16edo: Difference between revisions
Wikispaces>TallKite **Imported revision 602809000 - Original comment: ** |
Wikispaces>TallKite **Imported revision 602809012 - 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>2016-12-25 20: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-12-25 20:33:06 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>602809012</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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The second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". | The second approach is to preserve the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly". | ||
|| Degree || Cents ||= Approximate | ||= Degree ||= Cents ||= Approximate | ||
Ratios* ||||= Interval names, | Ratios* ||||= Interval names, | ||
major wider | major wider | ||
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Names | Names | ||
Simplified || | Simplified || | ||
|| 0 || 0 ||= 1/1 ||= unison ||= D ||= unison ||= D ||= Unison ||= | ||= 0 ||= 0 ||= 1/1 ||= unison ||= D ||= unison ||= D ||= Unison ||= Unison || | ||
|| 1 || 75 ||= 28/27, 27/26 ||= aug 1, dim 2nd ||= D#, Eb ||= dim 1, aug 2nd ||= Db, E# ||= Subminor 2nd ||= | ||= 1 ||= 75 ||= 28/27, 27/26 ||= aug 1, dim 2nd ||= D#, Eb ||= dim 1, aug 2nd ||= Db, E# ||= Subminor 2nd ||= Min 2nd || | ||
|| 2 || 150 ||= 12/11, 35/32 ||= min 2nd ||= E ||= maj 2nd ||= E ||= Neutral 2nd ||= | ||= 2 ||= 150 ||= 12/11, 35/32 ||= min 2nd ||= E ||= maj 2nd ||= E ||= Neutral 2nd ||= Maj 2nd || | ||
|| 3 || 225 ||= 8/7 ||= maj 2nd ||= E# ||= min 2nd ||= Eb ||= Supermajor 2nd, | ||= 3 ||= 225 ||= 8/7 ||= maj 2nd ||= E# ||= min 2nd ||= Eb ||= Supermajor 2nd, | ||
Septimal Whole-Tone ||= | Septimal Whole-Tone ||= Perf 2nd || | ||
|| 4 || 300 ||= 19/16, 32/27 ||= min 3rd ||= Fb ||= maj 3rd ||= F# ||= Minor 3rd ||= | ||= 4 ||= 300 ||= 19/16, 32/27 ||= min 3rd ||= Fb ||= maj 3rd ||= F# ||= Minor 3rd ||= Min 3rd || | ||
|| 5 || 375 ||= 5/4, 26/21 ||= maj 3rd ||= F ||= min 3rd ||= F ||= Major 3rd ||= | ||= 5 ||= 375 ||= 5/4, 26/21 ||= maj 3rd ||= F ||= min 3rd ||= F ||= Major 3rd ||= Maj 3rd || | ||
|| 6 || 450 ||= 13/10, 35/27 ||= aug 3rd, dim 4th ||= F#, Gb ||= dim 3rd, aug 4th ||= Fb, G# ||= Sub-4th, | ||= 6 ||= 450 ||= 13/10, 35/27 ||= aug 3rd, dim 4th ||= F#, Gb ||= dim 3rd, aug 4th ||= Fb, G# ||= Sub-4th, | ||
Supermajor 3rd ||= | Supermajor 3rd ||= Min 4th || | ||
|| 7 || 525 ||= 27/20, 52/35, 256/189 ||= perf 4th ||= G ||= perf 4th ||= G ||= Wide 4th ||= | ||= 7 ||= 525 ||= 27/20, 52/35, 256/189 ||= perf 4th ||= G ||= perf 4th ||= G ||= Wide 4th ||= Maj 4th || | ||
|| 8 || 600 ||= 7/5, 10/7 ||= aug 4th, dim 5th ||= G#, Ab ||= dim 4th, aug 5th ||= Gb, A# ||= Tritone ||= | ||= 8 ||= 600 ||= 7/5, 10/7 ||= aug 4th, dim 5th ||= G#, Ab ||= dim 4th, aug 5th ||= Gb, A# ||= Tritone ||= Aug 4th, | ||
Dim 5th || | |||
|| 9 || 675 ||= 40/27, 35/26, 189/128 ||= perf 5th ||= A ||= perf 5th ||= A ||= Narrow 5th ||= | ||= 9 ||= 675 ||= 40/27, 35/26, 189/128 ||= perf 5th ||= A ||= perf 5th ||= A ||= Narrow 5th ||= Min 5th || | ||
|| 10 || 750 ||= 20/13, 54/35 ||= aug 5th, dim 6th ||= A#, Bb ||= dim 5th, aug 6th ||= Ab, B# ||= Super-5th, | ||= 10 ||= 750 ||= 20/13, 54/35 ||= aug 5th, dim 6th ||= A#, Bb ||= dim 5th, aug 6th ||= Ab, B# ||= Super-5th, | ||
Subminor 6th ||= | Subminor 6th ||= Maj 5th || | ||
|| 11 || 825 ||= 8/5, 21/13 ||= min 6th ||= B ||= maj 6th ||= B ||= Minor 6th ||= | ||= 11 ||= 825 ||= 8/5, 21/13 ||= min 6th ||= B ||= maj 6th ||= B ||= Minor 6th ||= Min 6th || | ||
|| 12 || 900 ||= 27/16, 32/19 ||= maj 6th ||= B# ||= min 6th ||= Bb ||= Major 6th ||= | ||= 12 ||= 900 ||= 27/16, 32/19 ||= maj 6th ||= B# ||= min 6th ||= Bb ||= Major 6th ||= Maj 6th || | ||
|| 13 || 975 ||= 7/4 ||= min 7th ||= Cb ||= maj 7th ||= C# ||= Subminor 7th, | ||= 13 ||= 975 ||= 7/4 ||= min 7th ||= Cb ||= maj 7th ||= C# ||= Subminor 7th, | ||
Septimal Minor 7th ||= | Septimal Minor 7th ||= Perf 7th || | ||
|| 14 || 1050 ||= 11/6, 64/35 ||= maj 7th ||= C ||= min 7th ||= C ||= Neutral 7th ||= | ||= 14 ||= 1050 ||= 11/6, 64/35 ||= maj 7th ||= C ||= min 7th ||= C ||= Neutral 7th ||= Min 7th || | ||
|| 15 || 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# ||= Supermajor 7th ||= | ||= 15 ||= 1125 ||= 27/14, 52/27 ||= aug 7th, dim 8ve ||= C# ||= dim 7th, aug 8ve ||= D# ||= Supermajor 7th ||= Maj 7th || | ||
|| 16 || 1200 ||= 2/1 ||= 8ve ||= D ||= 8ve ||= D ||= | ||= 16 ||= 1200 ||= 2/1 ||= 8ve ||= D ||= 8ve ||= D ||= Octave ||= Octave || | ||
*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible. | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<td>Degree<br /> | <td style="text-align: center;">Degree<br /> | ||
</td> | </td> | ||
<td>Cents<br /> | <td style="text-align: center;">Cents<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">Approximate<br /> | <td style="text-align: center;">Approximate<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td>0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td>0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">1/1<br /> | <td style="text-align: center;">1/1<br /> | ||
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<td style="text-align: center;">Unison<br /> | <td style="text-align: center;">Unison<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Unison<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>1<br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td>75<br /> | <td style="text-align: center;">75<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">28/27, 27/26<br /> | <td style="text-align: center;">28/27, 27/26<br /> | ||
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<td style="text-align: center;">Subminor 2nd<br /> | <td style="text-align: center;">Subminor 2nd<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>2<br /> | <td style="text-align: center;">2<br /> | ||
</td> | </td> | ||
<td>150<br /> | <td style="text-align: center;">150<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">12/11, 35/32<br /> | <td style="text-align: center;">12/11, 35/32<br /> | ||
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<td style="text-align: center;">Neutral 2nd<br /> | <td style="text-align: center;">Neutral 2nd<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td>225<br /> | <td style="text-align: center;">225<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8/7<br /> | <td style="text-align: center;">8/7<br /> | ||
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Septimal Whole-Tone<br /> | Septimal Whole-Tone<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Perf 2nd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td>300<br /> | <td style="text-align: center;">300<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">19/16, 32/27<br /> | <td style="text-align: center;">19/16, 32/27<br /> | ||
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<td style="text-align: center;">Minor 3rd<br /> | <td style="text-align: center;">Minor 3rd<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 3rd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td>375<br /> | <td style="text-align: center;">375<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">5/4, 26/21<br /> | <td style="text-align: center;">5/4, 26/21<br /> | ||
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<td style="text-align: center;">Major 3rd<br /> | <td style="text-align: center;">Major 3rd<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 3rd<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td>450<br /> | <td style="text-align: center;">450<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">13/10, 35/27<br /> | <td style="text-align: center;">13/10, 35/27<br /> | ||
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Supermajor 3rd<br /> | Supermajor 3rd<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 4th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td>525<br /> | <td style="text-align: center;">525<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">27/20, 52/35, 256/189<br /> | <td style="text-align: center;">27/20, 52/35, 256/189<br /> | ||
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<td style="text-align: center;">Wide 4th<br /> | <td style="text-align: center;">Wide 4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 4th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>8<br /> | <td style="text-align: center;">8<br /> | ||
</td> | </td> | ||
<td>600<br /> | <td style="text-align: center;">600<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7/5, 10/7<br /> | <td style="text-align: center;">7/5, 10/7<br /> | ||
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<td style="text-align: center;">Tritone<br /> | <td style="text-align: center;">Tritone<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Aug 4th,<br /> | ||
Dim 5th<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>9<br /> | <td style="text-align: center;">9<br /> | ||
</td> | </td> | ||
<td>675<br /> | <td style="text-align: center;">675<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">40/27, 35/26, 189/128<br /> | <td style="text-align: center;">40/27, 35/26, 189/128<br /> | ||
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<td style="text-align: center;">Narrow 5th<br /> | <td style="text-align: center;">Narrow 5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 5th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>10<br /> | <td style="text-align: center;">10<br /> | ||
</td> | </td> | ||
<td>750<br /> | <td style="text-align: center;">750<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">20/13, 54/35<br /> | <td style="text-align: center;">20/13, 54/35<br /> | ||
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Subminor 6th<br /> | Subminor 6th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 5th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>11<br /> | <td style="text-align: center;">11<br /> | ||
</td> | </td> | ||
<td>825<br /> | <td style="text-align: center;">825<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8/5, 21/13<br /> | <td style="text-align: center;">8/5, 21/13<br /> | ||
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<td style="text-align: center;">Minor 6th<br /> | <td style="text-align: center;">Minor 6th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 6th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>12<br /> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td>900<br /> | <td style="text-align: center;">900<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">27/16, 32/19<br /> | <td style="text-align: center;">27/16, 32/19<br /> | ||
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<td style="text-align: center;">Major 6th<br /> | <td style="text-align: center;">Major 6th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 6th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>13<br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td>975<br /> | <td style="text-align: center;">975<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7/4<br /> | <td style="text-align: center;">7/4<br /> | ||
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Septimal Minor 7th<br /> | Septimal Minor 7th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Perf 7th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>14<br /> | <td style="text-align: center;">14<br /> | ||
</td> | </td> | ||
<td>1050<br /> | <td style="text-align: center;">1050<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11/6, 64/35<br /> | <td style="text-align: center;">11/6, 64/35<br /> | ||
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<td style="text-align: center;">Neutral 7th<br /> | <td style="text-align: center;">Neutral 7th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Min 7th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>15<br /> | <td style="text-align: center;">15<br /> | ||
</td> | </td> | ||
<td>1125<br /> | <td style="text-align: center;">1125<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">27/14, 52/27<br /> | <td style="text-align: center;">27/14, 52/27<br /> | ||
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<td style="text-align: center;">Supermajor 7th<br /> | <td style="text-align: center;">Supermajor 7th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Maj 7th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>16<br /> | <td style="text-align: center;">16<br /> | ||
</td> | </td> | ||
<td>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 /> | ||
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<td style="text-align: center;">D<br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Octave<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Octave<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||