User:Eufalesio/Harmonic Segments Along Fifths: Difference between revisions
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* Its prime mode always contains 4/3 and 3/2. | * 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 is repeated transposed by 3/2 until it reaches the octave. | * Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment is repeated transposed by 3/2 until it reaches the octave. | ||
* | * HSAF 1 is 6:8:9:12 with [[delta signature]] +2,+1,+2. Except for HSAF 1, its delta signature is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2. | ||
These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so | These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSAFs are a countable set of scales, and thus can be indexed. | ||
== Properties == | == Properties == | ||
Revision as of 08:41, 14 October 2026
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Harmonic Segments Along Fifths (HSAF) [sing. a Harmonic Segment along Fifths] are a type of octave-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).
Definitions
For a scale to be a HSAF, it must satisfy the following:
- Its period is 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 is repeated transposed by 3/2 until it reaches the octave.
- HSAF 1 is 6:8:9:12 with delta signature +2,+1,+2. Except for HSAF 1, its delta signature is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2.
These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSAFs are a countable set of scales, and thus can be indexed.
Properties
- Harmonic segments conforming HSAFs up to the 13th* are strictly proper constant structures.
- HSAF k is a subset of the 6k::12k harmonic segment.
- HSAF k contains harmonics k–2k as a subset.
- 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 HSAFs per se. They are only the chiral version of the HSAF, which you might call a SSAF, for Subharmonic Segment [...].
*except the first
Simple HSAFs can be visualized as edo or MOS detempers for simple HSAFs, which makes them particularly useful either for NEJI targets or essential tempering.
List of HSAFs 1-8
For every provided HSAF, the EFR and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity.
HSAF/SSAF 1 - Pythagorean Trial
6:8:9:12. Harmonic segment: 3:4.
First meaningful HSAF albeit a trivial case, and the only HSAF that is also a MOS scale (2L 1s), and thus achiral.
| 4/3 | 3/2 | 2 |
|---|---|---|
| 9/8 | 3/2 | |
| 4/3 | 16/9 |
HSAF 2 - Zontatonic
12:14:16:18:21:24. Harmonic segment: 6::8.
First usable HSAF, the first prime HSAF, 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 |
HSAF 3 - Íegmul
18:20:22:24:27:30:33:36. Harmonic segment: 9::12.
Second prime HSAF. 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 |
HSAF 4 - Ngwóoghe
24:26:28:30:32:34:36:39:42:45:48. Harmonic segment: 12::16.
First composite HSAF, containing HSAF 2 as a subset. It is naturally a 10edo detemper in the 2.3.5.7.13 subgroup, a negri detemper (1L 8s 4|4) if you exclude 17/12, and a pajarous detemper, 2L 8s 6|2 #7 scale.
| 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 |
HSAF 5
30:32:34:36:38:40:42:45:48:51:54:57:60. Harmonic segment: 12::16.
15::20. It is the third prime HSAF. 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.
| 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 |
HSAF 6
36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72. Harmonic segment: 18::24.
Last HSAF to be a constant structure, and first HSAF to contain two different (non-1) HSAFs as subsets. It is a 15edo detemper in the 2.3.5.7.11.19.23 subgroup, and nautilus detemper (1L 12s 6|6 and 14L 1s 6|8).
| 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 |
HSAF 7
42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84. Harmonic segment: 21::28.
The fourth prime HSAF. SSAF 7 is surprisingly a subharmonic interpolation of HSAF 3. It is naturally a 17edo detemper in the no-5,17,19,add-25-29-limit subgroup, surfacing as a strange liese extension detemper (2L 15s 15|1 10b).
| 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 |
HSAF 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 HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the 31-limit.
The raw scale or its extension could be seen as a wilsec detemper (negri weak extension) 19L 1s 10|9, but it is a very innacurate realization that only 19edo through patent val can faithfully represent (albeit badly as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of HSAF 4 in the 2.3.5.7.11.13.17.29.31 subgroup.
| 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 HSAF 8
There are infinite HSAFs as stated before, but HSAFs 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 HSAFs as their palettes fit within the 31-limit.