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	'''WIP'''		'''WIP'''

	Harmonic Segments Along Fifths (HSA5) are a type of [[Periodic scale\|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 [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).		Harmonic Segments Along Fifths (HSA5) are a type of [[Periodic scale\|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 [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).

	== Definitions ==		== Definitions ==
Line 13:		Line 15:

	== Properties ==		== Properties ==
	Except the first, the harmonic segments conforming HSA5s up to the 13th are [[strictly proper]] [[constant structure]]<nowiki/>s, so the		Except the first, the harmonic segments conforming HSA5s up to the 13th are [[strictly proper]] [[constant structure]]<nowiki/>s, 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.		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.

Eufalesio: Birth (WIP)

2026-08-23T20:10:48Z

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'''WIP'''

Harmonic Segments Along Fifths (HSA5) are a type of [[Periodic scale|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 [[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 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 structure]]<nowiki/>s, 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]]).
{| class="wikitable"
|-
!4/3
!3/2
! rowspan="3" |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]].
{| 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
|}

=== 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]].
{| 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
|}

=== 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.
{| class="wikitable" data-darkreader-inline-color=""
! colspan="10" |heHSA5 4
|-
!'''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
|}

=== 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.
{| class="wikitable"
|-
! colspan="12" |heHSA5 5
|-
!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
|-
! colspan="12" |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
! rowspan="12" |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.
{| class="wikitable"
|-
! colspan="15" |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
! 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
|}

User:Eufalesio/Harmonic Segments Along Fifths - Revision history

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