Xenharmonic Wiki

Tour of regular temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 146166319 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 146427673 - Original comment: **
Line 1: Line 1:
<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>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-01 04:38:20 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-02 04:36:12 UTC</tt>.<br>
: The original revision id was <tt>146166319</tt>.<br>
: The original revision id was <tt>146427673</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
Line 16: Line 16:
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 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 family]]==
===[[Meantone family]]===
The meantone family tempers out 81/80 and has a flattened fifth (or sharpened fourth) as generator. Four of these flattened fifths give the 5/1 interval. 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 Didymas comma (the 81/80 interval.)
The meantone family tempers out 81/80 and has a flattened fifth (or sharpened fourth) as generator. Four of these flattened fifths give the 5/1 interval. 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 Didymas comma (the 81/80 interval.)
===[[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 and a just diminished fourth. 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.


===[[Kleismic family]]===  
===[[Kleismic family]]===  
Line 24: Line 27:
===[[Magic family]]===  
===[[Magic family]]===  
The magic family tempers out 3125/3072, known as the magic comma or small diesis, has a generator which is a flattened major third.  
The magic family tempers out 3125/3072, known as the magic comma or small diesis, has a generator which is a flattened major third.  
===[[Diaschismic family]]===
The diaschismic family tempers out 2048/2025, the diaschisma. 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.
===[[Porcupine family]]===
The porcupine family tempers out 250/243, known as the maximal diesis or porcupine comma. It has a generator of a minor whole tone (10/9), three of which make up a fourth.


===Miracle[[#miracle]]===  
===Miracle[[#miracle]]===  
Line 31: Line 40:


So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.
So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.
===[[Diaschismic family]]===
The diaschismic family tempers out 2048/2025, the diaschisma. 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.
===[[Porcupine family]]===
The porcupine family tempers out 250/243, known as the maximal diesis or porcupine comma. It has a generator of a minor whole tone (10/9), three of which make up a fourth.
===[[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 and a just diminished fourth. 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.


==Rank 3 temperaments==  
==Rank 3 temperaments==  
Line 69: Line 69:
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 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;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Meantone family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;a class="wiki_link" href="/Meantone%20family"&gt;Meantone family&lt;/a&gt;&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Meantone family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;a class="wiki_link" href="/Meantone%20family"&gt;Meantone family&lt;/a&gt;&lt;/h3&gt;
The meantone family tempers out 81/80 and has a flattened fifth (or sharpened fourth) as generator. Four of these flattened fifths give the 5/1 interval. 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 Didymas comma (the 81/80 interval.)&lt;br /&gt;
The meantone family tempers out 81/80 and has a flattened fifth (or sharpened fourth) as generator. Four of these flattened fifths give the 5/1 interval. 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 Didymas comma (the 81/80 interval.)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Meantone family-Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;a class="wiki_link" href="/Kleismic%20family"&gt;Kleismic family&lt;/a&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&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 and a just diminished fourth. 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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&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 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 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;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-Meantone family-Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;a class="wiki_link" href="/Magic%20family"&gt;Magic 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="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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, 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, has a generator which is a flattened major third. &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="x-Meantone family-Miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Miracle&lt;!-- ws:start:WikiTextAnchorRule:31:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@miracle&amp;quot; title=&amp;quot;Anchor: miracle&amp;quot;/&amp;gt; --&gt;&lt;a name="miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:31 --&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Diaschismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&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. 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;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine family&lt;/a&gt;&lt;/h3&gt;
The porcupine family tempers out 250/243, known as the maximal diesis or porcupine comma. It has a generator of a minor whole tone (10/9), three of which make up a fourth.&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="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Miracle&lt;!-- ws:start:WikiTextAnchorRule:31:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@miracle&amp;quot; title=&amp;quot;Anchor: miracle&amp;quot;/&amp;gt; --&gt;&lt;a name="miracle"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:31 --&gt;&lt;/h3&gt;
  Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.&lt;br /&gt;
  Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.&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="x-Meantone family-Orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Orwell&lt;!-- ws:start:WikiTextAnchorRule:32:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@orwell&amp;quot; title=&amp;quot;Anchor: orwell&amp;quot;/&amp;gt; --&gt;&lt;a name="orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:32 --&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="x-Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Orwell&lt;!-- ws:start:WikiTextAnchorRule:32:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@orwell&amp;quot; title=&amp;quot;Anchor: orwell&amp;quot;/&amp;gt; --&gt;&lt;a name="orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:32 --&gt;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.&lt;br /&gt;
So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.&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="x-Meantone family-Diaschismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&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. 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;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc8"&gt;&lt;a name="x-Meantone family-Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine family&lt;/a&gt;&lt;/h3&gt;
The porcupine family tempers out 250/243, known as the maximal diesis or porcupine comma. It has a generator of a minor whole tone (10/9), three of which make up a fourth.&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="x-Meantone family-Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&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 and a just diminished fourth. 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.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x-Rank 3 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Rank 3 temperaments&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x-Rank 3 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Rank 3 temperaments&lt;/h2&gt;
Retrieved from "https://en.xen.wiki/w/Tour_of_regular_temperaments"