26edo: Difference between revisions
Wikispaces>xenwolf **Imported revision 589239972 - Original comment: ** |
Wikispaces>JosephRuhf **Imported revision 590261708 - 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>2016-08-27 14:51:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>590261708</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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1. In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately [[10_9|10/9]] or [[8_7|8/7]], but NOT [[9_8|9/8]]). | 1. In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately [[10_9|10/9]] or [[8_7|8/7]], but NOT [[9_8|9/8]]). | ||
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 [[tel:117649/117128|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 [[tel:117649/117128|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). | 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). | ||
5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone. | 5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives a 21:17; "mushtone". Mushtone is high in badness, but 26edo does it pretty well (and [[33edo]] even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone. | ||
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=Commas= | =Commas= | ||
26et tempers out the following commas. (Note: This assumes the val < 26 41 60 73 90 96 |.) | 26et tempers out the following commas. (Note: This assumes the val < [[tel:26 41 60 73 90 96|26 41 60 73 90 96]] |.) | ||
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || | ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 || | ||
||= 81/80 ||< | -4 4 -1 > ||> 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma || | ||= 81/80 ||< | -4 4 -1 > ||> 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma || | ||
||= 5696703/5695946 ||< | -17 62 -35 > ||> 0.23 ||= Senior ||= ||= || | ||= [[tel:5696703/5695946|5696703/5695946]] ||< | -17 62 -35 > ||> 0.23 ||= Senior ||= ||= || | ||
||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || | ||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= || | ||
||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||= || | ||= 50/49 ||< | 1 0 2 -2 > ||> 34.98 ||= Tritonic Diesis ||= Jubilisma ||= || | ||
| Line 138: | Line 138: | ||
||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||= || | ||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.07 ||= Orwellisma ||= Orwell Comma ||= || | ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= ||= || | ||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= ||= || | ||
||= 321489/320000 ||< | -9 8 -4 2 > ||> 8.04 ||= Varunisma ||= ||= || | ||= [[tel:321489/320000|321489/320000]] ||< | -9 8 -4 2 > ||> 8.04 ||= Varunisma ||= ||= || | ||
||= 1065875/1063543 ||< | -26 -1 1 9 > ||> 3.79 ||= Wadisma ||= ||= || | ||= [[tel:1065875/1063543|1065875/1063543]] ||< | -26 -1 1 9 > ||> 3.79 ||= Wadisma ||= ||= || | ||
||= 4375/4374 ||< | -1 -7 4 1 > ||> 0.40 ||= Ragisma ||= ||= || | ||= 4375/4374 ||< | -1 -7 4 1 > ||> 0.40 ||= Ragisma ||= ||= || | ||
||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||= || | ||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||= || | ||
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The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L 3s|4L 3s (mish)]]. | The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. [[MOSScales|MOS]] of type [[4L 3s|4L 3s (mish)]]. | ||
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200. | The 7-tone scale in cents: [[tel:0 231 323 554 646 877|0 231 323 554 646 877]] [[tel:969 1200|969 1200]]. | ||
The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. [[MOSScales|MOS]] of type [[4L 7s]]. | The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. [[MOSScales|MOS]] of type [[4L 7s]]. | ||
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200. | The 11-tone scale in cents: [[tel:0 92 231 323 415 554|0 92 231 323 415 554]] [[tel:646 785 877 969|646 785 877 969]] 1108, 1200. | ||
The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16_11|16:11]] and 3g approximates [[7_4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents. | The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates [[16_11|16:11]] and 3g approximates [[7_4|7:4]] (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents. | ||
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=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 ( | 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 (though 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). | ||
-Igs | -Igs | ||
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1. In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately <a class="wiki_link" href="/10_9">10/9</a> or <a class="wiki_link" href="/8_7">8/7</a>, but NOT <a class="wiki_link" href="/9_8">9/8</a>).<br /> | 1. In terms of more traditional chord types we have flattone, a variant of meantone with flat fifths, which yields interesting but to some unsatisfying results (due mainly to the dissonance of its thirds, and its major seconds of either approximately <a class="wiki_link" href="/10_9">10/9</a> or <a class="wiki_link" href="/8_7">8/7</a>, but NOT <a class="wiki_link" href="/9_8">9/8</a>).<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 /> | 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 [[tel:117649/117128|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 [[tel:117649/117128|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 /> | 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 /> | ||
5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives a 21:17; &quot;mushtone&quot;. Mushtone is high in badness, but 26edo does it pretty well (and <a class="wiki_link" href="/33edo">33edo</a> even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.<br /> | 5. It also has a pretty good 17th harmonic and tempers out the comma 459:448, thus three fifths gives a 17:14 and four gives a 21:17; &quot;mushtone&quot;. Mushtone is high in badness, but 26edo does it pretty well (and <a class="wiki_link" href="/33edo">33edo</a> even better). Because 26edo also tempers out 85:84, the septendecimal major and minor thirds are equivalent to their pental counterparts, making mushtone the same as flattone.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:11 -->Commas</h1> | <!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc5"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:11 -->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; <a class="wiki_link" href="http://tel.wikispaces.com/26%2041%2060%2073%2090%2096">26 41 60 73 90 96</a> |.)<br /> | ||
| Line 866: | Line 866: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">5696703/5695946<br /> | <td style="text-align: center;">[[tel:5696703/5695946|5696703/5695946]]<br /> | ||
</td> | </td> | ||
<td style="text-align: left;">| -17 62 -35 &gt;<br /> | <td style="text-align: left;">| -17 62 -35 &gt;<br /> | ||
| Line 964: | Line 964: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">321489/320000<br /> | <td style="text-align: center;">[[tel:321489/320000|321489/320000]]<br /> | ||
</td> | </td> | ||
<td style="text-align: left;">| -9 8 -4 2 &gt;<br /> | <td style="text-align: left;">| -9 8 -4 2 &gt;<br /> | ||
| Line 978: | Line 978: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">1065875/1063543<br /> | <td style="text-align: center;">[[tel:1065875/1063543|1065875/1063543]]<br /> | ||
</td> | </td> | ||
<td style="text-align: left;">| -26 -1 1 9 &gt;<br /> | <td style="text-align: left;">| -26 -1 1 9 &gt;<br /> | ||
| Line 1,124: | Line 1,124: | ||
<br /> | <br /> | ||
The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%203s">4L 3s (mish)</a>.<br /> | The 7-tone scale in degrees-in-between: 5 2 5 2 5 2 5. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%203s">4L 3s (mish)</a>.<br /> | ||
The 7-tone scale in cents: 0 231 323 554 646 877 969 1200.<br /> | The 7-tone scale in cents: <a class="wiki_link" href="http://tel.wikispaces.com/0%20231%20323%20554%20646%20877">0 231 323 554 646 877</a> <a class="wiki_link" href="http://tel.wikispaces.com/969%201200">969 1200</a>.<br /> | ||
<br /> | <br /> | ||
The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%207s">4L 7s</a>.<br /> | The 11-tone scale in degrees-in-between: 2 3 2 2 3 2 3 2 2 3 2. <a class="wiki_link" href="/MOSScales">MOS</a> of type <a class="wiki_link" href="/4L%207s">4L 7s</a>.<br /> | ||
The 11-tone scale in cents: 0 92 231 323 415 554 646 785 877 969 1108, 1200.<br /> | The 11-tone scale in cents: <a class="wiki_link" href="http://tel.wikispaces.com/0%2092%20231%20323%20415%20554">0 92 231 323 415 554</a> <a class="wiki_link" href="http://tel.wikispaces.com/646%20785%20877%20969">646 785 877 969</a> 1108, 1200.<br /> | ||
<br /> | <br /> | ||
The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates <a class="wiki_link" href="/16_11">16:11</a> and 3g approximates <a class="wiki_link" href="/7_4">7:4</a> (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.<br /> | The primary triad for orgone temperament is 8:11:14 and its subharmonic inversion, which these scales have in abundance. 2g approximates <a class="wiki_link" href="/16_11">16:11</a> and 3g approximates <a class="wiki_link" href="/7_4">7:4</a> (and I would call that the definition of Orgone Temperament). That also implies that g approximates the difference between 7:4 and 16:11, which is 77:64, about 320.1 cents.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc7"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:15 -->Additional Scalar Bases Available in 26-EDO:</h1> | <!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc7"><a name="Additional Scalar Bases Available in 26-EDO:"></a><!-- ws:end:WikiTextHeadingRule:15 -->Additional Scalar Bases Available in 26-EDO:</h1> | ||
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 ( | 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 (though 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 /> | ||