Consonant class scale: Difference between revisions

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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).

==Rank-2 consonant class scales==  

==Rank-3 consonant class scales==

===7-limit===  

=Links=  

http://tech.groups.yahoo.com/group/tuning-math/message/6056

http://tech.groups.yahoo.com/group/tuning-math/message/6060

http://tech.groups.yahoo.com/group/tuning-math/message/6062

http://tech.groups.yahoo.com/group/tuning-math/message/6064

http://launch.groups.yahoo.com/group/tuning/message/41360</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Consonant class scale&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;strong&gt;consonant class scale&lt;/strong&gt; is a scale with a &lt;strong&gt;consonant interval class&lt;/strong&gt;, that is, a number n such that all n-step intervals of the scale are &amp;quot;consonances&amp;quot; (of a certain odd limit).&lt;br /&gt;

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.&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Rank-2 consonant class scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Rank-2 consonant class scales&lt;/h2&gt;

&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Rank-2 consonant class scales-5-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;5-limit&lt;/h3&gt;

&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc2"&gt;&lt;a name="x-Rank-2 consonant class scales-5-limit-{6/5, 5/4}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;{6/5, 5/4}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Meantone[7] (2-step intervals consist of 4 6/5's and 3 5/4's)&lt;/li&gt;&lt;li&gt;Porcupine[7] (2-step intervals consist of 5 6/5's and 2 5/4's)&lt;/li&gt;&lt;li&gt;Mavila[7] (2-step intervals consist of 3 6/5's and 4 5/4's)&lt;/li&gt;&lt;li&gt;Srutal[10] (3-step intervals consist of 4 6/5's and 6 5/4's)&lt;/li&gt;&lt;li&gt;Negri[10] (3-step intervals consist of 3 6/5's and 7 5/4's)&lt;/li&gt;&lt;li&gt;Blackwood[10] (3-step intervals consist of 5 6/5's and 5 5/4's)&lt;/li&gt;&lt;li&gt;Sensi[11] (3-step intervals consist of 9 6/5's and 2 5/4's)&lt;/li&gt;&lt;li&gt;Orson[13] (4-step intervals consist of 3 6/5's and 10 5/4's)&lt;/li&gt;&lt;li&gt;Helmholtz[17] (5-step intervals consist of 8 6/5's and 9 5/4's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-Rank-2 consonant class scales-5-limit-{6/5, 4/3}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;{6/5, 4/3}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Mavila[5] (2-step intervals consist of 1 6/5 and 4 4/3's)&lt;/li&gt;&lt;li&gt;Hanson[7] (2-step intervals consist of 6 6/5's and 1 4/3)&lt;/li&gt;&lt;li&gt;Srutal[8] (3-step intervals consist of 2 6/5's and 6 4/3's)&lt;/li&gt;&lt;li&gt;Doublewide[10] (3-step intervals consist of 8 6/5's and 2 4/3's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc4"&gt;&lt;a name="x-Rank-2 consonant class scales-5-limit-{5/4, 4/3}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;{5/4, 4/3}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Meantone[5] (2-step intervals consist of 1 5/4 and 4 4/3's)&lt;/li&gt;&lt;li&gt;Porcupine[8] (3-step intervals consist of 3 5/4's and 5 4/3's)&lt;/li&gt;&lt;li&gt;Diminished[8] (3-step intervals consist of 4 5/4's and 4 4/3's)&lt;/li&gt;&lt;li&gt;Hanson[11] (4-step intervals consist of 6 5/4's and 5 4/3's)&lt;/li&gt;&lt;li&gt;Tetracot[13] (5-step intervals consist of 4 5/4's and 9 4/3's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;7-limit&lt;/h3&gt;

&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc6"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{8/7, 7/6}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;{8/7, 7/6}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Slendric[5] (1-step intervals consist of 4 8/7's and 1 7/6)&lt;/li&gt;&lt;li&gt;Superpyth[5] (1-step intervals consist of 3 8/7's and 2 7/6's)&lt;/li&gt;&lt;li&gt;Orwell[9] (2-step intervals consist of 1 8/7 and 8 7/6's)&lt;/li&gt;&lt;li&gt;Miracle[10] (2-step intervals consist of 8 8/7's and 2 7/6's)&lt;/li&gt;&lt;li&gt;Pajara[10] (2-step intervals consist of 6 8/7's and 4 7/6's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc7"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{8/7, 6/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;{8/7, 6/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Keemun[4] (aka &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/magical%20seventh%20chord"&gt;magical seventh chord&lt;/a&gt;)&lt;/li&gt;&lt;li&gt;Gorgo[5] (1-step intervals consist of 4 8/7's and 1 6/5)&lt;/li&gt;&lt;li&gt;Porcupine[8] (2-step intervals consist of 2 8/7's and 6 6/5's)&lt;/li&gt;&lt;li&gt;Armodue[9] (2-step intervals consist of 4 8/7's and 5 6/5's)&lt;/li&gt;&lt;li&gt;Astrology[10] (2-step intervals consist of 8 8/7's and 2 6/5's)&lt;/li&gt;&lt;li&gt;Flattone[12] (3-step intervals consist of 3 8/7's and 9 6/5's)&lt;/li&gt;&lt;li&gt;Magic[16] (4-step intervals consist of 4 8/7's and 12 6/5's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc8"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{7/6, 6/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;{7/6, 6/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Myna[4] (aka &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/starling%20tetrad"&gt;starling tetrad&lt;/a&gt;)&lt;/li&gt;&lt;li&gt;Sensi[8] (2-step intervals consist of 2 7/6's and 6 6/5's)&lt;/li&gt;&lt;li&gt;Superpelog[9] (2-step intervals consist of 8 7/6's and 1 6/5)&lt;/li&gt;&lt;li&gt;Meantone[12] (3-step intervals consist of 3 7/6's and 9 6/5's)&lt;/li&gt;&lt;li&gt;Augene[12] (3-step intervals consist of 3 7/6's and 9 6/5's)&lt;/li&gt;&lt;li&gt;Valentine[16] (4-step intervals consist of 4 7/6's and 12 6/5's)&lt;/li&gt;&lt;li&gt;Muggles[16] (4-step intervals consist of 4 7/6's and 12 6/5's)&lt;/li&gt;&lt;/ul&gt;{7/6, 5/4}&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;Godzilla[9] (2-step intervals consist of 8 7/6's and 1 5/4)&lt;/li&gt;&lt;li&gt;Deutone[7] (2-step intervals consist of 5 5/4's and 2 7/6's)&lt;/li&gt;&lt;/ul&gt;{7/6, 9/7}&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;Superpyth[7] (2-step intervals consist of 3 9/7's and 4 7/6's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc9"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{6/5, 9/7}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;{6/5, 9/7}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Godzilla[9] (3-step intervals consist of 3 6/5's and 6 9/7's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc10"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{9/7, 7/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;{9/7, 7/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Deutone[7] (3-step intervals consist of 4 7/5's and 3 9/7's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc11"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{4/3, 7/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;{4/3, 7/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Meantone[7] (3-step intervals consist of 6 4/3's and 1 7/5; both 2-step intervals are also consonant)&lt;/li&gt;&lt;li&gt;Negri[9] (4-step intervals consist of 5 4/3's and 4 7/5's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc12"&gt;&lt;a name="x-Rank-2 consonant class scales-7-limit-{4/3, 10/7}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;{4/3, 10/7}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Garibaldi[7] (3-step intervals consist of 6 4/3's and 1 10/7)&lt;/li&gt;&lt;li&gt;Dominant[7] (3-step intervals consist of 6 4/3's and 1 10/7)&lt;/li&gt;&lt;li&gt;Rodan[21] (9-step intervals consist of 18 4/3's and 3 10/7's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="x-Rank-2 consonant class scales-3.5.7-subgroup"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;3.5.7-subgroup&lt;/h3&gt;

&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc14"&gt;&lt;a name="x-Rank-2 consonant class scales-3.5.7-subgroup-{5/3, 7/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;{5/3, 7/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Rank-2 Bohlen-Pierce[5] (2-step intervals consist of 3 5/3's and 2 7/5's)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="x-Rank-3 consonant class scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Rank-3 consonant class scales&lt;/h2&gt;

&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="x-Rank-3 consonant class scales-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;7-limit&lt;/h3&gt;

&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc17"&gt;&lt;a name="x-Rank-3 consonant class scales-7-limit-{8/7, 6/5, 5/4}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;{8/7, 6/5, 5/4}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Marvel, &lt;a class="wiki_link" href="/marvel11max7a"&gt;marvel11max7a&lt;/a&gt; and marvel11max7b, 7 notes (2-step intervals consist of 1 8/7, 2 6/5's, and 4 5/4's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc18"&gt;&lt;a name="x-Rank-3 consonant class scales-7-limit-{7/6, 6/5, 5/4}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;{7/6, 6/5, 5/4}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Hewuermity or &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/zeus7"&gt;Zeus&lt;/a&gt;, &lt;a class="wiki_link" href="/zeus7tri"&gt;zeus7tri&lt;/a&gt;, 7 notes (2-step intervals consist of 2 7/6's, 1 6/5, and 4 5/4's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc19"&gt;&lt;a name="x-Rank-3 consonant class scales-7-limit-{5/4, 4/3, 7/5}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;{5/4, 4/3, 7/5}&lt;/h4&gt;

&lt;ul&gt;&lt;li&gt;Breed or Jove, 7 notes (3-step intervals consist of 2 5/4's, 1 4/3, and 4 7/5's)&lt;/li&gt;&lt;/ul&gt;&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc20"&gt;&lt;a name="x-Rank-3 consonant class scales-7-limit-{7/6, 6/5, 5/4}"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;{7/6, 6/5, 5/4}&lt;/h4&gt;

Portent; 4 5/4's 4 6/5's and 3 7/6's in the third class&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc21"&gt;&lt;a name="Links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Links&lt;/h1&gt;

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