21edo: Difference between revisions
Wikispaces>igliashon **Imported revision 242635235 - Original comment: ** |
Wikispaces>igliashon **Imported revision 243535909 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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- | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-30 18:09:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>243535909</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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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. | ||
|| **Degree** || **Cents** | || **Degree** || **Cents** | ||
**Value** || ** | **Value** || **7-EDO** | ||
**Notation** || **5L3s** | |||
**Notation** ||= **D.-R. Interval** | |||
**Types** ||= **Approximate** | |||
**Ratios*** || | **Ratios*** || | ||
|| 0 || 0 || C ||= Unison ||= 1/1 || | || 0 || 0 || C || C ||= Unison ||= 1/1 || | ||
|| 1 || 57.143 || C^/Dvv ||= Subminor 2nd ||= 28/27, 30/29 || | || 1 || 57.143 || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 || | ||
|| 2 || 114.286 || C^^/Dv ||= Minor 2nd ||= 16/15, 15/14, 29/27 || | || 2 || 114.286 || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 || | ||
|| 3 || 171.429 || D ||= Submajor 2nd ||= 10/9, 32/29 || | || 3 || 171.429 || D || D ||= Submajor 2nd ||= 10/9, 32/29 || | ||
|| 4 || 228.571 || D^/Evv ||= Supermajor 2nd ||= 8/7 || | || 4 || 228.571 || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 || | ||
|| 5 || 285.714 || D^^/Ev ||= Subminor 3rd ||= 27/23, 32/27 || | || 5 || 285.714 || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 || | ||
|| 6 || 342.857 || E ||= Neutral 3rd ||= 28/23 || | || 6 || 342.857 || E || E ||= Neutral 3rd ||= 28/23 || | ||
|| 7 || 400 || E^/Fvv ||= Major 3rd ||= 29/23 || | || 7 || 400 || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 || | ||
|| 8 || 457.143 || E^^/Fv ||= Third-Fourth ||= 30/23 || | || 8 || 457.143 || E^^/Fv || F ||= Third-Fourth ||= 30/23 || | ||
|| 9 || 514.286 || F ||= Acute 4th ||= 161/120, 256/189 || | || 9 || 514.286 || F || F# ||= Acute 4th ||= 161/120, 256/189 || | ||
|| 10 || 571.429 || F^/Gvv ||= Narrow Tritone ||= 32/23 || | || 10 || 571.429 || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 || | ||
|| 11 || 628.571 || F^^/Gv ||= Wide Tritone ||= 23/16 || | || 11 || 628.571 || F^^/Gv || G ||= Wide Tritone ||= 23/16 || | ||
|| 12 || 685.714 || G ||= Grave 5th ||= 189/128, 240/161 || | || 12 || 685.714 || G || G# ||= Grave 5th ||= 189/128, 240/161 || | ||
|| 13 || 742.857 || G^/Avv ||= Fifth-Sixth ||= 23/15 || | || 13 || 742.857 || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 || | ||
|| 14 || 800 || G^^/Av ||= Minor 6th ||= 46/29 || | || 14 || 800 || G^^/Av || H ||= Minor 6th ||= 46/29 || | ||
|| 15 || 857.143 || A ||= Neutral 6th ||= 23/14 || | || 15 || 857.143 || A || H#/Ab ||= Neutral 6th ||= 23/14 || | ||
|| 16 || 914.286 || A^/Bvv ||= Supermajor 6th ||= 27/16, 46/27 || | || 16 || 914.286 || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 || | ||
|| 17 || 971.429 || A^^/Bv ||= Subminor 7th ||= 7/4 || | || 17 || 971.429 || A^^/Bv || A# ||= Subminor 7th ||= 7/4 || | ||
|| 18 || 1028.571 || B ||= Supraminor 7th ||= 29/16, 9/5 || | || 18 || 1028.571 || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 || | ||
|| 19 || 1085.714 || B^/Cvv ||= Major 7th ||= 15/8 || | || 19 || 1085.714 || B^/Cvv || B ||= Major 7th ||= 15/8 || | ||
|| 20 || 1142.857 || B^^/Cv ||= Supermajor 7th ||= 27/14, 29/15 || | || 20 || 1142.857 || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 || | ||
|| 21 || 1200 || C ||= Octave ||= 2/1 || | || 21 || 1200 || C || C ||= Octave ||= 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; other approaches are possible. | ||
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==Moment-of-Symmetry Scales in 21-EDO:== | ==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. | 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 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. | ||
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<strong>Value</strong><br /> | <strong>Value</strong><br /> | ||
</td> | </td> | ||
<td><strong> | <td><strong>7-EDO</strong><br /> | ||
<strong>Notation</strong><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><strong>D.-R. | <td><strong>5L3s</strong><br /> | ||
<strong>Notation</strong><br /> | |||
</td> | |||
<td style="text-align: center;"><strong>D.-R. Interval</strong> <br /> | |||
<strong>Types</strong><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><strong>Approximate</strong><br /> | <td style="text-align: center;"><strong>Approximate</strong><br /> | ||
| Line 114: | Line 122: | ||
</td> | </td> | ||
<td>0<br /> | <td>0<br /> | ||
</td> | |||
<td>C<br /> | |||
</td> | </td> | ||
<td>C<br /> | <td>C<br /> | ||
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</td> | </td> | ||
<td>C^/Dvv<br /> | <td>C^/Dvv<br /> | ||
</td> | |||
<td>C#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Subminor 2nd<br /> | <td style="text-align: center;">Subminor 2nd<br /> | ||
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</td> | </td> | ||
<td>C^^/Dv<br /> | <td>C^^/Dv<br /> | ||
</td> | |||
<td>Db<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Minor 2nd<br /> | <td style="text-align: center;">Minor 2nd<br /> | ||
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</td> | </td> | ||
<td>171.429<br /> | <td>171.429<br /> | ||
</td> | |||
<td>D<br /> | |||
</td> | </td> | ||
<td>D<br /> | <td>D<br /> | ||
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</td> | </td> | ||
<td>D^/Evv<br /> | <td>D^/Evv<br /> | ||
</td> | |||
<td>D#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Supermajor 2nd<br /> | <td style="text-align: center;">Supermajor 2nd<br /> | ||
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</td> | </td> | ||
<td>D^^/Ev<br /> | <td>D^^/Ev<br /> | ||
</td> | |||
<td>Eb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Subminor 3rd<br /> | <td style="text-align: center;">Subminor 3rd<br /> | ||
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</td> | </td> | ||
<td>342.857<br /> | <td>342.857<br /> | ||
</td> | |||
<td>E<br /> | |||
</td> | </td> | ||
<td>E<br /> | <td>E<br /> | ||
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</td> | </td> | ||
<td>E^/Fvv<br /> | <td>E^/Fvv<br /> | ||
</td> | |||
<td>E#/Fb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Major 3rd<br /> | <td style="text-align: center;">Major 3rd<br /> | ||
| Line 212: | Line 236: | ||
</td> | </td> | ||
<td>E^^/Fv<br /> | <td>E^^/Fv<br /> | ||
</td> | |||
<td>F<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Third-Fourth<br /> | <td style="text-align: center;">Third-Fourth<br /> | ||
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</td> | </td> | ||
<td>F<br /> | <td>F<br /> | ||
</td> | |||
<td>F#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Acute 4th<br /> | <td style="text-align: center;">Acute 4th<br /> | ||
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</td> | </td> | ||
<td>F^/Gvv<br /> | <td>F^/Gvv<br /> | ||
</td> | |||
<td>Gb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Narrow Tritone<br /> | <td style="text-align: center;">Narrow Tritone<br /> | ||
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</td> | </td> | ||
<td>F^^/Gv<br /> | <td>F^^/Gv<br /> | ||
</td> | |||
<td>G<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Wide Tritone<br /> | <td style="text-align: center;">Wide Tritone<br /> | ||
| Line 260: | Line 292: | ||
</td> | </td> | ||
<td>G<br /> | <td>G<br /> | ||
</td> | |||
<td>G#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Grave 5th<br /> | <td style="text-align: center;">Grave 5th<br /> | ||
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</td> | </td> | ||
<td>G^/Avv<br /> | <td>G^/Avv<br /> | ||
</td> | |||
<td>Hb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Fifth-Sixth<br /> | <td style="text-align: center;">Fifth-Sixth<br /> | ||
| Line 284: | Line 320: | ||
</td> | </td> | ||
<td>G^^/Av<br /> | <td>G^^/Av<br /> | ||
</td> | |||
<td>H<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Minor 6th<br /> | <td style="text-align: center;">Minor 6th<br /> | ||
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</td> | </td> | ||
<td>A<br /> | <td>A<br /> | ||
</td> | |||
<td>H#/Ab<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Neutral 6th<br /> | <td style="text-align: center;">Neutral 6th<br /> | ||
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</td> | </td> | ||
<td>A^/Bvv<br /> | <td>A^/Bvv<br /> | ||
</td> | |||
<td>A<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Supermajor 6th<br /> | <td style="text-align: center;">Supermajor 6th<br /> | ||
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</td> | </td> | ||
<td>A^^/Bv<br /> | <td>A^^/Bv<br /> | ||
</td> | |||
<td>A#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Subminor 7th<br /> | <td style="text-align: center;">Subminor 7th<br /> | ||
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</td> | </td> | ||
<td>B<br /> | <td>B<br /> | ||
</td> | |||
<td>Bb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Supraminor 7th<br /> | <td style="text-align: center;">Supraminor 7th<br /> | ||
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</td> | </td> | ||
<td>B^/Cvv<br /> | <td>B^/Cvv<br /> | ||
</td> | |||
<td>B<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Major 7th<br /> | <td style="text-align: center;">Major 7th<br /> | ||
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</td> | </td> | ||
<td>B^^/Cv<br /> | <td>B^^/Cv<br /> | ||
</td> | |||
<td>B#/Cb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Supermajor 7th<br /> | <td style="text-align: center;">Supermajor 7th<br /> | ||
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</td> | </td> | ||
<td>1200<br /> | <td>1200<br /> | ||
</td> | |||
<td>C<br /> | |||
</td> | </td> | ||
<td>C<br /> | <td>C<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:4 -->Moment-of-Symmetry Scales in 21-EDO:</h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x21 equal divisions of the octave-Moment-of-Symmetry Scales in 21-EDO:"></a><!-- ws:end:WikiTextHeadingRule:4 -->Moment-of-Symmetry Scales in 21-EDO:</h2> | ||
<br /> | <br /> | ||
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. <br /> | 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.<br /> | ||
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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
| Line 595: | Line 647: | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Books / Literature:"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong>Books / Literature:</strong></h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Books / Literature:"></a><!-- ws:end:WikiTextHeadingRule:8 --><strong>Books / Literature:</strong></h1> | ||
Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.<br /> | Sword, Ron. &quot;Icosihenaphonic Scales for Guitar&quot;. IAAA Press. 1st ed: July 2009.<br /> | ||
<!-- ws:start:WikiTextRemoteImageRule: | <!-- ws:start:WikiTextRemoteImageRule:530:&lt;img src=&quot;http://www.ronsword.com/images/ron1.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 188px; width: 254px;&quot; /&gt; --><img src="http://www.ronsword.com/images/ron1.jpg" alt="external image ron1.jpg" title="external image ron1.jpg" style="height: 188px; width: 254px;" /><!-- ws:end:WikiTextRemoteImageRule:530 --><!-- ws:start:WikiTextRemoteImageRule:531:&lt;img src=&quot;http://www.swordguitars.com/21tetsm.JPG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 191px; width: 363px;&quot; /&gt; --><img src="http://www.swordguitars.com/21tetsm.JPG" alt="external image 21tetsm.JPG" title="external image 21tetsm.JPG" style="height: 191px; width: 363px;" /><!-- ws:end:WikiTextRemoteImageRule:531 --><br /> | ||
<strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br /> | <strong><em>21-edo Icosihenaphonic Acoustic Guitar (Ron Sword)</em></strong><br /> | ||
<br /> | <br /> | ||