Diaschismic family: Difference between revisions

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:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 23:27~~:22~~ UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 23:32:27 UTC</tt>.<br>

: The original revision id was <tt>~~213066378~~</tt>.<br>

: The original revision id was <tt>213067300</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>

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Map: [<2 0 11|, <0 1 -2|]

EDOs: 34, 46, 80, 286

Line 17:

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The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7.

==Pajara==

=Pajara=

Pajara, with wedgie <<2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10_7|10/7]] and a [[7_5|7/5]]. Aside from 22et, 34 with the val <34 54 79 96| and 56 with the val <56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

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Map: [<2 0 11 12|, <0 1 -2 -2|]

EDOs: 22, 34, 56

===11-limit===

==11-limit==

Commas: 50/49, 64/63, 99/98

Line 36:

Line 34:

Map: [<2 0 11 12 26|, <0 1 -2 -2 -6|]

EDOs: 22, 34, 56, 146

==Diaschismic==

=Diaschismic=

A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&58. However described, diaschismic has wedgie <<2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7_4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.

Line 49:

Line 46:

Map: [<2 0 11 31|, <0 1 -2 -8|]

EDOs: 46, 58, 162

===11-limit===

==11-limit==

Commas: 126/125, 176/175, 896/891

Line 58:

Line 54:

Map: [<2 0 11 31 45|, <0 1 -2 -8 -12|]

EDOs: 46, 58, 104, 162

===13-limit===

==13-limit==

Commas: 126/125, 196/195, 364/363, 2048/2025

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Map: [<2 0 11 31 45 55|, <0 1 -2 -8 -12 -15|]

EDOs: [[46edo]], [[58edo]], [[162edo]]

===17-limit===

==17-limit==

Commas: 126/125, 136/135, 176/175, 196/195, 256/255

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Map: [<2 0 11 31 45 55 5|, <0 1 -2 -8 -12 -15 1|]

EDOs: 46, 58, 104

==Keen==

=Keen=

Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie <<2 -4 18 -11 23 53||. It may also be described as the 22&56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, <<2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.

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Map: [<2 0 11 -23|, <0 1 -2 9|]

EDOs: 22, 56, 78

===11-limit===

==11-limit==

Commas: 100/99, 385/384, 1232/1215

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Line 88:

Map: [<2 0 11 -23 26|, <0 1 -2 9 -6|]

EDOs: 22, 56, 78

==Echidna==

=Echidna=

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<6 -12 10 -33 -1 57|| and may be called the 22&58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to <138 219 321 388|.

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Badness: 0.0580

===11-limit===

==11-limit==

Commas: 176/175, 896/891, 540/539

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Map: [<2 1 9 2 12|, <0 3 -6 5 -7|]

EDOs: 22, 58, 80, 138, 218

Badness: 0.0260

===13-limit===

==13-limit==

Commas: 176/175, 351/350, 364/363, 540/539

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Badness: 0.0237

===17-limit===

==17-limit==

Commas: 136/135, 176/175, 221/220, 256/255, 540/539

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Line 136:

Badness: 0.0203

==Echidnic==

=Echidnic=

Commas: 686/675, 1029/1024

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Badness: 0.0722

===11-limit===

==11-limit==

Commas: 385/384, 441/444, 686/675

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Badness: 0.0451

===13-limit===

==13-limit==

Commas: 91/90, 169/168, 385/384, 441/440

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Map: [<2 1 9 -2|, <0 2 -4 7|]

EDOs: 22, 46, 68, 250

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<br />

Map: [&lt;2 0 11|, &lt;0 1 -2|]<br />

~~<br />~~

EDOs: 34, 46, 80, 286<br />

<br />

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The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as <a class="wiki_link" href="/36_35">36/35</a>, the septimal quarter-tone) and echidna has a generator of 9/7.<br />

<br />

<h2 id="toc1"><a name="x-Pajara"></a>Pajara</h2>

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

Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>) but other tunings are possible. The 1/2 octave period serves as both a <a class="wiki_link" href="/10_7">10/7</a> and a <a class="wiki_link" href="/7_5">7/5</a>. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.<br />

<br />

Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.<br />

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<br />

Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]<br />

~~<br />~~

EDOs: 22, 34, 56<br />

<br />

<h3 id="toc2"><a name="x-Pajara-11-limit"></a>11-limit</h3>

<h2 id="toc2"><a name="Pajara-11-limit"></a>11-limit</h2>

Commas: 50/49, 64/63, 99/98<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 706.885<br />

<br />

Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]<br />

~~<br />~~

EDOs: 22, 34, 56, 146<br />

<br />

<h2 id="toc3"><a name="x-Diaschismic"></a>Diaschismic</h2>

<h1 id="toc3"><a name="Diaschismic"></a>Diaschismic</h1>

A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. <a class="wiki_link" href="/58edo">58et</a> provides an excellent tuning, but an alternative is to make <a class="wiki_link" href="/7_4">7/4</a> just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.<br />

<br />

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.<br />

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<br />

Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]<br />

~~<br />~~

EDOs: 46, 58, 162<br />

<br />

<h3 id="toc4"><a name="x-Diaschismic-11-limit"></a>11-limit</h3>

<h2 id="toc4"><a name="Diaschismic-11-limit"></a>11-limit</h2>

Commas: 126/125, 176/175, 896/891<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.714<br />

<br />

Map: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]<br />

~~<br />~~

EDOs: 46, 58, 104, 162<br />

<br />

<h3 id="toc5"><a name="x-Diaschismic-13-limit"></a>13-limit</h3>

<h2 id="toc5"><a name="Diaschismic-13-limit"></a>13-limit</h2>

Commas: 126/125, 196/195, 364/363, 2048/2025<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.704<br />

<br />

Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]<br />

~~<br />~~

EDOs: <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/58edo">58edo</a>, <a class="wiki_link" href="/162edo">162edo</a><br />

<br />

<h3 id="toc6"><a name="x-Diaschismic-17-limit"></a>17-limit</h3>

<h2 id="toc6"><a name="Diaschismic-17-limit"></a>17-limit</h2>

Commas: 126/125, 136/135, 176/175, 196/195, 256/255<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 703.812<br />

<br />

Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]<br />

~~<br />~~

EDOs: 46, 58, 104<br />

<br />

<h2 id="toc7"><a name="x-Keen"></a>Keen</h2>

<h1 id="toc7"><a name="Keen"></a>Keen</h1>

Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. <a class="wiki_link" href="/78edo">78et</a> is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.<br />

<br />

Commas: 2048/2025, 875/864<br />

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<br />

Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]<br />

~~<br />~~

EDOs: 22, 56, 78<br />

<br />

<h3 id="toc8"><a name="x-Keen-11-limit"></a>11-limit</h3>

<h2 id="toc8"><a name="Keen-11-limit"></a>11-limit</h2>

Commas: 100/99, 385/384, 1232/1215<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 707.609<br />

<br />

Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]<br />

~~<br />~~

EDOs: 22, 56, 78<br />

<br />

<h2 id="toc9"><a name="x-Echidna"></a>Echidna</h2>

<h1 id="toc9"><a name="Echidna"></a>Echidna</h1>

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. <a class="wiki_link" href="/58edo">58et</a> or <a class="wiki_link" href="/80edo">80et</a> make for good tunings, or their vals can be add to &lt;138 219 321 388|.<br />

<br />

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.<br />

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Badness: 0.0580<br />

<br />

<h3 id="toc10"><a name="x-Echidna-11-limit"></a>11-limit</h3>

<h2 id="toc10"><a name="Echidna-11-limit"></a>11-limit</h2>

Commas: 176/175, 896/891, 540/539<br />

<br />

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<br />

Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]<br />

~~<br />~~

EDOs: 22, 58, 80, 138, 218<br />

Badness: 0.0260<br />

<br />

<h3 id="toc11"><a name="x-Echidna-13-limit"></a>13-limit</h3>

<h2 id="toc11"><a name="Echidna-13-limit"></a>13-limit</h2>

Commas: 176/175, 351/350, 364/363, 540/539<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.756<br />

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Badness: 0.0237<br />

<br />

<h3 id="toc12"><a name="x-Echidna-17-limit"></a>17-limit</h3>

<h2 id="toc12"><a name="Echidna-17-limit"></a>17-limit</h2>

Commas: 136/135, 176/175, 221/220, 256/255, 540/539<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 434.816<br />

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Badness: 0.0203<br />

<br />

<h2 id="toc13"><a name="x-Echidnic"></a>Echidnic</h2>

<h1 id="toc13"><a name="Echidnic"></a>Echidnic</h1>

Commas: 686/675, 1029/1024<br />

<br />

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Badness: 0.0722<br />

<br />

<h3 id="toc14"><a name="x-Echidnic-11-limit"></a>11-limit</h3>

<h2 id="toc14"><a name="Echidnic-11-limit"></a>11-limit</h2>

Commas: 385/384, 441/444, 686/675<br />

<br />

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Badness: 0.0451<br />

<br />

<h3 id="toc15"><a name="x-Echidnic-13-limit"></a>13-limit</h3>

<h2 id="toc15"><a name="Echidnic-13-limit"></a>13-limit</h2>

Commas: 91/90, 169/168, 385/384, 441/440<br />

<br />

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<br />

Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]<br />

~~<br />~~

EDOs: 22, 46, 68, 250<br />

<br />

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 23:27:22 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-22 23:32:27 UTC</tt>.<br>
-: The original revision id was <tt>213066378</tt>.<br>
+: The original revision id was <tt>213067300</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 11: / Line 11: @@
 Map: [&lt;2 0 11|, &lt;0 1 -2|]
 EDOs: 34, 46, 80, 286
@@ Line 17: / Line 16: @@
 The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7.
-==Pajara==
+=Pajara=
 Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10_7|10/7]] and a [[7_5|7/5]]. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
@@ Line 27: / Line 26: @@
 Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
 EDOs: 22, 34, 56
-===11-limit===
+==11-limit==
 Commas: 50/49, 64/63, 99/98
@@ Line 36: / Line 34: @@
 Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
 EDOs: 22, 34, 56, 146
-==Diaschismic==
+=Diaschismic=
 A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7_4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.
@@ Line 49: / Line 46: @@
 Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
 EDOs: 46, 58, 162
-===11-limit===
+==11-limit==
 Commas: 126/125, 176/175, 896/891
@@ Line 58: / Line 54: @@
 Map: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]
 EDOs: 46, 58, 104, 162
-===13-limit===
+==13-limit==
 Commas: 126/125, 196/195, 364/363, 2048/2025
@@ Line 67: / Line 62: @@
 Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]
 EDOs: [[46edo]], [[58edo]], [[162edo]]
-===17-limit===
+==17-limit==
 Commas: 126/125, 136/135, 176/175, 196/195, 256/255
@@ Line 76: / Line 70: @@
 Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
 EDOs: 46, 58, 104
-==Keen==
+=Keen=
 Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.
@@ Line 87: / Line 80: @@
 Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]
 EDOs: 22, 56, 78
-===11-limit===
+==11-limit==
 Commas: 100/99, 385/384, 1232/1215
@@ Line 96: / Line 88: @@
 Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
 EDOs: 22, 56, 78
-==Echidna==
+=Echidna=
 Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to &lt;138 219 321 388|.
@@ Line 112: / Line 103: @@
 Badness: 0.0580
-===11-limit===
+==11-limit==
 Commas: 176/175, 896/891, 540/539
@@ Line 124: / Line 115: @@
 Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
 EDOs: 22, 58, 80, 138, 218
 Badness: 0.0260
-===13-limit===
+==13-limit==
 Commas: 176/175, 351/350, 364/363, 540/539
@@ Line 137: / Line 127: @@
 Badness: 0.0237
-===17-limit===
+==17-limit==
 Commas: 136/135, 176/175, 221/220, 256/255, 540/539
@@ Line 146: / Line 136: @@
 Badness: 0.0203
-==Echidnic==
+=Echidnic=
 Commas: 686/675, 1029/1024
@@ Line 155: / Line 145: @@
 Badness: 0.0722
-===11-limit===
+==11-limit==
 Commas: 385/384, 441/444, 686/675
@@ Line 164: / Line 154: @@
 Badness: 0.0451
-===13-limit===
+==13-limit==
 Commas: 91/90, 169/168, 385/384, 441/440
@@ Line 183: / Line 173: @@
 Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]
 EDOs: 22, 46, 68, 250
@@ Line 212: / Line 201: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11|, &amp;lt;0 1 -2|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 34, 46, 80, 286&lt;br /&gt;
 &lt;br /&gt;
@@ Line 218: / Line 206: @@
   The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;, the septimal quarter-tone) and echidna has a generator of 9/7.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Pajara&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Pajara"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Pajara&lt;/h1&gt;
- Pajara, with wedgie &amp;lt;&amp;lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;) but other tunings are possible. The 1/2 octave period serves as both a &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt; and a &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;. Aside from 22et, 34 with the val &amp;lt;34 54 79 96| and 56 with the val &amp;lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.&lt;br /&gt;
+Pajara, with wedgie &amp;lt;&amp;lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;) but other tunings are possible. The 1/2 octave period serves as both a &lt;a class="wiki_link" href="/10_7"&gt;10/7&lt;/a&gt; and a &lt;a class="wiki_link" href="/7_5"&gt;7/5&lt;/a&gt;. Aside from 22et, 34 with the val &amp;lt;34 54 79 96| and 56 with the val &amp;lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.&lt;br /&gt;
 &lt;br /&gt;
 Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.&lt;br /&gt;
@@ Line 228: / Line 216: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 12|, &amp;lt;0 1 -2 -2|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 34, 56&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Pajara-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Pajara-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
- Commas: 50/49, 64/63, 99/98&lt;br /&gt;
+Commas: 50/49, 64/63, 99/98&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 706.885&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 12 26|, &amp;lt;0 1 -2 -2 -6|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 34, 56, 146&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Diaschismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Diaschismic&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Diaschismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Diaschismic&lt;/h1&gt;
- A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;amp;58. However described, diaschismic has wedgie &amp;lt;&amp;lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. &lt;a class="wiki_link" href="/58edo"&gt;58et&lt;/a&gt; provides an excellent tuning, but an alternative is to make &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.&lt;br /&gt;
+A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;amp;58. However described, diaschismic has wedgie &amp;lt;&amp;lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. &lt;a class="wiki_link" href="/58edo"&gt;58et&lt;/a&gt; provides an excellent tuning, but an alternative is to make &lt;a class="wiki_link" href="/7_4"&gt;7/4&lt;/a&gt; just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.&lt;br /&gt;
 &lt;br /&gt;
 Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.&lt;br /&gt;
@@ Line 250: / Line 236: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 31|, &amp;lt;0 1 -2 -8|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 46, 58, 162&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Diaschismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Diaschismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;11-limit&lt;/h2&gt;
- Commas: 126/125, 176/175, 896/891&lt;br /&gt;
+Commas: 126/125, 176/175, 896/891&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.714&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 31 45|, &amp;lt;0 1 -2 -8 -12|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 46, 58, 104, 162&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Diaschismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Diaschismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h2&gt;
- Commas: 126/125, 196/195, 364/363, 2048/2025&lt;br /&gt;
+Commas: 126/125, 196/195, 364/363, 2048/2025&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.704&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 31 45 55|, &amp;lt;0 1 -2 -8 -12 -15|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt;, &lt;a class="wiki_link" href="/162edo"&gt;162edo&lt;/a&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Diaschismic-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;17-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Diaschismic-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;17-limit&lt;/h2&gt;
- Commas: 126/125, 136/135, 176/175, 196/195, 256/255&lt;br /&gt;
+Commas: 126/125, 136/135, 176/175, 196/195, 256/255&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 703.812&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 31 45 55 5|, &amp;lt;0 1 -2 -8 -12 -15 1|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 46, 58, 104&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="x-Keen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Keen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Keen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Keen&lt;/h1&gt;
- Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &amp;lt;&amp;lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;amp;56 temperament. &lt;a class="wiki_link" href="/78edo"&gt;78et&lt;/a&gt; is a good tuning choice, and remains a good one in the 11-limit, where keen, &amp;lt;&amp;lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.&lt;br /&gt;
+Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &amp;lt;&amp;lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;amp;56 temperament. &lt;a class="wiki_link" href="/78edo"&gt;78et&lt;/a&gt; is a good tuning choice, and remains a good one in the 11-limit, where keen, &amp;lt;&amp;lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.&lt;br /&gt;
 &lt;br /&gt;
 Commas: 2048/2025, 875/864&lt;br /&gt;
@@ Line 288: / Line 270: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 -23|, &amp;lt;0 1 -2 9|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 56, 78&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Keen-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Keen-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;11-limit&lt;/h2&gt;
- Commas: 100/99, 385/384, 1232/1215&lt;br /&gt;
+Commas: 100/99, 385/384, 1232/1215&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 707.609&lt;br /&gt;
 &lt;br /&gt;
 Map: [&amp;lt;2 0 11 -23 26|, &amp;lt;0 1 -2 9 -6|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 56, 78&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="x-Echidna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Echidna&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Echidna"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Echidna&lt;/h1&gt;
- Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &amp;lt;&amp;lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;amp;58 temperament. &lt;a class="wiki_link" href="/58edo"&gt;58et&lt;/a&gt; or &lt;a class="wiki_link" href="/80edo"&gt;80et&lt;/a&gt; make for good tunings, or their vals can be add to &amp;lt;138 219 321 388|.&lt;br /&gt;
+Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &amp;lt;&amp;lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;amp;58 temperament. &lt;a class="wiki_link" href="/58edo"&gt;58et&lt;/a&gt; or &lt;a class="wiki_link" href="/80edo"&gt;80et&lt;/a&gt; make for good tunings, or their vals can be add to &amp;lt;138 219 321 388|.&lt;br /&gt;
 &lt;br /&gt;
 Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.&lt;br /&gt;
@@ Line 313: / Line 293: @@
 Badness: 0.0580&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="x-Echidna-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Echidna-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;11-limit&lt;/h2&gt;
   Commas: 176/175, 896/891, 540/539&lt;br /&gt;
 &lt;br /&gt;
@@ Line 325: / Line 305: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 1 9 2 12|, &amp;lt;0 3 -6 5 -7|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 58, 80, 138, 218&lt;br /&gt;
 Badness: 0.0260&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="x-Echidna-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Echidna-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;13-limit&lt;/h2&gt;
- Commas: 176/175, 351/350, 364/363, 540/539&lt;br /&gt;
+Commas: 176/175, 351/350, 364/363, 540/539&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 434.756&lt;br /&gt;
@@ Line 338: / Line 317: @@
 Badness: 0.0237&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="x-Echidna-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;17-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Echidna-17-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;17-limit&lt;/h2&gt;
- Commas: 136/135, 176/175, 221/220, 256/255, 540/539&lt;br /&gt;
+Commas: 136/135, 176/175, 221/220, 256/255, 540/539&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 434.816&lt;br /&gt;
@@ Line 347: / Line 326: @@
 Badness: 0.0203&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x-Echidnic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Echidnic&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Echidnic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Echidnic&lt;/h1&gt;
   Commas: 686/675, 1029/1024&lt;br /&gt;
 &lt;br /&gt;
@@ Line 356: / Line 335: @@
 Badness: 0.0722&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x-Echidnic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;11-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Echidnic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;11-limit&lt;/h2&gt;
   Commas: 385/384, 441/444, 686/675&lt;br /&gt;
 &lt;br /&gt;
@@ Line 365: / Line 344: @@
 Badness: 0.0451&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="x-Echidnic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;13-limit&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Echidnic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;13-limit&lt;/h2&gt;
   Commas: 91/90, 169/168, 385/384, 441/440&lt;br /&gt;
 &lt;br /&gt;
@@ Line 384: / Line 363: @@
 &lt;br /&gt;
 Map: [&amp;lt;2 1 9 -2|, &amp;lt;0 2 -4 7|]&lt;br /&gt;
-&lt;br /&gt;
 EDOs: 22, 46, 68, 250&lt;br /&gt;
 &lt;br /&gt;