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User:Eufalesio/Harmonic Segments Along Fifths

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Revision as of 22:26, 3 September 2026 by Eufalesio (talk | contribs) (Final touches)

Harmonic Segments Along Fifths (HSA5) are a type of periodic scale devised by Eufalesio, built by taking a specific harmonic series segment that ends in 4/3 and repeating times 3/2. They are a unique mix of tetrachords and overtone scales that exhibit omnitetrachordality to some extent, and a type of NEJI (Near-equivalent Just Intonation). How to approach them musically is not within the scope of this article.

Definitions

For a scale to be a HSA5, it must satisfy the following:

  • Its period is always the octave.
  • Its prime mode always contains 4/3 and 3/2.
  • Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment repeats from 3/2 onwards.

These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSA5 are a countable set of scales, and thus can be indexed. Because they are formed from harmonic series segments, all except the first are are chiral scales, whose counterparts are built from subharmonic series segments.

Properties

Except the first, the harmonic segments conforming HSA5s up to the 13th are strictly proper constant structures.

The HSA5s beyond the 3rd will have noticeable gaps between 4/3 and 3/2. An HSA5 is said to need extensions if the smallest step of the harmonic series segment is smaller than (9/8)2/3 and this can be done two ways:

  • Insert intervals within the subgroup of the harmonic segment (Subgroup-extended HSA5) [seHSA5] so that the step variety is minimized. This is an arbitrary choice that depends on the HSA5.
  • Extend the main harmonic segment beyond 4/3 (Harmonically-extended HSA5) [heHSA5] until the segment between 4/3 and 3/2 has fairly well spaced steps. When the HSA5s is even, this is trivial.

Extensions are fairly arbitrary and not an absolute necessity, so the use of one over the other is not something that is entirely justifiable; it is as much as an artistic choice as the choice of HSA5 itself.

List of HSA5s

HSA5 1 - Pythagorean Trial

3::4. First meaningful HSA5 albeit a trivial case, and the only HSA5 that is also a MOS scale (2L 1s), and thus achiral.

4/3 3/2 2
9/8 3/2
4/3 16/9

HSA5 2 - Zontatonic

6::8. It is the first usable HSA5, the first prime HSA5, and the only one that is Strict variety 3 and a generator sequence. It is naturally a 5edo detemper in the 2.3.7 subgroup, or otherwise an archy detemper, 2L 3s 3|1 scale.

7/6 4/3 3/2 7/4 2
8/7 9/7 3/2 12/7
9/8 21/16 3/2 7/4
7/6 4/3 14/9 16/9
8/7 4/3 32/21 12/7

HSA5 3 - Íegmul

9::12. It is the second prime HSA5, and the last one to not need extensions. It is naturally a 7edo detemper in the 2.3.5.11 subgroup, or otherwise a dicot/mothra detemper, 3L 4s 5|1 #6 scale. (1-indexed).

10/9 11/9 4/3 3/2 5/3 11/6 2
11/10 6/5 27/20 3/2 33/20 9/5
12/11 27/22 15/11 3/2 18/11 20/11
9/8 5/4 11/8 3/2 5/3 11/6
10/9 11/9 4/3 40/27 44/27 16/9
11/10 6/5 4/3 22/15 8/5 9/5
12/11 40/33 4/3 16/11 18/11 20/11

HSA5 4 - Ngwóoghe

12::16. It is first composite HSA5, containing the 2nd as a subset, and the first one to need extensions. It is naturally a 10edo or detemper in the 2.3.5.7.13 subgroup, or otherwise a negri detemper, 1L 8s 4|4 scale. The only valid heSAH5 is 17/12. Reasonable seHSA5s include 7/5, 13/9, 45/32.

HSA5 4 heHSA5 4
13/12 7/6 5/4 4/3 3/2 13/8 7/4 15/8 2 13/12 7/6 5/4 4/3 17/12 3/2 13/8 7/4 15/8 2
14/13 15/13 16/13 18/13 3/2 21/13 45/26 24/13 14/13 15/13 16/13 17/13 18/13 3/2 21/13 45/26 24/13
15/14 8/7 9/7 39/28 3/2 45/28 12/7 13/7 15/14 8/7 17/14 9/7 39/28 3/2 45/28 12/7 13/7
16/15 6/5 13/10 7/5 3/2 8/5 26/15 28/15 16/15 17/15 6/5 13/10 7/5 3/2 8/5 26/15 28/15
9/8 39/32 21/16 45/32 3/2 13/8 7/4 15/8 17/16 9/8 39/32 21/16 45/32 3/2 13/8 7/4 15/8
13/12 7/6 5/4 4/3 13/9 14/9 5/3 16/9 18/17 39/34 21/17 45/34 24/17 26/17 28/17 30/17 32/17
14/13 15/13 16/13 4/3 56/39 20/13 64/39 24/13 13/12 7/6 5/4 4/3 13/9 14/9 5/3 16/9 17/9
15/14 8/7 26/21 4/3 10/7 32/21 12/7 13/7 14/13 15/13 16/13 4/3 56/39 20/13 64/39 68/39 24/13
16/15 52/45 56/45 4/3 64/45 8/5 26/15 28/15 15/14 8/7 26/21 4/3 10/7 32/21 34/21 12/7 13/7
16/15 52/45 56/45 4/3 64/45 68/45 8/5 26/15 28/15

HSA5 5

15::20. It is the third prime HSA5, and the first HSA5 not to be a constant structure without extensions. It is naturally a very close 12edo detemper in the 2.3.5.17.19 subgroup. The only valid heSAH5 is with 7/5. A very natural and reasonable seHSA5 is 17/12, which fits almost perfectly inside 4/3 and 3/2, though other usable seHSA5 is 64/45.

heHSA5 5
16/15 17/15 6/5 19/15 4/3 7/5 3/2 8/5 17/10 9/5 19/10 2
17/16 9/8 19/16 5/4 21/16 45/32 3/2 51/32 27/16 57/32 15/8
18/17 19/17 20/17 21/17 45/34 24/17 3/2 27/17 57/34 30/17 32/17
19/18 10/9 7/6 5/4 4/3 17/12 3/2 19/12 5/3 16/9 17/9
20/19 21/19 45/38 24/19 51/38 27/19 3/2 30/19 32/19 34/19 36/19
21/20 9/8 6/5 51/40 27/20 57/40 3/2 8/5 17/10 9/5 19/10
15/14 8/7 17/14 9/7 19/14 10/7 32/21 34/21 12/7 38/21 40/21
16/15 17/15 6/5 19/15 4/3 64/45 68/45 8/5 76/45 16/9 28/15
17/16 9/8 19/16 5/4 4/3 17/12 3/2 19/12 5/3 7/4 15/8
18/17 19/17 20/17 64/51 4/3 24/17 76/51 80/51 28/17 30/17 32/17
19/18 10/9 32/27 34/27 4/3 38/27 40/27 14/9 5/3 16/9 17/9
20/19 64/57 68/57 24/19 4/3 80/57 28/19 30/19 32/19 34/19 36/19
seHSA5 5 (17/12)
16/15 17/15 6/5 19/15 4/3 17/12 3/2 8/5 17/10 9/5 19/10 2
17/16 9/8 19/16 5/4 85/64 45/32 3/2 51/32 27/16 57/32 15/8
18/17 19/17 20/17 5/4 45/34 24/17 3/2 27/17 57/34 30/17 32/17
19/18 10/9 85/72 5/4 4/3 17/12 3/2 19/12 5/3 16/9 17/9
20/19 85/76 45/38 24/19 51/38 27/19 3/2 30/19 32/19 34/19 36/19
17/16 9/8 6/5 51/40 27/20 57/40 3/2 8/5 17/10 9/5 19/10
18/17 96/85 6/5 108/85 114/85 24/17 128/85 8/5 144/85 152/85 32/17
16/15 17/15 6/5 19/15 4/3 64/45 68/45 8/5 76/45 16/9 17/9
17/16 9/8 19/16 5/4 4/3 17/12 3/2 19/12 5/3 85/48 15/8
18/17 19/17 20/17 64/51 4/3 24/17 76/51 80/51 5/3 30/17 32/17
19/18 10/9 32/27 34/27 4/3 38/27 40/27 85/54 5/3 16/9 17/9
20/19 64/57 68/57 24/19 4/3 80/57 85/57 30/19 32/19 34/19 36/19

HSA5 6

18::24. It is the last HSA5 to be a constant structure without extensions. It is a 15edo detemper in the 2.3.5.7.11.19.23 subgroup.

heHSA5 6
19/18 10/9 7/6 11/9 23/18 4/3 25/18 13/9 3/2 19/12 5/3 7/4 11/6 23/12 2
20/19 21/19 22/19 23/19 24/19 25/19 26/19 27/19 3/2 30/19 63/38 33/19 69/38 36/19
21/20 11/10 23/20 6/5 5/4 13/10 27/20 57/40 3/2 63/40 33/20 69/40 9/5 19/10
22/21 23/21 8/7 25/21 26/21 9/7 19/14 10/7 3/2 11/7 23/14 12/7 38/21 40/21
23/22 12/11 25/22 13/11 27/22 57/44 15/11 63/44 3/2 69/44 18/11 19/11 20/11 21/11
24/23 25/23 26/23 27/23 57/46 30/23 63/46 33/23 3/2 36/23 38/23 40/23 42/23 44/23
25/24 13/12 9/8 19/16 5/4 21/16 11/8 23/16 3/2 19/12 5/3 7/4 11/6 23/12
26/25 27/25 57/50 6/5 63/50 33/25 69/50 36/25 38/25 8/5 42/25 44/25 46/25 48/25
27/26 57/52 15/13 63/52 33/26 69/52 18/13 19/13 20/13 21/13 22/13 23/13 24/13 25/13
19/18 10/9 7/6 11/9 23/18 4/3 38/27 40/27 14/9 44/27 46/27 16/9 50/27 52/27
20/19 21/19 22/19 23/19 24/19 4/3 80/57 28/19 88/57 92/57 32/19 100/57 104/57 36/19
21/20 11/10 23/20 6/5 19/15 4/3 7/5 22/15 23/15 8/5 5/3 26/15 9/5 19/10
22/21 23/21 8/7 76/63 80/63 4/3 88/63 92/63 32/21 100/63 104/63 12/7 38/21 40/21
23/22 12/11 38/33 40/33 14/11 4/3 46/33 16/11 50/33 52/33 18/11 19/11 20/11 21/11
24/23 76/69 80/69 28/23 88/69 4/3 32/23 100/69 104/69 36/23 38/23 40/23 42/23 44/23

HSA5 7

21::28. The fourth prime HSA5. The most reasonable heHSA5 with 29/21 and 10/7 contains 7::14 as a subset. It is naturally a 17edo detemper in the no-17,19-29-limit subgroup. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.

heHSA5 7
22/21 23/21 8/7 25/21 26/21 9/7 4/3 29/21 10/7 3/2 11/7 23/14 12/7 25/14 13/7 27/14 2
23/22 12/11 25/22 13/11 27/22 14/11 29/22 15/11 63/44 3/2 69/44 18/11 75/44 39/22 81/44 21/11
24/23 25/23 26/23 27/23 28/23 29/23 30/23 63/46 33/23 3/2 36/23 75/46 39/23 81/46 42/23 44/23
25/24 13/12 9/8 7/6 29/24 5/4 21/16 11/8 23/16 3/2 25/16 13/8 27/16 7/4 11/6 23/12
26/25 27/25 28/25 29/25 6/5 63/50 33/25 69/50 36/25 3/2 39/25 81/50 42/25 44/25 46/25 48/25
27/26 14/13 29/26 15/13 63/52 33/26 69/52 18/13 75/52 3/2 81/52 21/13 22/13 23/13 24/13 25/13
28/27 29/27 10/9 7/6 11/9 23/18 4/3 25/18 13/9 3/2 14/9 44/27 46/27 16/9 50/27 52/27
29/28 15/14 9/8 33/28 69/56 9/7 75/56 39/28 81/56 3/2 11/7 23/14 12/7 25/14 13/7 27/14
30/29 63/58 33/29 69/58 36/29 75/58 39/29 81/58 42/29 44/29 46/29 48/29 50/29 52/29 54/29 56/29
21/20 11/10 23/20 6/5 5/4 13/10 27/20 7/5 22/15 23/15 8/5 5/3 26/15 9/5 28/15 29/15
22/21 23/21 8/7 25/21 26/21 9/7 4/3 88/63 92/63 32/21 100/63 104/63 12/7 16/9 116/63 40/21
23/22 12/11 25/22 13/11 27/22 14/11 4/3 46/33 16/11 50/33 52/33 18/11 56/33 58/33 20/11 21/11
24/23 25/23 26/23 27/23 28/23 88/69 4/3 32/23 100/69 104/69 36/23 112/69 116/69 40/23 42/23 44/23
25/24 13/12 9/8 7/6 11/9 23/18 4/3 25/18 13/9 3/2 14/9 29/18 5/3 7/4 11/6 23/12
26/25 27/25 28/25 88/75 92/75 32/25 4/3 104/75 36/25 112/75 116/75 8/5 42/25 44/25 46/25 48/25
27/26 14/13 44/39 46/39 16/13 50/39 4/3 18/13 56/39 58/39 20/13 21/13 22/13 23/13 24/13 25/13
28/27 88/81 92/81 32/27 100/81 104/81 4/3 112/81 116/81 40/27 14/9 44/27 46/27 16/9 50/27 52/27

HSA5 8

24::32. It contains HSA5 2, HSA5 4 and 8afdo as subsets. As a superset of HSA5 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35. It is not an easy edo detemper, just a harmonic interpolation of HSA5 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSA5.

heHSA5 8
25/24 13/12 9/8 7/6 29/24 5/4 31/24 4/3 11/8 17/12 35/24 3/2 25/16 13/8 27/16 7/4 29/16 15/8 31/16 2
26/25 27/25 28/25 29/25 6/5 31/25 32/25 33/25 34/25 7/5 36/25 3/2 39/25 81/50 42/25 87/50 9/5 93/50 48/25
27/26 14/13 29/26 15/13 31/26 16/13 33/26 17/13 35/26 18/13 75/52 3/2 81/52 21/13 87/52 45/26 93/52 24/13 25/13
28/27 29/27 10/9 31/27 32/27 11/9 34/27 35/27 4/3 25/18 13/9 3/2 14/9 29/18 5/3 31/18 16/9 50/27 52/27
29/28 15/14 31/28 8/7 33/28 17/14 5/4 9/7 75/56 39/28 81/56 3/2 87/56 45/28 93/56 12/7 25/14 13/7 27/14
30/29 31/29 32/29 33/29 34/29 35/29 36/29 75/58 39/29 81/58 42/29 3/2 45/29 93/58 48/29 50/29 52/29 54/29 56/29
31/30 16/15 11/10 17/15 7/6 6/5 5/4 13/10 27/20 7/5 29/20 3/2 31/20 8/5 5/3 26/15 9/5 28/15 29/15
32/31 33/31 34/31 35/31 36/31 75/62 39/31 81/62 42/31 87/62 45/31 3/2 48/31 50/31 52/31 54/31 56/31 58/31 60/31
33/32 17/16 35/32 9/8 75/64 39/32 81/64 21/16 87/64 45/32 93/64 3/2 25/16 13/8 27/16 7/4 29/16 15/8 31/16
34/33 35/33 12/11 25/22 13/11 27/22 14/11 29/22 15/11 31/22 16/11 50/33 52/33 18/11 56/33 58/33 20/11 62/33 64/33
35/34 18/17 75/68 39/34 81/68 21/17 87/68 45/34 93/68 24/17 25/17 26/17 27/17 28/17 29/17 30/17 31/17 32/17 33/17
36/35 15/14 39/35 81/70 6/5 87/70 9/7 93/70 48/35 10/7 52/35 54/35 8/5 58/35 12/7 62/35 64/35 66/35 68/35
25/24 13/12 9/8 7/6 29/24 5/4 31/24 4/3 25/18 13/9 3/2 14/9 29/18 5/3 31/18 16/9 11/6 17/9 35/18
26/25 27/25 28/25 29/25 6/5 31/25 32/25 4/3 104/75 36/25 112/75 116/75 8/5 124/75 128/75 44/25 136/75 28/15 48/25
27/26 14/13 29/26 15/13 31/26 16/13 50/39 4/3 18/13 56/39 58/39 20/13 62/39 64/39 22/13 68/39 70/39 24/13 25/13
28/27 29/27 10/9 31/27 32/27 100/81 104/81 4/3 112/81 116/81 40/27 124/81 128/81 44/27 136/81 140/81 16/9 50/27 52/27
29/28 15/14 31/28 8/7 25/21 26/21 9/7 4/3 29/21 10/7 31/21 32/21 11/7 34/21 5/3 12/7 25/14 13/7 27/14
30/29 31/29 32/29 100/87 104/87 36/29 112/87 4/3 40/29 124/87 128/87 44/29 136/87 140/87 48/29 50/29 52/29 54/29 56/29
31/30 16/15 10/9 52/45 6/5 56/45 58/45 4/3 62/45 64/45 22/15 68/45 14/9 8/5 5/3 26/15 9/5 28/15 29/15
32/31 100/93 104/93 36/31 112/93 116/93 40/31 4/3 128/93 44/31 136/93 140/93 48/31 50/31 52/31 54/31 56/31 58/31 60/31

Beyond HSA5 8

There are infinite HSA5s as stated before, but HSA5s grow increasingly more complex in prime palette and in interval count the more times the fourth is split. As such, we present here only the first 8 HSA5s as their palettes fit within the 31-limit.

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