Eufalesio: Public birth, yet with things to polish

2026-10-15T21:52:26Z

Public birth, yet with things to polish

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Harmonic segments along fifths (HSAF) 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 [[tetrachord]]<nowiki/>s 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, the delta signature of any HSAF k 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.

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 structure]]<nowiki/>s.
* 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 [...].

<nowiki>*</nowiki>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 ===
<code>6:8:9:12</code>. 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]].
{| class="wikitable" data-darkreader-inline-color=""
!4/3
!3/2
! rowspan="3" |2
|-
|9/8
|3/2
|-
|4/3
|16/9
|}

=== HSAF 2 - Septimal minor pentatonic ===
<code>12:14:16:18:21:24</code>. 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).
{| class="wikitable" data-darkreader-inline-color=""
!7/6
!4/3
!3/2
!7/4
! rowspan="5" |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 ===
<code>18:20:22:24:27:30:33:36</code>. 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).
{| class="wikitable" data-darkreader-inline-color=""
!'''10/9'''
!'''11/9'''
!'''4/3'''
!'''3/2'''
!'''5/3'''
!'''11/6'''
! rowspan="7" |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 ===
<code>24:26:28:30:32:34:36:39:42:45:48</code>. 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]]<nowiki/>detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale.
{| class="wikitable" data-darkreader-inline-color=""
!'''13/12'''
!'''7/6'''
!'''5/4'''
!'''4/3'''
!'''17/12'''
!'''3/2'''
!'''13/8'''
!'''7/4'''
!'''15/8'''
! rowspan="10" |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 ===
<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>. 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.
{| class="wikitable" data-darkreader-inline-color=""
!16/15
!17/15
!6/5
!19/15
!4/3
!7/5
!3/2
!8/5
!17/10
!9/5
!19/10
! rowspan="12" |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 ===
<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>. 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]]<nowiki/>detemper in the 2.3.5.7.11.19.23 subgroup, and [[nautilus]] detemper ([[1L 12s]] 6|6 and [[14L 1s]] 6|8).
{| class="wikitable" data-darkreader-inline-color=""
!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
! rowspan="15" |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 ===
<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>'''.''' 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).
{| class="wikitable" data-darkreader-inline-color=""
!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
! rowspan="17" |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 ===
<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>. 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.
{| class="wikitable" data-darkreader-inline-color=""
!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
! rowspan="20" |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.

Harmonic segment along fifths - Revision history

Eufalesio: Public birth, yet with things to polish