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User:Eufalesio/Harmonic Segments Along Fifths

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Harmonic Segments Along Fifths (HSA5) are a type of periodic scales 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 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, they 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, so 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.

  • Insert intervals within the subgroup of the harmonic segment (Subgroup-extended HSA5) [seHSA5], which is an arbitrary choice that depending on the HSA5, will be easier or harder to justify.
  • 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.

List of HSA5s

HSA5 1 - Pyth Trial

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

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.

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.

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 detemper in the 2.3.5.7.13 subgroup. The only valid heSAH5 is with 17/12. Reasonable seHSA5s include 7/5, 13/9, 45/32.

heHSA5 4
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

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.

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, and the last heHSA5 to be a constant structure at all. 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
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