21edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~xenwolf~~|~~xenwolf~~]] and made on <tt>~~2013~~-07-~~05 09~~:21:39 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2014-06-28 15:08:52 UTC</tt>.

: The original revision id was <tt>~~440538192~~</tt>.

: The original revision id was <tt>515172790</tt>.

: The revision comment was: <tt>+1</tt>

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<h4>Original Wikitext content:</h4>

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In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

|| **Degree** || **Cents**

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**Notation** ||= **D.-R. Interval**

**Types** ||= **Approximate**

**Ratios*** ||

**Ratios*** ||= **Approximate**

|| 0 || 0 || C || C ||= Unison ||= 1/1 ||

**Ratios**** ||= **Approximate**

|| 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||

**Ratios***** ||

|| 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||

|| 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||

|| 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||

|| 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||

|| 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||

|| 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||

|| 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||

|| 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||

|| 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||

|| 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||

|| 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||

|| 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5 ||

|| 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||

|| 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||

|| 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||

|| 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 11/9, 32/25 ||

|| 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||

|| 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||

|| 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||

|| 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||

|| 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||

|| 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||

|| 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||

|| 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||

|| 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||

|| 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||

|| 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||

|| 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||

|| 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||

|| 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 18/11, 25/16 ||

|| 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||

|| 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||

|| 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||

|| 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3 ||

|| 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||

|| 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||

|| 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||

|| 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||

|| 21 || 1200 || C || C ||= Octave ||= 2/1 ||

|| 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||

|| 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||

|| 21 || 1200 || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||

*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament~~; other approaches are possible~~.

*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament

**based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament

***based on treating 21-EDO as 13-limit laconic temperament

**21-tone scales:**

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In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &quot;third-fourth&quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.

In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit <a class="wiki_link" href="/Laconic%20Family">laconic</a> temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments.

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit <a class="wiki_link" href="/Laconic%20Family">laconic</a> temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some &quot;interesting&quot; mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

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<td style="text-align: center;">Approximate

Ratios*

</td>

<td style="text-align: center;">Approximate

Ratios

</td>

<td style="text-align: center;">Approximate

Ratios*

</td>

</tr>

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</td>

<td style="text-align: center;">Unison

</td>

<td style="text-align: center;">1/1

</td>

<td style="text-align: center;">1/1

</td>

<td style="text-align: center;">1/1

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</td>

<td style="text-align: center;">28/27, 30/29

</td>

<td style="text-align: center;">35/34, 36/35

</td>

<td style="text-align: center;">64/63

</td>

</tr>

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</td>

<td style="text-align: center;">16/15, 15/14, 29/27

</td>

<td style="text-align: center;">18/17

</td>

<td style="text-align: center;">16/15, 25/24

</td>

</tr>

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</td>

<td style="text-align: center;">10/9, 32/29

</td>

<td style="text-align: center;">10/9,11/10

</td>

<td style="text-align: center;">9/8

</td>

</tr>

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</td>

<td style="text-align: center;">8/7

</td>

<td style="text-align: center;">8/7

</td>

<td style="text-align: center;">8/7, 10/9, 11/10

</td>

</tr>

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</td>

<td style="text-align: center;">27/23, 32/27

</td>

<td style="text-align: center;">13/11, 20/17

</td>

<td style="text-align: center;">6/5

</td>

</tr>

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</td>

<td style="text-align: center;">28/23

</td>

<td style="text-align: center;">11/9

</td>

<td style="text-align: center;">16/13

</td>

</tr>

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</td>

<td style="text-align: center;">29/23

</td>

<td style="text-align: center;">44/35

</td>

<td style="text-align: center;">5/4, 11/9, 32/25

</td>

</tr>

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</td>

<td style="text-align: center;">30/23

</td>

<td style="text-align: center;">13/10, 17/13, 22/17

</td>

<td style="text-align: center;">13/10

</td>

</tr>

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</td>

<td style="text-align: center;">161/120, 256/189

</td>

<td style="text-align: center;">35/26

</td>

<td style="text-align: center;">4/3, 18/13

</td>

</tr>

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</td>

<td style="text-align: center;">32/23

</td>

<td style="text-align: center;">18/13

</td>

<td style="text-align: center;">7/5, 11/8

</td>

</tr>

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</td>

<td style="text-align: center;">23/16

</td>

<td style="text-align: center;">13/9

</td>

<td style="text-align: center;">10/7, 16/11

</td>

</tr>

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</td>

<td style="text-align: center;">189/128, 240/161

</td>

<td style="text-align: center;">52/35

</td>

<td style="text-align: center;">3/2, 13/9

</td>

</tr>

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</td>

<td style="text-align: center;">23/15

</td>

<td style="text-align: center;">17/11, 20/13, 26/17

</td>

<td style="text-align: center;">20/13

</td>

</tr>

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</td>

<td style="text-align: center;">46/29

</td>

<td style="text-align: center;">35/22

</td>

<td style="text-align: center;">8/5, 18/11, 25/16

</td>

</tr>

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</td>

<td style="text-align: center;">23/14

</td>

<td style="text-align: center;">18/11

</td>

<td style="text-align: center;">13/8

</td>

</tr>

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</td>

<td style="text-align: center;">27/16, 46/27

</td>

<td style="text-align: center;">17/10, 22/13

</td>

<td style="text-align: center;">5/3

</td>

</tr>

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</td>

<td style="text-align: center;">7/4

</td>

<td style="text-align: center;">7/4

</td>

<td style="text-align: center;">7/4, 9/5, 20/11

</td>

</tr>

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</td>

<td style="text-align: center;">29/16, 9/5

</td>

<td style="text-align: center;">9/5, 20/11

</td>

<td style="text-align: center;">16/9

</td>

</tr>

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</td>

<td style="text-align: center;">15/8

</td>

<td style="text-align: center;">17/9

</td>

<td style="text-align: center;">15/8, 48/25

</td>

</tr>

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</td>

<td style="text-align: center;">27/14, 29/15

</td>

<td style="text-align: center;">35/18, 68/35

</td>

<td style="text-align: center;">63/32

</td>

</tr>

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</td>

<td style="text-align: center;">Octave

</td>

<td style="text-align: center;">2/1

</td>

<td style="text-align: center;">2/1

</td>

<td style="text-align: center;">2/1

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*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; ~~other approaches are possible~~.

*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament

based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament

*based on treating 21-EDO as 13-limit laconic temperament

21-tone scales:

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<h1 id="toc5"><a name="Books / Literature:"></a>Books / Literature:</h1>

Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" />

<img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" />

21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2013-07-05 09:21:39 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2014-06-28 15:08:52 UTC</tt>.<br>
-: The original revision id was <tt>440538192</tt>.<br>
+: The original revision id was <tt>515172790</tt>.<br>
-: The revision comment was: <tt>+1</tt><br>
+: The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 13: / Line 13: @@
 In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.
-In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.
+In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.
-The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments.
+The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
 || **Degree** || **Cents**
@@ Line 22: / Line 22: @@
 **Notation** ||= **D.-R. Interval**
 **Types** ||= **Approximate**
-**Ratios*** ||
+**Ratios*** ||= &lt;span style="text-align: center;"&gt;**Approximate**&lt;/span&gt;
-|| 0 || 0 || C || C ||= Unison ||= 1/1 ||
+&lt;span style="text-align: center;"&gt;**Ratios****&lt;/span&gt; ||= &lt;span style="text-align: center;"&gt;**Approximate**&lt;/span&gt;
-|| 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||
+&lt;span style="text-align: center;"&gt;**Ratios*****&lt;/span&gt; ||
-|| 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||
+|| 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
-|| 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||
+|| 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
-|| 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||
+|| 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
-|| 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||
+|| 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
-|| 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||
+|| 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
-|| 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||
+|| 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5 ||
-|| 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||
+|| 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
-|| 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||
+|| 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 11/9, 32/25 ||
-|| 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||
+|| 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
-|| 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||
+|| 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
-|| 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||
+|| 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
-|| 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||
+|| 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
-|| 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||
+|| 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
-|| 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||
+|| 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
-|| 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||
+|| 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 18/11, 25/16 ||
-|| 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||
+|| 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
-|| 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||
+|| 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3 ||
-|| 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||
+|| 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
-|| 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||
+|| 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
-|| 21 || 1200 || C || C ||= Octave ||= 2/1 ||
+|| 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
+|| 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
+|| 21 || 1200 || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||
-*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; other approaches are possible.
+*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament
+**based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
+***based on treating 21-EDO as 13-limit laconic temperament
 **21-tone scales:**
@@ Line 117: / Line 121: @@
   In diatonically-related terms, 21-EDO possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a &amp;quot;third-fourth&amp;quot; (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.&lt;br /&gt;
 &lt;br /&gt;
-In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit.&lt;br /&gt;
+In temperament terms, 21-EDO can be treated as a 13-limit temperament, but of harmonics 3, 5, 7, 11, and 13, the only harmonic 21-EDO approximates with anything approaching a near-Just flavor is the 7th harmonic. On the other hand, 21-EDO provides exceptionally accurate tunings of the 15th, 23rd, and 29th harmonics (within 3 cents or less), as well as a very reasonable approximation of the 27th harmonic (around 8 cents sharp). As such, treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament allows for a more accurate rationalization of the tuning, since almost every interval in 21-EDO can be described as a ratio within the 29-odd-limit. 21-EDO also works well on the 2.9/5.11/5.13/5.17/5.35/5 subgroup, which is possibly a more sensible way to treat it.&lt;br /&gt;
 &lt;br /&gt;
-The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit &lt;a class="wiki_link" href="/Laconic%20Family"&gt;laconic&lt;/a&gt; temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments.&lt;br /&gt;
+The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit &lt;a class="wiki_link" href="/Laconic%20Family"&gt;laconic&lt;/a&gt; temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic and spartan temperaments. These temperaments lead to some &amp;quot;interesting&amp;quot; mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 141: / Line 145: @@
          &lt;td style="text-align: center;"&gt;&lt;strong&gt;Approximate&lt;/strong&gt;&lt;br /&gt;
 &lt;strong&gt;Ratios&lt;/strong&gt;*&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="text-align: center;"&gt;&lt;strong&gt;Approximate&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="text-align: center;"&gt;&lt;strong&gt;Ratios&lt;/strong&gt;&lt;strong&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;span style="text-align: center;"&gt;&lt;/strong&gt;Approximate&lt;strong&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="text-align: center;"&gt;&lt;/strong&gt;Ratios*&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 153: / Line 163: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Unison&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
@@ Line 169: / Line 183: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;28/27, 30/29&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;35/34, 36/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 183: / Line 201: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;16/15, 15/14, 29/27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;18/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;16/15, 25/24&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 197: / Line 219: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;10/9, 32/29&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;10/9,11/10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 211: / Line 237: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;8/7, 10/9, 11/10&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 225: / Line 255: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/23, 32/27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13/11, 20/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;6/5&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 239: / Line 273: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;28/23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;11/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;16/13&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 253: / Line 291: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;29/23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;44/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;5/4, 11/9, 32/25&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 267: / Line 309: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;30/23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13/10, 17/13, 22/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13/10&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 281: / Line 327: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;161/120, 256/189&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;35/26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;4/3, 18/13&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 295: / Line 345: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;32/23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;18/13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7/5, 11/8&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 309: / Line 363: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;10/7, 16/11&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 323: / Line 381: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;189/128, 240/161&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;52/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;3/2, 13/9&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 337: / Line 399: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/11, 20/13, 26/17&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;20/13&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 351: / Line 417: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;46/29&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;35/22&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;8/5, 18/11, 25/16&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 365: / Line 435: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;18/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;13/8&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 379: / Line 453: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/16, 46/27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/10, 22/13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;5/3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 393: / Line 471: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7/4, 9/5, 20/11&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 407: / Line 489: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;29/16, 9/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;9/5, 20/11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;16/9&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 421: / Line 507: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;15/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;17/9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;15/8, 48/25&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 435: / Line 525: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/14, 29/15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;35/18, 68/35&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;63/32&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 447: / Line 541: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Octave&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
@@ Line 454: / Line 552: @@
 &lt;br /&gt;
-*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; other approaches are possible.&lt;br /&gt;
+*based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament&lt;br /&gt;
+&lt;strong&gt;based on treating 21-EDO as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament&lt;br /&gt;
+&lt;/strong&gt;*based on treating 21-EDO as 13-limit laconic temperament&lt;br /&gt;
 &lt;br /&gt;
 &lt;strong&gt;21-tone scales:&lt;/strong&gt;&lt;br /&gt;
@@ Line 770: / Line 870: @@
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
   Sword, Ron. &amp;quot;Icosihenaphonic Scales for Guitar&amp;quot;. IAAA Press. 1st ed: July 2009.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextRemoteImageRule:622:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:622 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:623:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:623 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:714:&amp;lt;img src=&amp;quot;http://www.ronsword.com/images/ron1.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 188px; width: 254px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:714 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:715:&amp;lt;img src=&amp;quot;http://www.swordguitars.com/21tetsm.JPG&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 191px; width: 363px;&amp;quot; /&amp;gt; --&gt;&lt;img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /&gt;&lt;!-- ws:end:WikiTextRemoteImageRule:715 --&gt;&lt;br /&gt;
 &lt;strong&gt;&lt;em&gt;21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)&lt;/em&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;

21edo: Difference between revisions

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