Dipentatonic scale

A dipentatonic scale is a 10-note octave-equivalent scale where every other note gives a fixed choice of pentatonic scale; hence a dipentatonic scale is a type of flought scale. Following from this, dipentatonic scales based on a MOS pentatonic scale have a maximum of two sizes for intervals that are an even number of steps. Many dipentatonic scales are generated by a diatonic-sized fifth (between the 7edo fifth and the 5edo fifth) of a fixed size. Some dipentatonic scales are MOSes, such as the Erlich decatonic in 22edo. Modulating by fifths is easy in dipentatonic scales where the interleaved pentatonics are generated by a fifth (i.e. 2L 3s and 3L 2s, depending on tuning).

The first part of this article classifies abstract dipentatonic scales by the pentatonics they interleave and by their rank (whether they are mosses or rank-3). 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.

Abstract dipentatonic scales

Dipentatonic mosses

Every 10 note mos except 5L 5s is dipentatonic. The degenerate case of a dipentatonic mos is 5L 5s, where the interleaved pentatonic is 5edo. The nondegenerate cases are:

1L 9s: Lsssssssss (pentatonic 1L 4s)
2L 8s: LssssLssss (pentatonic 2L 3s)
3L 7s: LssLssLsss (pentatonic 3L 2s)
4L 6s: LsLssLsLss (pentatonic 4L 1s)
6L 4s: LLsLsLLsLs (pentatonic 1L 4s)
7L 3s: LLLsLLsLLs (pentatonic 2L 3s)
8L 2s: LLLLsLLLLs (pentatonic 3L 2s)
9L 1s: LLLLLLLLLs (pentatonic 4L 1s)

Dipentatonic ternary scale patterns

A ternary dipentatonic scale with step sizes 5x ay (5-a)z has the form xYxYxYxYxY where the Y's are replaced with y and z steps 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. The following is a complete list of such abstract patterns, assuming octave equivalence:

1L 4M 5s, LsMsMsMsMs
2L 3M 5s, MsLsMsLsMs
3L 2M 5s, LsMsLsMsLs
4L 1M 5s, LsLsLsLsMs
1L 5M 4s, LMsMsMsMsM
Blackville (2L 5M 3s), sMLMsMLMsM
3L 5M 2s, LMsMLMsMLM
4L 5M 1s, LMLMLMLMsM
5L 1M 4s, LsLsLsLsLM
Blackdye (5L 2M 3s), LsLMLsLMLs
Onyxwood (5L 3M 2s), LMLsLMLsLM
5L 4M 1s, LMLMLMLMLs

Dipentic scales using the Pythagorean pentic and arbitrary offsets

A dipentic scale is a 10-note scale such that the even degrees form one copy of the pentic MOS 2L3s and the odd degrees form a second, shifted copy of 2L3s with the same generator tuning as the first copy. Dipentic scales are classified by the generator used by the pentic and the offset between the two copies of pentic. Here we classify dipythpentic scales, dipentic scales that use the 3/2-generated Pythagorean tuning for the two copies of pentic.

Assuming octave equivalence, the offsets δ and 1200 - δ behave the same. Taking that fact into account, the following offset ranges do not yield dipythpentic scales. With these offsets, the two pentic scales do not interleave because there is an s-step of one copy of pentic that is contained entirely within a pentic L-step of the second.

9/8 ≤ δ ≤ 32/27
81/64 ≤ δ ≤ 4/3

Offsets between 1/1 and 9/8 yield the three ternary dipythpentic scale patterns of the form ababacabac, where c > a. Outside of this range, ternary dipythpentic scales only occur with 3 values for offsets.

If δ < sqrt(9/8), we obtain the ternary scale 2L3m5s (smsmsLsmsL).
If δ = sqrt(9/8) = 101.955c, then we have a permutation of 2L8s (sssssLsssL).
If sqrt(9/8) < δ < sqrt(32/27), we have the ternary scale 2L5m3s (msmsmLmsmL).
If δ = sqrt(32/27) = 147.067c, we have a permutation of 7L3s (LsLsLLLsLL).
If sqrt(32/27) < δ < 9/8, we have 5L2m3s (LsLsLmLsLm).
In the degenerate case δ = 9/8, we have the ternary scale 5L2s diatonic MOS LLLsLLs.

Offsets between 32/27 and 81/64 yield dipentic scales of the pattern abcdcbabcb.

In the degenerate case δ = 32/27, then b > d > c > a = 0 and we have the non-dipentic ternary scale LsmsLLsL.
If 32/27 < δ < sqrt(729/512), then b > d > c > a > 0.
The offset δ = sqrt(729/512) = 305.865c implies b > d = c > a and we have the ternary scale sLmmmLsLmL.
If sqrt(729/512) < δ < sqrt(3/2), then b > c > d > a > 0.
The offset δ = sqrt(3/2) = 350.978c implies b = c > d = a > 0 and yields the 3L7s MOS sLLsLLsLLL.
If sqrt(3/2) < δ < sqrt(128/81), then c > b > a > d > 0.
The offset δ = sqrt(128/81) = 396.090c implies c > b = a > d > 0 and yields the ternary scale mmLsLmmmLm.
If sqrt(128/81) < δ < 81/64, then c > a > b > d > 0.
In the degenerate case δ = 81/64, c > a > b > d = 0 and we have msLLsmsLs.

Offsets between 4/3 and 1\\2 yield dipentic scales of the pattern ababcbabad.

In the degenerate case δ = 4/3, d > b > c > a = 0, and we obtain a non-dipentic ternary scale mmsmmL.
If 4/3 < δ < sqrt(243/128), then d > b > c > a > 0.
The offset δ = sqrt(243/128) = 554.888c implies d > b = c > a > 0 and yields the ternary scale smsmmmsmsL.
If sqrt(243/128) < δ < 1\\2, then d > c > b > a > 0.
If δ = 1\\2, then d = c > b = a > 0, and we get the 2L8s MOS ssssLssssL.

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