Kleismic 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:~~hstraub~~|~~hstraub~~]] and made on <tt>2014-08-~~13 05~~:55:35 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-21 20:22:16 UTC</tt>.<br>

: The original revision id was <tt>~~518397682~~</tt>.<br>

: The original revision id was <tt>519242608</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 10:

</span>

The [[5-limit]] parent comma for the **kleismic family** is 15625/15552, the kleisma. Its monzo is |-6 -5 6>, and flipping that yields <<6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called **Hanson**, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].

[[Tuning Ranges of Regular Temperaments|valid range]]: [300.000, 327.273] (4 to 11b)

nice range: [315.641, 317.263]

strict range: [315.641, 317.263]

[[POTE tuning|POTE generator]]: 317.007

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=Keemun=

[[Comma|Commas]]: 49/48, 126/125

valid range: [300.000, 327.273] (4 to 11b)

nice range: [308.744, 322.942]

strict range: [308.744, 322.942]

[[POTE tuning|POTE generator]]: ~6/5 = 316.473

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==11-limit==

[[Comma]]s: 49/48, 56/55, 100/99

valid range: [315.789, 320.000] (19 to 15)

nice range: [308.744, 324.341]

strict range: [315.789, 320.000]

[[POTE tuning|POTE generator]]: ~6/5 = 317.576

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==13-limit==

[[Comma]]s: 49/48, 56/55, 78/77, 100/99

valid range: 315.789 (19)

nice range: [303.597, 324.341]

strict range: 315.789

[[POTE tuning|POTE generator]]: ~6/5 = 316.611

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===Kema===

Commas: 49/48, 56/55, 91/90, 100/99

valid range: [315.789, 320.000] (19 to 15)

nice range: [308.744, 324.341]

strict range: [315.789, 320.000]

POTE generator: ~6/5 = 317.423

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=Catakleismic=

[[Comma|Commas]]: 225/224, 4375/4374

valid range: [315.789, 317.647] (19 to 34)

nice range: [315.641, 317.263]

strict range: [315.789, 317.263]

[[POTE tuning|POTE generator]]: 316.732

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==11-limit==

[[Comma|Commas]]: 225/224, 385/384, 4375/4374

valid range: [315.789, 316.981] (19 to 53)

nice range: [315.641, 317.263]

strict range: [315.789, 316.981]

[[POTE tuning|POTE generator]]: 316.719

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==13-limit==

[[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539

valid range: [315.789, 316.981] (19 to 53)

nice range: [315.641, 318.309]

strict range: [315.789, 316.981]

[[POTE tuning|POTE generator]]: 316.738

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</span><br />

The <a class="wiki_link" href="/5-limit">5-limit</a> parent comma for the <strong>kleismic family</strong> is 15625/15552, the kleisma. Its monzo is |-6 -5 6&gt;, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called <strong>Hanson</strong>, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a>.<br />

<br />

<a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [300.000, 327.273] (4 to 11b)<br />

nice range: [315.641, 317.263]<br />

strict range: [315.641, 317.263]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 317.007<br />

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

<a class="wiki_link" href="/Comma">Commas</a>: 49/48, 126/125<br />

<br />

valid range: [300.000, 327.273] (4 to 11b)<br />

nice range: [308.744, 322.942]<br />

strict range: [308.744, 322.942]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 316.473<br />

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

<a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 56/55, 100/99<br />

<br />

valid range: [315.789, 320.000] (19 to 15)<br />

nice range: [308.744, 324.341]<br />

strict range: [315.789, 320.000]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 317.576<br />

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<h2 id="toc3"><a name="Keemun-13-limit"></a>13-limit</h2>

<a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 56/55, 78/77, 100/99<br />

<br />

valid range: 315.789 (19)<br />

nice range: [303.597, 324.341]<br />

strict range: 315.789<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~6/5 = 316.611<br />

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<h3 id="toc4"><a name="Keemun-13-limit-Kema"></a>Kema</h3>

Commas: 49/48, 56/55, 91/90, 100/99<br />

<br />

valid range: [315.789, 320.000] (19 to 15)<br />

nice range: [308.744, 324.341]<br />

strict range: [315.789, 320.000] <br />

<br />

POTE generator: ~6/5 = 317.423<br />

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<h1 id="toc12"><a name="Catakleismic"></a>Catakleismic</h1>

<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 4375/4374<br />

<br />

valid range: [315.789, 317.647] (19 to 34)<br />

nice range: [315.641, 317.263]<br />

strict range: [315.789, 317.263]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.732<br />

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<h2 id="toc13"><a name="Catakleismic-11-limit"></a>11-limit</h2>

<a class="wiki_link" href="/Comma">Commas</a>: 225/224, 385/384, 4375/4374<br />

<br />

valid range: [315.789, 316.981] (19 to 53)<br />

nice range: [315.641, 317.263]<br />

strict range: [315.789, 316.981]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.719<br />

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<h2 id="toc14"><a name="Catakleismic-13-limit"></a>13-limit</h2>

<a class="wiki_link" href="/Comma">Commas</a>: 169/168, 225/224, 325/324, 540/539<br />

<br />

valid range: [315.789, 316.981] (19 to 53)<br />

nice range: [315.641, 318.309]<br />

strict range: [315.789, 316.981]<br />

<br />

<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 316.738<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:hstraub|hstraub]] and made on <tt>2014-08-13 05:55:35 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-08-21 20:22:16 UTC</tt>.<br>
-: The original revision id was <tt>518397682</tt>.<br>
+: The original revision id was <tt>519242608</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 10: / Line 10: @@
 &lt;/span&gt;
 The [[5-limit]] parent comma for the **kleismic family** is 15625/15552, the kleisma. Its monzo is |-6 -5 6&gt;, and flipping that yields &lt;&lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called **Hanson**, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include [[72edo]], [[87edo]] and [[140edo]].
+[[Tuning Ranges of Regular Temperaments|valid range]]: [300.000, 327.273] (4 to 11b)
+nice range: [315.641, 317.263]
+strict range: [315.641, 317.263]
 [[POTE tuning|POTE generator]]: 317.007
@@ Line 29: / Line 33: @@
 =Keemun=
 [[Comma|Commas]]: 49/48, 126/125
+valid range: [300.000, 327.273] (4 to 11b)
+nice range: [308.744, 322.942]
+strict range: [308.744, 322.942]
 [[POTE tuning|POTE generator]]: ~6/5 = 316.473
@@ Line 39: / Line 47: @@
 ==11-limit==
 [[Comma]]s: 49/48, 56/55, 100/99
+valid range: [315.789, 320.000] (19 to 15)
+nice range: [308.744, 324.341]
+strict range: [315.789, 320.000]
 [[POTE tuning|POTE generator]]: ~6/5 = 317.576
@@ Line 48: / Line 60: @@
 ==13-limit==
 [[Comma]]s: 49/48, 56/55, 78/77, 100/99
+valid range: 315.789 (19)
+nice range: [303.597, 324.341]
+strict range: 315.789
 [[POTE tuning|POTE generator]]: ~6/5 = 316.611
@@ Line 57: / Line 73: @@
 ===Kema===
 Commas: 49/48, 56/55, 91/90, 100/99
+valid range: [315.789, 320.000] (19 to 15)
+nice range: [308.744, 324.341]
+strict range: [315.789, 320.000]
 POTE generator: ~6/5 = 317.423
@@ Line 130: / Line 150: @@
 =Catakleismic=
 [[Comma|Commas]]: 225/224, 4375/4374
+valid range: [315.789, 317.647] (19 to 34)
+nice range: [315.641, 317.263]
+strict range: [315.789, 317.263]
 [[POTE tuning|POTE generator]]: 316.732
@@ Line 140: / Line 164: @@
 ==11-limit==
 [[Comma|Commas]]: 225/224, 385/384, 4375/4374
+valid range: [315.789, 316.981] (19 to 53)
+nice range: [315.641, 317.263]
+strict range: [315.789, 316.981]
 [[POTE tuning|POTE generator]]: 316.719
@@ Line 149: / Line 177: @@
 ==13-limit==
 [[Comma|Commas]]: 169/168, 225/224, 325/324, 540/539
+valid range: [315.789, 316.981] (19 to 53)
+nice range: [315.641, 318.309]
+strict range: [315.789, 316.981]
 [[POTE tuning|POTE generator]]: 316.738
@@ Line 572: / Line 604: @@
 &lt;/span&gt;&lt;br /&gt;
 The &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; parent comma for the &lt;strong&gt;kleismic family&lt;/strong&gt; is 15625/15552, the kleisma. Its monzo is |-6 -5 6&amp;gt;, and flipping that yields &amp;lt;&amp;lt;6 5 -6|| for the wedgie. This tells us the generator is a minor third, and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)^5 = 5/2 * 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called &lt;strong&gt;Hanson&lt;/strong&gt;, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments"&gt;valid range&lt;/a&gt;: [300.000, 327.273] (4 to 11b)&lt;br /&gt;
+nice range: [315.641, 317.263]&lt;br /&gt;
+strict range: [315.641, 317.263]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 317.007&lt;br /&gt;
@@ Line 591: / Line 627: @@
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Keemun"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Keemun&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 49/48, 126/125&lt;br /&gt;
+&lt;br /&gt;
+valid range: [300.000, 327.273] (4 to 11b)&lt;br /&gt;
+nice range: [308.744, 322.942]&lt;br /&gt;
+strict range: [308.744, 322.942]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~6/5 = 316.473&lt;br /&gt;
@@ Line 601: / Line 641: @@
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Keemun-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 49/48, 56/55, 100/99&lt;br /&gt;
+&lt;br /&gt;
+valid range: [315.789, 320.000] (19 to 15)&lt;br /&gt;
+nice range: [308.744, 324.341]&lt;br /&gt;
+strict range: [315.789, 320.000]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~6/5 = 317.576&lt;br /&gt;
@@ Line 610: / Line 654: @@
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Keemun-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;s: 49/48, 56/55, 78/77, 100/99&lt;br /&gt;
+&lt;br /&gt;
+valid range: 315.789 (19)&lt;br /&gt;
+nice range: [303.597, 324.341]&lt;br /&gt;
+strict range: 315.789&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~6/5 = 316.611&lt;br /&gt;
@@ Line 619: / Line 667: @@
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Keemun-13-limit-Kema"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Kema&lt;/h3&gt;
   Commas: 49/48, 56/55, 91/90, 100/99&lt;br /&gt;
+&lt;br /&gt;
+valid range: [315.789, 320.000] (19 to 15)&lt;br /&gt;
+nice range: [308.744, 324.341]&lt;br /&gt;
+strict range: [315.789, 320.000] &lt;br /&gt;
 &lt;br /&gt;
 POTE generator: ~6/5 = 317.423&lt;br /&gt;
@@ Line 692: / Line 744: @@
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Catakleismic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Catakleismic&lt;/h1&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 4375/4374&lt;br /&gt;
+&lt;br /&gt;
+valid range: [315.789, 317.647] (19 to 34)&lt;br /&gt;
+nice range: [315.641, 317.263]&lt;br /&gt;
+strict range: [315.789, 317.263]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 316.732&lt;br /&gt;
@@ Line 702: / Line 758: @@
 &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Catakleismic-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;11-limit&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 385/384, 4375/4374&lt;br /&gt;
+&lt;br /&gt;
+valid range: [315.789, 316.981] (19 to 53)&lt;br /&gt;
+nice range: [315.641, 317.263]&lt;br /&gt;
+strict range: [315.789, 316.981]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 316.719&lt;br /&gt;
@@ Line 711: / Line 771: @@
 &lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Catakleismic-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;13-limit&lt;/h2&gt;
   &lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 169/168, 225/224, 325/324, 540/539&lt;br /&gt;
+&lt;br /&gt;
+valid range: [315.789, 316.981] (19 to 53)&lt;br /&gt;
+nice range: [315.641, 318.309]&lt;br /&gt;
+strict range: [315.789, 316.981]&lt;br /&gt;
 &lt;br /&gt;
 &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 316.738&lt;br /&gt;