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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.
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 [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).


== Definitions ==
== Definitions ==
For a scale to be a HSAF, 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 the octave.
* 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 repeats from 3/2 onwards.
* 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 HSAF 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 HSAFs are a countable set of scales, and thus can be indexed.


== Properties ==
== Properties ==


* Harmonic segments conforming HSAFs 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 HSAF is a subset of the 6k::12k harmonic segment.
* 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 [...].
* 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
<nowiki>*</nowiki>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.  
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 ==
== List of HSAFs 1-8 ==
For every provided HSAF, the [[EFR]] and used segment are written out explicitly.
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 ===
=== HSAF/SSAF 1 - Pythagorean Trial ===
Line 48: Line 43:
|}
|}


=== HSAF 2 - ''Zontatonic'' ===
=== HSAF 2 - Septimal minor pentatonic ===
<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 HSAF, the first prime HSAF, 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]].
Line 82: Line 77:
|}
|}


=== HSAF 3 - ''Íegmul'' ===
=== HSAF 3 ===
<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 HSAF, and the last one to not need extensions.  
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).  
 
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=""
{| class="wikitable" data-darkreader-inline-color=""
!'''10/9'''
!'''10/9'''
Line 140: Line 133:
|}
|}


=== HSAF 4 - ''Ngwóoghe'' ===
=== HSAF 4 ===
<code>24:26:28:30:32:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.
<code>24:26:28:30:32:34:36:39:42:45:48</code>. 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.
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.
 
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" |HSAF 4
|-
|-
!13/12
!'''13/12'''
!7/6
!'''7/6'''
!5/4
!'''5/4'''
!4/3
!'''4/3'''
!3/2
!'''17/12'''
!13/8
!'''3/2'''
!7/4
!'''13/8'''
!15/8
!'''7/4'''
! rowspan="9" |2
!'''15/8'''
! rowspan="10" |2
|-
|-
|<small>14/13</small>
|<small>14/13</small>
|<small>15/13</small>
|<small>15/13</small>
|<small>16/13</small>
|<small>16/13</small>
|<small>17/13</small>
|<small>18/13</small>
|<small>18/13</small>
|<small>3/2</small>
|<small>3/2</small>
Line 170: Line 162:
|<small>15/14</small>
|<small>15/14</small>
|<small>8/7</small>
|<small>8/7</small>
|<small>17/14</small>
|<small>9/7</small>
|<small>9/7</small>
|<small>39/28</small>
|<small>39/28</small>
Line 178: Line 171:
|-
|-
|<small>16/15</small>
|<small>16/15</small>
|<small>17/15</small>
|<small>6/5</small>
|<small>6/5</small>
|<small>13/10</small>
|<small>13/10</small>
Line 186: Line 180:
|<small>28/15</small>
|<small>28/15</small>
|-
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>9/8</small>
|<small>39/32</small>
|<small>39/32</small>
Line 194: Line 189:
|<small>7/4</small>
|<small>7/4</small>
|<small>15/8</small>
|<small>15/8</small>
|-
|<small>18/17</small>
|<small>39/34</small>
|<small>21/17</small>
|<small>45/34</small>
|<small>24/17</small>
|<small>26/17</small>
|<small>28/17</small>
|<small>30/17</small>
|<small>32/17</small>
|-
|-
|<small>13/12</small>
|<small>13/12</small>
Line 203: Line 208:
|<small>5/3</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|-
|<small>14/13</small>
|<small>14/13</small>
Line 211: Line 217:
|<small>20/13</small>
|<small>20/13</small>
|<small>64/39</small>
|<small>64/39</small>
|<small>68/39</small>
|<small>24/13</small>
|<small>24/13</small>
|-
|-
Line 219: Line 226:
|<small>10/7</small>
|<small>10/7</small>
|<small>32/21</small>
|<small>32/21</small>
|<small>34/21</small>
|<small>12/7</small>
|<small>12/7</small>
|<small>13/7</small>
|<small>13/7</small>
Line 227: Line 235:
|<small>4/3</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>64/45</small>
|<small>68/45</small>
|<small>8/5</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>26/15</small>
|<small>28/15</small>
|<small>28/15</small>
|}
|}
{| class="wikitable" data-darkreader-inline-color=""
 
! colspan="10" |heHSAF 4 (17/12)
=== 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"
|-
|-
!'''13/12'''
!16/15
!'''7/6'''
!17/15
!'''5/4'''
!6/5
!'''4/3'''
!19/15
!'''17/12'''
!4/3
!'''3/2'''
!7/5
!'''13/8'''
!3/2
!'''7/4'''
!8/5
!'''15/8'''
!17/10
! rowspan="10" |2
!9/5
!19/10
! rowspan="12" |2
|-
|-
|<small>14/13</small>
|<small>17/16</small>
|<small>15/13</small>
|<small>9/8</small>
|<small>16/13</small>
|<small>19/16</small>
|<small>17/13</small>
|<small>5/4</small>
|<small>18/13</small>
|<small>21/16</small>
|<small>45/32</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>21/13</small>
|<small>51/32</small>
|<small>45/26</small>
|<small>27/16</small>
|<small>24/13</small>
|<small>57/32</small>
|<small>15/8</small>
|-
|-
|<small>15/14</small>
|<small>18/17</small>
|<small>8/7</small>
|<small>19/17</small>
|<small>17/14</small>
|<small>20/17</small>
|<small>9/7</small>
|<small>21/17</small>
|<small>39/28</small>
|<small>45/34</small>
|<small>24/17</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>45/28</small>
|<small>27/17</small>
|<small>12/7</small>
|<small>57/34</small>
|<small>13/7</small>
|<small>30/17</small>
|<small>32/17</small>
|-
|-
|<small>16/15</small>
|<small>19/18</small>
|<small>17/15</small>
|<small>10/9</small>
|<small>6/5</small>
|<small>7/6</small>
|<small>13/10</small>
|<small>5/4</small>
|<small>7/5</small>
|<small>4/3</small>
|<small>17/12</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>19/12</small>
|<small>26/15</small>
|<small>5/3</small>
|<small>28/15</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|-
|<small>17/16</small>
|<small>20/19</small>
|<small>9/8</small>
|<small>21/19</small>
|<small>39/32</small>
|<small>45/38</small>
|<small>21/16</small>
|<small>24/19</small>
|<small>45/32</small>
|<small>51/38</small>
|<small>27/19</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>13/8</small>
|<small>30/19</small>
|<small>7/4</small>
|<small>32/19</small>
|<small>15/8</small>
|<small>34/19</small>
|<small>36/19</small>
|-
|-
|<small>18/17</small>
|<small>21/20</small>
|<small>39/34</small>
|<small>9/8</small>
|<small>21/17</small>
|<small>6/5</small>
|<small>45/34</small>
|<small>51/40</small>
|<small>24/17</small>
|<small>27/20</small>
|<small>26/17</small>
|<small>57/40</small>
|<small>28/17</small>
|<small>3/2</small>
|<small>30/17</small>
|<small>8/5</small>
|<small>32/17</small>
|<small>17/10</small>
|-
|<small>9/5</small>
|<small>13/12</small>
|<small>19/10</small>
|<small>7/6</small>
|<small>5/4</small>
|<small>4/3</small>
|<small>13/9</small>
|<small>14/9</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|<small>14/13</small>
|<small>15/13</small>
|<small>16/13</small>
|<small>4/3</small>
|<small>56/39</small>
|<small>20/13</small>
|<small>64/39</small>
|<small>68/39</small>
|<small>24/13</small>
|-
|-
|<small>15/14</small>
|<small>15/14</small>
|<small>8/7</small>
|<small>8/7</small>
|<small>26/21</small>
|<small>17/14</small>
|<small>4/3</small>
|<small>9/7</small>
|<small>19/14</small>
|<small>10/7</small>
|<small>10/7</small>
|<small>32/21</small>
|<small>32/21</small>
|<small>34/21</small>
|<small>34/21</small>
|<small>12/7</small>
|<small>12/7</small>
|<small>13/7</small>
|<small>38/21</small>
|<small>40/21</small>
|-
|-
|<small>16/15</small>
|<small>16/15</small>
|<small>52/45</small>
|<small>17/15</small>
|<small>56/45</small>
|<small>6/5</small>
|<small>19/15</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>64/45</small>
|<small>68/45</small>
|<small>68/45</small>
|<small>8/5</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>76/45</small>
|<small>16/9</small>
|<small>28/15</small>
|<small>28/15</small>
|}
=== HSAF 5 ===
<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 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.
{| class="wikitable"
! colspan="12" |heHSAF 5 (7/5)
|-
|-
!16/15
|<small>17/16</small>
!17/15
|<small>9/8</small>
!6/5
|<small>19/16</small>
!19/15
!4/3
!7/5
!3/2
!8/5
!17/10
!9/5
!19/10
! rowspan="12" |2
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>19/16</small>
|<small>5/4</small>
|<small>5/4</small>
|<small>21/16</small>
|<small>4/3</small>
|<small>45/32</small>
|<small>17/12</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>51/32</small>
|<small>19/12</small>
|<small>27/16</small>
|<small>5/3</small>
|<small>57/32</small>
|<small>7/4</small>
|<small>15/8</small>
|<small>15/8</small>
|-
|-
Line 373: Line 359:
|<small>19/17</small>
|<small>19/17</small>
|<small>20/17</small>
|<small>20/17</small>
|<small>21/17</small>
|<small>64/51</small>
|<small>45/34</small>
|<small>4/3</small>
|<small>24/17</small>
|<small>24/17</small>
|<small>3/2</small>
|<small>76/51</small>
|<small>27/17</small>
|<small>80/51</small>
|<small>57/34</small>
|<small>28/17</small>
|<small>30/17</small>
|<small>30/17</small>
|<small>32/17</small>
|<small>32/17</small>
Line 384: Line 370:
|<small>19/18</small>
|<small>19/18</small>
|<small>10/9</small>
|<small>10/9</small>
|<small>7/6</small>
|<small>32/27</small>
|<small>5/4</small>
|<small>34/27</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>17/12</small>
|<small>38/27</small>
|<small>3/2</small>
|<small>40/27</small>
|<small>19/12</small>
|<small>14/9</small>
|<small>5/3</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>16/9</small>
Line 395: Line 381:
|-
|-
|<small>20/19</small>
|<small>20/19</small>
|<small>21/19</small>
|<small>64/57</small>
|<small>45/38</small>
|<small>68/57</small>
|<small>24/19</small>
|<small>24/19</small>
|<small>51/38</small>
|<small>4/3</small>
|<small>27/19</small>
|<small>80/57</small>
|<small>3/2</small>
|<small>28/19</small>
|<small>30/19</small>
|<small>32/19</small>
|<small>34/19</small>
|<small>36/19</small>
|-
|<small>21/20</small>
|<small>9/8</small>
|<small>6/5</small>
|<small>51/40</small>
|<small>27/20</small>
|<small>57/40</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>17/10</small>
|<small>9/5</small>
|<small>19/10</small>
|-
|<small>15/14</small>
|<small>8/7</small>
|<small>17/14</small>
|<small>9/7</small>
|<small>19/14</small>
|<small>10/7</small>
|<small>32/21</small>
|<small>34/21</small>
|<small>12/7</small>
|<small>38/21</small>
|<small>40/21</small>
|-
|<small>16/15</small>
|<small>17/15</small>
|<small>6/5</small>
|<small>19/15</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>68/45</small>
|<small>8/5</small>
|<small>76/45</small>
|<small>16/9</small>
|<small>28/15</small>
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>19/16</small>
|<small>5/4</small>
|<small>4/3</small>
|<small>17/12</small>
|<small>3/2</small>
|<small>19/12</small>
|<small>5/3</small>
|<small>7/4</small>
|<small>15/8</small>
|-
|<small>18/17</small>
|<small>19/17</small>
|<small>20/17</small>
|<small>64/51</small>
|<small>4/3</small>
|<small>24/17</small>
|<small>76/51</small>
|<small>80/51</small>
|<small>28/17</small>
|<small>30/17</small>
|<small>32/17</small>
|-
|<small>19/18</small>
|<small>10/9</small>
|<small>32/27</small>
|<small>34/27</small>
|<small>4/3</small>
|<small>38/27</small>
|<small>40/27</small>
|<small>14/9</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|<small>20/19</small>
|<small>64/57</small>
|<small>68/57</small>
|<small>24/19</small>
|<small>4/3</small>
|<small>80/57</small>
|<small>28/19</small>
|<small>30/19</small>
|<small>32/19</small>
|<small>34/19</small>
|<small>36/19</small>
|-
! colspan="12" |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
! rowspan="12" |2
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>19/16</small>
|<small>5/4</small>
|<small>85/64</small>
|<small>45/32</small>
|<small>3/2</small>
|<small>51/32</small>
|<small>27/16</small>
|<small>57/32</small>
|<small>15/8</small>
|-
|<small>18/17</small>
|<small>19/17</small>
|<small>20/17</small>
|<small>5/4</small>
|<small>45/34</small>
|<small>24/17</small>
|<small>3/2</small>
|<small>27/17</small>
|<small>57/34</small>
|<small>30/17</small>
|<small>32/17</small>
|-
|<small>19/18</small>
|<small>10/9</small>
|<small>85/72</small>
|<small>5/4</small>
|<small>4/3</small>
|<small>17/12</small>
|<small>3/2</small>
|<small>19/12</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|<small>20/19</small>
|<small>85/76</small>
|<small>45/38</small>
|<small>24/19</small>
|<small>51/38</small>
|<small>27/19</small>
|<small>3/2</small>
|<small>30/19</small>
|<small>32/19</small>
|<small>34/19</small>
|<small>36/19</small>
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>6/5</small>
|<small>51/40</small>
|<small>27/20</small>
|<small>57/40</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>17/10</small>
|<small>9/5</small>
|<small>19/10</small>
|-
|<small>18/17</small>
|<small>96/85</small>
|<small>6/5</small>
|<small>108/85</small>
|<small>114/85</small>
|<small>24/17</small>
|<small>128/85</small>
|<small>8/5</small>
|<small>144/85</small>
|<small>152/85</small>
|<small>32/17</small>
|-
|<small>16/15</small>
|<small>17/15</small>
|<small>6/5</small>
|<small>19/15</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>68/45</small>
|<small>8/5</small>
|<small>76/45</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|<small>17/16</small>
|<small>9/8</small>
|<small>19/16</small>
|<small>5/4</small>
|<small>4/3</small>
|<small>17/12</small>
|<small>3/2</small>
|<small>19/12</small>
|<small>5/3</small>
|<small>85/48</small>
|<small>15/8</small>
|-
|<small>18/17</small>
|<small>19/17</small>
|<small>20/17</small>
|<small>64/51</small>
|<small>4/3</small>
|<small>24/17</small>
|<small>76/51</small>
|<small>80/51</small>
|<small>5/3</small>
|<small>30/17</small>
|<small>32/17</small>
|-
|<small>19/18</small>
|<small>10/9</small>
|<small>32/27</small>
|<small>34/27</small>
|<small>4/3</small>
|<small>38/27</small>
|<small>40/27</small>
|<small>85/54</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>17/9</small>
|-
|<small>20/19</small>
|<small>64/57</small>
|<small>68/57</small>
|<small>24/19</small>
|<small>4/3</small>
|<small>80/57</small>
|<small>85/57</small>
|<small>30/19</small>
|<small>30/19</small>
|<small>32/19</small>
|<small>32/19</small>
Line 639: Line 394:


=== HSAF 6 ===
=== HSAF 6 ===
<code>36:38:40:42:44:46:48:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.
<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>. 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).
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).
{| class="wikitable"
{| class="wikitable"
! colspan="15" |heHSAF 6 (25/18, 13/9)
|-
|-
!19/18
!19/18
Line 873: Line 627:


=== HSAF 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>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>'''.''' 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. 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. Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15|1 10b).
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"
{| class="wikitable"
! colspan="17" |heHSAF 7 (29/21, 10/7)
|-
|-
!22/21
!22/21
Line 897: Line 648:
!13/7
!13/7
!27/14
!27/14
| rowspan="17" |2
! rowspan="17" |2
|-
|-
|<small>23/22</small>
|<small>23/22</small>
Line 1,172: Line 923:
|}
|}


=== HSAF 8 - ''Veljeuwof'' ===
=== 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.'''
<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, 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.
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 [[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.
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"
{| class="wikitable"
! colspan="20" |heHSAF 8 (11/8, 17/12, 35/24)
|-
|-
!25/24
!25/24
Line 1,200: Line 950:
!15/8
!15/8
!31/16
!31/16
| rowspan="20" |2
! rowspan="20" |2
|-
|-
|<small>26/25</small>
|<small>26/25</small>
Line 1,584: Line 1,334:


== Beyond HSAF 8 ==
== 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]].
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/w/User:Eufalesio/Harmonic_Segments_Along_Fifths"