Bipentatonic scale

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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. Following from this, dipentatonic scales have a maximum 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. The first part of 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. The second part of this article surveys dipentatonic scales in JI. Dipentatonic scales also exist in regular temperaments of course, but exploration of such scales is not included in the article for now at least.

Dipentatonic abstract rank-3 scale patterns

A rank-3 dipentatonic scale with step sizes 5x ay (5-a)z has the form xYxYxYxYxY where the Y's are the ys and zs arranged in the MOS pattern ay (5-a)z. The interlocking pentatonics are copies of the mos aL (5-a)s if y > z and (5-a)L as if z > y.

  • 1L 4M 5s
  • 2L 3M 5s
  • 3L 2M 5s
  • 4L 1M 5s
  • 1L 5M 4s
  • 2L 5M 3s
  • 3L 5M 2s
  • 4L 5M 1s
  • 5L 1M 4s
  • Blackdye (5L 2M 3s dipentatonic, LsLMLsLMLs)
  • 5L 3M 2s
  • 5L 4M 1s

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.) Similar scales are available in other edos that support this temperament, such as 37edo.

  • 3242323332: separation by 14\27 (~13/9) (more 5-limit harmony, with one 4:5:6:9:13; no wolf fifths)
  • 3242324232: 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.)
  • 3233323323: separation by 19\27 (~13/8) (neutral thirds MOS)
  • 3233323332: 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

Just dipentatonic scales

Every other note of a just dipentatonic scale gives a Pythagorean pentatonic. Dipentatonic scales that are also 3-SN scales can be constructed by placing the same interval above or below each step of a pentatonic scale. This interval defines the scale, and the logic behind the listing below.

These scales can be considered the minimum complexity rank-3 decatonic scales that are supersets of Pythagorean[5]. The can be thought of as Blackwood decatonics, but without 256/243 tempered out. Instead of 5 240c intervals in an octave as one generator and a 5/4 as the other, these scales have (one incstance of) a third generator of a prime > 3 along with Pythagorean[5], or two parallel Pythagorean[5]s, seperated by a prime or a prime to some power of three. They have form ABACABABAC (or CABABACABA, inverted, or beginning after ABA) where AB=9/8 and AC=32/27.

They are pentachordal, with pentachords ABAC and include four different tetrachordal scales as the subscale subtended by steps of 1212121. These scales have a quasi-sub-period of a third, and every second step obviously gives Pythagorean[5]. As there are three step sizes, there are also three sizes of the complement, the 10th. There are two sizes of each multiple of the quasi-sub-period, i.e. 3rd, 5th, 7th and 9th. Each remaining interval (4th, 6th, 8th) comes in three sizes. Examples will follow directly below.

The trivalent tetrachordal subset scales after steps of 1212121, with steps of A-9/8-C-9/8-A-9/8-C, or of C-9/8-A-9/8-C-9/8-A respectively, with tetrachords A-9/8-C or C-9/8-A respectively, are the notes not bracketed.

The first mode listed for each value of A can be considered 'major'; the second mode is it's inverse.

Adding prime 5:

A=10/9 (B=81/80, C=16/15) for SNS ((2/1, 3/2)[5], 10/9)[10]

10/9 (9/8) 5/4 4/3 (40/27) 3/2 5/3 (27/16) 15/8 2/1

16/15 (32/27) 6/5 4/3 (27/20) 3/2 8/5 (16/9) 9/5 2/1

A=16/15 (B=135/128, C=10/9) for SNS ((2/1, 3/2)[5], 16/15)[10]

16/15 (9/8) 6/5 4/3 (64/45) 3/2 8/5 (27/16) 9/5 2/1

10/9 (32/27) 5/4 4/3 (45/32) 3/2 5/3 (16/9) 15/8 2/1

A=25/24 (B=27/25, C=256/225) for SNS ((2/1, 3/2)[5], 25/24)[10]

25/24 (9/8) 75/64 4/3 (25/18) 3/2 25/16 (27/16) 225/128 2/1

256/225 (32/27) 32/25 4/3 (36/25) 3/2 128/75 (16/9) 48/25 2/1

Adding prime 7:

A=28/27 (B=243/224, C=8/7) for SNS ((2/1, 3/2[5]), 28/27)[10]

28/27 (9/8) 7/6 4/3 (112/81) 3/2 14/9 (27/16) 7/4 2/1

8/7 (32/27) 9/7 4/3 (81/56) 3/2 12/7 (16/9) 27/14 2/1

A=64/63 (B=567/512, C=7/6) for SNS ((2/1, 3/2[5]), 64/63)[10]

64/63 (9/8) 8/7 4/3 (256/189) 3/2 32/21 (27/16) 12/7 2/1

7/6 (32/27) 21/16 4/3 (189/128) 3/2 7/4 (16/9) 63/32 2/1

Adding prime 11:

A=12/11 (B=33/32, C=88/81) for SNS ((2/1, 3/2[5]), 12/11)[10]

12/11 (9/8) 27/22 4/3 (16/11) 3/2 18/11 (27/16) 81/44 2/1

88/81 (32/27) 11/9 4/3 (11/8) 3/2 44/27 (16/9) 11/6 2/1

Adding prime 13:

A=13/12 (B=27/26, C=128/117) for SNS ((2/1, 3/2[5]), 13/12)[10]

13/12 (9/8) 39/32 4/3 (13/9) 3/2 13/8 (27/16) 117/64 2/1

128/117 (32/27) 16/13 4/3 (18/13) 3/2 64/39 (16/9) 24/13 2/1

Adding prime 17:

A=17/16 (B=18/17, C=512/459) for SNS ((2/1, 3/2[5], 17/16)[10]

18/17 (9/8) 64/51 4/3 (24/17) 3/2 27/17 (27/16) 32/17 2/1

17/16 (32/27) 34/27 4/3 (17/12) 3/2 51/32 (16/9) 17/9 2/1

A=18/17 (B=17/16, C=272/243) for SNS ((2/1, 3/2[5]), 18/17)[10]

17/16 (9/8) 34/17 4/3 (17/12) 3/2 51/32 (27/16) 17/9 2/1

18/17 (32/27) 64/51 4/3 (24/17) 3/2 27/17 (16/9) 32/17 2/1

Adding prime 19:

A=19/18 (B=81/76, C=64/57) for SNS ((2/1, 3/2)[5], 19/18)[10]

19/18 (9/8) 19/16 4/3 (38/27) 3/2 19/12 (27/16) 57/32 2/1

64/57 (32/27) 24/19 4/3 (27/19) 3/2 32/19 (16/9) 36/19 2/1

Adding prime 23:

A=24/23 (B=69/64, C=92/81) for SNS ((2/1, 3/2)[5], 24/23)[10]

24/23 (9/8) 27/23 4/3 (32/23) 3/2 36/23 (27/16) 81/46 2/1

92/81 (32/27) 23/18 4/3 (23/16) 3/2 46/27 (16/9) 23/12 2/1

More complex subgroups:

A=11/10 (B=45/44, C=320/297) for SNS ((2/1, 3/2)[5], 11/10)[10]

11/10 (9/8) 99/80 4/3 (22/15) 3/2 33/20 (27/16) 297/160 2/1

320/297 (32/27) 40/33 4/3 (15/11) 3/2 160/99 (16/9) 20/11 2/1

A=14/13 (B=117/112, C=208/189) for SNS ((2/1, 3/2)[5], 14/13)[10]

14/13 (9/8) 63/52 4/3 (56/39) 3/2 21/13 (27/16) 189/104 2/1

208/189 (32/27) 26/21 4/3 (39/28) 3/2 104/63 (16/9) 13/7 2/1

A=15/14 (B=21/20, C=448/405) for SNS ((2/1, 3/2)[5], 15/14)[10]

15/14 (9/8) 135/112 4/3 (10/7) 3/2 45/28 (27/16) 40/21 2/1

21/20 (32/27) 56/45 4/3 (7/5) 3/2 224/135 (16/9) 28/15 2/1

A=20/19 (B=171/160, C=152/135) for SNS ((2/1, 3/2)[5], 20/19)[10]

20/19 (9/8) 45/38 4/3 (80/57) 3/2 30/19 (27/16) 135/76 2/1

152/135 (32/27) 4/3 (57/40) 3/2 45/19 (16/9) 40/19 2/1

A=21/20 (B=15/14, C=640/567) for SNS ((2/1, 3/2)[5], 21/20)[10]

21/20 (9/8) 189/160 4/3 (7/5) 3/2 63/40 (27/16) 567/320 2/1

640/567 (32/27) 80/63 4/3 (10/7) 3/2 320/189 (16/9) 40/21 2/1

A=22/21 (B=189/176, C=112/99) for SNS((2/1, 3/2)[5], 22/21)[10]

22/21 (9/8) 33/28 4/3 (88/63) 3/2 11/7 (27/16) 99/56 2/1

112/99 (32/27) 14/11 4/3 (63/44) 3/2 56/33 (16/9) 21/11 2/1

Dipentatonic scales in regular temperaments

Just a couple examples linked for now:

Marvel dipentatonic

Prodigy dipentatonic

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