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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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: This revision was by author [[User:Natebedell|Natebedell]] and made on <tt>2011-09-04 14:55:30 UTC</tt>.<br>
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As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.
As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.


Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, usually chosen to be smaller than the period, is referred to as the "generator".
Meantone is a familiar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as "rank 2" temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the "period", and another interval, usually chosen to be smaller than the period, is referred to as the "generator".


===[[Meantone family]]===  
===[[Meantone family]]===  
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===[[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), or alternatively put, the difference between a just major third (5/4) and a just diminished fourth ((4/3)^8, divided by 8, or 8192/6561). 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 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).


===[[Kleismic family]]===  
===[[Kleismic family]]===  
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===[[Magic family]]===  
===[[Magic family]]===  
The magic family tempers out 3125/3072, which is the difference between five 5/4's (3125/2048) and a 3/2, and is known as the magic comma or small diesis. It has a generator which is a flattened major third.
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.


===[[Diaschismic family]]===  
===[[Diaschismic family]]===  
The diaschismic family tempers out 2048/2025, the diaschisma, typically defined as the diference between three octaves and four perfect fifths plus two major thirds, or the difference between an augmented sixth and a pythagorean minor seventh. It has a period of half an octave and a generator of a fifth, usually a sharpened one. 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. 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.


===[[Porcupine family]]===  
===[[Porcupine family]]===  
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===[[Augmented family]]===  
===[[Augmented family]]===  
The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major 3rds and a 2/1 octave, and so identifies the major third with 1/3 octave. Hence it has the same 400 cent thirds as [[12edo]], which is an excellent tuning for augmented.
The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented.


===[[Dicot family]]===  
===[[Dicot family]]===  
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As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; of the various temperaments, where a &lt;strong&gt;comma&lt;/strong&gt; is a small interval, not a square or cube or other power, which is tempered out by the temperament.&lt;br /&gt;
As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; of the various temperaments, where a &lt;strong&gt;comma&lt;/strong&gt; is a small interval, not a square or cube or other power, which is tempered out by the temperament.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &amp;quot;rank 2&amp;quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &amp;quot;period&amp;quot;, and another interval, usually chosen to be smaller than the period, is referred to as the &amp;quot;generator&amp;quot;.&lt;br /&gt;
Meantone is a familiar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &amp;quot;rank 2&amp;quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &amp;quot;period&amp;quot;, and another interval, usually chosen to be smaller than the period, is referred to as the &amp;quot;generator&amp;quot;.&lt;br /&gt;
&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="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;
&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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&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), or alternatively put, the difference between a just major third (5/4) and a just diminished fourth ((4/3)^8, divided by 8, or 8192/6561). 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 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;
&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;
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&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, which is the difference between five 5/4's (3125/2048) and a 3/2, and is known as the magic comma or small diesis. It has a generator which is a flattened major third.&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;
&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 diaschisma, typically defined as the diference between three octaves and four perfect fifths plus two major thirds, or the difference between an augmented sixth and a pythagorean minor seventh. It has a period of half an octave and a generator of a fifth, usually a sharpened one. 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 pajara temperament, where the intervals 50/49 and 64/63 are tempered out.&lt;br /&gt;
  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 &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;
&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--Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine 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--Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine family&lt;/a&gt;&lt;/h3&gt;
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&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="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Augmented family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;a class="wiki_link" href="/Augmented%20family"&gt;Augmented family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Augmented family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;a class="wiki_link" href="/Augmented%20family"&gt;Augmented family&lt;/a&gt;&lt;/h3&gt;
  The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major 3rds and a 2/1 octave, and so identifies the major third with 1/3 octave. Hence it has the same 400 cent thirds as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, which is an excellent tuning for augmented.&lt;br /&gt;
  The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, which is an excellent tuning for augmented.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Dicot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;a class="wiki_link" href="/Dicot%20family"&gt;Dicot family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Dicot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;a class="wiki_link" href="/Dicot%20family"&gt;Dicot family&lt;/a&gt;&lt;/h3&gt;
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