User:Eufalesio/Harmonic Segments Along Fifths: Difference between revisions
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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 | 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 == | ||
* 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 == | == 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 | <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 47: | ||
=== HSA5 2 - ''Zontatonic'' === | === HSA5 2 - ''Zontatonic'' === | ||
'''6::8'''. | <code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''. | ||
First usable HSA5, the first prime HSA5, and the only one that is [[Strict variety]] 3 and a [[generator sequence]]. It is naturally a [[5edo]] detemper in the [[2.3.7 subgroup]], or otherwise an [[archy]] detemper, [[2L 3s]] 3|1 scale. | |||
{| class="wikitable" data-darkreader-inline-color="" | {| class="wikitable" data-darkreader-inline-color="" | ||
!7/6 | !7/6 | ||
| Line 68: | Line 79: | ||
=== HSA5 3 - ''Íegmul'' === | === HSA5 3 - ''Íegmul'' === | ||
'''9::12'''. | <code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''. | ||
Second prime HSA5, and the last one to not need extensions. 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). | |||
{| class="wikitable" data-darkreader-inline-color="" | {| class="wikitable" data-darkreader-inline-color="" | ||
!'''10/9''' | !'''10/9''' | ||
| Line 122: | Line 135: | ||
=== HSA5 4 - ''Ngwóoghe'' === | === HSA5 4 - ''Ngwóoghe'' === | ||
'''12::16'''. | <code>24:26:28:30:32:36:39:42:45:48</code>. Harmonic segment: '''12::16'''. | ||
First composite HSA5, containing the 2nd as a subset, and the first one to need extensions. 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. | |||
{| class="wikitable" data-darkreader-inline-color="" | {| class="wikitable" data-darkreader-inline-color="" | ||
! colspan="9" |HSA5 4 | ! colspan="9" |HSA5 4 | ||
|- | |- | ||
!13/12 | !13/12 | ||
| Line 136: | Line 150: | ||
!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''' | ||
!'''3/2''' | !'''3/2''' | ||
!'''13/8''' | !'''13/8''' | ||
| Line 150: | Line 240: | ||
|<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> | ||
|<small>21/13</small> | |<small>21/13</small> | ||
| Line 167: | Line 249: | ||
|<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>3/2</small> | ||
|<small>45/28</small> | |<small>45/28</small> | ||
| Line 184: | Line 258: | ||
|- | |- | ||
|<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>3/2</small> | ||
|<small>8/5</small> | |<small>8/5</small> | ||
| Line 201: | Line 267: | ||
|<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>3/2</small> | ||
|<small>13/8</small> | |<small>13/8</small> | ||
| Line 219: | Line 277: | ||
|<small>15/8</small> | |<small>15/8</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> | |<small>24/17</small> | ||
|<small>26/17</small> | |<small>26/17</small> | ||
|<small>28/17</small> | |<small>28/17</small> | ||
| Line 237: | Line 287: | ||
|<small>32/17</small> | |<small>32/17</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> | |<small>13/9</small> | ||
|<small>14/9</small> | |<small>14/9</small> | ||
|<small>5/3</small> | |<small>5/3</small> | ||
| Line 255: | Line 297: | ||
|<small>17/9</small> | |<small>17/9</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> | |<small>56/39</small> | ||
|<small>20/13</small> | |<small>20/13</small> | ||
|<small>64/39</small> | |<small>64/39</small> | ||
| Line 273: | Line 307: | ||
|<small>24/13</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>26/21</small> | ||
|<small>4/3</small> | |<small>4/3</small> | ||
|<small> | |<small>10/7</small> | ||
|<small>32/21</small> | |<small>32/21</small> | ||
|<small>34/21</small> | |<small>34/21</small> | ||
| Line 291: | Line 317: | ||
|<small>13/7</small> | |<small>13/7</small> | ||
|- | |- | ||
|<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> | |<small>64/45</small> | ||
|<small>68/45</small> | |<small>68/45</small> | ||
|<small>8/5</small> | |<small>8/5</small> | ||
| Line 304: | Line 329: | ||
=== HSA5 5 === | === HSA5 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 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. | '''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. | ||
{| class="wikitable" | {| class="wikitable" | ||
! colspan="12" |heHSA5 5 | ! colspan="12" |heHSA5 5 (7/5) | ||
|- | |- | ||
!16/15 | !16/15 | ||
| Line 602: | Line 629: | ||
=== HSA5 6 === | === HSA5 6 === | ||
'''18::24 | <code>36:38:40:42:44:46:48:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''. | ||
Last HSA5 to be a constant structure without extensions. It is a [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup. | |||
{| class="wikitable" | {| class="wikitable" | ||
! colspan="15" |heHSA5 6 | ! colspan="15" |heHSA5 6 (25/18, 13/9) | ||
|- | |- | ||
!19/18 | !19/18 | ||
| Line 834: | Line 863: | ||
=== 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. 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. | |||
{| class="wikitable" | {| class="wikitable" | ||
! colspan="17" |heHSA5 7 | ! colspan="17" |heHSA5 7 (29/21, 10/7) | ||
|- | |- | ||
!22/21 | !22/21 | ||
| Line 1,130: | Line 1,161: | ||
=== HSA5 8 === | === HSA5 8 === | ||
'''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 [[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. | |||
{| class="wikitable" | {| class="wikitable" | ||
! colspan="20" |heHSA5 8 | ! colspan="20" |heHSA5 8 (11/8, 17/12, 35/24) | ||
|- | |- | ||
!25/24 | !25/24 | ||