Tour of regular temperaments: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>2011-07-05 21:01:45 UTC</tt>.<br>

: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-05 23:54:43 UTC</tt>.<br>

: The original revision id was <tt>~~240127623~~</tt>.<br>

: The original revision id was <tt>240147837</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>

Line 29:

===[[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, 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.)

===[[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.

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).

===[[Kleismic family]]===

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 [[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 (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.

===[[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, 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.

===[[Diaschismic family]]===

Line 44:

===[[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.

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, 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.

===[[Wuerschmidt family]]===

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8>. Wuerschmidt itself has a generator of a major third, eight of which give a 6; that is, (5/4)^8 * (393216/390625) = 6. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8>. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.

===[[Augmented family]]===

The augmented family tempers out the diesis of 128/125, 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 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.

===[[Dicot family]]===

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24, and hence identify major and minor thirds. [[7edo]] makes for a good dicot tuning.

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2), and hence identify major and minor thirds. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2.

===[[Tetracot family]]===

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma.

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.

===[[Sensipent family]]===

Line 291:

<br />

<h3 id="toc3"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Meantone family"></a><a class="wiki_link" href="/Meantone%20family">Meantone family</a></h3>

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 <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/43edo">43edo</a>, <a class="wiki_link" href="/50edo">50edo</a>, <a class="wiki_link" href="/55edo">55edo</a> and <a class="wiki_link" href="/81edo">81edo</a>. 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.)<br />

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 <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/43edo">43edo</a>, <a class="wiki_link" href="/50edo">50edo</a>, <a class="wiki_link" href="/55edo">55edo</a> and <a class="wiki_link" href="/81edo">81edo</a>. 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.)<br />

<br />

<h3 id="toc4"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Schismatic family"></a><a class="wiki_link" href="/Schismatic%20family">Schismatic family</a></h3>

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 <a class="wiki_link" href="/Microtempering">microtemperament</a> 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.<br />

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 <a class="wiki_link" href="/Microtempering">microtemperament</a> 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).<br />

<br />

<h3 id="toc5"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Kleismic family"></a><a class="wiki_link" href="/Kleismic%20family">Kleismic family</a></h3>

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 <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/49edo">49edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a> among its possible tunings.<br />

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 <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/49edo">49edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a> among its possible tunings.<br />

<br />

<h3 id="toc6"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Magic family"></a><a class="wiki_link" href="/Magic%20family">Magic family</a></h3>

The magic family tempers out 3125/3072, known as the magic comma or small diesis, has a generator which is a flattened major third.<br />

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.<br />

<br />

<h3 id="toc7"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Diaschismic family"></a><a class="wiki_link" href="/Diaschismic%20family">Diaschismic family</a></h3>

Line 306:

<br />

<h3 id="toc8"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Porcupine family"></a><a class="wiki_link" href="/Porcupine%20family">Porcupine family</a></h3>

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.<br />

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, 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.<br />

<br />

<h3 id="toc9"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Wuerschmidt family"></a><a class="wiki_link" href="/Wuerschmidt%20family">Wuerschmidt family</a></h3>

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6; that is, (5/4)^8 * (393216/390625) = 6. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as wuerschmit tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br />

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as wuerschmit tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br />

<br />

<h3 id="toc10"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Augmented family"></a><a class="wiki_link" href="/Augmented%20family">Augmented family</a></h3>

The augmented family tempers out the diesis of 128/125, and so identifies the major third with 1/3 octave. Hence it has the same 400 cent thirds as <a class="wiki_link" href="/12edo">12edo</a>, which is an excellent tuning for augmented.<br />

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 <a class="wiki_link" href="/12edo">12edo</a>, which is an excellent tuning for augmented.<br />

<br />

<h3 id="toc11"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Dicot family"></a><a class="wiki_link" href="/Dicot%20family">Dicot family</a></h3>

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24, and hence identify major and minor thirds. <a class="wiki_link" href="/7edo">7edo</a> makes for a good dicot tuning.<br />

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2), and hence identify major and minor thirds. <a class="wiki_link" href="/7edo">7edo</a> makes for a &quot;good&quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &quot;neutral&quot; dicot 3rds span a 3/2.<br />

<br />

<h3 id="toc12"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Tetracot family"></a><a class="wiki_link" href="/Tetracot%20family">Tetracot family</a></h3>

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma.<br />

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.<br />

<br />

<h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments--Sensipent family"></a><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-07-05 21:01:45 UTC</tt>.<br>
+: This revision was by author [[User:igliashon|igliashon]] and made on <tt>2011-07-05 23:54:43 UTC</tt>.<br>
-: The original revision id was <tt>240127623</tt>.<br>
+: The original revision id was <tt>240147837</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>
@@ Line 29: / Line 29: @@
 ===[[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, 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.)
 ===[[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.
+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).
 ===[[Kleismic family]]===
-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 [[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 (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.
 ===[[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, 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.
 ===[[Diaschismic family]]===
@@ Line 44: / Line 44: @@
 ===[[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.
+The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, 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.
 ===[[Wuerschmidt family]]===
-The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6; that is, (5/4)^8 * (393216/390625) = 6. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.
+The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.
 ===[[Augmented family]]===
-The augmented family tempers out the diesis of 128/125, 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 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.
 ===[[Dicot family]]===
-The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24, and hence identify major and minor thirds. [[7edo]] makes for a good dicot tuning.
+The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2), and hence identify major and minor thirds. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2.
 ===[[Tetracot family]]===
-The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma.
+The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.
 ===[[Sensipent family]]===
@@ Line 291: / Line 291: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Meantone family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&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, 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;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&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;
+  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;
 &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--Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&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 (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;
 &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--Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&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, 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;
 &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--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;
@@ Line 306: / Line 306: @@
 &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--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;
+  The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, 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="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments--Wuerschmidt family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;a class="wiki_link" href="/Wuerschmidt%20family"&gt;Wuerschmidt family&lt;/a&gt;&lt;/h3&gt;
-  The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6; that is, (5/4)^8 * (393216/390625) = 6. Both &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; and &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; can be used as wuerschmit tunings, as can &lt;a class="wiki_link" href="/65edo"&gt;65edo&lt;/a&gt;, which is quite accurate.&lt;br /&gt;
+  The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. Both &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; and &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; can be used as wuerschmit tunings, as can &lt;a class="wiki_link" href="/65edo"&gt;65edo&lt;/a&gt;, which is quite accurate.&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--Augmented family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&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, 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 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;
 &lt;br /&gt;
 &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--Dicot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;a class="wiki_link" href="/Dicot%20family"&gt;Dicot family&lt;/a&gt;&lt;/h3&gt;
-  The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24, and hence identify major and minor thirds. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a good dicot tuning.&lt;br /&gt;
+  The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2), and hence identify major and minor thirds. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a &amp;quot;good&amp;quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &amp;quot;neutral&amp;quot; dicot 3rds span a 3/2.&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--Tetracot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;a class="wiki_link" href="/Tetracot%20family"&gt;Tetracot family&lt;/a&gt;&lt;/h3&gt;
-  The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma.&lt;br /&gt;
+  The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by 20000/19683, the minimal diesis or tetracot comma. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.&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--Sensipent family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;a class="wiki_link" href="/Sensipent%20family"&gt;Sensipent family&lt;/a&gt;&lt;/h3&gt;