13edo: Difference between revisions
Wikispaces>clamengh **Imported revision 537922188 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 545187078 - 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:JosephRuhf|JosephRuhf]] and made on <tt>2015-03-24 16:06:50 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>545187078</tt>.<br> | ||
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As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size. | As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size. | ||
|| **Degree** || Cents ||= Approximated 21-limit Ratios* || Erv Wilson || Archaeotonic || Oneirotonic || [[26edo]] names || Kentaku || | || **Degree** || Cents | ||
<span style="background-color: #ffffff;">DMS</span> ||= Approximated 21-limit Ratios* || Erv Wilson || Archaeotonic || Oneirotonic || [[26edo]] names || Kentaku || | |||
|| 0 || 0 ||= 1/1 || H || C || C || C || J || | || 0 || 0 ||= 1/1 || H || C || C || C || J || | ||
|| 1 || 92.3077 ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">β</span> || C#/Db || C#/Db || Cx/Dbb || J#/Kb || | || 1 || 92.3077 | ||
|| 2 || 184.6154 ||= 9/8, 10/9, 11/10, 19/17, 21/19 || A || D || D || D || K || | 27º41'32" ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">β</span> || C#/Db || C#/Db || Cx/Dbb || J#/Kb || | ||
|| 3 || 276.9231 ||= 7/6, 13/11, 20/17, 19/16, 22/19 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">δ</span> || D#/Eb || D#/Eb || Dx/Ebb || K#/Lb || | || 2 || 184.6154 | ||
|| 4 || 369.2308 ||= 5/4, 11/9, 16/13, 26/21 || C || E || E || E || L || | 55°22'5" ||= 9/8, 10/9, 11/10, 19/17, 21/19 || A || D || D || D || K || | ||
|| 5 || 461.5385 ||= 13/10, 17/13, 21/16, 22/17 || B || E#/Fb || F || Ex/Fb || M || | || 3 || 276.9231 | ||
|| 6 || 553.84 ||= 11/8, 18/13, 26/19 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">ε</span> || F || F#/Gb || F# || M#/Nb || | 83°4'37" ||= 7/6, 13/11, 20/17, 19/16, 22/19 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">δ</span> || D#/Eb || D#/Eb || Dx/Ebb || K#/Lb || | ||
|| 7 || 646.15 ||= 16/11, 13/9, 19/13 || D || F#/Gb || G || Gb || N || | || 4 || 369.2308 | ||
|| 8 || 738.46 ||= 17/11, 20/13, 26/17, 32/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">γ</span> || G || G#/Hb || G# || N#/Ob || | 110°46'9" ||= 5/4, 11/9, 16/13, 26/21 || C || E || E || E || L || | ||
|| 9 || 830.77 ||= 8/5, 13/8, 18/11, 21/13 || F || G#/Ab || H || Ab || O || | || 5 || 461.5385 | ||
|| 10 || 923.08 ||= 12/7, 17/10, 22/13, 19/11 || E || A || A || A# || P || | 138°27'42" ||= 13/10, 17/13, 21/16, 22/17 || B || E#/Fb || F || Ex/Fb || M || | ||
|| 11 || 1015.38 ||= 9/5, 16/9, 20/11, 34/19, 38/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">α</span> || A#/Bb || A#/Bb || Bb || P#/Qb || | || 6 || 553.84 | ||
|| 12 || 1107.69 ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 || G || B/Cb || B || B#/Cbb || Q || | 166°9'14" ||= 11/8, 18/13, 26/19 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">ε</span> || F || F#/Gb || F# || M#/Nb || | ||
|| 13 || 1200 ||= 2/1 || H || C/B# || C || C || J || | || 7 || 646.15 | ||
193°50'46" ||= 16/11, 13/9, 19/13 || D || F#/Gb || G || Gb || N || | |||
|| 8 || 738.46 | |||
221°32'18" ||= 17/11, 20/13, 26/17, 32/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">γ</span> || G || G#/Hb || G# || N#/Ob || | |||
|| 9 || 830.77 | |||
249°13'51" ||= 8/5, 13/8, 18/11, 21/13 || F || G#/Ab || H || Ab || O || | |||
|| 10 || 923.08 | |||
276°55'37" ||= 12/7, 17/10, 22/13, 19/11 || E || A || A || A# || P || | |||
|| 11 || 1015.38 | |||
304°37'55" ||= 9/5, 16/9, 20/11, 34/19, 38/21 || <span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;">α</span> || A#/Bb || A#/Bb || Bb || P#/Qb || | |||
|| 12 || 1107.69 | |||
332°18'28" ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 || G || B/Cb || B || B#/Cbb || Q || | |||
|| 13 || 1200 | |||
360° ||= 2/1 || H || C/B# || C || C || J || | |||
*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible. | ||
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</td> | </td> | ||
<td>Cents<br /> | <td>Cents<br /> | ||
<span style="background-color: #ffffff;">DMS</span><br /> | |||
</td> | </td> | ||
<td style="text-align: center;">Approximated 21-limit Ratios*<br /> | <td style="text-align: center;">Approximated 21-limit Ratios*<br /> | ||
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</td> | </td> | ||
<td>92.3077<br /> | <td>92.3077<br /> | ||
27º41'32&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">17/16, 18/17, 19/18, 20/19, 21/20, 22/21<br /> | <td style="text-align: center;">17/16, 18/17, 19/18, 20/19, 21/20, 22/21<br /> | ||
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</td> | </td> | ||
<td>184.6154<br /> | <td>184.6154<br /> | ||
55°22'5&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">9/8, 10/9, 11/10, 19/17, 21/19<br /> | <td style="text-align: center;">9/8, 10/9, 11/10, 19/17, 21/19<br /> | ||
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</td> | </td> | ||
<td>276.9231<br /> | <td>276.9231<br /> | ||
83°4'37&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">7/6, 13/11, 20/17, 19/16, 22/19<br /> | <td style="text-align: center;">7/6, 13/11, 20/17, 19/16, 22/19<br /> | ||
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</td> | </td> | ||
<td>369.2308<br /> | <td>369.2308<br /> | ||
110°46'9&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">5/4, 11/9, 16/13, 26/21<br /> | <td style="text-align: center;">5/4, 11/9, 16/13, 26/21<br /> | ||
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</td> | </td> | ||
<td>461.5385<br /> | <td>461.5385<br /> | ||
138°27'42&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">13/10, 17/13, 21/16, 22/17<br /> | <td style="text-align: center;">13/10, 17/13, 21/16, 22/17<br /> | ||
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</td> | </td> | ||
<td>553.84<br /> | <td>553.84<br /> | ||
166°9'14&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">11/8, 18/13, 26/19<br /> | <td style="text-align: center;">11/8, 18/13, 26/19<br /> | ||
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</td> | </td> | ||
<td>646.15<br /> | <td>646.15<br /> | ||
193°50'46&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">16/11, 13/9, 19/13<br /> | <td style="text-align: center;">16/11, 13/9, 19/13<br /> | ||
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</td> | </td> | ||
<td>738.46<br /> | <td>738.46<br /> | ||
221°32'18&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">17/11, 20/13, 26/17, 32/21<br /> | <td style="text-align: center;">17/11, 20/13, 26/17, 32/21<br /> | ||
| Line 365: | Line 388: | ||
</td> | </td> | ||
<td>830.77<br /> | <td>830.77<br /> | ||
249°13'51&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br /> | <td style="text-align: center;">8/5, 13/8, 18/11, 21/13<br /> | ||
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</td> | </td> | ||
<td>923.08<br /> | <td>923.08<br /> | ||
276°55'37&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">12/7, 17/10, 22/13, 19/11<br /> | <td style="text-align: center;">12/7, 17/10, 22/13, 19/11<br /> | ||
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</td> | </td> | ||
<td>1015.38<br /> | <td>1015.38<br /> | ||
304°37'55&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">9/5, 16/9, 20/11, 34/19, 38/21<br /> | <td style="text-align: center;">9/5, 16/9, 20/11, 34/19, 38/21<br /> | ||
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</td> | </td> | ||
<td>1107.69<br /> | <td>1107.69<br /> | ||
332°18'28&quot;<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">17/9, 19/10, 21/11, 32/17, 36/19, 40/21<br /> | <td style="text-align: center;">17/9, 19/10, 21/11, 32/17, 36/19, 40/21<br /> | ||
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</td> | </td> | ||
<td>1200<br /> | <td>1200<br /> | ||
360°<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">2/1<br /> | <td style="text-align: center;">2/1<br /> | ||