13edo: Difference between revisions
Wikispaces>hstraub **Imported revision 243200999 - Original comment: ** |
Wikispaces>igliashon **Imported revision 243300347 - 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: | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-28 17:10:31 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>243300347</tt>.<br> | ||
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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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=13 tone equal temperament / 13edo= | =13 tone equal temperament / 13edo= | ||
13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | 13edo refers to a tuning system which divides the octave (frequency ratio 2:1) into 13 equal parts. The steps are of similar size to those of 12edo (albeit squashed), while the intervals between a minor third & major sixth are xenharmonic (not similar to anything available in 12edo). | ||
|| Degree || Cents ||= Approximate Ratios* || | || Degree || Cents ||= Approximate Ratios* || 6L1s Names || | ||
|| 0 || 0 ||= 1/1 || C || | || 0 || 0 ||= 1/1 || C || | ||
|| 1 || 92.3077 ||= 55/52, 117/110, 26/25 || C#/Db || | || 1 || 92.3077 ||= 22/21, 55/52, 117/110, 26/25 || C#/Db || | ||
|| 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || | || 2 || 184.6154 ||= 10/9, 9/8, 11/10 || D || | ||
|| 3 || 276.9231 ||= 13/11 || D#/Eb || | || 3 || 276.9231 ||= 13/11, 7/6 || D#/Eb || | ||
|| 4 || 369.2308 ||= 5/4, 16/13, 11/9 || E || | || 4 || 369.2308 ||= 5/4, 16/13, 11/9, 26/21 || E || | ||
|| 5 || 461.5385 ||= 13/10 || E#/Fb || | || 5 || 461.5385 ||= 13/10, 21/16 || E#/Fb || | ||
|| 6 || 553.84 ||= 11/8, 18/13 || F || | || 6 || 553.84 ||= 11/8, 18/13 || F || | ||
|| 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || | || 7 || 646.15 ||= 16/11, 13/9 || F#/Gb || | ||
|| 8 || 738.46 ||= 20/13 || G || | || 8 || 738.46 ||= 20/13, 32/21 || G || | ||
|| 9 || 830.77 ||= 8/5, 13/8, 18/11 || G#/Ab || | || 9 || 830.77 ||= 8/5, 13/8, 18/11, 21/13 || G#/Ab || | ||
|| 10 || 923.08 ||= 22/13 || A || | || 10 || 923.08 ||= 22/13, 12/7 || A || | ||
|| 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || | || 11 || 1015.38 ||= 9/5, 16/9, 20/11 || A#/Bb || | ||
|| 12 || 1107.69 ||= 25/13, 104/55 || B/Cb || | || 12 || 1107.69 ||= 21/11, 25/13, 104/55 || B/Cb || | ||
|| 13 || 1200 ||= 2/1 || C/B# || | || 13 || 1200 ||= 2/1 || C/B# || | ||
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | ||
F- | |||
=Harmony in 13edo= | =Harmony in 13edo= | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a "stack of 3rds" the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the [[k*N subgroups|2*13 subgroup]] 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et. | ||
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<td style="text-align: center;">Approximate Ratios*<br /> | <td style="text-align: center;">Approximate Ratios*<br /> | ||
</td> | </td> | ||
<td> | <td>6L1s Names<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 106: | Line 106: | ||
<td>92.3077<br /> | <td>92.3077<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">55/52, 117/110, 26/25<br /> | <td style="text-align: center;">22/21, 55/52, 117/110, 26/25<br /> | ||
</td> | </td> | ||
<td>C#/Db<br /> | <td>C#/Db<br /> | ||
| Line 126: | Line 126: | ||
<td>276.9231<br /> | <td>276.9231<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13/11<br /> | <td style="text-align: center;">13/11, 7/6<br /> | ||
</td> | </td> | ||
<td>D#/Eb<br /> | <td>D#/Eb<br /> | ||
| Line 136: | Line 136: | ||
<td>369.2308<br /> | <td>369.2308<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5/4, 16/13, 11/9<br /> | <td style="text-align: center;">5/4, 16/13, 11/9, 26/21<br /> | ||
</td> | </td> | ||
<td>E<br /> | <td>E<br /> | ||
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<td>461.5385<br /> | <td>461.5385<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13/10<br /> | <td style="text-align: center;">13/10, 21/16<br /> | ||
</td> | </td> | ||
<td>E#/Fb<br /> | <td>E#/Fb<br /> | ||
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<td>738.46<br /> | <td>738.46<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">20/13<br /> | <td style="text-align: center;">20/13, 32/21<br /> | ||
</td> | </td> | ||
<td>G<br /> | <td>G<br /> | ||
| Line 186: | Line 186: | ||
<td>830.77<br /> | <td>830.77<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8/5, 13/8, 18/11<br /> | <td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br /> | ||
</td> | </td> | ||
<td>G#/Ab<br /> | <td>G#/Ab<br /> | ||
| Line 196: | Line 196: | ||
<td>923.08<br /> | <td>923.08<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">22/13<br /> | <td style="text-align: center;">22/13, 12/7<br /> | ||
</td> | </td> | ||
<td>A<br /> | <td>A<br /> | ||
| Line 216: | Line 216: | ||
<td>1107.69<br /> | <td>1107.69<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">25/13, 104/55<br /> | <td style="text-align: center;">21/11, 25/13, 104/55<br /> | ||
</td> | </td> | ||
<td>B/Cb<br /> | <td>B/Cb<br /> | ||
| Line 233: | Line 233: | ||
</table> | </table> | ||
*based on treating 13-EDO as a 2.5.9.11.13 temperament; other approaches are possible | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.<br /> | ||
<br /> | <br /> | ||
F-<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmony in 13edo"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmony in 13edo</h1> | ||
Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | Contrary to popular belief, consonant harmony is possible in 13-EDO, but it requires a radically different approach than that used in 12-EDO (or other Pythagorean or Meantone-based tunings). Trying to approximate the usual major and minor triads of 12-EDO within 13-EDO is usually a disappointment if consonance is the goal; 0-3-7, 0-4-7, 0-3-8, and 0-4-8 are all rather rough in 13-EDO. Typically, the most consonant harmonies do not use a &quot;stack of 3rds&quot; the way they do in 12-TET, since the strongest dissonances in 13-EDO are near the middle of the octave (degrees 6, 7, and 8). Instead, a stack of whole-tones, or a mixture of whole-tones and minor 3rds, often yields good results. For example, one way to view 13-EDO is as a subgroup temperament of harmonics 2.5.9.11.13. It actually performs quite admirably in this regard, and a chord of 0-4-15-19-22 (approximating 4:5:9:11:13) sounds very convincing. An even larger subgroup is the <a class="wiki_link" href="/k%2AN%20subgroups">2*13 subgroup</a> 2.9.5.21.11.13, on which 13 has the same tuning and commas as 26et.<br /> | ||