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Tour of regular temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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===[[Meantone family]]===  
===[[Meantone family]]===  
The meantone family tempers out 81/80, the difference between 81/16 (3*3*3*3/2*2*2*2, a stack of four perfect fifths) and 80/16 (aka 5/1, the fifth harmonic) and has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from equal divisions of the octave, regular tunings include flattening by 1/3, 2/7, 1/4, 1/5 and 1/6 the Syntonic comma (the 81/80 interval.)
The meantone family tempers out 81/80, also called the syntonic comma. 81/80 manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as some sort of average or "mean" of the two tones. It has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of syntonic comma (the 81/80 interval.)


===[[Schismatic family]]===  
===[[Schismatic family]]===  
The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymas comma (81/80). The 5-limit version of the temperament is a [[Microtempering|microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity (it takes eight schismatic-tempered 3/2's to reach 5/4, as opposed to four meantone-tempered 3/2's).
The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80). The 5-limit version of the temperament is a [[Microtempering|microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity - whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4.


===[[Kleismic family]]===  
===[[Kleismic family]]===  
The kleismic family of temperaments tempers out the kleisma of 15625/15552 (the difference between six 6/5's--23328/15625--and 3/2) and has a slightly sharpened minor third as a generator. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.
The kleismic family of temperaments tempers out the kleisma of 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.


===[[Magic family]]===  
===[[Magic family]]===  
The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.
The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.
The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]]  among its possible tunings, with the latter being near-optimal.


===[[Diaschismic family]]===  
===[[Diaschismic family]]===  
The diaschismic family tempers out 2048/2025, the [[diaschisma]]. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. Using [[22edo]] as a tuning is associated with [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out.
The diaschismic family tempers out 2048/2025, the [[diaschisma]], which is the difference between the Pythagorean major third of 81/64 and the 5-limit diminished fourth of 32/25. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]]temperament, where the intervals 50/49 and 64/63 are tempered out, and of which [[22edo]] is an excellent tuning.


===[[Pelogic family]]===  
===[[Pelogic family]]===  
This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma.
This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, the accumulated flatness leads you to a minor third + 2 octaves instead of a major third + 2 octaves, and one consequence of this is that it generates 2L5s "anti-diatonic" scales. Mavila and Armodue are the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].


===[[Porcupine family]]===  
===[[Porcupine family]]===  
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&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Meantone family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;&lt;a class="wiki_link" href="/Meantone%20family"&gt;Meantone family&lt;/a&gt;&lt;/h3&gt;
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  The meantone family tempers out 81/80, the difference between 81/16 (3*3*3*3/2*2*2*2, a stack of four perfect fifths) and 80/16 (aka 5/1, the fifth harmonic) and has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43edo&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;, &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; and &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;. Aside from equal divisions of the octave, regular tunings include flattening by 1/3, 2/7, 1/4, 1/5 and 1/6 the Syntonic comma (the 81/80 interval.)&lt;br /&gt;
  The meantone family tempers out 81/80, also called the syntonic comma. 81/80 manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as some sort of average or &amp;quot;mean&amp;quot; of the two tones. It has a flattened fifth (or sharpened fourth) as generator. Some meantone tunings are &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43edo&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;, &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; and &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of syntonic comma (the 81/80 interval.)&lt;br /&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="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;a class="wiki_link" href="/Schismatic%20family"&gt;Schismatic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;&lt;a class="wiki_link" href="/Schismatic%20family"&gt;Schismatic family&lt;/a&gt;&lt;/h3&gt;
  The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymas comma (81/80). The 5-limit version of the temperament is a &lt;a class="wiki_link" href="/Microtempering"&gt;microtemperament&lt;/a&gt; which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity (it takes eight schismatic-tempered 3/2's to reach 5/4, as opposed to four meantone-tempered 3/2's).&lt;br /&gt;
  The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the syntonic comma (81/80). The 5-limit version of the temperament is a &lt;a class="wiki_link" href="/Microtempering"&gt;microtemperament&lt;/a&gt; which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity - whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;a class="wiki_link" href="/Kleismic%20family"&gt;Kleismic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;a class="wiki_link" href="/Kleismic%20family"&gt;Kleismic family&lt;/a&gt;&lt;/h3&gt;
  The kleismic family of temperaments tempers out the kleisma of 15625/15552 (the difference between six 6/5's--23328/15625--and 3/2) and has a slightly sharpened minor third as a generator. The kleismic family includes &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, &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; among its possible tunings.&lt;br /&gt;
  The kleismic family of temperaments tempers out the kleisma of 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, &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; among its possible tunings.&lt;br /&gt;
&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="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/Magic%20family"&gt;Magic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/Magic%20family"&gt;Magic family&lt;/a&gt;&lt;/h3&gt;
  The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.&lt;br /&gt;
  The magic family tempers out 3125/3072, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.&lt;br /&gt;
The magic family includes &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;  among its possible tunings, with the latter being near-optimal.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Diaschismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Diaschismic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Diaschismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Diaschismic family&lt;/a&gt;&lt;/h3&gt;
  The diaschismic family tempers out 2048/2025, the &lt;a class="wiki_link" href="/diaschisma"&gt;diaschisma&lt;/a&gt;. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. Using &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; as a tuning is associated with &lt;a class="wiki_link" href="/pajara"&gt;pajara&lt;/a&gt; temperament, where the intervals 50/49 and 64/63 are tempered out.&lt;br /&gt;
  The diaschismic family tempers out 2048/2025, the &lt;a class="wiki_link" href="/diaschisma"&gt;diaschisma&lt;/a&gt;, which is the difference between the Pythagorean major third of 81/64 and the 5-limit diminished fourth of 32/25. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. A noted 7-limit extension to diaschismic is&lt;a class="wiki_link" href="/pajara"&gt; pajara&lt;/a&gt;temperament, where the intervals 50/49 and 64/63 are tempered out, and of which &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is an excellent tuning.&lt;br /&gt;
&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="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Pelogic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Pelogic%20family"&gt;Pelogic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Pelogic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Pelogic%20family"&gt;Pelogic family&lt;/a&gt;&lt;/h3&gt;
  This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma.&lt;br /&gt;
  This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, the accumulated flatness leads you to a minor third + 2 octaves instead of a major third + 2 octaves, and one consequence of this is that it generates 2L5s &amp;quot;anti-diatonic&amp;quot; scales. Mavila and Armodue are the most notable temperaments associated with the pelogic comma. Tunings include &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine family&lt;/a&gt;&lt;/h3&gt;
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