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:igliashon|igliashon]] and made on <tt>2011-07-24 15:28:53 UTC</tt>.

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-24 15:44:01 UTC</tt>.

: The original revision id was <tt>~~242634011~~</tt>.

: The original revision id was <tt>242635235</tt>.

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

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==Harmony in 21-EDO:==

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.

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.

|| Degree || Cents

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

Value || Name ||= Approximate

**Value** || **Letter** ||= **D.-R. Name** ||= **Approximate**

Ratios* ||

**Ratios*** ||

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

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

|| 1 || 57.143 || C^/Dvv ||= 28/27, 30/29 ||

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

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

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

|| 3 || 171.429 || D ||= 10/9, 32/29 ||

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

|| 4 || 228.571 || D^/Evv ||= 8/7 ||

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

|| 5 || 285.714 || D^^/Ev ||= 27/23, 32/27 ||

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

|| 6 || 342.857 || E ||= 28/23 ||

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

|| 7 || 400 || E^/Fvv ||= 29/23 ||

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

|| 8 || 457.143 || E^^/Fv ||= 30/23 ||

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

|| 9 || 514.286 || F ||= 161/120, 256/189 ||

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

|| 10 || 571.429 || F^/Gvv ||= 32/23 ||

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

|| 11 || 628.571 || F^^/Gv ||= 23/16 ||

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

|| 12 || 685.714 || G ||= 189/128, 240/161 ||

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

|| 13 || 742.857 || G^/Avv ||= 23/15 ||

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

|| 14 || 800 || G^^/Av ||= 46/29 ||

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

|| 15 || 857.143 || A ||= 23/14 ||

|| 15 || 857.143 || A ||= Neutral 6th ||= 23/14 ||

|| 16 || 914.286 || A^/Bvv ||= 27/16, 46/27 ||

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

|| 17 || 971.429 || A^^/Bv ||= 7/4 ||

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

|| 18 || 1028.571 || B ||= 29/16, 9/5 ||

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

|| 19 || 1085.714 || B^/Cvv ||= 15/8 ||

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

|| 20 || 1142.857 || B^^/Cv ||= 27/14, 29/15 ||

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

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

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

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

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[[augment12]]

==Moment-of-Symmetry Scales in 21-EDO:==

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.

For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.

For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

==13-limit Commas==

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<h2 id="toc1"><a name="x21 equal divisions of the octave-Harmony in 21-EDO:"></a>Harmony in 21-EDO:</h2>

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.

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.

<tr>

<td>Degree

<td>Degree

</td>

<td>Cents

<td>Cents

Value

Value

</td>

<td>~~Name~~

<td>Letter

</td>

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

<td style="text-align: center;">D.-R. Name

Ratios*

</td>

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

Ratios*

</td>

</tr>

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</td>

<td>C

</td>

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

</td>

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

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Line 128:

</td>

<td>C^/Dvv

</td>

<td style="text-align: center;">Subminor 2nd

</td>

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

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Line 140:

</td>

<td>C^^/Dv

</td>

<td style="text-align: center;">Minor 2nd

</td>

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

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Line 152:

</td>

<td>D

</td>

<td style="text-align: center;">Submajor 2nd

</td>

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

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Line 164:

</td>

<td>D^/Evv

</td>

<td style="text-align: center;">Supermajor 2nd

</td>

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

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</td>

<td>D^^/Ev

</td>

<td style="text-align: center;">Subminor 3rd

</td>

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

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</td>

<td>E

</td>

<td style="text-align: center;">Neutral 3rd

</td>

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

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</td>

<td>E^/Fvv

</td>

<td style="text-align: center;">Major 3rd

</td>

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

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Line 212:

</td>

<td>E^^/Fv

</td>

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

</td>

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

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</td>

<td>F

</td>

<td style="text-align: center;">Acute 4th

</td>

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

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Line 236:

</td>

<td>F^/Gvv

</td>

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

</td>

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

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</td>

<td>F^^/Gv

</td>

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

</td>

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

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</td>

<td>G

</td>

<td style="text-align: center;">Grave 5th

</td>

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

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Line 272:

</td>

<td>G^/Avv

</td>

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

</td>

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

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</td>

<td>G^^/Av

</td>

<td style="text-align: center;">Minor 6th

</td>

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

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Line 296:

</td>

<td>A

</td>

<td style="text-align: center;">Neutral 6th

</td>

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

Line 267:

Line 308:

</td>

<td>A^/Bvv

</td>

<td style="text-align: center;">Supermajor 6th

</td>

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

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Line 320:

</td>

<td>A^^/Bv

</td>

<td style="text-align: center;">Subminor 7th

</td>

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

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Line 332:

</td>

<td>B

</td>

<td style="text-align: center;">Supraminor 7th

</td>

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

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Line 344:

</td>

<td>B^/Cvv

</td>

<td style="text-align: center;">Major 7th

</td>

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

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Line 356:

</td>

<td>B^^/Cv

</td>

<td style="text-align: center;">Supermajor 7th

</td>

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

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</td>

<td>C

</td>

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

</td>

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

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<a class="wiki_link" href="/augment12">augment12</a>

<h2 id="toc2"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a>Moment-of-Symmetry Scales in 21-EDO:</h2>

Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.

For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.

For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.

<h2 id="~~toc2~~"><a name="x21 equal divisions of the octave-13-limit Commas"></a>13-limit Commas</h2>

<h2 id="toc3"><a name="x21 equal divisions of the octave-13-limit Commas"></a>13-limit Commas</h2>

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)

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<h1 id="~~toc3~~"><a name="Books / Literature:"></a>Books / Literature:</h1>

<h1 id="toc4"><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)

<h1 id="~~toc4~~"><a name="Compositions/Listening:"></a>Compositions/Listening:</h1>

<h1 id="toc5"><a name="Compositions/Listening:"></a>Compositions/Listening:</h1>

<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank">Short Clip of 21-edo Acoustic</a> by <a class="wiki_link" href="/Ron%20Sword">Ron Sword</a>

<a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank">Open tuning Drone Improvisation in 21-edo</a> by Ron Sword

<a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933715" rel="nofollow">Anomalous Readings</a> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow">play</a> by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a></body></html></pre></div>

@@ 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:igliashon|igliashon]] and made on <tt>2011-07-24 15:28:53 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-24 15:44:01 UTC</tt>.<br>
-: The original revision id was <tt>242634011</tt>.<br>
+: The original revision id was <tt>242635235</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>
@@ Line 11: / Line 11: @@
 ==Harmony in 21-EDO:==
+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.
--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.
-|| Degree || Cents
+|| **Degree** || **Cents**
-Value || Name ||= Approximate
+**Value** || **Letter** ||= **D.-R. Name** ||= **Approximate**
-Ratios* ||
+**Ratios*** ||
-|| 0 || 0 || C ||= 1/1 ||
+|| 0 || 0 || C ||= Unison ||= 1/1 ||
-|| 1 || 57.143 || C^/Dvv ||= 28/27, 30/29 ||
+|| 1 || 57.143 || C^/Dvv ||= Subminor 2nd ||= 28/27, 30/29 ||
-|| 2 || 114.286 || C^^/Dv ||= 16/15, 15/14, 29/27 ||
+|| 2 || 114.286 || C^^/Dv ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||
-|| 3 || 171.429 || D ||= 10/9, 32/29 ||
+|| 3 || 171.429 || D ||= Submajor 2nd ||= 10/9, 32/29 ||
-|| 4 || 228.571 || D^/Evv ||= 8/7 ||
+|| 4 || 228.571 || D^/Evv ||= Supermajor 2nd ||= 8/7 ||
-|| 5 || 285.714 || D^^/Ev ||= 27/23, 32/27 ||
+|| 5 || 285.714 || D^^/Ev ||= Subminor 3rd ||= 27/23, 32/27 ||
-|| 6 || 342.857 || E ||= 28/23 ||
+|| 6 || 342.857 || E ||= Neutral 3rd ||= 28/23 ||
-|| 7 || 400 || E^/Fvv ||= 29/23 ||
+|| 7 || 400 || E^/Fvv ||= Major 3rd ||= 29/23 ||
-|| 8 || 457.143 || E^^/Fv ||= 30/23 ||
+|| 8 || 457.143 || E^^/Fv ||= Third-Fourth ||= 30/23 ||
-|| 9 || 514.286 || F ||= 161/120, 256/189 ||
+|| 9 || 514.286 || F ||= Acute 4th ||= 161/120, 256/189 ||
-|| 10 || 571.429 || F^/Gvv ||= 32/23 ||
+|| 10 || 571.429 || F^/Gvv ||= Narrow Tritone ||= 32/23 ||
-|| 11 || 628.571 || F^^/Gv ||= 23/16 ||
+|| 11 || 628.571 || F^^/Gv ||= Wide Tritone ||= 23/16 ||
-|| 12 || 685.714 || G ||= 189/128, 240/161 ||
+|| 12 || 685.714 || G ||= Grave 5th ||= 189/128, 240/161 ||
-|| 13 || 742.857 || G^/Avv ||= 23/15 ||
+|| 13 || 742.857 || G^/Avv ||= Fifth-Sixth ||= 23/15 ||
-|| 14 || 800 || G^^/Av ||= 46/29 ||
+|| 14 || 800 || G^^/Av ||= Minor 6th ||= 46/29 ||
-|| 15 || 857.143 || A ||= 23/14 ||
+|| 15 || 857.143 || A ||= Neutral 6th ||= 23/14 ||
-|| 16 || 914.286 || A^/Bvv ||= 27/16, 46/27 ||
+|| 16 || 914.286 || A^/Bvv ||= Supermajor 6th ||= 27/16, 46/27 ||
-|| 17 || 971.429 || A^^/Bv ||= 7/4 ||
+|| 17 || 971.429 || A^^/Bv ||= Subminor 7th ||= 7/4 ||
-|| 18 || 1028.571 || B ||= 29/16, 9/5 ||
+|| 18 || 1028.571 || B ||= Supraminor 7th ||= 29/16, 9/5 ||
-|| 19 || 1085.714 || B^/Cvv ||= 15/8 ||
+|| 19 || 1085.714 || B^/Cvv ||= Major 7th ||= 15/8 ||
-|| 20 || 1142.857 || B^^/Cv ||= 27/14, 29/15 ||
+|| 20 || 1142.857 || B^^/Cv ||= Supermajor 7th ||= 27/14, 29/15 ||
-|| 21 || 1200 || C ||= 2/1 ||
+|| 21 || 1200 || C ||= Octave ||= 2/1 ||
 *based on treating 21-EDO as a 2.7.15.23.27.29 subgroup temperament; other approaches are possible.
@@ Line 46: / Line 47: @@
 [[augment12]]
+==Moment-of-Symmetry Scales in 21-EDO:==
+Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period.
+For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.
+For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.
 ==13-limit Commas==
@@ Line 84: / Line 90: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x21 equal divisions of the octave-Harmony in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Harmony in 21-EDO:&lt;/h2&gt;
-  &lt;br /&gt;
+  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;
--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;
+&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;
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;Degree&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Cents&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;Cents&lt;/strong&gt;&lt;br /&gt;
-Value&lt;br /&gt;
+&lt;strong&gt;Value&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Name&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;Letter&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Approximate&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;D.-R. Name&lt;/strong&gt;&lt;br /&gt;
-Ratios*&lt;br /&gt;
+&lt;/td&gt;
+        &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;/tr&gt;
@@ Line 107: / Line 116: @@
 &lt;/td&gt;
          &lt;td&gt;C&lt;br /&gt;
+&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;
@@ Line 117: / Line 128: @@
 &lt;/td&gt;
          &lt;td&gt;C^/Dvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Subminor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;28/27, 30/29&lt;br /&gt;
@@ Line 127: / Line 140: @@
 &lt;/td&gt;
          &lt;td&gt;C^^/Dv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Minor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;16/15, 15/14, 29/27&lt;br /&gt;
@@ Line 137: / Line 152: @@
 &lt;/td&gt;
          &lt;td&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Submajor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;10/9, 32/29&lt;br /&gt;
@@ Line 147: / Line 164: @@
 &lt;/td&gt;
          &lt;td&gt;D^/Evv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Supermajor 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;8/7&lt;br /&gt;
@@ Line 157: / Line 176: @@
 &lt;/td&gt;
          &lt;td&gt;D^^/Ev&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Subminor 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/23, 32/27&lt;br /&gt;
@@ Line 167: / Line 188: @@
 &lt;/td&gt;
          &lt;td&gt;E&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Neutral 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;28/23&lt;br /&gt;
@@ Line 177: / Line 200: @@
 &lt;/td&gt;
          &lt;td&gt;E^/Fvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Major 3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;29/23&lt;br /&gt;
@@ Line 187: / Line 212: @@
 &lt;/td&gt;
          &lt;td&gt;E^^/Fv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Third-Fourth&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;30/23&lt;br /&gt;
@@ Line 197: / Line 224: @@
 &lt;/td&gt;
          &lt;td&gt;F&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Acute 4th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;161/120, 256/189&lt;br /&gt;
@@ Line 207: / Line 236: @@
 &lt;/td&gt;
          &lt;td&gt;F^/Gvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Narrow Tritone&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;32/23&lt;br /&gt;
@@ Line 217: / Line 248: @@
 &lt;/td&gt;
          &lt;td&gt;F^^/Gv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Wide Tritone&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/16&lt;br /&gt;
@@ Line 227: / Line 260: @@
 &lt;/td&gt;
          &lt;td&gt;G&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Grave 5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;189/128, 240/161&lt;br /&gt;
@@ Line 237: / Line 272: @@
 &lt;/td&gt;
          &lt;td&gt;G^/Avv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Fifth-Sixth&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/15&lt;br /&gt;
@@ Line 247: / Line 284: @@
 &lt;/td&gt;
          &lt;td&gt;G^^/Av&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Minor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;46/29&lt;br /&gt;
@@ Line 257: / Line 296: @@
 &lt;/td&gt;
          &lt;td&gt;A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Neutral 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;23/14&lt;br /&gt;
@@ Line 267: / Line 308: @@
 &lt;/td&gt;
          &lt;td&gt;A^/Bvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Supermajor 6th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/16, 46/27&lt;br /&gt;
@@ Line 277: / Line 320: @@
 &lt;/td&gt;
          &lt;td&gt;A^^/Bv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Subminor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
@@ Line 287: / Line 332: @@
 &lt;/td&gt;
          &lt;td&gt;B&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Supraminor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;29/16, 9/5&lt;br /&gt;
@@ Line 297: / Line 344: @@
 &lt;/td&gt;
          &lt;td&gt;B^/Cvv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Major 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;15/8&lt;br /&gt;
@@ Line 307: / Line 356: @@
 &lt;/td&gt;
          &lt;td&gt;B^^/Cv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Supermajor 7th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;27/14, 29/15&lt;br /&gt;
@@ Line 317: / Line 368: @@
 &lt;/td&gt;
          &lt;td&gt;C&lt;br /&gt;
+&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;
@@ Line 331: / Line 384: @@
 &lt;a class="wiki_link" href="/augment12"&gt;augment12&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Moment-of-Symmetry Scales in 21-EDO:&lt;/h2&gt;
+ &lt;br /&gt;
+Since 21-EDO contains sub-EDOs of 3 and 7, it contains no heptatonic MOS scales (other than 7-EDO) and a wealth of scales that repeat at a 1/3-octave period. &lt;br /&gt;
+For 7-limit harmony (based on a chord of 0-7-12-17 approximating 4:5:6:7), using 1/3-octave period scales (i.e. those related to augmented temperament) yields the most harmonically-efficient scales. The 9-note 3L6s scale (related to Tcherpnin's scale in 12-TET) is an excellent example.&lt;br /&gt;
 &lt;br /&gt;
+For scales with a full-octave period, only 6 degrees of 21-EDO generate unique scales: 1\21, 2\21, 4\21, 5\21, 8\21, and 10\21. Other degrees generate either 7-EDO, 3-EDO, or a repetition of one of the other scales.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;13-limit Commas&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit Commas&lt;/h2&gt;
 EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78 |.)&lt;br /&gt;
@@ Line 535: / Line 593: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;strong&gt;Books / Literature:&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Books / Literature:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&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:436:&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:436 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:437:&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:437 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextRemoteImageRule:484:&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:484 --&gt;&lt;!-- ws:start:WikiTextRemoteImageRule:485:&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:485 --&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;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Compositions/Listening:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;strong&gt;Compositions/Listening:&lt;/strong&gt;&lt;/h1&gt;
   &lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/21_improv.mp3" rel="nofollow" target="_blank"&gt;Short Clip of 21-edo Acoustic&lt;/a&gt; by &lt;a class="wiki_link" href="/Ron%20Sword"&gt;Ron Sword&lt;/a&gt;&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron_Sword_21_Tone_improv.mp3" rel="nofollow" target="_blank"&gt;Open tuning Drone Improvisation in 21-edo&lt;/a&gt; by Ron Sword&lt;br /&gt;
 &lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933715" rel="nofollow"&gt;Anomalous Readings&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+anomalousreadingsin21tet.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Andrew%20Heathwaite"&gt;Andrew Heathwaite&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>