26edo: Difference between revisions
Wikispaces>igliashon **Imported revision 242824431 - Original comment: ** |
Wikispaces>igliashon **Imported revision 242831105 - 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>2011-07-25 | : This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-25 22:23:40 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>242831105</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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2. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo. | 2. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo. | ||
3. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and 117649/117128. The 65536/65219 comma, the orgonisma, leads to [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14. | 3. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and 117649/117128. The 65536/65219 comma, the orgonisma, leads to [[Orgonia|orgone temperament]] with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14. | ||
4. We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs). | |||
=<span style="font-size: 1.4em;">Intervals</span>= | |||
|| degree || [[cent]]s ||= Approximate | |||
Ratios* || | |||
|| 0 || 0 ||= 1/1 || | |||
|| 1 || 46.154 ||= 33/32, 49/48, 36/35, 25/24 || | |||
|| 2 || 92.308 ||= 21/20 || | |||
|| 3 || 138.46 ||= 14/13, 16/15 || | |||
|| 4 || 184.62 ||= 9/8, 10/9, 11/10 || | |||
|| 5 || 230.77 ||= 8/7 || | |||
|| 6 || 276.92 ||= 7/6, 13/11, 33/28 || | |||
|| 7 || 323.08 ||= 6/5 || | |||
|| 8 || 369.23 ||= 5/4, 16/13 || | |||
|| 9 || 415.38 ||= 9/7, 14/11, 33/26 || | |||
|| 10 || 461.54 ||= 21/16, 13/10 || | |||
|| 11 || 507.69 ||= 4/3 || | |||
|| 12 || 553.85 ||= 11/8, 18/13 || | |||
|| 13 || 600.00 ||= 7/5, 10/7 || | |||
|| 14 || 646.15 ||= 16/11, 13/9 || | |||
|| 15 || 692.31 ||= 3/2 || | |||
|| 16 || 738.46 ||= 32/21, 20/13 || | |||
|| 17 || 784.62 ||= 11/7, 14/9 || | |||
|| 18 || 830.77 ||= 13/8, 8/5 || | |||
|| 19 || 876.92 ||= 5/3 || | |||
|| 20 || 923.08 ||= 12/7, 22/13 || | |||
|| 21 || 969.23 ||= 7/4 || | |||
|| 22 || 1015.4 ||= 9/5, 16/9, 20/11 || | |||
|| 23 || 1061.5 ||= 13/7, 15/8 || | |||
|| 24 || 1107.7 ||= 40/21 || | |||
|| 25 || 1153.8 ||= 64/33, 96/49, 35/18, 48/25 || | |||
|| 26 || 1200 ||= 2/1 || | |||
*based on treating 26-EDO as a 13-limit temperament; other approaches are possible. | |||
=Commas= | =Commas= | ||
| Line 72: | Line 106: | ||
[[image:orgone_heptatonic.jpg]] | [[image:orgone_heptatonic.jpg]] | ||
= | = = | ||
=Additional Scalar Bases Available in 26-EDO:= | =Additional Scalar Bases Available in 26-EDO:= | ||
Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval). | Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval). | ||
| Line 128: | Line 131: | ||
2. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&amp;26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo.<br /> | 2. As two chains of meantone fifths half an octave apart, it supports injera temperament. The generator for this is an interval which can be called either 21/20 or 15/14, and which represents two steps of 26, and hence one step of 13. Hence in 26edo (as opposed to, for instance, 38edo) it can be viewed as two parallel 13edo scales, and from that point of view we can consider it as supporting the 13b&amp;26 temperament, allowing the two chains be shifted slightly and which can be used for more atonal melodies. In this way its internal dynamics resemble those of 14edo.<br /> | ||
3. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and 117649/117128. The 65536/65219 comma, the orgonisma, leads to <a class="wiki_link" href="/Orgonia">orgone temperament</a> with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14.<br /> | 3. 26edo nearly perfectly approximates the 7th and 11th harmonics, and an entire system may be constructed analogous to that based on the 3rd and 5th harmonics. In terms of subgroups, this is the 2.7.11 subgroup, and on this 26 tempers out the pair of commas 65536/65219 and 117649/117128. The 65536/65219 comma, the orgonisma, leads to <a class="wiki_link" href="/Orgonia">orgone temperament</a> with an approximate 77/64 generator of 7\26, with MOS scales of size 7, 11 and 15. The 117649/117128 comma leads to a half-octave period and an approximate 49/44 generator of 4\26, leading to MOS of size 8 and 14.<br /> | ||
4. We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 --><span style="font-size: 1.4em;">Intervals</span></h1> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>degree<br /> | |||
</td> | |||
<td><a class="wiki_link" href="/cent">cent</a>s<br /> | |||
</td> | |||
<td style="text-align: center;">Approximate<br /> | |||
Ratios*<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>0<br /> | |||
</td> | |||
<td>0<br /> | |||
</td> | |||
<td style="text-align: center;">1/1<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>1<br /> | |||
</td> | |||
<td>46.154<br /> | |||
</td> | |||
<td style="text-align: center;">33/32, 49/48, 36/35, 25/24<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>2<br /> | |||
</td> | |||
<td>92.308<br /> | |||
</td> | |||
<td style="text-align: center;">21/20<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>3<br /> | |||
</td> | |||
<td>138.46<br /> | |||
</td> | |||
<td style="text-align: center;">14/13, 16/15<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>4<br /> | |||
</td> | |||
<td>184.62<br /> | |||
</td> | |||
<td style="text-align: center;">9/8, 10/9, 11/10<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>5<br /> | |||
</td> | |||
<td>230.77<br /> | |||
</td> | |||
<td style="text-align: center;">8/7<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>6<br /> | |||
</td> | |||
<td>276.92<br /> | |||
</td> | |||
<td style="text-align: center;">7/6, 13/11, 33/28<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>7<br /> | |||
</td> | |||
<td>323.08<br /> | |||
</td> | |||
<td style="text-align: center;">6/5<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>8<br /> | |||
</td> | |||
<td>369.23<br /> | |||
</td> | |||
<td style="text-align: center;">5/4, 16/13<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>9<br /> | |||
</td> | |||
<td>415.38<br /> | |||
</td> | |||
<td style="text-align: center;">9/7, 14/11, 33/26<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>10<br /> | |||
</td> | |||
<td>461.54<br /> | |||
</td> | |||
<td style="text-align: center;">21/16, 13/10<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>11<br /> | |||
</td> | |||
<td>507.69<br /> | |||
</td> | |||
<td style="text-align: center;">4/3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>12<br /> | |||
</td> | |||
<td>553.85<br /> | |||
</td> | |||
<td style="text-align: center;">11/8, 18/13<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>13<br /> | |||
</td> | |||
<td>600.00<br /> | |||
</td> | |||
<td style="text-align: center;">7/5, 10/7<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>14<br /> | |||
</td> | |||
<td>646.15<br /> | |||
</td> | |||
<td style="text-align: center;">16/11, 13/9<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>15<br /> | |||
</td> | |||
<td>692.31<br /> | |||
</td> | |||
<td style="text-align: center;">3/2<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>16<br /> | |||
</td> | |||
<td>738.46<br /> | |||
</td> | |||
<td style="text-align: center;">32/21, 20/13<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>17<br /> | |||
</td> | |||
<td>784.62<br /> | |||
</td> | |||
<td style="text-align: center;">11/7, 14/9<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>18<br /> | |||
</td> | |||
<td>830.77<br /> | |||
</td> | |||
<td style="text-align: center;">13/8, 8/5<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>19<br /> | |||
</td> | |||
<td>876.92<br /> | |||
</td> | |||
<td style="text-align: center;">5/3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>20<br /> | |||
</td> | |||
<td>923.08<br /> | |||
</td> | |||
<td style="text-align: center;">12/7, 22/13<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>21<br /> | |||
</td> | |||
<td>969.23<br /> | |||
</td> | |||
<td style="text-align: center;">7/4<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>22<br /> | |||
</td> | |||
<td>1015.4<br /> | |||
</td> | |||
<td style="text-align: center;">9/5, 16/9, 20/11<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>23<br /> | |||
</td> | |||
<td>1061.5<br /> | |||
</td> | |||
<td style="text-align: center;">13/7, 15/8<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>24<br /> | |||
</td> | |||
<td>1107.7<br /> | |||
</td> | |||
<td style="text-align: center;">40/21<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>25<br /> | |||
</td> | |||
<td>1153.8<br /> | |||
</td> | |||
<td style="text-align: center;">64/33, 96/49, 35/18, 48/25<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>26<br /> | |||
</td> | |||
<td>1200<br /> | |||
</td> | |||
<td style="text-align: center;">2/1<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
*based on treating 26-EDO as a 13-limit temperament; other approaches are possible.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:4 -->Commas</h1> | ||
26et tempers out the following commas. (Note: This assumes the val &lt; 26 41 60 73 90 96 |.)<br /> | 26et tempers out the following commas. (Note: This assumes the val &lt; 26 41 60 73 90 96 |.)<br /> | ||
| Line 402: | Line 640: | ||
</table> | </table> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Orgone Temperament"></a><!-- ws:end:WikiTextHeadingRule:6 --><a class="wiki_link" href="/Orgonia">Orgone Temperament</a></h1> | ||
<br /> | <br /> | ||
<a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:<br /> | <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> first proposed orgone temperament to take advantage of 26edo's excellent 11 and 7 approximations. 7 degrees of 26edo is a wide minor third of approximately 323.077 cents, and that interval taken as a generator produces 7-tone and 11-tone MOS scales:<br /> | ||
| Line 628: | Line 866: | ||
Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.<br /> | Orgone has a minimax tuning which sharpens both 7 and 11 by 1/5 of an orgonisma, or 1.679 cents. This makes the generator g a 77/64 sharp by 2/5 of the orgonisma. From this we may conclude that 24/89 or 31/115 would be reasonable alternatives to the 7/26 generator.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:716:&lt;img src=&quot;/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/orgone_heptatonic.jpg/155606933/orgone_heptatonic.jpg" alt="orgone_heptatonic.jpg" title="orgone_heptatonic.jpg" /><!-- ws:end:WikiTextLocalImageRule:716 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --> </h1> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:10 -->Additional Scalar Bases Available in 26-EDO:</h1> | |||
<!-- ws:start:WikiTextHeadingRule: | |||
Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).<br /> | Since the perfect 5th in 26-EDO spans 15 degrees, it can be divided into three equal parts (each approximately an 8/7) as well as five equal parts (each approximately a 13/12). The former approach produces MOS at 1L+4s, 5L+1s, and 5L+6s (5 5 5 5 6, 5 5 5 5 5 1, and 4 1 4 1 4 1 4 1 4 1 1 respectively), and is excellent for 4:6:7 triads. The latter produces MOS at 1L+7s and 8L+1s (3 3 3 3 3 3 3 5 and 3 3 3 3 3 3 3 3 2 respectively), and is fairly well-supplied with 4:6:7:11:13 pentads. It also works well for more conventional (thought further from Just) 6:7:9 triads, as well as 4:5:6 triads that use the worse mapping for 5 (making 5/4 the 415.38-cent interval).<br /> | ||
<br /> | <br /> | ||
-Igs<br /> | -Igs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Literature"></a><!-- ws:end:WikiTextHeadingRule:12 -->Literature</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow">Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.</a><br /> | <a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow">Sword, Ron. **Icosihexaphonic Scales for Guitar**. IAAA Press. 2010 - A Guitar-scale thesaurus for 26-EDO.</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Compositions"></a><!-- ws:end:WikiTextHeadingRule:14 -->Compositions</h1> | ||
<br /> | <br /> | ||
<a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3" rel="nofollow">A Time-Yellowed Photo of the Cliffs Hangs on the Wall </a> by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a><br /> | <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20A%20Time-Yellowed%20Photograph%20of%20Cliffs%20Hangs%20in%20the%20Hall.mp3" rel="nofollow">A Time-Yellowed Photo of the Cliffs Hangs on the Wall </a> by <a class="wiki_link" href="/IgliashonJones">Igliashon Jones</a><br /> | ||