26edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2016-02-~~18 11~~:35:26 UTC</tt>.

: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-02-23 10:49:00 UTC</tt>.

: The original revision id was <tt>~~575176839~~</tt>.

: The original revision id was <tt>575604951</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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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).

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. (I propose the name //mushtone// for this temperament since it's flatter than flattone, and results in "mushy" neutral-sounding major and minor thirds. Also there is a street in my hometown called Mushtown Road). 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.

=Intervals=

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The following table shows how [[Just-24|some prominent just intervals]] are represented in 26edo (ordered by absolute error).

|| **Interval, complement** || **Error (abs., in [[cent|cents]])** ||

||= [[13_12|13/12]], [[24_13|24/13]] ||= 0.111 ||

||= [[8_7|8/7]], [[7_4|7/4]] ||= 0.405 ||

||= [[14_11|14/11]], [[11_7|11/7]] ||= 2.123 ||

||= [[10_9|10/9]], [[9_5|9/5]] ||= 2.212 ||

||= [[11_8|11/8]], [[16_11|16/11]] ||= 2.528 ||

||= [[13_10|13/10]], [[20_13|20/13]] ||= 7.325 ||

||= [[6_5|6/5]], [[5_3|5/3]] ||= 7.436 ||

||= [[18_13|18/13]], [[13_9|13/9]] ||= 9.536 ||

||= [[4_3|4/3]], [[3_2|3/2]] ||= 9.647 ||

||= [[16_13|16/13]], [[13_8|13/8]] ||= 9.758 ||

||= [[7_6|7/6]], [[12_7|12/7]] ||= 10.052 ||

||= [[14_13|14/13]], [[13_7|13/7]] ||= 10.163 ||

||= [[12_11|12/11]], [[11_6|11/6]] ||= 12.176 ||

||= [[13_11|13/11]], [[22_13|22/13]] ||= 12.287 ||

||= [[15_11|15/11]], [[22_15|22/15]] ||= 16.895 ||

||= [[15_13|15/13]], [[26_15|26/15]] ||= 16.972 ||

||= [[5_4|5/4]], [[8_5|8/5]] ||= 17.083 ||

||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 ||

||= [[15_14|15/14]], [[28_15|28/15]] ||= 19.019 ||

||= [[9_8|9/8]], [[16_9|16/9]] ||= 19.295 ||

||= [[16_15|16/15]], [[15_8|15/8]] ||= 19.424 ||

||= [[11_10|11/10]], [[20_11|20/11]] ||= 19.611 ||

||= [[9_7|9/7]], [[14_9|14/9]] ||= 19.699 ||

||= [[11_9|11/9]], [[18_11|18/11]] ||= 21.823 ||

=Rank two temperaments=

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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.

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. (I propose the name mushtone for this temperament since it's flatter than flattone, and results in &quot;mushy&quot; neutral-sounding major and minor thirds. Also there is a street in my hometown called Mushtown Road). 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.

<h1 id="toc1"><a name="Intervals"></a>Intervals</h1>

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</tr>

<tr>

</td>

<td style="text-align: center;">0.405

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</tr>

<tr>

</td>

<td style="text-align: center;">2.212

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</tr>

<tr>

</td>

<td style="text-align: center;">2.528

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</tr>

<tr>

</td>

<td style="text-align: center;">7.436

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</tr>

<tr>

</td>

<td style="text-align: center;">9.647

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</tr>

<tr>

</td>

<td style="text-align: center;">10.052

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</tr>

<tr>

</td>

<td style="text-align: center;">17.083

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</tr>

<tr>

</td>

<td style="text-align: center;">17.488

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</tr>

<tr>

</td>

<td style="text-align: center;">19.295

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</tr>

<tr>

</td>

<td style="text-align: center;">19.699

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</tr>

<tr>

</td>

<td style="text-align: center;">21.823

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-02-18 11:35:26 UTC</tt>.<br>
+: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2016-02-23 10:49:00 UTC</tt>.<br>
-: The original revision id was <tt>575176839</tt>.<br>
+: The original revision id was <tt>575604951</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>
@@ Line 16: / Line 16: @@
 . 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.
 . We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).
+. 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. (I propose the name //mushtone// for this temperament since it's flatter than flattone, and results in "mushy" neutral-sounding major and minor thirds. Also there is a street in my hometown called Mushtown Road). 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.
 =&lt;span style="font-size: 1.4em;"&gt;Intervals&lt;/span&gt;=
@@ Line 79: / Line 80: @@
 The following table shows how [[Just-24|some prominent just intervals]] are represented in 26edo (ordered by absolute error).
 || **Interval, complement** || **Error (abs., in [[cent|cents]])** ||
-||= [[13_12|13/12]], [[24_13|24/13]] ||=  0.111 ||
+||= [[13_12|13/12]], [[24_13|24/13]] ||= 0.111 ||
-||= [[8_7|8/7]],     [[7_4|7/4]]     ||=  0.405 ||
+||= [[8_7|8/7]], [[7_4|7/4]] ||= 0.405 ||
-||= [[14_11|14/11]], [[11_7|11/7]]   ||=  2.123 ||
+||= [[14_11|14/11]], [[11_7|11/7]] ||= 2.123 ||
-||= [[10_9|10/9]],   [[9_5|9/5]]     ||=  2.212 ||
+||= [[10_9|10/9]], [[9_5|9/5]] ||= 2.212 ||
-||= [[11_8|11/8]],   [[16_11|16/11]] ||=  2.528 ||
+||= [[11_8|11/8]], [[16_11|16/11]] ||= 2.528 ||
-||= [[13_10|13/10]], [[20_13|20/13]] ||=  7.325 ||
+||= [[13_10|13/10]], [[20_13|20/13]] ||= 7.325 ||
-||= [[6_5|6/5]],     [[5_3|5/3]]     ||=  7.436 ||
+||= [[6_5|6/5]], [[5_3|5/3]] ||= 7.436 ||
-||= [[18_13|18/13]], [[13_9|13/9]]   ||=  9.536 ||
+||= [[18_13|18/13]], [[13_9|13/9]] ||= 9.536 ||
-||= [[4_3|4/3]],     [[3_2|3/2]]     ||=  9.647 ||
+||= [[4_3|4/3]], [[3_2|3/2]] ||= 9.647 ||
-||= [[16_13|16/13]], [[13_8|13/8]]   ||=  9.758 ||
+||= [[16_13|16/13]], [[13_8|13/8]] ||= 9.758 ||
-||= [[7_6|7/6]],     [[12_7|12/7]]   ||= 10.052 ||
+||= [[7_6|7/6]], [[12_7|12/7]] ||= 10.052 ||
-||= [[14_13|14/13]], [[13_7|13/7]]   ||= 10.163 ||
+||= [[14_13|14/13]], [[13_7|13/7]] ||= 10.163 ||
-||= [[12_11|12/11]], [[11_6|11/6]]   ||= 12.176 ||
+||= [[12_11|12/11]], [[11_6|11/6]] ||= 12.176 ||
 ||= [[13_11|13/11]], [[22_13|22/13]] ||= 12.287 ||
 ||= [[15_11|15/11]], [[22_15|22/15]] ||= 16.895 ||
 ||= [[15_13|15/13]], [[26_15|26/15]] ||= 16.972 ||
-||= [[5_4|5/4]],     [[8_5|8/5]]     ||= 17.083 ||
+||= [[5_4|5/4]], [[8_5|8/5]] ||= 17.083 ||
-||= [[7_5|7/5]],     [[10_7|10/7]]   ||= 17.488 ||
+||= [[7_5|7/5]], [[10_7|10/7]] ||= 17.488 ||
 ||= [[15_14|15/14]], [[28_15|28/15]] ||= 19.019 ||
-||= [[9_8|9/8]],     [[16_9|16/9]]   ||= 19.295 ||
+||= [[9_8|9/8]], [[16_9|16/9]] ||= 19.295 ||
-||= [[16_15|16/15]], [[15_8|15/8]]   ||= 19.424 ||
+||= [[16_15|16/15]], [[15_8|15/8]] ||= 19.424 ||
 ||= [[11_10|11/10]], [[20_11|20/11]] ||= 19.611 ||
-||= [[9_7|9/7]],     [[14_9|14/9]]   ||= 19.699 ||
+||= [[9_7|9/7]], [[14_9|14/9]] ||= 19.699 ||
-||= [[11_9|11/9]],   [[18_11|18/11]] ||= 21.823 ||
+||= [[11_9|11/9]], [[18_11|18/11]] ||= 21.823 ||
 =Rank two temperaments=
@@ Line 229: / Line 230: @@
 . 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 &lt;a class="wiki_link" href="/Orgonia"&gt;orgone temperament&lt;/a&gt; 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.&lt;br /&gt;
 . We can also treat 26-EDO as a full 13-limit temperament, since it is consistent on the 13-limit (unlike all lower EDOs).&lt;br /&gt;
+. 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. (I propose the name &lt;em&gt;mushtone&lt;/em&gt; for this temperament since it's flatter than flattone, and results in &amp;quot;mushy&amp;quot; neutral-sounding major and minor thirds. Also there is a street in my hometown called Mushtown Road). Mushtone is high in badness, but 26edo does it pretty well (and &lt;a class="wiki_link" href="/33edo"&gt;33edo&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;&lt;span style="font-size: 1.4em;"&gt;Intervals&lt;/span&gt;&lt;/h1&gt;
@@ Line 564: / Line 566: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;,     &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/8_7"&gt;8/7&lt;/a&gt;, &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;0.405&lt;br /&gt;
@@ Line 576: / Line 578: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;,   &lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;, &lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;2.212&lt;br /&gt;
@@ Line 582: / Line 584: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;,   &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_8"&gt;11/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_11"&gt;16/11&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;2.528&lt;br /&gt;
@@ Line 594: / Line 596: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;,     &lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/6_5"&gt;6/5&lt;/a&gt;, &lt;a class="wiki_link" href="/5_3"&gt;5/3&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;7.436&lt;br /&gt;
@@ Line 606: / Line 608: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;,     &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/4_3"&gt;4/3&lt;/a&gt;, &lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;9.647&lt;br /&gt;
@@ Line 618: / Line 620: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;,     &lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_6"&gt;7/6&lt;/a&gt;, &lt;a class="wiki_link" href="/12_7"&gt;12/7&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;10.052&lt;br /&gt;
@@ Line 654: / Line 656: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;,     &lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;, &lt;a class="wiki_link" href="/8_5"&gt;8/5&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17.083&lt;br /&gt;
@@ Line 660: / Line 662: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;,     &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17.488&lt;br /&gt;
@@ Line 672: / Line 674: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;,     &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_8"&gt;9/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_9"&gt;16/9&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19.295&lt;br /&gt;
@@ Line 690: / Line 692: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;,     &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/9_7"&gt;9/7&lt;/a&gt;, &lt;a class="wiki_link" href="/14_9"&gt;14/9&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;19.699&lt;br /&gt;
@@ Line 696: / Line 698: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_9"&gt;11/9&lt;/a&gt;,   &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="/11_9"&gt;11/9&lt;/a&gt;, &lt;a class="wiki_link" href="/18_11"&gt;18/11&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;21.823&lt;br /&gt;