User:Eufalesio/Harmonic Segments Along Fifths
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Harmonic Segments Along Fifths (HSA5) [sing. a Harmonic Segment along Fifths] are a type of periodic scale 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.
Properties
- Harmonic segments conforming HSA5s up to the 13th* are strictly proper constant structures.
- The kth HSA5 is a subset of the 6k::12k harmonic segment.
- Because they are formed from harmonic series segments, they are chiral scales*, whose left-handed counterparts are built from subharmonic series segments. The left-handed versions are not HSA5s per se. They are only the chiral version of the HSA5.
*except the first
The HSA5s beyond the 3rd will have noticeable gaps between 4/3 and 3/2. A HSA5 has the 9/8 step between 4/3 and 3/2 that never shrinks in size, creating very lopsided scales with fairly complex HSA5s down the line. The gap between 4/3 and 3/2 can be filled in 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. When the HSA5 is odd, you have two "valid" choices.
These are called extensions. They are fairly arbitrary and not an absolute necessity, so the use of one over the other is not something that is entirely justifiable; it may be as much as an artistic choice as the choice of HSA5 itself.
Simple HSA5s can be visualized as edo or MOS detempers for simple HSA5s, which makes them particularly useful either for NEJI targets or essential tempering.
List of HSA5s 1-8
For every provided HSA5, the EFR and used segment are written out explicitly.
HSA5 1 - Pythagorean Trial
6:8:9:12. Harmonic segment: 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
12:14:16:18:21:24. Harmonic segment: 6::8.
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, and either a semaphore (1L 4s 2|2) or archy detemper (2L 3s 3|1).
| 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
18:20:22:24:27:30:33:36. Harmonic segment: 9::12.
Second prime HSA5, and the last one to not need extensions.
It is naturally a 7edo detemper in the 2.3.5.11 subgroup, and either a porcupine (1L 6s 3|3) or dicot/mothra detemper (3L 4s 5|1 #5).
| 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
24:26:28:30:32:36:39:42:45:48. Harmonic segment: 12::16.
First composite HSA5, containing the 2nd as a subset, and the first one to need extensions. The only valid heSAH5 is 17/12. Reasonable seHSA5s include 7/5, 13/9, 45/32.
The raw scale is naturally a 10edo detemper in the 2.3.5.7.13 subgroup, and a negri detemper (1L 8s 4|4). The extensions are naturally a 10edo detemper and a pajarous detemper, 2L 8s 6|2 #7 scale.
| HSA5 4 | ||||||||
|---|---|---|---|---|---|---|---|---|
| 13/12 | 7/6 | 5/4 | 4/3 | 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 | |
| 15/14 | 8/7 | 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 | |
| 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 | |
| 14/13 | 15/13 | 16/13 | 4/3 | 56/39 | 20/13 | 64/39 | 24/13 | |
| 15/14 | 8/7 | 26/21 | 4/3 | 10/7 | 32/21 | 12/7 | 13/7 | |
| 16/15 | 52/45 | 56/45 | 4/3 | 64/45 | 8/5 | 26/15 | 28/15 | |
| heHSA5 4 (17/12) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| 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 | 17/13 | 18/13 | 3/2 | 21/13 | 45/26 | 24/13 | |
| 15/14 | 8/7 | 17/14 | 9/7 | 39/28 | 3/2 | 45/28 | 12/7 | 13/7 | |
| 16/15 | 17/15 | 6/5 | 13/10 | 7/5 | 3/2 | 8/5 | 26/15 | 28/15 | |
| 17/16 | 9/8 | 39/32 | 21/16 | 45/32 | 3/2 | 13/8 | 7/4 | 15/8 | |
| 18/17 | 39/34 | 21/17 | 45/34 | 24/17 | 26/17 | 28/17 | 30/17 | 32/17 | |
| 13/12 | 7/6 | 5/4 | 4/3 | 13/9 | 14/9 | 5/3 | 16/9 | 17/9 | |
| 14/13 | 15/13 | 16/13 | 4/3 | 56/39 | 20/13 | 64/39 | 68/39 | 24/13 | |
| 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
30:32:34:36:38:40:45:48:51:54:57:60. Harmonic segment: 12::16.
15::20. It is the third prime HSA5, and the first HSA5 not to be a constant structure without extensions. 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 seHSA5s are 17/12, 45/32, 64/45.
Its extension is naturally a very close 12edo detemper in the 2.3.5.17.19 subgroup, and a diaschismic detemper (10L 2s 8|2 b4) scale.
| heHSA5 5 (7/5) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 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
36:38:40:42:44:46:48:54:57:60:63:66:69:72. Harmonic segment: 18::24.
Last HSA5 to be a constant structure without extensions. It and its heHSA5 are 15edo detempers in the 2.3.5.7.11.19.23 subgroup, and nautilus detempers (1L 12s 6|6 and 14L 1s 6|8).
| heHSA5 6 (25/18, 13/9) | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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
42:44:46:48:50:52:54:56:63:66:69:72:75:78:81:84. Harmonic segment: 21::28.
The fourth prime HSA5. The most reasonable heHSA5 with 29/21 and 10/7 contains 7::14 as a subset. A good seHSA5s is with 11/8 and 13/9. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.
It is naturally a 17edo detemper in the no-5,17,19,add-25-29-limit subgroup. Its extension can be seen as a strange liese extension detemper (2L 15s 15|1 10b).
| heHSA5 7 (29/21, 10/7) | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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 - Veljeuwof
48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96. Harmonic segment: 24::32.
It contains HSA5 2, HSA5 4 and as subsets, and the heHSA5 also contains 8afdo. As a superset of HSA5 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes.
The raw scale or its extension could be seen as wilsec detempers (negri weak extension), 1L 16 8|8 and 19L 1s 10|9 respectively, but it is a very innacurate realization that only 19edo through patent val can faithfully represent. Thus, it's better seen as just a linear 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 (11/8, 17/12, 35/24) | |||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 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.