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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).
Tetrachordal otonal scales (TO(S)) are a type of octave-[[periodic scale]] built by taking concatenating two harmonic series segments that overlap at some points. 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).
 
The concept is an extensio of what had already been theorized by [[Erv Wilson]], which he named ''diaphonic cycle''.<ref>1. WILSON, Erv. ''Anaphoria'' [online]. [Accessed 26 October 2026 19:00 UTC+2]. Available from: https://www.anaphoria.com/diaphonicset.pdf</ref>


== Definitions ==
== Definitions ==
For a scale to be a HSAF, it must satisfy the following:
To build a TO scale, do 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.
* Take two harmonic series segments (rows): 2k::4k and 3k::6k. The 3k::6k is called the '''bottom row.'''
* The rows will overlap exactly at the points of the harmonic series segment k::2k. The set of all TO scales for each k is called '''category k.''' Category 0 and 1 do not have any meaningful TO scales.
* Any point where the two rows overlap is called a '''transition point'''. A transition point at 4/3 or 3/2 is called '''perfect''', between 4/3 and 3/2 '''imperfect'''. Outside 4/3 ≤ c ≤ 3/2 the scale the transition is '''non-tetrachordal,''' and between that range, the TO scale can be built and thus they are optionally called '''tetrachordal''' transition points. In this article, all transition points are tetrachordal.
* To build a TO scale, build a scale by starting on the bottom row and transitioning to the top row at a transition point between 4/3 and 3/2 inclusive.


== Properties ==
== Properties ==


* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
* Category x's amount of transition points can be derived with the following formula:<math>J\left(x\right)=1+\operatorname{floor}\left(\frac{x}{6}\right)-I\left(\frac{\operatorname{mod}\left(x,6\right)-1}{6}\right) \text{for} \space I\left(x\right)=1-\operatorname{ceil}\left(x\right)+\operatorname{floor}\left(x\right)</math> Which for each category x = 2,3,4… the list of tetrachordial transition points starts with 1,1,1,1,2,1,2,2,2…
* HSAF k is a subset of the 6k::12k harmonic segment.
* The amount of perfect transition points of a scale can be derived with <math>K\left(x\right)=I\left(\frac{x}{2}\right)+I\left(\frac{x}{3}\right)</math>. Thus the amount of imperfect transition points is <math>J\left(x\right)-K\left(x\right)</math>.
* HSAF k contains harmonics k–2k as a subset.
* '''Clean scales''' are scales with that transition only once, from the bottom to the top row, and their amount is equal to the number of transition points <math>J(x)</math>.
* 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 [...].
* The total number of scales per category is <math>2^{J\left(x\right)-1}</math>, and the amount of non-clean scales (otherwise called ''convoluted'') is thus <math>2^{J\left(x\right)-1}-J\left(x\right)</math>.
* Category 0 and 1 are excluded, as they get nonsensical results like category 1 having 0.5 scales.
* The amount of notes a scale of a given category x has <math>S(x)</math> falls between the ranges of these functions:<math>2x+1+\operatorname{floor}\left(\frac{x-1}{3}\right) ≤ S(x) ≤ 2x+1+\operatorname{floor}\left(\frac{x-2}{2}\right) </math> And the amount of notes are closely related to the scale tree of diatonic scales.
* The prime limits of the TOs of a category k sit between the biggest primes smaller than the 3 times the smallest and biggest transition point ratios.
* All TOs are chiral, for which the right hand version is TO, the left hand is TU.
The data can be summarized in this table:
{| class="wikitable"
|-
! <math>x</math> !! Transition points
Number of clean scales
! Perfect TP !! Imperfect TP !! Total scales !! Convoluted scales !! colspan="2" | Note ranges
! colspan="2" |p-limits
|-
| 2 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 5
| colspan="2" |7
|-
| 3 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 7
| colspan="2" |11
|-
| 4 || 1 || 1 || 0 || 1 || 0 || colspan="2" | 10
| colspan="2" |17
|-
| 5 || 1 || 0 || 1 || 1 || 0 || colspan="2" | 12
| colspan="2" |19
|-
| 6 || 2 || 2 || 0 || 2 || 0 || 14 || 15
| colspan="2" |23
|-
| 7 || 1 || 0 || 1 || 1 || 0 || colspan="2" | 17
| colspan="2" |29
|-
| 8 || 2 || 1 || 1 || 2 || 0 || 19 || 20
| colspan="2" |31
|-
| 9 || 2 || 1 || 1 || 2 || 0 || 21 || 22
|31
|37
|-
| 10 || 2 || 1 || 1 || 2 || 0 || 24 || 25
|37
|43
|-
| 11 || 2 || 0 || 2 || 2 || 0 || 26 || 27
|43
|47
|-
| 12 || 3 || 2 || 1 || 4 || 1 || 28 || 30
|47
|53
|-
| 13 || 2 || 0 || 2 || 2 || 0 || 31 || 32
|47
|53
|-
| 14 || 3 || 1 || 2 || 4 || 1 || 33 || 35
|53
|61
|-
| 15 || 3 || 1 || 2 || 4 || 1 || 35 || 37
|59
|61
|-
| 16 || 3 || 1 || 2 || 4 || 1 || 38 || 40
|61
|71
|}
Simple TOs can be visualized as edo or MOS detempers, which makes them particularly useful either for NEJI targets or essential tempering.  


<nowiki>*</nowiki>except the first
== List of categories ==
For every provided TO, the [[EFR]] and used segment are written out explicitly. Names in italic are names never seen before on the wiki, declared here for specificity. Only TOs are listed: to get the respective TUs, retrovert the scale.


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.  
For categories with more than one scale, append letters from a up to whatever is necessary, which works for categories from 2 to 31 except 30 (using letters a-p). After that, append numbers.


== List of HSAFs 1-8 ==
=== Category 2 - Septimal minor pentatonic ===
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.
<code>12:14:16:18:21:24</code>: TO 2.  


=== HSAF/SSAF 1 - Pythagorean Trial ===
First usable TO, the first prime TO, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
<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]].
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=""
{| class="wikitable" data-darkreader-inline-color=""
!7/6
!4/3
!4/3
!3/2
!3/2
! rowspan="3" |2
!7/4
! rowspan="5" |2
|-
|-
|<small>9/8</small>
|<small>8/7</small>
|<small>9/7</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>12/7</small>
|-
|-
|<small>4/3</small>
|<small>9/8</small>
|<small>16/9</small>
|}
 
=== 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
|-
|<small>8/7</small>
|<small>9/7</small>
|<small>3/2</small>
|<small>12/7</small>
|-
|<small>9/8</small>
|<small>21/16</small>
|<small>21/16</small>
|<small>3/2</small>
|<small>3/2</small>
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|}
|}


=== HSAF 3 ===
=== Category 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>: TO 3.


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).  
Second prime TO. 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'''
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|}
|}


=== HSAF 4 ===
=== Category 4 ===
<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.
<code>24:26:28:30:32:34:36:39:42:45:48</code>: TO 4.


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.
First composite TO, containing TO 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=""
{| class="wikitable" data-darkreader-inline-color=""
!'''13/12'''
!'''13/12'''
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|}
|}


=== HSAF 5 ===
=== Category 5 ===
<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>. Harmonic segment: '''15::20'''.
<code>30:32:34:36:38:40:42:45:48:51:54:57:60</code>: TO 5.  


'''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.
'''15::20'''. Third prime TO. 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=""
{| class="wikitable" data-darkreader-inline-color=""
!16/15
!16/15
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|}
|}


=== HSAF 6 ===
=== Category 6 ===
<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.
It is the last category to feature no imperfect transition points and the first category to feature two transition points, and thus two scales. Last category to feature TO [[constant structure]]<nowiki/>s. It is a [[14edo|14c-edo]] and [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup (add-17 or add-13), and [[nautilus]] detemper ([[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]]<nowiki/>detemper in the 2.3.5.7.11.19.23 subgroup, and [[nautilus]] detemper ([[1L 12s]] 6|6 and [[14L 1s]] 6|8).
==== TO 6a ====
<code>36:38:40:42:44:46:48:51:54:57:60:63:66:69:72</code>.  
{| class="wikitable" data-darkreader-inline-color=""
{| class="wikitable" data-darkreader-inline-color=""
|-
!19/18
!19/18
!10/9
!10/9
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!23/18
!23/18
!4/3
!4/3
!25/18
!17/12
!13/9
!3/2
!3/2
!19/12
!19/12
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!11/6
!11/6
!23/12
!23/12
! rowspan="15" |2
! rowspan="14" |2
|-
|-
|<small>20/19</small>
|<small>20/19</small>
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|<small>23/19</small>
|<small>23/19</small>
|<small>24/19</small>
|<small>24/19</small>
|<small>25/19</small>
|<small>51/38</small>
|<small>26/19</small>
|<small>27/19</small>
|<small>27/19</small>
|<small>3/2</small>
|<small>3/2</small>
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|<small>23/20</small>
|<small>23/20</small>
|<small>6/5</small>
|<small>6/5</small>
|<small>5/4</small>
|<small>51/40</small>
|<small>13/10</small>
|<small>27/20</small>
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|<small>57/40</small>
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|<small>23/21</small>
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|<small>8/7</small>
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|<small>17/14</small>
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|-
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|<small>17/16</small>
|<small>13/12</small>
|<small>9/8</small>
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|<small>19/16</small>
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|<small>23/12</small>
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|-
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|<small>26/25</small>
|<small>18/17</small>
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|<small>19/17</small>
|<small>57/50</small>
|<small>20/17</small>
|<small>6/5</small>
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|<small>63/50</small>
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|<small>23/17</small>
|<small>69/50</small>
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|<small>32/17</small>
|<small>48/25</small>
|-
|<small>27/26</small>
|<small>57/52</small>
|<small>15/13</small>
|<small>63/52</small>
|<small>33/26</small>
|<small>69/52</small>
|<small>18/13</small>
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|-
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|<small>19/18</small>
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|<small>46/27</small>
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|-
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|-
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|}
|}


=== HSAF 7 ===
==== TO 6b ====
<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>'''.''' Harmonic segment: '''21::28'''.
<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>.
 
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=""
{| class="wikitable" data-darkreader-inline-color=""
!22/21
!19/18
!23/21
!10/9
!8/7
!7/6
!25/21
!11/9
!26/21
!23/18
!9/7
!4/3
!4/3
!29/21
!25/18
!10/7
!13/9
!3/2
!3/2
!11/7
!19/12
!23/14
!5/3
!12/7
!7/4
!25/14
!11/6
!13/7
!23/12
!27/14
! rowspan="15" |2
! rowspan="17" |2
|-
|-
|<small>23/22</small>
|<small>20/19</small>
|<small>12/11</small>
|<small>21/19</small>
|<small>25/22</small>
|<small>22/19</small>
|<small>13/11</small>
|<small>23/19</small>
|<small>27/22</small>
|<small>24/19</small>
|<small>14/11</small>
|<small>25/19</small>
|<small>29/22</small>
|<small>26/19</small>
|<small>15/11</small>
|<small>27/19</small>
|<small>63/44</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>69/44</small>
|<small>30/19</small>
|<small>18/11</small>
|<small>63/38</small>
|<small>75/44</small>
|<small>33/19</small>
|<small>39/22</small>
|<small>69/38</small>
|<small>81/44</small>
|<small>36/19</small>
|<small>21/11</small>
|-
|-
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|-
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|-
|-
|<small>26/25</small>
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|<small>69/50</small>
|<small>63/44</small>
|<small>36/25</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>39/25</small>
|<small>69/44</small>
|<small>81/50</small>
|<small>18/11</small>
|<small>42/25</small>
|<small>19/11</small>
|<small>44/25</small>
|<small>20/11</small>
|<small>46/25</small>
|<small>21/11</small>
|<small>48/25</small>
|-
|-
|<small>27/26</small>
|<small>24/23</small>
|<small>14/13</small>
|<small>25/23</small>
|<small>29/26</small>
|<small>26/23</small>
|<small>15/13</small>
|<small>27/23</small>
|<small>63/52</small>
|<small>57/46</small>
|<small>33/26</small>
|<small>30/23</small>
|<small>69/52</small>
|<small>63/46</small>
|<small>18/13</small>
|<small>33/23</small>
|<small>75/52</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>81/52</small>
|<small>36/23</small>
|<small>21/13</small>
|<small>38/23</small>
|<small>22/13</small>
|<small>40/23</small>
|<small>23/13</small>
|<small>42/23</small>
|<small>24/13</small>
|<small>44/23</small>
|<small>25/13</small>
|-
|-
|<small>28/27</small>
|<small>25/24</small>
|<small>29/27</small>
|<small>13/12</small>
|<small>10/9</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>19/16</small>
|<small>11/9</small>
|<small>5/4</small>
|<small>23/18</small>
|<small>21/16</small>
|<small>4/3</small>
|<small>11/8</small>
|<small>25/18</small>
|<small>23/16</small>
|<small>13/9</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>19/12</small>
|<small>44/27</small>
|<small>5/3</small>
|<small>46/27</small>
|<small>7/4</small>
|<small>16/9</small>
|<small>11/6</small>
|<small>50/27</small>
|<small>23/12</small>
|<small>52/27</small>
|-
|-
|<small>29/28</small>
|<small>26/25</small>
|<small>15/14</small>
|<small>27/25</small>
|<small>9/8</small>
|<small>57/50</small>
|<small>33/28</small>
|<small>6/5</small>
|<small>69/56</small>
|<small>63/50</small>
|<small>9/7</small>
|<small>33/25</small>
|<small>75/56</small>
|<small>69/50</small>
|<small>39/28</small>
|<small>36/25</small>
|<small>81/56</small>
|<small>38/25</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>11/7</small>
|<small>42/25</small>
|<small>23/14</small>
|<small>44/25</small>
|<small>12/7</small>
|<small>46/25</small>
|<small>25/14</small>
|<small>48/25</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|-
|<small>30/29</small>
|<small>27/26</small>
|<small>63/58</small>
|<small>57/52</small>
|<small>33/29</small>
|<small>15/13</small>
|<small>69/58</small>
|<small>63/52</small>
|<small>36/29</small>
|<small>33/26</small>
|<small>75/58</small>
|<small>69/52</small>
|<small>39/29</small>
|<small>18/13</small>
|<small>81/58</small>
|<small>19/13</small>
|<small>42/29</small>
|<small>20/13</small>
|<small>44/29</small>
|<small>21/13</small>
|<small>46/29</small>
|<small>22/13</small>
|<small>48/29</small>
|<small>23/13</small>
|<small>50/29</small>
|<small>24/13</small>
|<small>52/29</small>
|<small>25/13</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|-
|<small>21/20</small>
|<small>19/18</small>
|<small>11/10</small>
|<small>10/9</small>
|<small>23/20</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>11/9</small>
|<small>5/4</small>
|<small>23/18</small>
|<small>13/10</small>
|<small>4/3</small>
|<small>27/20</small>
|<small>38/27</small>
|<small>7/5</small>
|<small>40/27</small>
|<small>22/15</small>
|<small>14/9</small>
|<small>23/15</small>
|<small>44/27</small>
|<small>8/5</small>
|<small>46/27</small>
|<small>5/3</small>
|<small>16/9</small>
|<small>26/15</small>
|<small>50/27</small>
|<small>9/5</small>
|<small>52/27</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|-
|<small>22/21</small>
|<small>20/19</small>
|<small>23/21</small>
|<small>21/19</small>
|<small>8/7</small>
|<small>22/19</small>
|<small>25/21</small>
|<small>23/19</small>
|<small>26/21</small>
|<small>24/19</small>
|<small>9/7</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>88/63</small>
|<small>80/57</small>
|<small>92/63</small>
|<small>28/19</small>
|<small>32/21</small>
|<small>88/57</small>
|<small>100/63</small>
|<small>92/57</small>
|<small>104/63</small>
|<small>32/19</small>
|<small>12/7</small>
|<small>100/57</small>
|<small>16/9</small>
|<small>104/57</small>
|<small>116/63</small>
|<small>36/19</small>
|<small>40/21</small>
|-
|-
|<small>23/22</small>
|<small>21/20</small>
|<small>12/11</small>
|<small>11/10</small>
|<small>25/22</small>
|<small>23/20</small>
|<small>13/11</small>
|<small>6/5</small>
|<small>27/22</small>
|<small>19/15</small>
|<small>14/11</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>46/33</small>
|<small>7/5</small>
|<small>16/11</small>
|<small>22/15</small>
|<small>50/33</small>
|<small>23/15</small>
|<small>52/33</small>
|<small>8/5</small>
|<small>18/11</small>
|<small>5/3</small>
|<small>56/33</small>
|<small>26/15</small>
|<small>58/33</small>
|<small>9/5</small>
|<small>20/11</small>
|<small>19/10</small>
|<small>21/11</small>
|-
|-
|<small>24/23</small>
|<small>22/21</small>
|<small>25/23</small>
|<small>23/21</small>
|<small>26/23</small>
|<small>8/7</small>
|<small>27/23</small>
|<small>76/63</small>
|<small>28/23</small>
|<small>80/63</small>
|<small>88/69</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>32/23</small>
|<small>88/63</small>
|<small>100/69</small>
|<small>92/63</small>
|<small>104/69</small>
|<small>32/21</small>
|<small>36/23</small>
|<small>100/63</small>
|<small>112/69</small>
|<small>104/63</small>
|<small>116/69</small>
|<small>12/7</small>
|<small>40/23</small>
|<small>38/21</small>
|<small>42/23</small>
|<small>40/21</small>
|<small>44/23</small>
|-
|-
|<small>25/24</small>
|<small>23/22</small>
|<small>13/12</small>
|<small>12/11</small>
|<small>9/8</small>
|<small>38/33</small>
|<small>7/6</small>
|<small>40/33</small>
|<small>11/9</small>
|<small>14/11</small>
|<small>23/18</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>46/33</small>
|<small>13/9</small>
|<small>16/11</small>
|<small>3/2</small>
|<small>50/33</small>
|<small>14/9</small>
|<small>52/33</small>
|<small>29/18</small>
|<small>18/11</small>
|<small>5/3</small>
|<small>19/11</small>
|<small>7/4</small>
|<small>20/11</small>
|<small>11/6</small>
|<small>21/11</small>
|<small>23/12</small>
|-
|-
|<small>26/25</small>
|<small>24/23</small>
|<small>27/25</small>
|<small>76/69</small>
|<small>28/25</small>
|<small>80/69</small>
|<small>88/75</small>
|<small>28/23</small>
|<small>92/75</small>
|<small>88/69</small>
|<small>32/25</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>104/75</small>
|<small>32/23</small>
|<small>36/25</small>
|<small>100/69</small>
|<small>112/75</small>
|<small>104/69</small>
|<small>116/75</small>
|<small>36/23</small>
|<small>8/5</small>
|<small>38/23</small>
|<small>42/25</small>
|<small>40/23</small>
|<small>44/25</small>
|<small>42/23</small>
|<small>46/25</small>
|<small>44/23</small>
|<small>48/25</small>
|}
|-
 
|<small>27/26</small>
=== Category 7 ===
|<small>14/13</small>
<code>42:44:46:48:50:52:54:56:58:60:63:66:69:72:75:78:81:84</code>''':''' TO 7. The last category to feature only one scale.
|<small>44/39</small>
 
|<small>46/39</small>
Fourth prime TO. TU 7 is surprisingly a subharmonic interpolation of TO 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).
|<small>16/13</small>
 
|<small>50/39</small>
It is also the biggest category to feature only one scale.
|<small>4/3</small>
{| class="wikitable" data-darkreader-inline-color=""
|<small>18/13</small>
!22/21
|<small>56/39</small>
!23/21
|<small>58/39</small>
!8/7
|<small>20/13</small>
!25/21
|<small>21/13</small>
!26/21
|<small>22/13</small>
!9/7
|<small>23/13</small>
!4/3
|<small>24/13</small>
!29/21
|<small>25/13</small>
!10/7
|-
!3/2
|<small>28/27</small>
!11/7
|<small>88/81</small>
!23/14
|<small>92/81</small>
!12/7
|<small>32/27</small>
!25/14
|<small>100/81</small>
!13/7
|<small>104/81</small>
!27/14
|<small>4/3</small>
! rowspan="17" |2
|<small>112/81</small>
|-
|<small>116/81</small>
|<small>23/22</small>
|<small>40/27</small>
|<small>12/11</small>
|<small>14/9</small>
|<small>25/22</small>
|<small>44/27</small>
|<small>13/11</small>
|<small>46/27</small>
|<small>27/22</small>
|<small>16/9</small>
|<small>14/11</small>
|<small>50/27</small>
|<small>29/22</small>
|<small>52/27</small>
|<small>15/11</small>
|}
|<small>63/44</small>
 
|<small>3/2</small>
=== HSAF 8 ===
|<small>69/44</small>
<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.'''
|<small>18/11</small>
 
|<small>75/44</small>
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]].
|<small>39/22</small>
 
|<small>81/44</small>
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.
|<small>21/11</small>
{| class="wikitable" data-darkreader-inline-color=""
|-
!25/24
|<small>24/23</small>
!13/12
|<small>25/23</small>
!9/8
|<small>26/23</small>
!7/6
|<small>27/23</small>
!29/24
|<small>28/23</small>
!5/4
|<small>29/23</small>
!31/24
|<small>30/23</small>
!4/3
|<small>63/46</small>
!11/8
|<small>33/23</small>
!17/12
|<small>3/2</small>
!35/24
|<small>36/23</small>
!3/2
|<small>75/46</small>
!25/16
|<small>39/23</small>
!13/8
|<small>81/46</small>
!27/16
|<small>42/23</small>
!7/4
|<small>44/23</small>
!29/16
!15/8
!31/16
! rowspan="20" |2
|-
|-
|<small>26/25</small>
|<small>25/24</small>
|<small>27/25</small>
|<small>13/12</small>
|<small>28/25</small>
|<small>9/8</small>
|<small>29/25</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>29/24</small>
|<small>31/25</small>
|<small>5/4</small>
|<small>32/25</small>
|<small>21/16</small>
|<small>11/8</small>
|<small>23/16</small>
|<small>3/2</small>
|<small>25/16</small>
|<small>13/8</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>11/6</small>
|<small>23/12</small>
|-
|<small>26/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>6/5</small>
|<small>63/50</small>
|<small>33/25</small>
|<small>33/25</small>
|<small>34/25</small>
|<small>69/50</small>
|<small>7/5</small>
|<small>36/25</small>
|<small>36/25</small>
|<small>3/2</small>
|<small>3/2</small>
Line 960: Line 961:
|<small>81/50</small>
|<small>81/50</small>
|<small>42/25</small>
|<small>42/25</small>
|<small>87/50</small>
|<small>44/25</small>
|<small>9/5</small>
|<small>46/25</small>
|<small>93/50</small>
|<small>48/25</small>
|<small>48/25</small>
|-
|-
Line 969: Line 969:
|<small>29/26</small>
|<small>29/26</small>
|<small>15/13</small>
|<small>15/13</small>
|<small>31/26</small>
|<small>63/52</small>
|<small>16/13</small>
|<small>33/26</small>
|<small>33/26</small>
|<small>17/13</small>
|<small>69/52</small>
|<small>35/26</small>
|<small>18/13</small>
|<small>18/13</small>
|<small>75/52</small>
|<small>75/52</small>
Line 979: Line 977:
|<small>81/52</small>
|<small>81/52</small>
|<small>21/13</small>
|<small>21/13</small>
|<small>87/52</small>
|<small>22/13</small>
|<small>45/26</small>
|<small>23/13</small>
|<small>93/52</small>
|<small>24/13</small>
|<small>24/13</small>
|<small>25/13</small>
|<small>25/13</small>
Line 988: Line 985:
|<small>29/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>7/6</small>
|<small>32/27</small>
|<small>11/9</small>
|<small>11/9</small>
|<small>34/27</small>
|<small>23/18</small>
|<small>35/27</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>25/18</small>
Line 998: Line 993:
|<small>3/2</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>14/9</small>
|<small>29/18</small>
|<small>44/27</small>
|<small>5/3</small>
|<small>46/27</small>
|<small>31/18</small>
|<small>16/9</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>50/27</small>
Line 1,007: Line 1,001:
|<small>29/28</small>
|<small>29/28</small>
|<small>15/14</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>9/8</small>
|<small>8/7</small>
|<small>33/28</small>
|<small>33/28</small>
|<small>17/14</small>
|<small>69/56</small>
|<small>5/4</small>
|<small>9/7</small>
|<small>9/7</small>
|<small>75/56</small>
|<small>75/56</small>
Line 1,017: Line 1,009:
|<small>81/56</small>
|<small>81/56</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>87/56</small>
|<small>11/7</small>
|<small>45/28</small>
|<small>23/14</small>
|<small>93/56</small>
|<small>12/7</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>25/14</small>
Line 1,026: Line 1,017:
|-
|-
|<small>30/29</small>
|<small>30/29</small>
|<small>31/29</small>
|<small>63/58</small>
|<small>32/29</small>
|<small>33/29</small>
|<small>33/29</small>
|<small>34/29</small>
|<small>69/58</small>
|<small>35/29</small>
|<small>36/29</small>
|<small>36/29</small>
|<small>75/58</small>
|<small>75/58</small>
Line 1,036: Line 1,025:
|<small>81/58</small>
|<small>81/58</small>
|<small>42/29</small>
|<small>42/29</small>
|<small>3/2</small>
|<small>44/29</small>
|<small>45/29</small>
|<small>46/29</small>
|<small>93/58</small>
|<small>48/29</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>50/29</small>
Line 1,045: Line 1,033:
|<small>56/29</small>
|<small>56/29</small>
|-
|-
|<small>31/30</small>
|<small>21/20</small>
|<small>16/15</small>
|<small>11/10</small>
|<small>11/10</small>
|<small>17/15</small>
|<small>23/20</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>6/5</small>
|<small>5/4</small>
|<small>5/4</small>
Line 1,055: Line 1,041:
|<small>27/20</small>
|<small>27/20</small>
|<small>7/5</small>
|<small>7/5</small>
|<small>29/20</small>
|<small>22/15</small>
|<small>3/2</small>
|<small>23/15</small>
|<small>31/20</small>
|<small>8/5</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>5/3</small>
Line 1,065: Line 1,050:
|<small>29/15</small>
|<small>29/15</small>
|-
|-
|<small>32/31</small>
|<small>22/21</small>
|<small>33/31</small>
|<small>23/21</small>
|<small>34/31</small>
|<small>8/7</small>
|<small>35/31</small>
|<small>25/21</small>
|<small>36/31</small>
|<small>26/21</small>
|<small>75/62</small>
|<small>9/7</small>
|<small>39/31</small>
|<small>4/3</small>
|<small>81/62</small>
|<small>88/63</small>
|<small>42/31</small>
|<small>92/63</small>
|<small>87/62</small>
|<small>32/21</small>
|<small>45/31</small>
|<small>100/63</small>
|<small>3/2</small>
|<small>104/63</small>
|<small>48/31</small>
|<small>12/7</small>
|<small>50/31</small>
|<small>16/9</small>
|<small>52/31</small>
|<small>116/63</small>
|<small>54/31</small>
|<small>40/21</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|-
|-
|<small>33/32</small>
|<small>23/22</small>
|<small>17/16</small>
|<small>12/11</small>
|<small>35/32</small>
|<small>25/22</small>
|<small>9/8</small>
|<small>13/11</small>
|<small>75/64</small>
|<small>27/22</small>
|<small>39/32</small>
|<small>14/11</small>
|<small>81/64</small>
|<small>4/3</small>
|<small>21/16</small>
|<small>46/33</small>
|<small>87/64</small>
|<small>16/11</small>
|<small>45/32</small>
|<small>50/33</small>
|<small>93/64</small>
|<small>3/2</small>
|<small>25/16</small>
|<small>13/8</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>29/16</small>
|<small>15/8</small>
|<small>31/16</small>
|-
|<small>34/33</small>
|<small>35/33</small>
|<small>12/11</small>
|<small>25/22</small>
|<small>13/11</small>
|<small>27/22</small>
|<small>14/11</small>
|<small>29/22</small>
|<small>15/11</small>
|<small>31/22</small>
|<small>16/11</small>
|<small>50/33</small>
|<small>52/33</small>
|<small>52/33</small>
|<small>18/11</small>
|<small>18/11</small>
Line 1,122: Line 1,082:
|<small>58/33</small>
|<small>58/33</small>
|<small>20/11</small>
|<small>20/11</small>
|<small>62/33</small>
|<small>21/11</small>
|<small>64/33</small>
|-
|-
|<small>35/34</small>
|<small>24/23</small>
|<small>18/17</small>
|<small>25/23</small>
|<small>75/68</small>
|<small>26/23</small>
|<small>39/34</small>
|<small>27/23</small>
|<small>81/68</small>
|<small>28/23</small>
|<small>21/17</small>
|<small>88/69</small>
|<small>87/68</small>
|<small>4/3</small>
|<small>45/34</small>
|<small>32/23</small>
|<small>93/68</small>
|<small>100/69</small>
|<small>24/17</small>
|<small>104/69</small>
|<small>25/17</small>
|<small>36/23</small>
|<small>26/17</small>
|<small>112/69</small>
|<small>27/17</small>
|<small>116/69</small>
|<small>28/17</small>
|<small>40/23</small>
|<small>29/17</small>
|<small>42/23</small>
|<small>30/17</small>
|<small>44/23</small>
|<small>31/17</small>
|<small>32/17</small>
|<small>33/17</small>
|-
|<small>36/35</small>
|<small>15/14</small>
|<small>39/35</small>
|<small>81/70</small>
|<small>6/5</small>
|<small>87/70</small>
|<small>9/7</small>
|<small>93/70</small>
|<small>48/35</small>
|<small>10/7</small>
|<small>52/35</small>
|<small>54/35</small>
|<small>8/5</small>
|<small>58/35</small>
|<small>12/7</small>
|<small>62/35</small>
|<small>64/35</small>
|<small>66/35</small>
|<small>68/35</small>
|-
|-
|<small>25/24</small>
|<small>25/24</small>
Line 1,169: Line 1,105:
|<small>9/8</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>7/6</small>
|<small>29/24</small>
|<small>11/9</small>
|<small>5/4</small>
|<small>23/18</small>
|<small>31/24</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>25/18</small>
Line 1,179: Line 1,114:
|<small>29/18</small>
|<small>29/18</small>
|<small>5/3</small>
|<small>5/3</small>
|<small>31/18</small>
|<small>7/4</small>
|<small>16/9</small>
|<small>11/6</small>
|<small>11/6</small>
|<small>17/9</small>
|<small>23/12</small>
|<small>35/18</small>
|-
|-
|<small>26/25</small>
|<small>26/25</small>
|<small>27/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>88/75</small>
|<small>6/5</small>
|<small>92/75</small>
|<small>31/25</small>
|<small>32/25</small>
|<small>32/25</small>
|<small>4/3</small>
|<small>4/3</small>
Line 1,198: Line 1,130:
|<small>116/75</small>
|<small>116/75</small>
|<small>8/5</small>
|<small>8/5</small>
|<small>124/75</small>
|<small>42/25</small>
|<small>128/75</small>
|<small>44/25</small>
|<small>44/25</small>
|<small>136/75</small>
|<small>46/25</small>
|<small>28/15</small>
|<small>48/25</small>
|<small>48/25</small>
|-
|-
|<small>27/26</small>
|<small>27/26</small>
|<small>14/13</small>
|<small>14/13</small>
|<small>29/26</small>
|<small>44/39</small>
|<small>15/13</small>
|<small>46/39</small>
|<small>31/26</small>
|<small>16/13</small>
|<small>16/13</small>
|<small>50/39</small>
|<small>50/39</small>
Line 1,217: Line 1,146:
|<small>58/39</small>
|<small>58/39</small>
|<small>20/13</small>
|<small>20/13</small>
|<small>62/39</small>
|<small>21/13</small>
|<small>64/39</small>
|<small>22/13</small>
|<small>22/13</small>
|<small>68/39</small>
|<small>23/13</small>
|<small>70/39</small>
|<small>24/13</small>
|<small>24/13</small>
|<small>25/13</small>
|<small>25/13</small>
|-
|-
|<small>28/27</small>
|<small>28/27</small>
|<small>29/27</small>
|<small>88/81</small>
|<small>10/9</small>
|<small>92/81</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>32/27</small>
|<small>100/81</small>
|<small>100/81</small>
Line 1,236: Line 1,162:
|<small>116/81</small>
|<small>116/81</small>
|<small>40/27</small>
|<small>40/27</small>
|<small>124/81</small>
|<small>14/9</small>
|<small>128/81</small>
|<small>44/27</small>
|<small>44/27</small>
|<small>136/81</small>
|<small>46/27</small>
|<small>140/81</small>
|<small>16/9</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>50/27</small>
|<small>52/27</small>
|<small>52/27</small>
|}
=== Category 8 ===
It contains TO 2 and TO 4. With supersets of TO 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 category to fit completely within the [[31-limit]]. TO 8a adds 23, TO 8b adds 17.
The scales can be seen [[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 (well enough for TO 8a albeit badly for TO 8b as it tempers together 17/12 and 35/24). Thus, it's better seen as just a linear interpolation of TO 4 in the 2.3.5.7.11.13.17.29.31 subgroup.
==== TO 8a ====
<code>48:50:52:54:56:58:60:62:64:66:69:72:75:78:81:84:87:90:93:96</code>. Last TO to be a constant structure
{| class="wikitable"
|-
|-
|<small>29/28</small>
!25/24
|<small>15/14</small>
!13/12
|<small>31/28</small>
!9/8
|<small>8/7</small>
!7/6
|<small>25/21</small>
!29/24
|<small>26/21</small>
!5/4
|<small>9/7</small>
!31/24
|<small>4/3</small>
!4/3
|<small>29/21</small>
!11/8
|<small>10/7</small>
!23/16
|<small>31/21</small>
!3/2
|<small>32/21</small>
!25/16
|<small>11/7</small>
!13/8
|<small>34/21</small>
!27/16
|<small>5/3</small>
!7/4
|<small>12/7</small>
!29/16
|<small>25/14</small>
!15/8
|<small>13/7</small>
!31/16
|<small>27/14</small>
! rowspan="19" |2
|-
|-
|<small>30/29</small>
|<small>26/25</small>
|<small>31/29</small>
|<small>27/25</small>
|<small>32/29</small>
|<small>28/25</small>
|<small>100/87</small>
|<small>29/25</small>
|<small>104/87</small>
|<small>36/29</small>
|<small>112/87</small>
|<small>4/3</small>
|<small>40/29</small>
|<small>124/87</small>
|<small>128/87</small>
|<small>44/29</small>
|<small>136/87</small>
|<small>140/87</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>10/9</small>
|<small>52/45</small>
|<small>6/5</small>
|<small>6/5</small>
|<small>56/45</small>
|<small>31/25</small>
|<small>58/45</small>
|<small>32/25</small>
|<small>4/3</small>
|<small>33/25</small>
|<small>62/45</small>
|<small>69/50</small>
|<small>64/45</small>
|<small>36/25</small>
|<small>22/15</small>
|<small>3/2</small>
|<small>68/45</small>
|<small>39/25</small>
|<small>14/9</small>
|<small>81/50</small>
|<small>8/5</small>
|<small>42/25</small>
|<small>5/3</small>
|<small>87/50</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>93/50</small>
|<small>29/15</small>
|<small>48/25</small>
|-
|-
|<small>32/31</small>
|<small>27/26</small>
|<small>100/93</small>
|<small>14/13</small>
|<small>104/93</small>
|<small>29/26</small>
|<small>36/31</small>
|<small>15/13</small>
|<small>112/93</small>
|<small>31/26</small>
|<small>116/93</small>
|<small>16/13</small>
|<small>40/31</small>
|<small>33/26</small>
|<small>69/52</small>
|<small>18/13</small>
|<small>75/52</small>
|<small>3/2</small>
|<small>81/52</small>
|<small>21/13</small>
|<small>87/52</small>
|<small>45/26</small>
|<small>93/52</small>
|<small>24/13</small>
|<small>25/13</small>
|-
|<small>28/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>11/9</small>
|<small>23/18</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>128/93</small>
|<small>25/18</small>
|<small>44/31</small>
|<small>13/9</small>
|<small>136/93</small>
|<small>3/2</small>
|<small>140/93</small>
|<small>14/9</small>
|<small>48/31</small>
|<small>29/18</small>
|<small>50/31</small>
|<small>5/3</small>
|<small>52/31</small>
|<small>31/18</small>
|<small>54/31</small>
|<small>16/9</small>
|<small>56/31</small>
|<small>50/27</small>
|<small>52/27</small>
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>33/28</small>
|<small>69/56</small>
|<small>9/7</small>
|<small>75/56</small>
|<small>39/28</small>
|<small>81/56</small>
|<small>3/2</small>
|<small>87/56</small>
|<small>45/28</small>
|<small>93/56</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>33/29</small>
|<small>69/58</small>
|<small>36/29</small>
|<small>75/58</small>
|<small>39/29</small>
|<small>81/58</small>
|<small>42/29</small>
|<small>3/2</small>
|<small>45/29</small>
|<small>93/58</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>11/10</small>
|<small>23/20</small>
|<small>6/5</small>
|<small>5/4</small>
|<small>13/10</small>
|<small>27/20</small>
|<small>7/5</small>
|<small>29/20</small>
|<small>3/2</small>
|<small>31/20</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>33/31</small>
|<small>69/62</small>
|<small>36/31</small>
|<small>75/62</small>
|<small>39/31</small>
|<small>81/62</small>
|<small>42/31</small>
|<small>87/62</small>
|<small>45/31</small>
|<small>3/2</small>
|<small>48/31</small>
|<small>50/31</small>
|<small>52/31</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>58/31</small>
|<small>60/31</small>
|<small>60/31</small>
|-
|<small>33/32</small>
|<small>69/64</small>
|<small>9/8</small>
|<small>75/64</small>
|<small>39/32</small>
|<small>81/64</small>
|<small>21/16</small>
|<small>87/64</small>
|<small>45/32</small>
|<small>93/64</small>
|<small>3/2</small>
|<small>25/16</small>
|<small>13/8</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>29/16</small>
|<small>15/8</small>
|<small>31/16</small>
|-
|<small>23/22</small>
|<small>12/11</small>
|<small>25/22</small>
|<small>13/11</small>
|<small>27/22</small>
|<small>14/11</small>
|<small>29/22</small>
|<small>15/11</small>
|<small>31/22</small>
|<small>16/11</small>
|<small>50/33</small>
|<small>52/33</small>
|<small>18/11</small>
|<small>56/33</small>
|<small>58/33</small>
|<small>20/11</small>
|<small>62/33</small>
|<small>64/33</small>
|-
|<small>24/23</small>
|<small>25/23</small>
|<small>26/23</small>
|<small>27/23</small>
|<small>28/23</small>
|<small>29/23</small>
|<small>30/23</small>
|<small>31/23</small>
|<small>32/23</small>
|<small>100/69</small>
|<small>104/69</small>
|<small>36/23</small>
|<small>112/69</small>
|<small>116/69</small>
|<small>40/23</small>
|<small>124/69</small>
|<small>128/69</small>
|<small>44/23</small>
|-
|<small>25/24</small>
|<small>13/12</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>29/24</small>
|<small>5/4</small>
|<small>31/24</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>13/9</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>29/18</small>
|<small>5/3</small>
|<small>31/18</small>
|<small>16/9</small>
|<small>11/6</small>
|<small>23/12</small>
|-
|<small>26/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>6/5</small>
|<small>31/25</small>
|<small>32/25</small>
|<small>4/3</small>
|<small>104/75</small>
|<small>36/25</small>
|<small>112/75</small>
|<small>116/75</small>
|<small>8/5</small>
|<small>124/75</small>
|<small>128/75</small>
|<small>44/25</small>
|<small>46/25</small>
|<small>48/25</small>
|-
|<small>27/26</small>
|<small>14/13</small>
|<small>29/26</small>
|<small>15/13</small>
|<small>31/26</small>
|<small>16/13</small>
|<small>50/39</small>
|<small>4/3</small>
|<small>18/13</small>
|<small>56/39</small>
|<small>58/39</small>
|<small>20/13</small>
|<small>62/39</small>
|<small>64/39</small>
|<small>22/13</small>
|<small>23/13</small>
|<small>24/13</small>
|<small>25/13</small>
|-
|<small>28/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>100/81</small>
|<small>104/81</small>
|<small>4/3</small>
|<small>112/81</small>
|<small>116/81</small>
|<small>40/27</small>
|<small>124/81</small>
|<small>128/81</small>
|<small>44/27</small>
|<small>46/27</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>52/27</small>
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>25/21</small>
|<small>26/21</small>
|<small>9/7</small>
|<small>4/3</small>
|<small>29/21</small>
|<small>10/7</small>
|<small>31/21</small>
|<small>32/21</small>
|<small>11/7</small>
|<small>23/14</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>100/87</small>
|<small>104/87</small>
|<small>36/29</small>
|<small>112/87</small>
|<small>4/3</small>
|<small>40/29</small>
|<small>124/87</small>
|<small>128/87</small>
|<small>44/29</small>
|<small>46/29</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>10/9</small>
|<small>52/45</small>
|<small>6/5</small>
|<small>56/45</small>
|<small>58/45</small>
|<small>4/3</small>
|<small>62/45</small>
|<small>64/45</small>
|<small>22/15</small>
|<small>23/15</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>100/93</small>
|<small>104/93</small>
|<small>36/31</small>
|<small>112/93</small>
|<small>116/93</small>
|<small>40/31</small>
|<small>4/3</small>
|<small>128/93</small>
|<small>44/31</small>
|<small>46/31</small>
|<small>48/31</small>
|<small>50/31</small>
|<small>52/31</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|}
==== TO 8b ====
<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>.
{| 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
|-
|<small>26/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>6/5</small>
|<small>31/25</small>
|<small>32/25</small>
|<small>33/25</small>
|<small>34/25</small>
|<small>7/5</small>
|<small>36/25</small>
|<small>3/2</small>
|<small>39/25</small>
|<small>81/50</small>
|<small>42/25</small>
|<small>87/50</small>
|<small>9/5</small>
|<small>93/50</small>
|<small>48/25</small>
|-
|<small>27/26</small>
|<small>14/13</small>
|<small>29/26</small>
|<small>15/13</small>
|<small>31/26</small>
|<small>16/13</small>
|<small>33/26</small>
|<small>17/13</small>
|<small>35/26</small>
|<small>18/13</small>
|<small>75/52</small>
|<small>3/2</small>
|<small>81/52</small>
|<small>21/13</small>
|<small>87/52</small>
|<small>45/26</small>
|<small>93/52</small>
|<small>24/13</small>
|<small>25/13</small>
|-
|<small>28/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>11/9</small>
|<small>34/27</small>
|<small>35/27</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>13/9</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>29/18</small>
|<small>5/3</small>
|<small>31/18</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>52/27</small>
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>33/28</small>
|<small>17/14</small>
|<small>5/4</small>
|<small>9/7</small>
|<small>75/56</small>
|<small>39/28</small>
|<small>81/56</small>
|<small>3/2</small>
|<small>87/56</small>
|<small>45/28</small>
|<small>93/56</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>33/29</small>
|<small>34/29</small>
|<small>35/29</small>
|<small>36/29</small>
|<small>75/58</small>
|<small>39/29</small>
|<small>81/58</small>
|<small>42/29</small>
|<small>3/2</small>
|<small>45/29</small>
|<small>93/58</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>11/10</small>
|<small>17/15</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>5/4</small>
|<small>13/10</small>
|<small>27/20</small>
|<small>7/5</small>
|<small>29/20</small>
|<small>3/2</small>
|<small>31/20</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>33/31</small>
|<small>34/31</small>
|<small>35/31</small>
|<small>36/31</small>
|<small>75/62</small>
|<small>39/31</small>
|<small>81/62</small>
|<small>42/31</small>
|<small>87/62</small>
|<small>45/31</small>
|<small>3/2</small>
|<small>48/31</small>
|<small>50/31</small>
|<small>52/31</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|-
|<small>33/32</small>
|<small>17/16</small>
|<small>35/32</small>
|<small>9/8</small>
|<small>75/64</small>
|<small>39/32</small>
|<small>81/64</small>
|<small>21/16</small>
|<small>87/64</small>
|<small>45/32</small>
|<small>93/64</small>
|<small>3/2</small>
|<small>25/16</small>
|<small>13/8</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>29/16</small>
|<small>15/8</small>
|<small>31/16</small>
|-
|<small>34/33</small>
|<small>35/33</small>
|<small>12/11</small>
|<small>25/22</small>
|<small>13/11</small>
|<small>27/22</small>
|<small>14/11</small>
|<small>29/22</small>
|<small>15/11</small>
|<small>31/22</small>
|<small>16/11</small>
|<small>50/33</small>
|<small>52/33</small>
|<small>18/11</small>
|<small>56/33</small>
|<small>58/33</small>
|<small>20/11</small>
|<small>62/33</small>
|<small>64/33</small>
|-
|<small>35/34</small>
|<small>18/17</small>
|<small>75/68</small>
|<small>39/34</small>
|<small>81/68</small>
|<small>21/17</small>
|<small>87/68</small>
|<small>45/34</small>
|<small>93/68</small>
|<small>24/17</small>
|<small>25/17</small>
|<small>26/17</small>
|<small>27/17</small>
|<small>28/17</small>
|<small>29/17</small>
|<small>30/17</small>
|<small>31/17</small>
|<small>32/17</small>
|<small>33/17</small>
|-
|<small>36/35</small>
|<small>15/14</small>
|<small>39/35</small>
|<small>81/70</small>
|<small>6/5</small>
|<small>87/70</small>
|<small>9/7</small>
|<small>93/70</small>
|<small>48/35</small>
|<small>10/7</small>
|<small>52/35</small>
|<small>54/35</small>
|<small>8/5</small>
|<small>58/35</small>
|<small>12/7</small>
|<small>62/35</small>
|<small>64/35</small>
|<small>66/35</small>
|<small>68/35</small>
|-
|<small>25/24</small>
|<small>13/12</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>29/24</small>
|<small>5/4</small>
|<small>31/24</small>
|<small>4/3</small>
|<small>25/18</small>
|<small>13/9</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>29/18</small>
|<small>5/3</small>
|<small>31/18</small>
|<small>16/9</small>
|<small>11/6</small>
|<small>17/9</small>
|<small>35/18</small>
|-
|<small>26/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>6/5</small>
|<small>31/25</small>
|<small>32/25</small>
|<small>4/3</small>
|<small>104/75</small>
|<small>36/25</small>
|<small>112/75</small>
|<small>116/75</small>
|<small>8/5</small>
|<small>124/75</small>
|<small>128/75</small>
|<small>44/25</small>
|<small>136/75</small>
|<small>28/15</small>
|<small>48/25</small>
|-
|<small>27/26</small>
|<small>14/13</small>
|<small>29/26</small>
|<small>15/13</small>
|<small>31/26</small>
|<small>16/13</small>
|<small>50/39</small>
|<small>4/3</small>
|<small>18/13</small>
|<small>56/39</small>
|<small>58/39</small>
|<small>20/13</small>
|<small>62/39</small>
|<small>64/39</small>
|<small>22/13</small>
|<small>68/39</small>
|<small>70/39</small>
|<small>24/13</small>
|<small>25/13</small>
|-
|<small>28/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>100/81</small>
|<small>104/81</small>
|<small>4/3</small>
|<small>112/81</small>
|<small>116/81</small>
|<small>40/27</small>
|<small>124/81</small>
|<small>128/81</small>
|<small>44/27</small>
|<small>136/81</small>
|<small>140/81</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>52/27</small>
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>25/21</small>
|<small>26/21</small>
|<small>9/7</small>
|<small>4/3</small>
|<small>29/21</small>
|<small>10/7</small>
|<small>31/21</small>
|<small>32/21</small>
|<small>11/7</small>
|<small>34/21</small>
|<small>5/3</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>100/87</small>
|<small>104/87</small>
|<small>36/29</small>
|<small>112/87</small>
|<small>4/3</small>
|<small>40/29</small>
|<small>124/87</small>
|<small>128/87</small>
|<small>44/29</small>
|<small>136/87</small>
|<small>140/87</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>10/9</small>
|<small>52/45</small>
|<small>6/5</small>
|<small>56/45</small>
|<small>58/45</small>
|<small>4/3</small>
|<small>62/45</small>
|<small>64/45</small>
|<small>22/15</small>
|<small>68/45</small>
|<small>14/9</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>100/93</small>
|<small>104/93</small>
|<small>36/31</small>
|<small>112/93</small>
|<small>116/93</small>
|<small>40/31</small>
|<small>4/3</small>
|<small>128/93</small>
|<small>44/31</small>
|<small>136/93</small>
|<small>140/93</small>
|<small>48/31</small>
|<small>50/31</small>
|<small>52/31</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|}
== Beyond Category 8 ==
There are infinite categories with infinitely many TOs, but TOs grow increasingly more complex in prime palette and in interval count. As such, we present here only the TOs whose palettes fit within the 31-limit.
==== TO 9a ====
<code>54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96:99:102:105:108</code>. The biggest TO to feature a palette below the 31-limit, with [[harmonics 9-18]] as a subset, as well as TO 3.
No patent val supports an equalized tuning; 19edo, the closest match, tempers together 34/27 and 35/27, as well as 17/9 and 35/128. The closest match is 22edo, which either of 22 or 22f vals provide the basic tuning for [[1L 19s]].
{| class="wikitable"
|-
!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
!11/6
!17/9
!35/18
! rowspan="21" |2
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>33/28</small>
|<small>17/14</small>
|<small>5/4</small>
|<small>9/7</small>
|<small>75/56</small>
|<small>39/28</small>
|<small>81/56</small>
|<small>3/2</small>
|<small>87/56</small>
|<small>45/28</small>
|<small>93/56</small>
|<small>12/7</small>
|<small>99/56</small>
|<small>51/28</small>
|<small>15/8</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>33/29</small>
|<small>34/29</small>
|<small>35/29</small>
|<small>36/29</small>
|<small>75/58</small>
|<small>39/29</small>
|<small>81/58</small>
|<small>42/29</small>
|<small>3/2</small>
|<small>45/29</small>
|<small>93/58</small>
|<small>48/29</small>
|<small>99/58</small>
|<small>51/29</small>
|<small>105/58</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>11/10</small>
|<small>17/15</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>5/4</small>
|<small>13/10</small>
|<small>27/20</small>
|<small>7/5</small>
|<small>29/20</small>
|<small>3/2</small>
|<small>31/20</small>
|<small>8/5</small>
|<small>33/20</small>
|<small>17/10</small>
|<small>7/4</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>33/31</small>
|<small>34/31</small>
|<small>35/31</small>
|<small>36/31</small>
|<small>75/62</small>
|<small>39/31</small>
|<small>81/62</small>
|<small>42/31</small>
|<small>87/62</small>
|<small>45/31</small>
|<small>3/2</small>
|<small>48/31</small>
|<small>99/62</small>
|<small>51/31</small>
|<small>105/62</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|-
|<small>33/32</small>
|<small>17/16</small>
|<small>35/32</small>
|<small>9/8</small>
|<small>75/64</small>
|<small>39/32</small>
|<small>81/64</small>
|<small>21/16</small>
|<small>87/64</small>
|<small>45/32</small>
|<small>93/64</small>
|<small>3/2</small>
|<small>99/64</small>
|<small>51/32</small>
|<small>105/64</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>29/16</small>
|<small>15/8</small>
|<small>31/16</small>
|-
|<small>34/33</small>
|<small>35/33</small>
|<small>12/11</small>
|<small>25/22</small>
|<small>13/11</small>
|<small>27/22</small>
|<small>14/11</small>
|<small>29/22</small>
|<small>15/11</small>
|<small>31/22</small>
|<small>16/11</small>
|<small>3/2</small>
|<small>17/11</small>
|<small>35/22</small>
|<small>18/11</small>
|<small>56/33</small>
|<small>58/33</small>
|<small>20/11</small>
|<small>62/33</small>
|<small>64/33</small>
|-
|<small>35/34</small>
|<small>18/17</small>
|<small>75/68</small>
|<small>39/34</small>
|<small>81/68</small>
|<small>21/17</small>
|<small>87/68</small>
|<small>45/34</small>
|<small>93/68</small>
|<small>24/17</small>
|<small>99/68</small>
|<small>3/2</small>
|<small>105/68</small>
|<small>27/17</small>
|<small>28/17</small>
|<small>29/17</small>
|<small>30/17</small>
|<small>31/17</small>
|<small>32/17</small>
|<small>33/17</small>
|-
|<small>36/35</small>
|<small>15/14</small>
|<small>39/35</small>
|<small>81/70</small>
|<small>6/5</small>
|<small>87/70</small>
|<small>9/7</small>
|<small>93/70</small>
|<small>48/35</small>
|<small>99/70</small>
|<small>51/35</small>
|<small>3/2</small>
|<small>54/35</small>
|<small>8/5</small>
|<small>58/35</small>
|<small>12/7</small>
|<small>62/35</small>
|<small>64/35</small>
|<small>66/35</small>
|<small>68/35</small>
|-
|<small>25/24</small>
|<small>13/12</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>29/24</small>
|<small>5/4</small>
|<small>31/24</small>
|<small>4/3</small>
|<small>11/8</small>
|<small>17/12</small>
|<small>35/24</small>
|<small>3/2</small>
|<small>14/9</small>
|<small>29/18</small>
|<small>5/3</small>
|<small>31/18</small>
|<small>16/9</small>
|<small>11/6</small>
|<small>17/9</small>
|<small>35/18</small>
|-
|<small>26/25</small>
|<small>27/25</small>
|<small>28/25</small>
|<small>29/25</small>
|<small>6/5</small>
|<small>31/25</small>
|<small>32/25</small>
|<small>33/25</small>
|<small>34/25</small>
|<small>7/5</small>
|<small>36/25</small>
|<small>112/75</small>
|<small>116/75</small>
|<small>8/5</small>
|<small>124/75</small>
|<small>128/75</small>
|<small>44/25</small>
|<small>136/75</small>
|<small>28/15</small>
|<small>48/25</small>
|-
|<small>27/26</small>
|<small>14/13</small>
|<small>29/26</small>
|<small>15/13</small>
|<small>31/26</small>
|<small>16/13</small>
|<small>33/26</small>
|<small>17/13</small>
|<small>35/26</small>
|<small>18/13</small>
|<small>56/39</small>
|<small>58/39</small>
|<small>20/13</small>
|<small>62/39</small>
|<small>64/39</small>
|<small>22/13</small>
|<small>68/39</small>
|<small>70/39</small>
|<small>24/13</small>
|<small>25/13</small>
|-
|<small>28/27</small>
|<small>29/27</small>
|<small>10/9</small>
|<small>31/27</small>
|<small>32/27</small>
|<small>11/9</small>
|<small>34/27</small>
|<small>35/27</small>
|<small>4/3</small>
|<small>112/81</small>
|<small>116/81</small>
|<small>40/27</small>
|<small>124/81</small>
|<small>128/81</small>
|<small>44/27</small>
|<small>136/81</small>
|<small>140/81</small>
|<small>16/9</small>
|<small>50/27</small>
|<small>52/27</small>
|-
|<small>29/28</small>
|<small>15/14</small>
|<small>31/28</small>
|<small>8/7</small>
|<small>33/28</small>
|<small>17/14</small>
|<small>5/4</small>
|<small>9/7</small>
|<small>4/3</small>
|<small>29/21</small>
|<small>10/7</small>
|<small>31/21</small>
|<small>32/21</small>
|<small>11/7</small>
|<small>34/21</small>
|<small>5/3</small>
|<small>12/7</small>
|<small>25/14</small>
|<small>13/7</small>
|<small>27/14</small>
|-
|<small>30/29</small>
|<small>31/29</small>
|<small>32/29</small>
|<small>33/29</small>
|<small>34/29</small>
|<small>35/29</small>
|<small>36/29</small>
|<small>112/87</small>
|<small>4/3</small>
|<small>40/29</small>
|<small>124/87</small>
|<small>128/87</small>
|<small>44/29</small>
|<small>136/87</small>
|<small>140/87</small>
|<small>48/29</small>
|<small>50/29</small>
|<small>52/29</small>
|<small>54/29</small>
|<small>56/29</small>
|-
|<small>31/30</small>
|<small>16/15</small>
|<small>11/10</small>
|<small>17/15</small>
|<small>7/6</small>
|<small>6/5</small>
|<small>56/45</small>
|<small>58/45</small>
|<small>4/3</small>
|<small>62/45</small>
|<small>64/45</small>
|<small>22/15</small>
|<small>68/45</small>
|<small>14/9</small>
|<small>8/5</small>
|<small>5/3</small>
|<small>26/15</small>
|<small>9/5</small>
|<small>28/15</small>
|<small>29/15</small>
|-
|<small>32/31</small>
|<small>33/31</small>
|<small>34/31</small>
|<small>35/31</small>
|<small>36/31</small>
|<small>112/93</small>
|<small>116/93</small>
|<small>40/31</small>
|<small>4/3</small>
|<small>128/93</small>
|<small>44/31</small>
|<small>136/93</small>
|<small>140/93</small>
|<small>48/31</small>
|<small>50/31</small>
|<small>52/31</small>
|<small>54/31</small>
|<small>56/31</small>
|<small>58/31</small>
|<small>60/31</small>
|-
|<small>33/32</small>
|<small>17/16</small>
|<small>35/32</small>
|<small>9/8</small>
|<small>7/6</small>
|<small>29/24</small>
|<small>5/4</small>
|<small>31/24</small>
|<small>4/3</small>
|<small>11/8</small>
|<small>17/12</small>
|<small>35/24</small>
|<small>3/2</small>
|<small>25/16</small>
|<small>13/8</small>
|<small>27/16</small>
|<small>7/4</small>
|<small>29/16</small>
|<small>15/8</small>
|<small>31/16</small>
|-
|<small>34/33</small>
|<small>35/33</small>
|<small>12/11</small>
|<small>112/99</small>
|<small>116/99</small>
|<small>40/33</small>
|<small>124/99</small>
|<small>128/99</small>
|<small>4/3</small>
|<small>136/99</small>
|<small>140/99</small>
|<small>16/11</small>
|<small>50/33</small>
|<small>52/33</small>
|<small>18/11</small>
|<small>56/33</small>
|<small>58/33</small>
|<small>20/11</small>
|<small>62/33</small>
|<small>64/33</small>
|-
|<small>35/34</small>
|<small>18/17</small>
|<small>56/51</small>
|<small>58/51</small>
|<small>20/17</small>
|<small>62/51</small>
|<small>64/51</small>
|<small>22/17</small>
|<small>4/3</small>
|<small>70/51</small>
|<small>24/17</small>
|<small>25/17</small>
|<small>26/17</small>
|<small>27/17</small>
|<small>28/17</small>
|<small>29/17</small>
|<small>30/17</small>
|<small>31/17</small>
|<small>32/17</small>
|<small>33/17</small>
|-
|<small>36/35</small>
|<small>16/15</small>
|<small>116/105</small>
|<small>8/7</small>
|<small>124/105</small>
|<small>128/105</small>
|<small>44/35</small>
|<small>136/105</small>
|<small>4/3</small>
|<small>48/35</small>
|<small>10/7</small>
|<small>52/35</small>
|<small>54/35</small>
|<small>8/5</small>
|<small>58/35</small>
|<small>12/7</small>
|<small>62/35</small>
|<small>64/35</small>
|<small>66/35</small>
|<small>68/35</small>
|}
|}


== Beyond HSAF 8 ==
== References ==
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.
<references />
Retrieved from "https://en.xen.wiki/w/Tetrachordal_otonal_scale"