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* 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. Because they are formed from harmonic series segments, all except the first are are chiral scales, whose counterparts are built from [[subharmonic series]] segments.
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.


== Properties ==
== Properties ==
Except the first, the harmonic segments conforming HSA5s up to the 13th are [[strictly proper]] [[constant structure]]<nowiki/>s.
 
* Harmonic segments conforming HSA5s up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
* The kth HSA5 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.
 
<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 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:


* 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 HSA5) [seHSA5] so that the step variety is minimized. This is an arbitrary choice that depends on the HSA5.
* 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.
* 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.


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 HSA5 itself.


== List of HSA5s ==
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.
 
== List of HSA5s 1-8 ==
For every provided HSA5, the [[EFR]] and used segment are written out explicitly.
 
=== HSA5 1 - Pythagorean Trial ===
=== HSA5 1 - Pythagorean Trial ===
'''3::4'''. First meaningful HSA5 albeit a trivial case, and the only HSA5 that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]].
<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]].
{| class="wikitable"
{| class="wikitable"
!4/3
!4/3
Line 38: Line 49:


=== HSA5 2 - ''Zontatonic'' ===
=== HSA5 2 - ''Zontatonic'' ===
'''6::8'''. It is the first usable HSA5, the first prime HSA5, and the only one that is [[Strict variety]] 3 and a [[generator sequence]]. It is naturally a [[5edo]] detemper in the [[2.3.7 subgroup]], or otherwise an [[archy]] detemper, [[2L 3s]] 3|1 scale.
<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]].
 
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
!7/6
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=== HSA5 3 - ''Íegmul'' ===
=== HSA5 3 - ''Íegmul'' ===
'''9::12'''. It is the second prime HSA5, and the last one to not need extensions. It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], or otherwise a [[dicot]]/[[mothra]] detemper, [[3L 4s]] 5|1 #6 scale. (1-indexed).
<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.  
 
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 122: Line 141:


=== HSA5 4 - ''Ngwóoghe'' ===
=== HSA5 4 - ''Ngwóoghe'' ===
'''12::16'''. It is the first composite HSA5, containing the 2nd as a subset, and the first one to need extensions. It is naturally a [[10edo]] or detemper in the [[2.3.5.7.13 subgroup]], or otherwise a [[negri]] detemper, [[1L 8s]] 4|4 scale. The only valid heSAH5 is [[17/12]]. Reasonable seHSA5s include 7/5, 13/9, 45/32.
<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.
 
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" |HSA5 4
! colspan="10" |heHSA5 4
|-
|-
!13/12
!13/12
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!15/8
!15/8
! rowspan="9" |2
! rowspan="9" |2
|-
|<small>14/13</small>
|<small>15/13</small>
|<small>16/13</small>
|<small>18/13</small>
|<small>3/2</small>
|<small>21/13</small>
|<small>45/26</small>
|<small>24/13</small>
|-
|<small>15/14</small>
|<small>8/7</small>
|<small>9/7</small>
|<small>39/28</small>
|<small>3/2</small>
|<small>45/28</small>
|<small>12/7</small>
|<small>13/7</small>
|-
|<small>16/15</small>
|<small>6/5</small>
|<small>13/10</small>
|<small>7/5</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>28/15</small>
|-
|<small>9/8</small>
|<small>39/32</small>
|<small>21/16</small>
|<small>45/32</small>
|<small>3/2</small>
|<small>13/8</small>
|<small>7/4</small>
|<small>15/8</small>
|-
|<small>13/12</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>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>24/13</small>
|-
|<small>15/14</small>
|<small>8/7</small>
|<small>26/21</small>
|<small>4/3</small>
|<small>10/7</small>
|<small>32/21</small>
|<small>12/7</small>
|<small>13/7</small>
|-
|<small>16/15</small>
|<small>52/45</small>
|<small>56/45</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>28/15</small>
|}
{| class="wikitable" data-darkreader-inline-color=""
! colspan="10" |heHSA5 4 (17/12)
|-
!'''13/12'''
!'''13/12'''
!'''7/6'''
!'''7/6'''
!'''5/4'''
!'''5/4'''
!'''4/3'''
!'''4/3'''
!'''''17/12'''''
!'''17/12'''
!'''3/2'''
!'''3/2'''
!'''13/8'''
!'''13/8'''
Line 150: Line 248:
|<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>21/13</small>
|<small>45/26</small>
|<small>24/13</small>
|<small>14/13</small>
|<small>15/13</small>
|<small>16/13</small>
|<small>17/13</small>
|<small>''18/13''</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>21/13</small>
|<small>21/13</small>
Line 167: Line 257:
|<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>
|<small>3/2</small>
|<small>45/28</small>
|<small>12/7</small>
|<small>13/7</small>
|<small>15/14</small>
|<small>8/7</small>
|<small>17/14</small>
|<small>9/7</small>
|<small>''39/28''</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>45/28</small>
|<small>45/28</small>
Line 184: Line 266:
|-
|-
|<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>
|<small>7/5</small>
|<small>7/5</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>28/15</small>
|<small>16/15</small>
|<small>17/15</small>
|<small>6/5</small>
|<small>13/10</small>
|<small>''7/5''</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>8/5</small>
|<small>8/5</small>
Line 201: Line 275:
|<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>
|<small>21/16</small>
|<small>21/16</small>
|<small>45/32</small>
|<small>45/32</small>
|<small>3/2</small>
|<small>13/8</small>
|<small>7/4</small>
|<small>15/8</small>
|<small>17/16</small>
|<small>9/8</small>
|<small>39/32</small>
|<small>21/16</small>
|<small>''45/32''</small>
|<small>3/2</small>
|<small>3/2</small>
|<small>13/8</small>
|<small>13/8</small>
Line 219: Line 285:
|<small>15/8</small>
|<small>15/8</small>
|-
|-
|<small>13/12</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>18/17</small>
|<small>18/17</small>
|<small>39/34</small>
|<small>39/34</small>
|<small>21/17</small>
|<small>21/17</small>
|<small>45/34</small>
|<small>45/34</small>
|<small>''24/17''</small>
|<small>24/17</small>
|<small>26/17</small>
|<small>26/17</small>
|<small>28/17</small>
|<small>28/17</small>
Line 237: Line 295:
|<small>32/17</small>
|<small>32/17</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>24/13</small>
|<small>13/12</small>
|<small>13/12</small>
|<small>7/6</small>
|<small>7/6</small>
|<small>5/4</small>
|<small>5/4</small>
|<small>4/3</small>
|<small>4/3</small>
|<small>''13/9''</small>
|<small>13/9</small>
|<small>14/9</small>
|<small>14/9</small>
|<small>5/3</small>
|<small>5/3</small>
Line 255: Line 305:
|<small>17/9</small>
|<small>17/9</small>
|-
|-
|<small>15/14</small>
|<small>8/7</small>
|<small>26/21</small>
|<small>4/3</small>
|<small>10/7</small>
|<small>32/21</small>
|<small>12/7</small>
|<small>13/7</small>
|<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>4/3</small>
|<small>4/3</small>
|<small>''56/39''</small>
|<small>56/39</small>
|<small>20/13</small>
|<small>20/13</small>
|<small>64/39</small>
|<small>64/39</small>
Line 273: Line 315:
|<small>24/13</small>
|<small>24/13</small>
|-
|-
|<small>16/15</small>
|<small>52/45</small>
|<small>56/45</small>
|<small>4/3</small>
|<small>64/45</small>
|<small>8/5</small>
|<small>26/15</small>
|<small>28/15</small>
|<small>15/14</small>
|<small>15/14</small>
|<small>8/7</small>
|<small>8/7</small>
|<small>26/21</small>
|<small>26/21</small>
|<small>4/3</small>
|<small>4/3</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>
Line 291: Line 325:
|<small>13/7</small>
|<small>13/7</small>
|-
|-
| colspan="9" |
|<small>16/15</small>
|<small>16/15</small>
|<small>52/45</small>
|<small>52/45</small>
|<small>56/45</small>
|<small>56/45</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>
Line 304: Line 337:


=== HSA5 5 ===
=== HSA5 5 ===
'''15::20'''. It is the third prime HSA5, and the first HSA5 not to be a constant structure without extensions. It is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]]. The only valid heSAH5 is with [[7/5]]. A very natural and reasonable seHSA5 is 17/12, which fits almost perfectly inside 4/3 and 3/2, though other usable seHSA5 is 64/45.
<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.
 
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
! colspan="12" |heHSA5 5 (7/5)
|-
|-
!16/15
!16/15
Line 602: Line 639:


=== HSA5 6 ===
=== HSA5 6 ===
'''18::24.''' It is the last HSA5 to be a constant structure without extensions. It is a [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup.
<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).
{| class="wikitable"
{| class="wikitable"
! colspan="15" |heHSA5 6
! colspan="15" |heHSA5 6 (25/18, 13/9)
|-
|-
!19/18
!19/18
Line 834: Line 873:


=== HSA5 7 ===
=== HSA5 7 ===
'''21::28'''. The fourth prime HSA5. The most reasonable heHSA5 with 29/21 and 10/7 contains 7::14 as a subset. It is naturally a 17edo detemper in the no-17,19-29-limit subgroup. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.
<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.
 
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
! colspan="17" |heHSA5 7 (29/21, 10/7)
|-
|-
!22/21
!22/21
Line 1,129: Line 1,172:
|}
|}


=== HSA5 8 ===
=== HSA5 8 - ''Veljeuwof'' ===
'''24::32.''' It contains HSA5 2, HSA5 4 and [[8afdo]] as subsets. As a superset of HSA5 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35. It is not an easy edo detemper, just a harmonic interpolation of HSA5 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSA5.
<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.
 
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.
{| class="wikitable"
{| class="wikitable"
! colspan="20" |heHSA5 8
! colspan="20" |heHSA5 8 (11/8, 17/12, 35/24)
|-
|-
!25/24
!25/24
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