23edo: Difference between revisions
Wikispaces>TallKite **Imported revision 602739272 - Original comment: ** |
Wikispaces>TallKite **Imported revision 602808722 - 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:TallKite|TallKite]] and made on <tt>2016-12- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-25 19:46:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>602808722</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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[[image:Ciclo Icositrifonía.png width="491" height="490" link="Harmony of 23edo"]] | [[image:Ciclo Icositrifonía.png width="491" height="490" link="Harmony of 23edo"]] | ||
Like 16edo, 23edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first is with major wider than minor, and sharp higher than flat, to preserve melodic contour. The second way is with major narrower than minor, and sharp lower than flat, to preserve interval arithmetic and chord names. | |||
||= <span style="color: #660000;">[[Degree]]</span> ||= [[Cent]]s ||= Approximate | ||= <span style="color: #660000;">[[Degree]]</span> ||= [[Cent]]s ||= Approximate | ||
Ratios* ||= Armodue | Ratios* ||||= major wider | ||
than minor ||||= major narrower | |||
than minor ||= Armodue | |||
Notation || Notes || | Notation || Notes || | ||
||= 0 ||= 0 ||= 1/1 ||= 1 || || | ||= 0 ||= 0 ||= 1/1 ||= P1 ||= D ||= P1 ||= D ||= 1 || || | ||
||= 1 ||= 52.174 ||= 33/32, 34/33 ||= 2b || || | ||= 1 ||= 52.174 ||= 33/32, 34/33 ||= A1 ||= D# ||= d1 ||= Db ||= 2b || || | ||
||= 2 ||= 104.348 ||= 17/16, 16/15, 18/17 ||= 1# || Less than 1-cent off 17/16 || | ||= 2 ||= 104.348 ||= 17/16, 16/15, 18/17 ||= d2 ||= Eb ||= A2 ||= Ex ||= 1# || Less than 1-cent off 17/16 || | ||
||= 3 ||= 156.522 ||= 11/10, 12/11, 35/32 ||= 2 || || | ||= 3 ||= 156.522 ||= 11/10, 12/11, 35/32 ||= m2 ||= E ||= M2 ||= E ||= 2 || || | ||
||= 4· ||= 208.696 ||= 9/8, 44/39 ||= 3b || || | ||= 4· ||= 208.696 ||= 9/8, 44/39 ||= M2 ||= E# ||= m2 ||= Eb ||= 3b || || | ||
||= 5 ||= 260.870 ||= 7/6, 15/13, 29/25 ||= 2# || || | ||= 5 ||= 260.870 ||= 7/6, 15/13, 29/25 ||= A2, d3 ||= Ex, Fbb ||= d2, A3 ||= Ebb, Fx ||= 2# || || | ||
||= 6 ||= 313.043 ||= 6/5 ||= 3 || Much better | ||= 6 ||= 313.043 ||= 6/5 ||= m3 ||= Fb ||= M3 ||= F# ||= 3 || Much better 6/5 than 12-edo || | ||
||= 7· ||= 365.217 ||= 16/13, 21/17, 26/21 ||= 4b || || | ||= 7· ||= 365.217 ||= 16/13, 21/17, 26/21 ||= M3 ||= F ||= m3 ||= F ||= 4b || || | ||
||= 8 ||= 417.391 ||= 14/11, 33/26 ||= 3# || Practically just | ||= 8 ||= 417.391 ||= 14/11, 33/26 ||= A3 ||= F# ||= d3 ||= Fb ||= 3# || Practically just 14/11 || | ||
||= 9 ||= 469.565 ||= 21/16, 17/13 ||= 4 || || | ||= 9 ||= 469.565 ||= 21/16, 17/13 ||= d4 ||= Gb ||= A4 ||= G# ||= 4 || || | ||
||= 10· ||= 521.739 ||= 23/17, 88/65, 256/189 ||= 5 || || | ||= 10· ||= 521.739 ||= 23/17, 88/65, 256/189 ||= P4 ||= G ||= P4 ||= G ||= 5 || || | ||
||= 11 ||= 573.913 ||= 7/5, 32/23, 46/33 ||= 6b || || | ||= 11 ||= 573.913 ||= 7/5, 32/23, 46/33 ||= A4 ||= G# ||= d4 ||= Gb ||= 6b || || | ||
||= 12 ||= 626.087 ||= 10/7, 23/16, 33/23 ||= 5# || || | ||= 12 ||= 626.087 ||= 10/7, 23/16, 33/23 ||= d5 ||= Ab ||= A5 ||= A# ||= 5# || || | ||
||= 13· ||= 678.261 ||= 34/23, 65/44, 189/128 ||= 6 || Great generator | ||= 13· ||= 678.261 ||= 34/23, 65/44, 189/128 ||= P5 ||= A ||= P5 ||= A ||= 6 || Great Hornbostel generator || | ||
||= 14 ||= 730.435 ||= 32/21, 26/17 ||= 7b || || | ||= 14 ||= 730.435 ||= 32/21, 26/17 ||= A5 ||= A# ||= d5 ||= Ab ||= 7b || || | ||
||= 15 ||= 782.609 ||= 11/7, 52/33 ||= 6# || || | ||= 15 ||= 782.609 ||= 11/7, 52/33 ||= d6 ||= Bb ||= A6 ||= B# ||= 6# || || | ||
||= 16· ||= 834.783 ||= 13/8, 34/21, 21/13 ||= 7 || || | ||= 16· ||= 834.783 ||= 13/8, 34/21, 21/13 ||= m6 ||= B ||= M6 ||= B ||= 7 || || | ||
||= 17 ||= 886.957 ||= 5/3 ||= 8b || || | ||= 17 ||= 886.957 ||= 5/3 ||= M6 ||= B# ||= m6 ||= Bb ||= 8b || || | ||
||= 18 ||= 939.130 ||= 12/7, 26/15, 50/29 ||= 7# || || | ||= 18 ||= 939.130 ||= 12/7, 26/15, 50/29 ||= A6, d7 ||= Bx, Cbb ||= d6, A7 ||= Bbb, Cx ||= 7# || || | ||
||= 19· ||= 991.304 ||= 16/9, 39/22 ||= 8 || || | ||= 19· ||= 991.304 ||= 16/9, 39/22 ||= m7 ||= Cb ||= M7 ||= C# ||= 8 || || | ||
||= 20 ||= 1043.478 ||= 11/6, 20/11, 64/35 ||= 9b || || | ||= 20 ||= 1043.478 ||= 11/6, 20/11, 64/35 ||= M7 ||= C ||= m7 ||= C ||= 9b || || | ||
||= 21 ||= 1095.652 ||= 15/8, 17/9, 32/17 ||= 8# || || | ||= 21 ||= 1095.652 ||= 15/8, 17/9, 32/17 ||= A7 ||= C# ||= d7 ||= Cb ||= 8# || || | ||
||= 22 ||= 1147.826 ||= 33/17, 64/33 ||= 9 || || | ||= 22 ||= 1147.826 ||= 33/17, 64/33 ||= d8 ||= Db ||= A8 ||= D# ||= 9 || || | ||
||= | ||= 23 ||= 1200 ||= 2/1 ||= P8 ||= D ||= P8 ||= D ||= 1 || || | ||
*based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible. | *based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible. | ||
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However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29|Rapoport]]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale. | However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a [[7-limit]] temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to [[13edo|13-EDO]] and [[18edo|18-EDO]] and produces [[MOSScales|MOS scales]] of 5 and 8 notes: 5 5 4 5 4 (the [[3L 2s|"anti-pentatonic"]]) and 4 1 4 1 4 4 1 4 (the "quarter-tone" version of the Blackwood/[[http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29|Rapoport]]/Wilson 13-EDO "subminor" scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-"4/3", we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale. | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc1"><a name="x23 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="font-size: 1.4em;">Intervals</span></h2> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc1"><a name="x23 tone equal temperament-Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 --><span style="font-size: 1.4em;">Intervals</span></h2> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1012:&lt;a href=&quot;/Harmony%20of%2023edo&quot;&gt;&lt;img src=&quot;/file/view/Ciclo%20Icositrifon%C3%ADa.png/390051610/491x490/Ciclo%20Icositrifon%C3%ADa.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 490px; width: 491px;&quot; /&gt;&lt;/a&gt; --><a href="/Harmony%20of%2023edo"><img src="/file/view/Ciclo%20Icositrifon%C3%ADa.png/390051610/491x490/Ciclo%20Icositrifon%C3%ADa.png" alt="Ciclo Icositrifonía.png" title="Ciclo Icositrifonía.png" style="height: 490px; width: 491px;" /></a><!-- ws:end:WikiTextLocalImageRule:1012 --><br /> | ||
<br /> | <br /> | ||
Like 16edo, 23edo can be notated with conventional notation, including the staff, note names, chord names, relative notation, etc. in two ways. The first is with major wider than minor, and sharp higher than flat, to preserve melodic contour. The second way is with major narrower than minor, and sharp lower than flat, to preserve interval arithmetic and chord names.<br /> | |||
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<td style="text-align: center;">Approximate<br /> | <td style="text-align: center;">Approximate<br /> | ||
Ratios*<br /> | Ratios*<br /> | ||
</td> | |||
<td colspan="2" style="text-align: center;">major wider<br /> | |||
than minor<br /> | |||
</td> | |||
<td colspan="2" style="text-align: center;">major narrower<br /> | |||
than minor<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Armodue<br /> | <td style="text-align: center;">Armodue<br /> | ||
| Line 250: | Line 249: | ||
</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
</td> | |||
<td style="text-align: center;">P1<br /> | |||
</td> | |||
<td style="text-align: center;">D<br /> | |||
</td> | |||
<td style="text-align: center;">P1<br /> | |||
</td> | |||
<td style="text-align: center;">D<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
| Line 262: | Line 269: | ||
</td> | </td> | ||
<td style="text-align: center;">33/32, 34/33<br /> | <td style="text-align: center;">33/32, 34/33<br /> | ||
</td> | |||
<td style="text-align: center;">A1<br /> | |||
</td> | |||
<td style="text-align: center;">D#<br /> | |||
</td> | |||
<td style="text-align: center;">d1<br /> | |||
</td> | |||
<td style="text-align: center;">Db<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">2b<br /> | <td style="text-align: center;">2b<br /> | ||
| Line 274: | Line 289: | ||
</td> | </td> | ||
<td style="text-align: center;">17/16, 16/15, 18/17<br /> | <td style="text-align: center;">17/16, 16/15, 18/17<br /> | ||
</td> | |||
<td style="text-align: center;">d2<br /> | |||
</td> | |||
<td style="text-align: center;">Eb<br /> | |||
</td> | |||
<td style="text-align: center;">A2<br /> | |||
</td> | |||
<td style="text-align: center;">Ex<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">1#<br /> | <td style="text-align: center;">1#<br /> | ||
| Line 286: | Line 309: | ||
</td> | </td> | ||
<td style="text-align: center;">11/10, 12/11, 35/32<br /> | <td style="text-align: center;">11/10, 12/11, 35/32<br /> | ||
</td> | |||
<td style="text-align: center;">m2<br /> | |||
</td> | |||
<td style="text-align: center;">E<br /> | |||
</td> | |||
<td style="text-align: center;">M2<br /> | |||
</td> | |||
<td style="text-align: center;">E<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">2<br /> | <td style="text-align: center;">2<br /> | ||
| Line 298: | Line 329: | ||
</td> | </td> | ||
<td style="text-align: center;">9/8, 44/39<br /> | <td style="text-align: center;">9/8, 44/39<br /> | ||
</td> | |||
<td style="text-align: center;">M2<br /> | |||
</td> | |||
<td style="text-align: center;">E#<br /> | |||
</td> | |||
<td style="text-align: center;">m2<br /> | |||
</td> | |||
<td style="text-align: center;">Eb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">3b<br /> | <td style="text-align: center;">3b<br /> | ||
| Line 310: | Line 349: | ||
</td> | </td> | ||
<td style="text-align: center;">7/6, 15/13, 29/25<br /> | <td style="text-align: center;">7/6, 15/13, 29/25<br /> | ||
</td> | |||
<td style="text-align: center;">A2, d3<br /> | |||
</td> | |||
<td style="text-align: center;">Ex, Fbb<br /> | |||
</td> | |||
<td style="text-align: center;">d2, A3<br /> | |||
</td> | |||
<td style="text-align: center;">Ebb, Fx<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">2#<br /> | <td style="text-align: center;">2#<br /> | ||
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</td> | </td> | ||
<td style="text-align: center;">6/5<br /> | <td style="text-align: center;">6/5<br /> | ||
</td> | |||
<td style="text-align: center;">m3<br /> | |||
</td> | |||
<td style="text-align: center;">Fb<br /> | |||
</td> | |||
<td style="text-align: center;">M3<br /> | |||
</td> | |||
<td style="text-align: center;">F#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td>Much better | <td>Much better 6/5 than 12-edo<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 334: | Line 389: | ||
</td> | </td> | ||
<td style="text-align: center;">16/13, 21/17, 26/21<br /> | <td style="text-align: center;">16/13, 21/17, 26/21<br /> | ||
</td> | |||
<td style="text-align: center;">M3<br /> | |||
</td> | |||
<td style="text-align: center;">F<br /> | |||
</td> | |||
<td style="text-align: center;">m3<br /> | |||
</td> | |||
<td style="text-align: center;">F<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">4b<br /> | <td style="text-align: center;">4b<br /> | ||
| Line 346: | Line 409: | ||
</td> | </td> | ||
<td style="text-align: center;">14/11, 33/26<br /> | <td style="text-align: center;">14/11, 33/26<br /> | ||
</td> | |||
<td style="text-align: center;">A3<br /> | |||
</td> | |||
<td style="text-align: center;">F#<br /> | |||
</td> | |||
<td style="text-align: center;">d3<br /> | |||
</td> | |||
<td style="text-align: center;">Fb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">3#<br /> | <td style="text-align: center;">3#<br /> | ||
</td> | </td> | ||
<td>Practically just | <td>Practically just 14/11<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 358: | Line 429: | ||
</td> | </td> | ||
<td style="text-align: center;">21/16, 17/13<br /> | <td style="text-align: center;">21/16, 17/13<br /> | ||
</td> | |||
<td style="text-align: center;">d4<br /> | |||
</td> | |||
<td style="text-align: center;">Gb<br /> | |||
</td> | |||
<td style="text-align: center;">A4<br /> | |||
</td> | |||
<td style="text-align: center;">G#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
| Line 370: | Line 449: | ||
</td> | </td> | ||
<td style="text-align: center;">23/17, 88/65, 256/189<br /> | <td style="text-align: center;">23/17, 88/65, 256/189<br /> | ||
</td> | |||
<td style="text-align: center;">P4<br /> | |||
</td> | |||
<td style="text-align: center;">G<br /> | |||
</td> | |||
<td style="text-align: center;">P4<br /> | |||
</td> | |||
<td style="text-align: center;">G<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
| Line 382: | Line 469: | ||
</td> | </td> | ||
<td style="text-align: center;">7/5, 32/23, 46/33<br /> | <td style="text-align: center;">7/5, 32/23, 46/33<br /> | ||
</td> | |||
<td style="text-align: center;">A4<br /> | |||
</td> | |||
<td style="text-align: center;">G#<br /> | |||
</td> | |||
<td style="text-align: center;">d4<br /> | |||
</td> | |||
<td style="text-align: center;">Gb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">6b<br /> | <td style="text-align: center;">6b<br /> | ||
| Line 394: | Line 489: | ||
</td> | </td> | ||
<td style="text-align: center;">10/7, 23/16, 33/23<br /> | <td style="text-align: center;">10/7, 23/16, 33/23<br /> | ||
</td> | |||
<td style="text-align: center;">d5<br /> | |||
</td> | |||
<td style="text-align: center;">Ab<br /> | |||
</td> | |||
<td style="text-align: center;">A5<br /> | |||
</td> | |||
<td style="text-align: center;">A#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">5#<br /> | <td style="text-align: center;">5#<br /> | ||
| Line 406: | Line 509: | ||
</td> | </td> | ||
<td style="text-align: center;">34/23, 65/44, 189/128<br /> | <td style="text-align: center;">34/23, 65/44, 189/128<br /> | ||
</td> | |||
<td style="text-align: center;">P5<br /> | |||
</td> | |||
<td style="text-align: center;">A<br /> | |||
</td> | |||
<td style="text-align: center;">P5<br /> | |||
</td> | |||
<td style="text-align: center;">A<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td>Great generator | <td>Great Hornbostel generator<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 418: | Line 529: | ||
</td> | </td> | ||
<td style="text-align: center;">32/21, 26/17<br /> | <td style="text-align: center;">32/21, 26/17<br /> | ||
</td> | |||
<td style="text-align: center;">A5<br /> | |||
</td> | |||
<td style="text-align: center;">A#<br /> | |||
</td> | |||
<td style="text-align: center;">d5<br /> | |||
</td> | |||
<td style="text-align: center;">Ab<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">7b<br /> | <td style="text-align: center;">7b<br /> | ||
| Line 430: | Line 549: | ||
</td> | </td> | ||
<td style="text-align: center;">11/7, 52/33<br /> | <td style="text-align: center;">11/7, 52/33<br /> | ||
</td> | |||
<td style="text-align: center;">d6<br /> | |||
</td> | |||
<td style="text-align: center;">Bb<br /> | |||
</td> | |||
<td style="text-align: center;">A6<br /> | |||
</td> | |||
<td style="text-align: center;">B#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">6#<br /> | <td style="text-align: center;">6#<br /> | ||
| Line 442: | Line 569: | ||
</td> | </td> | ||
<td style="text-align: center;">13/8, 34/21, 21/13<br /> | <td style="text-align: center;">13/8, 34/21, 21/13<br /> | ||
</td> | |||
<td style="text-align: center;">m6<br /> | |||
</td> | |||
<td style="text-align: center;">B<br /> | |||
</td> | |||
<td style="text-align: center;">M6<br /> | |||
</td> | |||
<td style="text-align: center;">B<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
| Line 454: | Line 589: | ||
</td> | </td> | ||
<td style="text-align: center;">5/3<br /> | <td style="text-align: center;">5/3<br /> | ||
</td> | |||
<td style="text-align: center;">M6<br /> | |||
</td> | |||
<td style="text-align: center;">B#<br /> | |||
</td> | |||
<td style="text-align: center;">m6<br /> | |||
</td> | |||
<td style="text-align: center;">Bb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">8b<br /> | <td style="text-align: center;">8b<br /> | ||
| Line 466: | Line 609: | ||
</td> | </td> | ||
<td style="text-align: center;">12/7, 26/15, 50/29<br /> | <td style="text-align: center;">12/7, 26/15, 50/29<br /> | ||
</td> | |||
<td style="text-align: center;">A6, d7<br /> | |||
</td> | |||
<td style="text-align: center;">Bx, Cbb<br /> | |||
</td> | |||
<td style="text-align: center;">d6, A7<br /> | |||
</td> | |||
<td style="text-align: center;">Bbb, Cx<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">7#<br /> | <td style="text-align: center;">7#<br /> | ||
| Line 478: | Line 629: | ||
</td> | </td> | ||
<td style="text-align: center;">16/9, 39/22<br /> | <td style="text-align: center;">16/9, 39/22<br /> | ||
</td> | |||
<td style="text-align: center;">m7<br /> | |||
</td> | |||
<td style="text-align: center;">Cb<br /> | |||
</td> | |||
<td style="text-align: center;">M7<br /> | |||
</td> | |||
<td style="text-align: center;">C#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">8<br /> | <td style="text-align: center;">8<br /> | ||
| Line 490: | Line 649: | ||
</td> | </td> | ||
<td style="text-align: center;">11/6, 20/11, 64/35<br /> | <td style="text-align: center;">11/6, 20/11, 64/35<br /> | ||
</td> | |||
<td style="text-align: center;">M7<br /> | |||
</td> | |||
<td style="text-align: center;">C<br /> | |||
</td> | |||
<td style="text-align: center;">m7<br /> | |||
</td> | |||
<td style="text-align: center;">C<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">9b<br /> | <td style="text-align: center;">9b<br /> | ||
| Line 502: | Line 669: | ||
</td> | </td> | ||
<td style="text-align: center;">15/8, 17/9, 32/17<br /> | <td style="text-align: center;">15/8, 17/9, 32/17<br /> | ||
</td> | |||
<td style="text-align: center;">A7<br /> | |||
</td> | |||
<td style="text-align: center;">C#<br /> | |||
</td> | |||
<td style="text-align: center;">d7<br /> | |||
</td> | |||
<td style="text-align: center;">Cb<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">8#<br /> | <td style="text-align: center;">8#<br /> | ||
| Line 514: | Line 689: | ||
</td> | </td> | ||
<td style="text-align: center;">33/17, 64/33<br /> | <td style="text-align: center;">33/17, 64/33<br /> | ||
</td> | |||
<td style="text-align: center;">d8<br /> | |||
</td> | |||
<td style="text-align: center;">Db<br /> | |||
</td> | |||
<td style="text-align: center;">A8<br /> | |||
</td> | |||
<td style="text-align: center;">D#<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">9<br /> | <td style="text-align: center;">9<br /> | ||
| Line 521: | Line 704: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">23<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1200<br /> | <td style="text-align: center;">1200<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">2/1<br /> | <td style="text-align: center;">2/1<br /> | ||
</td> | |||
<td style="text-align: center;">P8<br /> | |||
</td> | |||
<td style="text-align: center;">D<br /> | |||
</td> | |||
<td style="text-align: center;">P8<br /> | |||
</td> | |||
<td style="text-align: center;">D<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
| Line 534: | Line 725: | ||
</table> | </table> | ||
*based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.<br /> | *based on treating 23-EDO as a 2.9.15.21.33.13.17 subgroup temperament; other approaches are possible.<br /> | ||
<br /> | <br /> | ||
The chart below shows some of the <a class="wiki_link" href="/MOSScales">Moment of Symmetry (MOS)</a> modes of <a class="wiki_link" href="/Mavila">Mavila</a> available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note MOSs:<br /> | The chart below shows some of the <a class="wiki_link" href="/MOSScales">Moment of Symmetry (MOS)</a> modes of <a class="wiki_link" href="/Mavila">Mavila</a> available in 23edo, mainly Pentatonic(5-note), anti-diatonic(7-note), 9- and 16-note MOSs. Here the outer ring represents individual step of 23edo itself, while the rings moving inward represent 16, 9, 7 and 5 note MOSs:<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
23-EDO was proposed by ethnomusicologist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erich_von_Hornbostel" rel="nofollow">Erich von Hornbostel</a> as the result of continuing a circle of &quot;blown&quot; fifths of ~678-cent fifths that (he argued) resulted from &quot;overblowing&quot; a bamboo pipe.<br /> | 23-EDO was proposed by ethnomusicologist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erich_von_Hornbostel" rel="nofollow">Erich von Hornbostel</a> as the result of continuing a circle of &quot;blown&quot; fifths of ~678-cent fifths that (he argued) resulted from &quot;overblowing&quot; a bamboo pipe.<br /> | ||
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<br /> | <br /> | ||
However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a <a class="wiki_link" href="/7-limit">7-limit</a> temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to <a class="wiki_link" href="/13edo">13-EDO</a> and <a class="wiki_link" href="/18edo">18-EDO</a> and produces <a class="wiki_link" href="/MOSScales">MOS scales</a> of 5 and 8 notes: 5 5 4 5 4 (the <a class="wiki_link" href="/3L%202s">&quot;anti-pentatonic&quot;</a>) and 4 1 4 1 4 4 1 4 (the &quot;quarter-tone&quot; version of the Blackwood/<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow">Rapoport</a>/Wilson 13-EDO &quot;subminor&quot; scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-&quot;4/3&quot;, we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.<br /> | However, one can also map 3/2 to 14 degrees of 23-EDO without significantly increasing the error, taking us to a <a class="wiki_link" href="/7-limit">7-limit</a> temperament where two 'broad 3/2's equals 7/3, meaning 28/27 is tempered out, and six 4/3's octave-reduced equals 5/4, meaning 4096/3645 is tempered out. Both of these are very large commas, so this is not at all an accurate temperament, but it is related to <a class="wiki_link" href="/13edo">13-EDO</a> and <a class="wiki_link" href="/18edo">18-EDO</a> and produces <a class="wiki_link" href="/MOSScales">MOS scales</a> of 5 and 8 notes: 5 5 4 5 4 (the <a class="wiki_link" href="/3L%202s">&quot;anti-pentatonic&quot;</a>) and 4 1 4 1 4 4 1 4 (the &quot;quarter-tone&quot; version of the Blackwood/<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Paul_Rapoport_%28music_critic%29" rel="nofollow">Rapoport</a>/Wilson 13-EDO &quot;subminor&quot; scale). Alternatively we can treat this temperament as a 2.9.21 subgroup, and instead of calling 9 degrees of 23-EDO a Sub-&quot;4/3&quot;, we can call it 21/16. Here three 21/16's gets us to 9/4, meaning 1029/1024 is tempered out. This allows us to treat a triad of 0-4-9 degrees of 23-EDO as an approximation to 16:18:21, and 0-5-9 as 1/(16:18:21); both of these triads are abundant in the 8-note MOS scale.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc6"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:14 -->Books</h1> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc6"><a name="Books"></a><!-- ws:end:WikiTextHeadingRule:14 -->Books</h1> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc7"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:16 -->Instruments</h1> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc7"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:16 -->Instruments</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1015:&lt;img src=&quot;/file/view/Icositriphonic_Guitar.PNG/206712964/601x305/Icositriphonic_Guitar.PNG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 305px; width: 601px;&quot; /&gt; --><img src="/file/view/Icositriphonic_Guitar.PNG/206712964/601x305/Icositriphonic_Guitar.PNG" alt="Icositriphonic_Guitar.PNG" title="Icositriphonic_Guitar.PNG" style="height: 305px; width: 601px;" /><!-- ws:end:WikiTextLocalImageRule:1015 --><br /> | ||
<em>An Icositriphonic 8-string Guitar. 23-ED2 Guitar by Ron Sword.</em><br /> | <em>An Icositriphonic 8-string Guitar. 23-ED2 Guitar by Ron Sword.</em><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1016:&lt;img src=&quot;/file/view/Icositriphonic_Bass.JPG/206711470/640x193/Icositriphonic_Bass.JPG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 193px; width: 640px;&quot; /&gt; --><img src="/file/view/Icositriphonic_Bass.JPG/206711470/640x193/Icositriphonic_Bass.JPG" alt="Icositriphonic_Bass.JPG" title="Icositriphonic_Bass.JPG" style="height: 193px; width: 640px;" /><!-- ws:end:WikiTextLocalImageRule:1016 --><br /> | ||
<em>An Icositriphonic P-Bass. 23-ED2 Bass by Tútim Dennsuul Wafiil.</em><br /> | <em>An Icositriphonic P-Bass. 23-ED2 Bass by Tútim Dennsuul Wafiil.</em><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1017:&lt;img src=&quot;/file/view/Icositriphonic%20Gitar%201.png/358559595/640x216/Icositriphonic%20Gitar%201.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 216px; width: 640px;&quot; /&gt; --><img src="/file/view/Icositriphonic%20Gitar%201.png/358559595/640x216/Icositriphonic%20Gitar%201.png" alt="Icositriphonic Gitar 1.png" title="Icositriphonic Gitar 1.png" style="height: 216px; width: 640px;" /><!-- ws:end:WikiTextLocalImageRule:1017 --><br /> | ||
<em>An Icositriphonic 'Hendrix' TravelGuitar. 23-ED2 'Gitar' by Tútim Dennsuul Wafiil.</em><br /> | <em>An Icositriphonic 'Hendrix' TravelGuitar. 23-ED2 'Gitar' by Tútim Dennsuul Wafiil.</em><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1018:&lt;img src=&quot;/file/view/Icositritar%201.png/358559793/640x271/Icositritar%201.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 271px; width: 640px;&quot; /&gt; --><img src="/file/view/Icositritar%201.png/358559793/640x271/Icositritar%201.png" alt="Icositritar 1.png" title="Icositritar 1.png" style="height: 271px; width: 640px;" /><!-- ws:end:WikiTextLocalImageRule:1018 --><br /> | ||
<em>The 'Icositritar', 23-ED2 5-string Acoustic Guitar by Tútim Dennsuul Wafiil.</em><br /> | <em>The 'Icositritar', 23-ED2 5-string Acoustic Guitar by Tútim Dennsuul Wafiil.</em><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1019:&lt;img src=&quot;/file/view/Teclado%20Icositrif%C3%B3nico.PNG/390051980/607x323/Teclado%20Icositrif%C3%B3nico.PNG&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 323px; width: 607px;&quot; /&gt; --><img src="/file/view/Teclado%20Icositrif%C3%B3nico.PNG/390051980/607x323/Teclado%20Icositrif%C3%B3nico.PNG" alt="Teclado Icositrifónico.PNG" title="Teclado Icositrifónico.PNG" style="height: 323px; width: 607px;" /><!-- ws:end:WikiTextLocalImageRule:1019 --><br /> | ||
<em>An illustrative image of a 23-ED2 Keyboard</em><br /> | <em>An illustrative image of a 23-ED2 Keyboard</em><br /> | ||
<br /> | <br /> | ||
Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.<br /> | Chris Vaisvil made a do it yourself 23 edo electric guitar out of less than $50 of material. Here he is playing it.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1020:&lt;img src=&quot;/file/view/playing.jpg/241873781/playing.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/playing.jpg/241873781/playing.jpg" alt="playing.jpg" title="playing.jpg" /><!-- ws:end:WikiTextLocalImageRule:1020 --><br /> | ||
Here is a still shot of the completed instrument.<br /> | Here is a still shot of the completed instrument.<br /> | ||
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This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion.<br /> | This movie is a series of still shots Chris took during the process of making a 23 edo guitar in a stick like form. At the end the guitar is played without effects etc. and the open string tuning is sounded - which starts with a normal E and then adjusted to the 9th / 7th fret unison, like a typical 12edo guitar fashion.<br /> | ||
<!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/youtube/K4iO7k152og?h=349&amp;w=560&quot; class=&quot;WikiMedia WikiMediaYoutube&quot; id=&quot;wikitext@@media@@type=&amp;quot;youtube&amp;quot; key=&amp;quot;K4iO7k152og&amp;quot; height=&amp;quot;349&amp;quot; width=&amp;quot;560&amp;quot;&quot; title=&quot;YouTube Video&quot;height=&quot;349&quot; width=&quot;560&quot;/&gt; --><iframe width="560" height="349" src="//www.youtube.com/embed/K4iO7k152og" frameborder="0" allowfullscreen></iframe><!-- ws:end:WikiTextMediaRule:1 --></body></html></pre></div> | <!-- ws:start:WikiTextMediaRule:1:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/youtube/K4iO7k152og?h=349&amp;w=560&quot; class=&quot;WikiMedia WikiMediaYoutube&quot; id=&quot;wikitext@@media@@type=&amp;quot;youtube&amp;quot; key=&amp;quot;K4iO7k152og&amp;quot; height=&amp;quot;349&amp;quot; width=&amp;quot;560&amp;quot;&quot; title=&quot;YouTube Video&quot;height=&quot;349&quot; width=&quot;560&quot;/&gt; --><iframe width="560" height="349" src="//www.youtube.com/embed/K4iO7k152og" frameborder="0" allowfullscreen></iframe><!-- ws:end:WikiTextMediaRule:1 --></body></html></pre></div> | ||