Consonant class scale
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A **consonant class scale** is a scale with a **consonant interval class**, that is, a number n such that all n-step intervals of the scale are "consonances" (of a certain odd limit).
The models for such scales are meantone[5] (in which the 2-step intervals are 5/4 and 4/3), and even better known, meantone[7] (in which the 2-step intervals are 6/5 and 5/4). Harmonizing a diatonic melody by parallel thirds is a commonly used formula that reliably produces beautiful harmony. Consonant class scales are an attempt to generalize this property of the diatonic scale.
==Rank-2 consonant class scales==
===5-limit===
===={6/5, 5/4}====
* Meantone[7] (2-step intervals consist of 4 6/5's and 3 5/4's)
* Porcupine[7] (2-step intervals consist of 5 6/5's and 2 5/4's)
* Mavila[7] (2-step intervals consist of 3 6/5's and 4 5/4's)
* Srutal[10] (3-step intervals consist of 4 6/5's and 6 5/4's)
* Negri[10] (3-step intervals consist of 3 6/5's and 7 5/4's)
* Blackwood[10] (3-step intervals consist of 5 6/5's and 5 5/4's)
* Sensi[11] (3-step intervals consist of 9 6/5's and 2 5/4's)
* Orson[13] (4-step intervals consist of 3 6/5's and 10 5/4's)
* Helmholtz[17] (5-step intervals consist of 8 6/5's and 9 5/4's)
===={6/5, 4/3}====
* Mavila[5] (2-step intervals consist of 1 6/5 and 4 4/3's)
* Hanson[7] (2-step intervals consist of 6 6/5's and 1 4/3)
* Srutal[8] (3-step intervals consist of 2 6/5's and 6 4/3's)
* Doublewide[10] (3-step intervals consist of 8 6/5's and 2 4/3's)
===={5/4, 4/3}====
* Meantone[5] (2-step intervals consist of 1 5/4 and 4 4/3's)
* Porcupine[8] (3-step intervals consist of 3 5/4's and 5 4/3's)
* Diminished[8] (3-step intervals consist of 4 5/4's and 4 4/3's)
* Hanson[11] (4-step intervals consist of 6 5/4's and 5 4/3's)
* Tetracot[13] (5-step intervals consist of 4 5/4's and 9 4/3's)
===7-limit===
===={8/7, 7/6}====
* Slendric[5] (1-step intervals consist of 4 8/7's and 1 7/6)
* Superpyth[5] (1-step intervals consist of 3 8/7's and 2 7/6's)
* Orwell[9] (2-step intervals consist of 1 8/7 and 8 7/6's)
* Miracle[10] (2-step intervals consist of 8 8/7's and 2 7/6's)
* Pajara[10] (2-step intervals consist of 6 8/7's and 4 7/6's)
===={8/7, 6/5}====
* Keemun[4] (aka [[xenharmonic/magical seventh chord|magical seventh chord]])
* Gorgo[5] (1-step intervals consist of 4 8/7's and 1 6/5)
* Porcupine[8] (2-step intervals consist of 2 8/7's and 6 6/5's)
* Armodue[9] (2-step intervals consist of 4 8/7's and 5 6/5's)
* Astrology[10] (2-step intervals consist of 8 8/7's and 2 6/5's)
* Flattone[12] (3-step intervals consist of 3 8/7's and 9 6/5's)
* Magic[16] (4-step intervals consist of 4 8/7's and 12 6/5's)
===={7/6, 6/5}====
* Myna[4] (aka [[xenharmonic/starling tetrad|starling tetrad]])
* Sensi[8] (2-step intervals consist of 2 7/6's and 6 6/5's)
* Superpelog[9] (2-step intervals consist of 8 7/6's and 1 6/5)
* Meantone[12] (3-step intervals consist of 3 7/6's and 9 6/5's)
* Augene[12] (3-step intervals consist of 3 7/6's and 9 6/5's)
* Valentine[16] (4-step intervals consist of 4 7/6's and 12 6/5's)
* Muggles[16] (4-step intervals consist of 4 7/6's and 12 6/5's)
{7/6, 5/4}
* Godzilla[9] (2-step intervals consist of 8 7/6's and 1 5/4)
* Deutone[7] (2-step intervals consist of 5 5/4's and 2 7/6's)
{7/6, 9/7}
* Superpyth[7] (2-step intervals consist of 3 9/7's and 4 7/6's)
===={6/5, 9/7}====
* Godzilla[9] (3-step intervals consist of 3 6/5's and 6 9/7's)
===={9/7, 7/5}====
* Deutone[7] (3-step intervals consist of 4 7/5's and 3 9/7's)
===={4/3, 7/5}====
* Meantone[7] (3-step intervals consist of 6 4/3's and 1 7/5; both 2-step intervals are also consonant)
* Negri[9] (4-step intervals consist of 5 4/3's and 4 7/5's)
===={4/3, 10/7}====
* Garibaldi[7] (3-step intervals consist of 6 4/3's and 1 10/7)
* Dominant[7] (3-step intervals consist of 6 4/3's and 1 10/7)
* Rodan[21] (9-step intervals consist of 18 4/3's and 3 10/7's)
===3.5.7-subgroup===
===={5/3, 7/5}====
* Rank-2 Bohlen-Pierce[5] (2-step intervals consist of 3 5/3's and 2 7/5's)
==Rank-3 consonant class scales==
===7-limit===
===={8/7, 6/5, 5/4}====
* Marvel, 7 notes (2-step intervals consist of 1 8/7, 2 6/5's, and 4 5/4's)
===={7/6, 6/5, 5/4}====
* Hewuermity or [[xenharmonic/zeus7|Zeus]], 7 notes (2-step intervals consist of 2 7/6's, 1 6/5, and 4 5/4's)
===={5/4, 4/3, 7/5}====
* Breed or Jove, 7 notes (3-step intervals consist of 2 5/4's, 1 4/3, and 4 7/5's)
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http://launch.groups.yahoo.com/group/tuning/message/41360Original HTML content:
<html><head><title>Consonant class scale</title></head><body>A <strong>consonant class scale</strong> is a scale with a <strong>consonant interval class</strong>, that is, a number n such that all n-step intervals of the scale are "consonances" (of a certain odd limit).<br />
<br />
The models for such scales are meantone[5] (in which the 2-step intervals are 5/4 and 4/3), and even better known, meantone[7] (in which the 2-step intervals are 6/5 and 5/4). Harmonizing a diatonic melody by parallel thirds is a commonly used formula that reliably produces beautiful harmony. Consonant class scales are an attempt to generalize this property of the diatonic scale.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Rank-2 consonant class scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank-2 consonant class scales</h2>
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Rank-2 consonant class scales-5-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->5-limit</h3>
<!-- ws:start:WikiTextHeadingRule:4:<h4> --><h4 id="toc2"><a name="x-Rank-2 consonant class scales-5-limit-{6/5, 5/4}"></a><!-- ws:end:WikiTextHeadingRule:4 -->{6/5, 5/4}</h4>
<ul><li>Meantone[7] (2-step intervals consist of 4 6/5's and 3 5/4's)</li><li>Porcupine[7] (2-step intervals consist of 5 6/5's and 2 5/4's)</li><li>Mavila[7] (2-step intervals consist of 3 6/5's and 4 5/4's)</li><li>Srutal[10] (3-step intervals consist of 4 6/5's and 6 5/4's)</li><li>Negri[10] (3-step intervals consist of 3 6/5's and 7 5/4's)</li><li>Blackwood[10] (3-step intervals consist of 5 6/5's and 5 5/4's)</li><li>Sensi[11] (3-step intervals consist of 9 6/5's and 2 5/4's)</li><li>Orson[13] (4-step intervals consist of 3 6/5's and 10 5/4's)</li><li>Helmholtz[17] (5-step intervals consist of 8 6/5's and 9 5/4's)</li></ul><!-- ws:start:WikiTextHeadingRule:6:<h4> --><h4 id="toc3"><a name="x-Rank-2 consonant class scales-5-limit-{6/5, 4/3}"></a><!-- ws:end:WikiTextHeadingRule:6 -->{6/5, 4/3}</h4>
<ul><li>Mavila[5] (2-step intervals consist of 1 6/5 and 4 4/3's)</li><li>Hanson[7] (2-step intervals consist of 6 6/5's and 1 4/3)</li><li>Srutal[8] (3-step intervals consist of 2 6/5's and 6 4/3's)</li><li>Doublewide[10] (3-step intervals consist of 8 6/5's and 2 4/3's)</li></ul><!-- ws:start:WikiTextHeadingRule:8:<h4> --><h4 id="toc4"><a name="x-Rank-2 consonant class scales-5-limit-{5/4, 4/3}"></a><!-- ws:end:WikiTextHeadingRule:8 -->{5/4, 4/3}</h4>
<ul><li>Meantone[5] (2-step intervals consist of 1 5/4 and 4 4/3's)</li><li>Porcupine[8] (3-step intervals consist of 3 5/4's and 5 4/3's)</li><li>Diminished[8] (3-step intervals consist of 4 5/4's and 4 4/3's)</li><li>Hanson[11] (4-step intervals consist of 6 5/4's and 5 4/3's)</li><li>Tetracot[13] (5-step intervals consist of 4 5/4's and 9 4/3's)</li></ul><!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Rank-2 consonant class scales-7-limit"></a><!-- ws:end:WikiTextHeadingRule:10 -->7-limit</h3>
<!-- ws:start:WikiTextHeadingRule:12:<h4> --><h4 id="toc6"><a name="x-Rank-2 consonant class scales-7-limit-{8/7, 7/6}"></a><!-- ws:end:WikiTextHeadingRule:12 -->{8/7, 7/6}</h4>
<ul><li>Slendric[5] (1-step intervals consist of 4 8/7's and 1 7/6)</li><li>Superpyth[5] (1-step intervals consist of 3 8/7's and 2 7/6's)</li><li>Orwell[9] (2-step intervals consist of 1 8/7 and 8 7/6's)</li><li>Miracle[10] (2-step intervals consist of 8 8/7's and 2 7/6's)</li><li>Pajara[10] (2-step intervals consist of 6 8/7's and 4 7/6's)</li></ul><!-- ws:start:WikiTextHeadingRule:14:<h4> --><h4 id="toc7"><a name="x-Rank-2 consonant class scales-7-limit-{8/7, 6/5}"></a><!-- ws:end:WikiTextHeadingRule:14 -->{8/7, 6/5}</h4>
<ul><li>Keemun[4] (aka <a class="wiki_link" href="http://xenharmonic.wikispaces.com/magical%20seventh%20chord">magical seventh chord</a>)</li><li>Gorgo[5] (1-step intervals consist of 4 8/7's and 1 6/5)</li><li>Porcupine[8] (2-step intervals consist of 2 8/7's and 6 6/5's)</li><li>Armodue[9] (2-step intervals consist of 4 8/7's and 5 6/5's)</li><li>Astrology[10] (2-step intervals consist of 8 8/7's and 2 6/5's)</li><li>Flattone[12] (3-step intervals consist of 3 8/7's and 9 6/5's)</li><li>Magic[16] (4-step intervals consist of 4 8/7's and 12 6/5's)</li></ul><!-- ws:start:WikiTextHeadingRule:16:<h4> --><h4 id="toc8"><a name="x-Rank-2 consonant class scales-7-limit-{7/6, 6/5}"></a><!-- ws:end:WikiTextHeadingRule:16 -->{7/6, 6/5}</h4>
<ul><li>Myna[4] (aka <a class="wiki_link" href="http://xenharmonic.wikispaces.com/starling%20tetrad">starling tetrad</a>)</li><li>Sensi[8] (2-step intervals consist of 2 7/6's and 6 6/5's)</li><li>Superpelog[9] (2-step intervals consist of 8 7/6's and 1 6/5)</li><li>Meantone[12] (3-step intervals consist of 3 7/6's and 9 6/5's)</li><li>Augene[12] (3-step intervals consist of 3 7/6's and 9 6/5's)</li><li>Valentine[16] (4-step intervals consist of 4 7/6's and 12 6/5's)</li><li>Muggles[16] (4-step intervals consist of 4 7/6's and 12 6/5's)</li></ul>{7/6, 5/4}<br />
<ul><li>Godzilla[9] (2-step intervals consist of 8 7/6's and 1 5/4)</li><li>Deutone[7] (2-step intervals consist of 5 5/4's and 2 7/6's)</li></ul>{7/6, 9/7}<br />
<ul><li>Superpyth[7] (2-step intervals consist of 3 9/7's and 4 7/6's)</li></ul><!-- ws:start:WikiTextHeadingRule:18:<h4> --><h4 id="toc9"><a name="x-Rank-2 consonant class scales-7-limit-{6/5, 9/7}"></a><!-- ws:end:WikiTextHeadingRule:18 -->{6/5, 9/7}</h4>
<ul><li>Godzilla[9] (3-step intervals consist of 3 6/5's and 6 9/7's)</li></ul><!-- ws:start:WikiTextHeadingRule:20:<h4> --><h4 id="toc10"><a name="x-Rank-2 consonant class scales-7-limit-{9/7, 7/5}"></a><!-- ws:end:WikiTextHeadingRule:20 -->{9/7, 7/5}</h4>
<ul><li>Deutone[7] (3-step intervals consist of 4 7/5's and 3 9/7's)</li></ul><!-- ws:start:WikiTextHeadingRule:22:<h4> --><h4 id="toc11"><a name="x-Rank-2 consonant class scales-7-limit-{4/3, 7/5}"></a><!-- ws:end:WikiTextHeadingRule:22 -->{4/3, 7/5}</h4>
<ul><li>Meantone[7] (3-step intervals consist of 6 4/3's and 1 7/5; both 2-step intervals are also consonant)</li><li>Negri[9] (4-step intervals consist of 5 4/3's and 4 7/5's)</li></ul><!-- ws:start:WikiTextHeadingRule:24:<h4> --><h4 id="toc12"><a name="x-Rank-2 consonant class scales-7-limit-{4/3, 10/7}"></a><!-- ws:end:WikiTextHeadingRule:24 -->{4/3, 10/7}</h4>
<ul><li>Garibaldi[7] (3-step intervals consist of 6 4/3's and 1 10/7)</li><li>Dominant[7] (3-step intervals consist of 6 4/3's and 1 10/7)</li><li>Rodan[21] (9-step intervals consist of 18 4/3's and 3 10/7's)</li></ul><!-- ws:start:WikiTextHeadingRule:26:<h3> --><h3 id="toc13"><a name="x-Rank-2 consonant class scales-3.5.7-subgroup"></a><!-- ws:end:WikiTextHeadingRule:26 -->3.5.7-subgroup</h3>
<!-- ws:start:WikiTextHeadingRule:28:<h4> --><h4 id="toc14"><a name="x-Rank-2 consonant class scales-3.5.7-subgroup-{5/3, 7/5}"></a><!-- ws:end:WikiTextHeadingRule:28 -->{5/3, 7/5}</h4>
<ul><li>Rank-2 Bohlen-Pierce[5] (2-step intervals consist of 3 5/3's and 2 7/5's)</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="x-Rank-3 consonant class scales"></a><!-- ws:end:WikiTextHeadingRule:30 -->Rank-3 consonant class scales</h2>
<!-- ws:start:WikiTextHeadingRule:32:<h3> --><h3 id="toc16"><a name="x-Rank-3 consonant class scales-7-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->7-limit</h3>
<!-- ws:start:WikiTextHeadingRule:34:<h4> --><h4 id="toc17"><a name="x-Rank-3 consonant class scales-7-limit-{8/7, 6/5, 5/4}"></a><!-- ws:end:WikiTextHeadingRule:34 -->{8/7, 6/5, 5/4}</h4>
<ul><li>Marvel, 7 notes (2-step intervals consist of 1 8/7, 2 6/5's, and 4 5/4's)</li></ul><!-- ws:start:WikiTextHeadingRule:36:<h4> --><h4 id="toc18"><a name="x-Rank-3 consonant class scales-7-limit-{7/6, 6/5, 5/4}"></a><!-- ws:end:WikiTextHeadingRule:36 -->{7/6, 6/5, 5/4}</h4>
<ul><li>Hewuermity or <a class="wiki_link" href="http://xenharmonic.wikispaces.com/zeus7">Zeus</a>, 7 notes (2-step intervals consist of 2 7/6's, 1 6/5, and 4 5/4's)</li></ul><!-- ws:start:WikiTextHeadingRule:38:<h4> --><h4 id="toc19"><a name="x-Rank-3 consonant class scales-7-limit-{5/4, 4/3, 7/5}"></a><!-- ws:end:WikiTextHeadingRule:38 -->{5/4, 4/3, 7/5}</h4>
<ul><li>Breed or Jove, 7 notes (3-step intervals consist of 2 5/4's, 1 4/3, and 4 7/5's)</li></ul><br />
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