31edo: 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-~~16 17~~:31:43 UTC</tt>.

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-02-18 07:46:25 UTC</tt>.

: The original revision id was <tt>~~575003751~~</tt>.

: The original revision id was <tt>575153305</tt>.

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

: The revision comment was: <tt>added some links</tt>

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

<h4>Original Wikitext content:</h4>

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[[toc|flat]]

----

//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]].

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|| [[11_7|11/7]], [[14_11|14/11]] || 8.298 ||

|| [[9_7|9/7]], [[14_9|14/9]] || 9.278 ||

|| 11/8, 16/11 || 9.382 ||

|| [[11_8|11/8]], [[16_11|16/11]] || 9.382 ||

|| 11/10, 20/11 || 10.166 ||

|| [[11_10|11/10]], [[20_11|20/11]] || 10.166 ||

|| 13/10, 20/13 || 10.302 ||

|| [[13_10|13/10]], [[20_13|20/13]] || 10.302 ||

|| 9/8, 16/9 || 10.362 ||

|| [[9_8|9/8]], [[16_9|16/9]] || 10.362 ||

|| 13/8, 16/13 || 11.085 ||

|| [[13_8|13/8]], [[16_13|16/13]] || 11.085 ||

|| 9/5, 10/9 || 11.145 ||

|| [[9_5|9/5]], [[10_9|10/9]] || 11.145 ||

|| 13/7, 14/13 || 12.169 ||

|| [[13_7|13/7]], [[14_13|14/13]] || 12.169 ||

|| 15/13, 26/15 || 15.483 ||

|| [[15_13|15/13]], [[26_15|26/15]] || 15.483 ||

|| 13/12, 24/13 || 16.266 ||

|| [[13_12|13/12]], [[24_13|24/13]] || 16.266 ||

|| 13/9, 18/13 || 17.263 ||

|| [[13_9|13/9]], [[18_13|18/13]] || 17.263 ||

|| 13/11, 22/13 || 18.242 ||

|| [[13_11|13/11]], [[22_13|22/13]] || 18.242 ||

===1\31 octave - approx. 38.71¢ - Diesis===

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<a href="#Linear temperaments">Linear temperaments</a> | <a href="#Intervals">Intervals</a> | <a href="#Harmonic Scale">Harmonic Scale</a> | <a href="#toc34"> </a> | <a href="#Commas">Commas</a> | <a href="#Modes">Modes</a> | <a href="#Music in 31-edo">Music in 31-edo</a> | <a href="#Practical Theory / Books">Practical Theory / Books</a> | <a href="#Other Articles">Other Articles</a>

<hr />

Thirty-one tone equal temperament, also called 31-tET, 31-EDO, 31-et, or tricesimoprimal meantone temperament, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

Thirty-one tone equal temperament, also called 31-tET, 31-EDO, 31-et, or tricesimoprimal meantone temperament, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many <a class="wiki_link" href="/7-limit">7-limit</a> JI scales are well-approximated in 31 (with tempering, of course). It also deals with the <a class="wiki_link" href="/11-limit">11-limit</a> fairly well, and is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

31edo is the 11th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/29edo">29edo</a> and coming before <a class="wiki_link" href="/37edo">37edo</a>.

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</tr>

<tr>

<td>11/8, 16/11

</td>

<td>9.382

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</tr>

<tr>

<td>11/10, 20/11

</td>

<td>10.166

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</tr>

<tr>

<td>13/10, 20/13

</td>

<td>10.302

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</tr>

<tr>

<td>9/8, 16/9

</td>

<td>10.362

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</tr>

<tr>

<td>13/8, 16/13

</td>

<td>11.085

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</tr>

<tr>

<td>9/5, 10/9

</td>

<td>11.145

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</tr>

<tr>

<td>13/7, 14/13

</td>

<td>12.169

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</tr>

<tr>

<td>15/13, 26/15

</td>

<td>15.483

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</tr>

<tr>

<td>13/12, 24/13

</td>

<td>16.266

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</tr>

<tr>

<td>13/9, 18/13

</td>

<td>17.263

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</tr>

<tr>

<td>13/11, 22/13

</td>

<td>18.242

@@ 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>2016-02-16 17:31:43 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-02-18 07:46:25 UTC</tt>.<br>
-: The original revision id was <tt>575003751</tt>.<br>
+: The original revision id was <tt>575153305</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>added some links</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 10: / Line 10: @@
 [[toc|flat]]
 ----
-//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
+//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
 edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]].
@@ Line 53: / Line 53: @@
 || [[11_7|11/7]], [[14_11|14/11]] || 8.298 ||
 || [[9_7|9/7]], [[14_9|14/9]] || 9.278 ||
-|| 11/8, 16/11 || 9.382 ||
+|| [[11_8|11/8]], [[16_11|16/11]] || 9.382 ||
-|| 11/10, 20/11 || 10.166 ||
+|| [[11_10|11/10]], [[20_11|20/11]] || 10.166 ||
-|| 13/10, 20/13 || 10.302 ||
+|| [[13_10|13/10]], [[20_13|20/13]] || 10.302 ||
-|| 9/8, 16/9 || 10.362 ||
+|| [[9_8|9/8]], [[16_9|16/9]] || 10.362 ||
-|| 13/8, 16/13 || 11.085 ||
+|| [[13_8|13/8]], [[16_13|16/13]] || 11.085 ||
-|| 9/5, 10/9 || 11.145 ||
+|| [[9_5|9/5]], [[10_9|10/9]] || 11.145 ||
-|| 13/7, 14/13 || 12.169 ||
+|| [[13_7|13/7]], [[14_13|14/13]] || 12.169 ||
-|| 15/13, 26/15 || 15.483 ||
+|| [[15_13|15/13]], [[26_15|26/15]] || 15.483 ||
-|| 13/12, 24/13 || 16.266 ||
+|| [[13_12|13/12]], [[24_13|24/13]] || 16.266 ||
-|| 13/9, 18/13 || 17.263 ||
+|| [[13_9|13/9]], [[18_13|18/13]] || 17.263 ||
-|| 13/11, 22/13 || 18.242 ||
+|| [[13_11|13/11]], [[22_13|22/13]] || 18.242 ||
 ===1\31 octave - approx. 38.71¢ - Diesis===
@@ Line 415: / Line 415: @@
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-&lt;em&gt;Thirty-one tone equal temperament&lt;/em&gt;, also called &lt;em&gt;31-tET&lt;/em&gt;, &lt;em&gt;31-EDO&lt;/em&gt;, &lt;em&gt;31-et&lt;/em&gt;, or &lt;em&gt;tricesimoprimal meantone temperament&lt;/em&gt;, is the scale derived by dividing the octave into 31 &lt;a class="wiki_link" href="/equal"&gt;equally&lt;/a&gt; large steps. The term 'Tricesimoprimal' was first used by &lt;a class="wiki_link" href="/Adriaan%20Fokker"&gt;Adriaan Fokker&lt;/a&gt;. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt;. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It also deals with the 11-limit fairly well, and is consistent through it, but is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.&lt;br /&gt;
+&lt;em&gt;Thirty-one tone equal temperament&lt;/em&gt;, also called &lt;em&gt;31-tET&lt;/em&gt;, &lt;em&gt;31-EDO&lt;/em&gt;, &lt;em&gt;31-et&lt;/em&gt;, or &lt;em&gt;tricesimoprimal meantone temperament&lt;/em&gt;, is the scale derived by dividing the octave into 31 &lt;a class="wiki_link" href="/equal"&gt;equally&lt;/a&gt; large steps. The term 'Tricesimoprimal' was first used by &lt;a class="wiki_link" href="/Adriaan%20Fokker"&gt;Adriaan Fokker&lt;/a&gt;. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 &lt;a class="wiki_link" href="/cents"&gt;cents&lt;/a&gt;. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;. Many &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; JI scales are well-approximated in 31 (with tempering, of course). It also deals with the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; fairly well, and is consistent through it, but is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.&lt;br /&gt;
 &lt;br /&gt;
 edo is the 11th &lt;a class="wiki_link" href="/prime%20numbers"&gt;prime&lt;/a&gt; edo, following &lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt; and coming before &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;.&lt;br /&gt;
@@ Line 649: / Line 649: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;11/8, 16/11&lt;br /&gt;
+         &lt;td&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&gt;9.382&lt;br /&gt;
@@ Line 655: / Line 655: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;11/10, 20/11&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/11_10"&gt;11/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_11"&gt;20/11&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;10.166&lt;br /&gt;
@@ Line 661: / Line 661: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/10, 20/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_10"&gt;13/10&lt;/a&gt;, &lt;a class="wiki_link" href="/20_13"&gt;20/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;10.302&lt;br /&gt;
@@ Line 667: / Line 667: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;9/8, 16/9&lt;br /&gt;
+         &lt;td&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&gt;10.362&lt;br /&gt;
@@ Line 673: / Line 673: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/8, 16/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_8"&gt;13/8&lt;/a&gt;, &lt;a class="wiki_link" href="/16_13"&gt;16/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;11.085&lt;br /&gt;
@@ Line 679: / Line 679: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;9/5, 10/9&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/9_5"&gt;9/5&lt;/a&gt;, &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;11.145&lt;br /&gt;
@@ Line 685: / Line 685: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/7, 14/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_7"&gt;13/7&lt;/a&gt;, &lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;12.169&lt;br /&gt;
@@ Line 691: / Line 691: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;15/13, 26/15&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/15_13"&gt;15/13&lt;/a&gt;, &lt;a class="wiki_link" href="/26_15"&gt;26/15&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;15.483&lt;br /&gt;
@@ Line 697: / Line 697: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/12, 24/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_12"&gt;13/12&lt;/a&gt;, &lt;a class="wiki_link" href="/24_13"&gt;24/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;16.266&lt;br /&gt;
@@ Line 703: / Line 703: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/9, 18/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_9"&gt;13/9&lt;/a&gt;, &lt;a class="wiki_link" href="/18_13"&gt;18/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;17.263&lt;br /&gt;
@@ Line 709: / Line 709: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;13/11, 22/13&lt;br /&gt;
+         &lt;td&gt;&lt;a class="wiki_link" href="/13_11"&gt;13/11&lt;/a&gt;, &lt;a class="wiki_link" href="/22_13"&gt;22/13&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;18.242&lt;br /&gt;