User:Eufalesio/Harmonic Segments Along Fifths: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Eufalesio (talk | contribs)
autopatrolled
885 edits
Hotfixes!
Tags: Visual edit Mobile edit Mobile web edit Advanced mobile edit
Eufalesio (talk | contribs)
autopatrolled
885 edits
HSA5 to HSAF
Line 1: Line 1:
{{Editable user page}}
{{Editable user page}}


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 [[tetrachord]]<nowiki/>s 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.
Harmonic Segments Along Fifths (HSAF) [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 [[tetrachord]]<nowiki/>s 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 ==
== Definitions ==
For a scale to be a HSA5, it must satisfy the following:
For a scale to be a HSAF, it must satisfy the following:


* Its [[period]] is always the octave.
* Its [[period]] is always the octave.
Line 10: Line 10:
* Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment repeats from 3/2 onwards.
* 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.
These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSAF are a countable set of scales, and thus can be indexed.


== Properties ==
== Properties ==


* Harmonic segments conforming HSA5s up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
* The kth HSA5 is a subset of the 6k::12k harmonic segment.
* The kth HSAF 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.
* 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.


<nowiki>*</nowiki>except the first
<nowiki>*</nowiki>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:
The HSAFs beyond the 3rd will have noticeable gaps between 4/3 and 3/2. A HSAF has the 9/8 step between 4/3 and 3/2 that never shrinks in size, creating very lopsided scales with fairly complex HSAFs 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.
* Insert intervals within the subgroup of the harmonic segment (Subgroup-extended HSAF) [seHSAF] so that the step variety is minimized. This is an arbitrary choice that depends on the HSAF.
* 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.
* Extend the main harmonic segment beyond 4/3 (Harmonically-extended HSAF) [heHSAF] until the segment between 4/3 and 3/2 has fairly well spaced steps. When the HSAFs is even, this is trivial. When the HSAF 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.
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 HSAF 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.  
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 HSA5s 1-8 ==
== List of HSAFs 1-8 ==
For every provided HSA5, the [[EFR]] and used segment are written out explicitly.
For every provided HSAF, the [[EFR]] and used segment are written out explicitly.


=== HSA5 1 - Pythagorean Trial ===
=== HSAF 1 - Pythagorean Trial ===
<code>6:8:9:12</code>. Harmonic segment: '''3:4'''.  
<code>6:8:9:12</code>. 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]].
First meaningful HSAF albeit a trivial case, and the only HSAF that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]].
{| class="wikitable"
{| class="wikitable"
!4/3
!4/3
Line 48: Line 48:
|}
|}


=== HSA5 2 - ''Zontatonic'' ===
=== HSAF 2 - ''Zontatonic'' ===
<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.  
<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.  


First usable HSA5, the first prime HSA5, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
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).
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).
Line 82: Line 82:
|}
|}


=== HSA5 3 - ''Íegmul'' ===
=== HSAF 3 - ''Íegmul'' ===
<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.  
<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.  


Second prime HSA5, and the last one to not need extensions.  
Second prime HSAF, 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).
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).
Line 140: Line 140:
|}
|}


=== HSA5 4 - ''Ngwóoghe'' ===
=== HSAF 4 - ''Ngwóoghe'' ===
<code>24:26:28:30:32:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.
<code>24:26:28:30:32:36:39:42:45:48</code>. 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.
First composite HSAF, containing the 2nd as a subset, and the first one to need extensions. The only valid heSAH5 is [[17/12]]. Reasonable seHSAFs 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.
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.
{| class="wikitable" data-darkreader-inline-color=""
{| class="wikitable" data-darkreader-inline-color=""
! colspan="9" |HSA5 4
! colspan="9" |HSAF 4
|-
|-
!13/12
!13/12
Line 232: Line 232:
|}
|}
{| class="wikitable" data-darkreader-inline-color=""
{| class="wikitable" data-darkreader-inline-color=""
! colspan="10" |heHSA5 4 (17/12)
! colspan="10" |heHSAF 4 (17/12)
|-
|-
!'''13/12'''
!'''13/12'''
Line 336: Line 336:
|}
|}


=== HSA5 5 ===
=== HSAF 5 ===
<code>30:32:34:36:38:40:45:48:51:54:57:60</code>. Harmonic segment: '''12::16'''.
<code>30:32:34:36:38:40:45:48:51:54:57:60</code>. 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.
'''15::20'''. It is the third prime HSAF, and the first HSAF not to be a constant structure without extensions. The only valid heSAH5 is with [[7/5]]. A very natural and reasonable seHSAF is 17/12, which fits almost perfectly inside 4/3 and 3/2, though other usable seHSAFs 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.
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"
{| class="wikitable"
! colspan="12" |heHSA5 5 (7/5)
! colspan="12" |heHSAF 5 (7/5)
|-
|-
!16/15
!16/15
Line 490: Line 490:
|<small>36/19</small>
|<small>36/19</small>
|-
|-
! colspan="12" |seHSA5 5 (17/12)
! colspan="12" |seHSAF 5 (17/12)
|-
|-
!16/15
!16/15
Line 638: Line 638:
|}
|}


=== HSA5 6 ===
=== HSAF 6 ===
<code>36:38:40:42:44:46:48:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.
<code>36:38:40:42:44:46:48:54:57:60:63:66:69:72</code>. 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).
Last HSAF to be a constant structure without extensions. It and its heHSAF 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).
{| class="wikitable"
{| class="wikitable"
! colspan="15" |heHSA5 6 (25/18, 13/9)
! colspan="15" |heHSAF 6 (25/18, 13/9)
|-
|-
!19/18
!19/18
Line 872: Line 872:
|}
|}


=== HSA5 7 ===
=== HSAF 7 ===
<code>42:44:46:48:50:52:54:56:63:66:69:72:75:78:81:84</code>'''.''' Harmonic segment: '''21::28'''.  
<code>42:44:46:48:50:52:54:56:63:66:69:72:75:78:81:84</code>'''.''' 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.  
The fourth prime HSAF. The most reasonable heHSAF with 29/21 and 10/7 contains 7::14 as a subset. A good seHSAFs is with 11/8 and 13/9. Its chiral version 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. Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15|1 10b).
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).
{| class="wikitable"
{| class="wikitable"
! colspan="17" |heHSA5 7 (29/21, 10/7)
! colspan="17" |heHSAF 7 (29/21, 10/7)
|-
|-
!22/21
!22/21
Line 1,172: Line 1,172:
|}
|}


=== HSA5 8 - ''Veljeuwof'' ===
=== HSAF 8 - ''Veljeuwof'' ===
<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.'''
<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 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.
It contains HSAF 2, HSAF 4 and as subsets, and the heHSAF also contains [[8afdo]]. 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.


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.
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 HSAF 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSAF.
{| class="wikitable"
{| class="wikitable"
! colspan="20" |heHSA5 8 (11/8, 17/12, 35/24)
! colspan="20" |heHSAF 8 (11/8, 17/12, 35/24)
|-
|-
!25/24
!25/24
Line 1,583: Line 1,583:
|}
|}


== Beyond HSA5 8 ==
== Beyond HSAF 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]].
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]].

Revision as of 10:53, 11 October 2026

This user page is editable by any wiki editor.

As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages.

However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits).

Harmonic Segments Along Fifths (HSAF) [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 HSAF, 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 HSAF 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.
  • The kth HSAF 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 HSAFs per se. They are only the chiral version of the HSAF.

*except the first

The HSAFs beyond the 3rd will have noticeable gaps between 4/3 and 3/2. A HSAF has the 9/8 step between 4/3 and 3/2 that never shrinks in size, creating very lopsided scales with fairly complex HSAFs 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 HSAF) [seHSAF] so that the step variety is minimized. This is an arbitrary choice that depends on the HSAF.
  • Extend the main harmonic segment beyond 4/3 (Harmonically-extended HSAF) [heHSAF] until the segment between 4/3 and 3/2 has fairly well spaced steps. When the HSAFs is even, this is trivial. When the HSAF 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 HSAF itself.

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.

HSAF 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, 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

HSAF 4 - Ngwóoghe

24:26:28:30:32:36:39:42:45:48. Harmonic segment: 12::16.

First composite HSAF, containing the 2nd as a subset, and the first one to need extensions. The only valid heSAH5 is 17/12. Reasonable seHSAFs 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.

HSAF 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
heHSAF 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

HSAF 5

30:32:34:36:38:40:45:48:51:54:57:60. Harmonic segment: 12::16.

15::20. It is the third prime HSAF, and the first HSAF not to be a constant structure without extensions. The only valid heSAH5 is with 7/5. A very natural and reasonable seHSAF is 17/12, which fits almost perfectly inside 4/3 and 3/2, though other usable seHSAFs 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.

heHSAF 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
seHSAF 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

HSAF 6

36:38:40:42:44:46:48:54:57:60:63:66:69:72. Harmonic segment: 18::24.

Last HSAF to be a constant structure without extensions. It and its heHSAF 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).

heHSAF 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

HSAF 7

42:44:46:48:50:52:54:56:63:66:69:72:75:78:81:84. Harmonic segment: 21::28.

The fourth prime HSAF. The most reasonable heHSAF with 29/21 and 10/7 contains 7::14 as a subset. A good seHSAFs is with 11/8 and 13/9. Its chiral version 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. Its extension can be seen as a strange liese extension detemper (2L 15s 15|1 10b).

heHSAF 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

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, HSAF 4 and as subsets, and the heHSAF also contains 8afdo. 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.

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 HSAF 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSAF.

heHSAF 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 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.

Retrieved from "https://en.xen.wiki/index.php?title=User:Eufalesio/Harmonic_Segments_Along_Fifths&oldid=239348"