-5-9-11-14-16-20-22-25-27: separation by 25\27 (~15/8) (more 5-limit harmony, with one 4:5:6:7:9:13; though there's a flat "wolf fifth" it doesn't occur as the same interval class as a regular fifth.)
  • 0-3-5-8-11-14-16-19-22-24-27: separation by 19\27 (~13/8) (neutral thirds MOS)
  • 0-3-5-8-11-14-16-19-22-25-27: separation by 3\27 (~13/12) (neutral thirds MODMOS)

29edo

Flat-fifth edos

For edos with fifths flatter than 12edo, let's loosen the definition of a dipentatonic a bit: the pentatonics don't have to be every other note.

...

19edo

26edo

31edo