Tour of regular temperaments: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<h4>Original Wikitext content:</h4>

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=[[edo|Equal temperaments]]=

[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.

[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.

=Rank 2 (including "linear") temperaments[[#lineartemperaments]]=

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Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].

P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the ~~Syntonic~~ comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.

P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.

==Families==

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===[[Schismatic family]]===

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.

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. Tunings include [[xenharmonic/12edo|12edo]], [[xenharmonic/29edo|29edo]], [[xenharmonic/41edo|41edo]], [[xenharmonic/53edo|53edo]], and [[xenharmonic/118edo|118edo]].

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

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===[[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.

The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.

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

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===[[Würschmidt family]]===

The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8>. Würschmidt 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. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.

The würschmidt (or wuerschmidt) family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8>. Würschmidt 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. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.

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

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. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L6s).

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. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "[[@http://www.tcherepnin.com/alex/basic_elem1.htm#9step|Tcherepnin's scale]]" (3L6s).

===[[Dimipent family]]===

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===[[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. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.

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]]===

This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.

This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include [[xenharmonic/8edo|8edo]], [[xenharmonic/19edo|19edo]], [[xenharmonic/46edo|46edo]], and [[xenharmonic/65edo|65edo]].

===[[Semicomma family|Orwell and the semicomma family]]===

The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7> is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.

The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7> is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to [[orwell]] temperament.

===[[Pythagorean family]]===

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===[[Fifive family]]===

This tempers out the fifive ~~comme~~, 9765625/9565938.

This tempers out the fifive comma, 9765625/9565938.

===[[Maja family]]===

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===[[Starling temperaments]]===

Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">~~The~~ difference between three 6/5s, one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

===[[Marvel temperaments]]===

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===[[Turkish maqam music temperaments]]===

Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments.

~~Vsrious~~ theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|maqam music]] in a systematic way. This includes, in effect, certain linear temperaments.

===[[Very low accuracy temperaments]]===

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===[[Breed family]]===

Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, ~~2749et~~ will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.

Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.

===[[Ragisma family]]===

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===[[Porcupine rank three family]]===

These are the rank three temperaments tempering out the porcupine comma or ~~aximal~~ diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

===[[Archytas family]]===

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==[[Orgonia]]==

~~By //~~Orgonia// is ~~meant~~ the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3>, the orgonisma.

Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3>, the orgonisma.

==[[The Biosphere]]==

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<h1 id="toc3"><a name="Equal temperaments"></a><a class="wiki_link" href="/edo">Equal temperaments</a></h1>

<br />

<a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.<br />

<a class="wiki_link" href="/Equal%20Temperaments">Equal temperaments</a> (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &quot;Rank 1&quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &quot;equal division&quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &quot;fun&quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.<br />

<br />

<h1 id="toc4"><a name="Rank 2 (including &quot;linear&quot;) temperaments"></a>Rank 2 (including &quot;linear&quot;) temperaments<a name="lineartemperaments"></a></h1>

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Regular temperaments of ranks two and three are cataloged <a class="wiki_link" href="/Optimal%20patent%20val">here</a>. Other pages listing them are <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>, and a <a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments">page</a> listing higher limit rank two temperaments. There is also <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>'s <a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow">giant list of regular temperaments</a>.<br />

<br />

P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the ~~Syntonic~~ comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.<br />

P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.<br />

<br />

<h2 id="toc5"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families"></a>Families</h2>

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<br />

<h3 id="toc7"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-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 syntonic comma (81/80). 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 - 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.<br />

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 <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 - 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. Tunings include <a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo">12edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/29edo">29edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/53edo">53edo</a>, and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/118edo">118edo</a>.<br />

<br />

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

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<h3 id="toc9"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-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, which is the difference between five 5/4's (3125/2048) and a 3/2. The magic generator is itself an approximate 5/4.~~<br />~~

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 <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/25edo">25edo</a>, and <a class="wiki_link" href="/41edo">41edo</a> among its possible tunings, with the latter being near-optimal.<br />

The magic family includes <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/25edo">25edo</a>, and <a class="wiki_link" href="/41edo">41edo</a> among its possible tunings, with the latter being near-optimal.<br />

<br />

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

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<h3 id="toc13"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Würschmidt family"></a><a class="wiki_link" href="/W%C3%BCrschmidt%20family">Würschmidt family</a></h3>

The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&gt;. Würschmidt 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. It tends to generate the same MOS's as <a class="wiki_link" href="/magic%20family">magic temperament</a>, but is tuned slightly more accurately. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as würschmidt tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br />

The würschmidt (or wuerschmidt) family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&gt;. Würschmidt 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. It tends to generate the same MOS's as <a class="wiki_link" href="/magic%20family">magic temperament</a>, but is tuned slightly more accurately. Both <a class="wiki_link" href="/31edo">31edo</a> and <a class="wiki_link" href="/34edo">34edo</a> can be used as würschmidt tunings, as can <a class="wiki_link" href="/65edo">65edo</a>, which is quite accurate.<br />

<br />

<h3 id="toc14"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Augmented family"></a><a class="wiki_link" href="/Augmented%20family">Augmented family</a></h3>

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 <a class="wiki_link" href="/12edo">12edo</a>, which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the &quot;augmented scale&quot; (3L3s) in common 12-based music theory, as well as what is commonly called &quot;Tcherepnin's scale&quot; (3L6s).<br />

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 <a class="wiki_link" href="/12edo">12edo</a>, which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the &quot;augmented scale&quot; (3L3s) in common 12-based music theory, as well as what is commonly called &quot;<a class="wiki_link_ext" href="http://www.tcherepnin.com/alex/basic_elem1.htm#9step" rel="nofollow" target="_blank">Tcherepnin's scale</a>&quot; (3L6s).<br />

<br />

<h3 id="toc15"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Dimipent family"></a><a class="wiki_link" href="/Dimipent%20family">Dimipent family</a></h3>

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<h3 id="toc17"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-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. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.<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. <a class="wiki_link" href="/7edo">7edo</a> can also be considered a tetracot tuning, as can <a class="wiki_link" href="/20edo">20edo</a>, <a class="wiki_link" href="/27edo">27edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, and <a class="wiki_link" href="/41edo">41edo</a>.<br />

<br />

<h3 id="toc18"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Sensipent family"></a><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3>

This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.<br />

This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include <a class="wiki_link" href="http://xenharmonic.wikispaces.com/8edo">8edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo">19edo</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/46edo">46edo</a>, and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo">65edo</a>.<br />

<br />

<h3 id="toc19"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Orwell and the semicomma family"></a><a class="wiki_link" href="/Semicomma%20family">Orwell and the semicomma family</a></h3>

The semicomma (also known as <strong>Fokker's comma)</strong> 2109375/2097152 = |-21 3 7&gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.<br />

The semicomma (also known as <strong>Fokker's comma)</strong> 2109375/2097152 = |-21 3 7&gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to <a class="wiki_link" href="/orwell">orwell</a> temperament.<br />

<br />

<h3 id="toc20"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Pythagorean family"></a><a class="wiki_link" href="/Pythagorean%20family">Pythagorean family</a></h3>

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<br />

<h3 id="toc38"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Fifive family"></a><a class="wiki_link" href="/Fifive%20family">Fifive family</a></h3>

This tempers out the fifive ~~comme~~, 9765625/9565938.<br />

This tempers out the fifive comma, 9765625/9565938.<br />

<br />

<h3 id="toc39"><a name="Rank 2 (including &quot;linear&quot;) temperaments-Families-Maja family"></a><a class="wiki_link" href="/Maja%20family">Maja family</a></h3>

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<br />

<h3 id="toc57"><a name="Temperaments for a given comma--Starling temperaments"></a><a class="wiki_link" href="/Starling%20temperaments">Starling temperaments</a></h3>

Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">~~The~~ difference between three 6/5s, one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br />

Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave) </span>is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br />

<br />

<h3 id="toc58"><a name="Temperaments for a given comma--Marvel temperaments"></a><a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a></h3>

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<br />

<h3 id="toc70"><a name="Temperaments for a given comma--Turkish maqam music temperaments"></a><a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments">Turkish maqam music temperaments</a></h3>

~~<br />~~

Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">makam (maqam) music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br />

~~Vsrious~~ theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish <a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian">maqam music</a> in a systematic way. This includes, in effect, certain linear temperaments.<br />

<br />

<h3 id="toc71"><a name="Temperaments for a given comma--Very low accuracy temperaments"></a><a class="wiki_link" href="/Very%20low%20accuracy%20temperaments">Very low accuracy temperaments</a></h3>

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<br />

<h3 id="toc79"><a name="Rank 3 temperaments--Breed family"></a><a class="wiki_link" href="/Breed%20family">Breed family</a></h3>

Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, ~~2749et~~ will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.<br />

Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.<br />

<br />

<h3 id="toc80"><a name="Rank 3 temperaments--Ragisma family"></a><a class="wiki_link" href="/Ragisma%20family">Ragisma family</a></h3>

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<br />

<h3 id="toc99"><a name="Rank 3 temperaments--Porcupine rank three family"></a><a class="wiki_link" href="/Porcupine%20rank%20three%20family">Porcupine rank three family</a></h3>

These are the rank three temperaments tempering out the porcupine comma or ~~aximal~~ diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br />

These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.<br />

<br />

<h3 id="toc100"><a name="Rank 3 temperaments--Archytas family"></a><a class="wiki_link" href="/Archytas%20family">Archytas family</a></h3>

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<br />

<h2 id="toc112"><a name="Commatic realms-Orgonia"></a><a class="wiki_link" href="/Orgonia">Orgonia</a></h2>

~~By <em>~~Orgonia~~</em>~~ is ~~meant~~ the commatic realm of the <a class="wiki_link" href="/11-limit">11-limit</a> comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.<br />

Orgonia is the commatic realm of the <a class="wiki_link" href="/11-limit">11-limit</a> comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.<br />

<br />

<h2 id="toc113"><a name="Commatic realms-The Biosphere"></a><a class="wiki_link" href="/The%20Biosphere">The Biosphere</a></h2>

@@ 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:hstraub|hstraub]] and made on <tt>2013-08-16 05:34:36 UTC</tt>.<br>
+: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2013-12-03 20:58:57 UTC</tt>.<br>
-: The original revision id was <tt>445080536</tt>.<br>
+: The original revision id was <tt>474462244</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>added a few links and corrected some typos</tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
@@ Line 29: / Line 29: @@
 =[[edo|Equal temperaments]]=
-[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.
+[[Equal Temperaments|Equal temperaments]] (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a "Rank 1") temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an "equal division" of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve "fun" results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.
 =Rank 2 (including "linear") temperaments[[#lineartemperaments]]=
@@ Line 35: / Line 35: @@
 Regular temperaments of ranks two and three are cataloged [[Optimal patent val|here]]. Other pages listing them are [[Paul Erlich]]'s [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]], and a [[Proposed names for rank 2 temperaments|page]] listing higher limit rank two temperaments. There is also [[Graham Breed]]'s [[http://x31eq.com/catalog2.html|giant list of regular temperaments]].
-P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the Syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.
+P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.
 ==Families==
@@ Line 47: / Line 47: @@
 ===[[Schismatic family]]===
-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.
+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. Tunings include [[xenharmonic/12edo|12edo]], [[xenharmonic/29edo|29edo]], [[xenharmonic/41edo|41edo]], [[xenharmonic/53edo|53edo]], and [[xenharmonic/118edo|118edo]].
 ===[[Kleismic family]]===
@@ Line 53: / Line 53: @@
 ===[[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.
-The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.
 ===[[Diaschismic family]]===
@@ Line 66: / Line 65: @@
 ===[[Würschmidt family]]===
-The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&gt;. Würschmidt 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. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.
+The würschmidt (or wuerschmidt) family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&gt;. Würschmidt 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. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.
 ===[[Augmented family]]===
-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. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L6s).
+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. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "[[@http://www.tcherepnin.com/alex/basic_elem1.htm#9step|Tcherepnin's scale]]" (3L6s).
 ===[[Dimipent family]]===
@@ Line 78: / Line 77: @@
 ===[[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. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.
+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]]===
-This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.
+This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include [[xenharmonic/8edo|8edo]], [[xenharmonic/19edo|19edo]], [[xenharmonic/46edo|46edo]], and [[xenharmonic/65edo|65edo]].
 ===[[Semicomma family|Orwell and the semicomma family]]===
-The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7&gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.
+The semicomma (also known as **Fokker's comma)** 2109375/2097152 = |-21 3 7&gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to [[orwell]] temperament.
 ===[[Pythagorean family]]===
@@ Line 141: / Line 140: @@
 ===[[Fifive family]]===
-This tempers out the fifive comme, 9765625/9565938.
+This tempers out the fifive comma, 9765625/9565938.
 ===[[Maja family]]===
@@ Line 200: / Line 199: @@
 ===[[Starling temperaments]]===
-Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;The difference between three 6/5s, one 7/6, and an octave) &lt;/span&gt;is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.
+Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;the difference between three 6/5s plus one 7/6, and an octave) &lt;/span&gt;is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.
 ===[[Marvel temperaments]]===
@@ Line 239: / Line 238: @@
 ===[[Turkish maqam music temperaments]]===
+Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|makam (maqam) music]] in a systematic way. This includes, in effect, certain linear temperaments.
-Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish [[Arabic, Turkish, Persian|maqam music]] in a systematic way. This includes, in effect, certain linear temperaments.
 ===[[Very low accuracy temperaments]]===
@@ Line 267: / Line 265: @@
 ===[[Breed family]]===
-Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.
+Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.
 ===[[Ragisma family]]===
@@ Line 327: / Line 325: @@
 ===[[Porcupine rank three family]]===
-These are the rank three temperaments tempering out the porcupine comma or aximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
+These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
 ===[[Archytas family]]===
@@ Line 369: / Line 367: @@
 ==[[Orgonia]]==
-By //Orgonia// is meant the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.
+Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = |16 0 0 -2 -3&gt;, the orgonisma.
 ==[[The Biosphere]]==
@@ Line 406: / Line 404: @@
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;a class="wiki_link" href="/edo"&gt;Equal temperaments&lt;/a&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;Equal temperaments&lt;/a&gt; (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &amp;quot;Rank 1&amp;quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &amp;quot;equal division&amp;quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &amp;quot;fun&amp;quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Equal%20Temperaments"&gt;Equal temperaments&lt;/a&gt; (abbreviated ET or tET) and equal divisions of the octave (abbreviated EDO or ED2) are similar concepts, although there are distinctions in the way these terms are used. A p-limit ET is simply a p-limit temperament that uses a single generator (making it a &amp;quot;Rank 1&amp;quot;) temperament, which thus maps the set of n-limit JI intervals using one-dimensional coordinates. An ET thus does not have to be thought of as an &amp;quot;equal division&amp;quot; of any interval (let alone the octave), and in fact many ETs do not divide the pure octave at all. On the other hand, an n-EDO is a division of the octave into n equal parts, with no consideration given to mapping of JI intervals. An EDO can be treated as an ET by applying a temperament mapping to the intervals of the EDO, typically by using a val for a temperament supported by that EDO (although one can also use unsupported vals or poorly-supported vals to achieve &amp;quot;fun&amp;quot; results). The familiar 12-note equal temperament (12-ET) reduces the size of the perfect fifth (about 701.955 cents) by 1/12 of the Pythagorean comma, resulting in a fifth of 700.0 cents, although there are other temperaments supported by 12-tET.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Rank 2 (including &amp;quot;linear&amp;quot;) temperaments&lt;!-- ws:start:WikiTextAnchorRule:350:&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@@lineartemperaments&amp;quot; title=&amp;quot;Anchor: lineartemperaments&amp;quot;/&amp;gt; --&gt;&lt;a name="lineartemperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:350 --&gt;&lt;/h1&gt;
@@ Line 412: / Line 410: @@
 Regular temperaments of ranks two and three are cataloged &lt;a class="wiki_link" href="/Optimal%20patent%20val"&gt;here&lt;/a&gt;. Other pages listing them are &lt;a class="wiki_link" href="/Paul%20Erlich"&gt;Paul Erlich&lt;/a&gt;'s &lt;a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments"&gt;catalog of 5-limit rank two temperaments&lt;/a&gt;, and a &lt;a class="wiki_link" href="/Proposed%20names%20for%20rank%202%20temperaments"&gt;page&lt;/a&gt; listing higher limit rank two temperaments. There is also &lt;a class="wiki_link" href="/Graham%20Breed"&gt;Graham Breed&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://x31eq.com/catalog2.html" rel="nofollow"&gt;giant list of regular temperaments&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the Syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.&lt;br /&gt;
+P-limit Rank 2 temperaments map all intervals of p-limit JI using a set of 2-dimensional coordinates, thus rank-2 temperaments are said to have two generators (though they may have any number of step-sizes). This means that a rank-2 temperament is defined by a set of 2 vals, one val for each generator. Rank-2 temperaments can be reduced to a related rank-1 temperaments by tempering out an additional comma that is not already tempered out. For example, 5-limit meantone temperament, which is rank-2 (defined by tempering the syntonic comma of 81/80 out of 3-dimensional 5-limit JI), can be reduced to 12-TET by tempering out the Pythagorean comma.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Families&lt;/h2&gt;
@@ Line 424: / Line 422: @@
 &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-Families-Schismatic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&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 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;
+  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. Tunings include &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo"&gt;41edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/53edo"&gt;53edo&lt;/a&gt;, and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/118edo"&gt;118edo&lt;/a&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-Families-Kleismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;a class="wiki_link" href="/Kleismic%20family"&gt;Kleismic family&lt;/a&gt;&lt;/h3&gt;
@@ Line 430: / Line 428: @@
 &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-Families-Magic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&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. 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;
-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;!-- 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-Families-Diaschismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Diaschismic family&lt;/a&gt;&lt;/h3&gt;
@@ Line 443: / Line 440: @@
 &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-Families-Würschmidt family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;a class="wiki_link" href="/W%C3%BCrschmidt%20family"&gt;Würschmidt family&lt;/a&gt;&lt;/h3&gt;
-  The wuerschmidt family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Würschmidt 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. It tends to generate the same MOS's as &lt;a class="wiki_link" href="/magic%20family"&gt;magic temperament&lt;/a&gt;, but is tuned slightly more accurately. 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 würschmidt tunings, as can &lt;a class="wiki_link" href="/65edo"&gt;65edo&lt;/a&gt;, which is quite accurate.&lt;br /&gt;
+  The würschmidt (or wuerschmidt) family tempers out Würschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Würschmidt 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. It tends to generate the same MOS's as &lt;a class="wiki_link" href="/magic%20family"&gt;magic temperament&lt;/a&gt;, but is tuned slightly more accurately. 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 würschmidt 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:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Augmented family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&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 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. It is the temperament that results in what is commonly called the &amp;quot;augmented scale&amp;quot; (3L3s) in common 12-based music theory, as well as what is commonly called &amp;quot;Tcherepnin's scale&amp;quot; (3L6s).&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. It is the temperament that results in what is commonly called the &amp;quot;augmented scale&amp;quot; (3L3s) in common 12-based music theory, as well as what is commonly called &amp;quot;&lt;a class="wiki_link_ext" href="http://www.tcherepnin.com/alex/basic_elem1.htm#9step" rel="nofollow" target="_blank"&gt;Tcherepnin's scale&lt;/a&gt;&amp;quot; (3L6s).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Dimipent family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;a class="wiki_link" href="/Dimipent%20family"&gt;Dimipent family&lt;/a&gt;&lt;/h3&gt;
@@ Line 455: / Line 452: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Tetracot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&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. 7edo can also be considered a tetracot tuning, as can 20edo, 27edo, 34edo, and 41edo.&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. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; can also be considered a tetracot tuning, as can &lt;a class="wiki_link" href="/20edo"&gt;20edo&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Sensipent family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;a class="wiki_link" href="/Sensipent%20family"&gt;Sensipent family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma.&lt;br /&gt;
+  This tempers out the sensipent comma, 78732/78125, also known as the medium semicomma. Tunings include &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8edo"&gt;8edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/46edo"&gt;46edo&lt;/a&gt;, and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/65edo"&gt;65edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Orwell and the semicomma family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;a class="wiki_link" href="/Semicomma%20family"&gt;Orwell and the semicomma family&lt;/a&gt;&lt;/h3&gt;
-  The semicomma (also known as &lt;strong&gt;Fokker's comma)&lt;/strong&gt; 2109375/2097152 = |-21 3 7&amp;gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to orwell temperament.&lt;br /&gt;
+  The semicomma (also known as &lt;strong&gt;Fokker's comma)&lt;/strong&gt; 2109375/2097152 = |-21 3 7&amp;gt; is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as 7/6, leading to &lt;a class="wiki_link" href="/orwell"&gt;orwell&lt;/a&gt; temperament.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc20"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Pythagorean family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;a class="wiki_link" href="/Pythagorean%20family"&gt;Pythagorean family&lt;/a&gt;&lt;/h3&gt;
@@ Line 518: / Line 515: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc38"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Fifive family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;&lt;a class="wiki_link" href="/Fifive%20family"&gt;Fifive family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the fifive comme, 9765625/9565938.&lt;br /&gt;
+  This tempers out the fifive comma, 9765625/9565938.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc39"&gt;&lt;a name="Rank 2 (including &amp;quot;linear&amp;quot;) temperaments-Families-Maja family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;&lt;a class="wiki_link" href="/Maja%20family"&gt;Maja family&lt;/a&gt;&lt;/h3&gt;
@@ Line 577: / Line 574: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:114:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc57"&gt;&lt;a name="Temperaments for a given comma--Starling temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:114 --&gt;&lt;a class="wiki_link" href="/Starling%20temperaments"&gt;Starling temperaments&lt;/a&gt;&lt;/h3&gt;
-  Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;The difference between three 6/5s, one 7/6, and an octave) &lt;/span&gt;is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.&lt;br /&gt;
+  Not a family or clan, but related by the fact that 126/125, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;the difference between three 6/5s plus one 7/6, and an octave) &lt;/span&gt;is tempered out, are myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:116:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc58"&gt;&lt;a name="Temperaments for a given comma--Marvel temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:116 --&gt;&lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;Marvel temperaments&lt;/a&gt;&lt;/h3&gt;
@@ Line 616: / Line 613: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:140:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc70"&gt;&lt;a name="Temperaments for a given comma--Turkish maqam music temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:140 --&gt;&lt;a class="wiki_link" href="/Turkish%20maqam%20music%20temperaments"&gt;Turkish maqam music temperaments&lt;/a&gt;&lt;/h3&gt;
-  &lt;br /&gt;
+  Various theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish &lt;a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian"&gt;makam (maqam) music&lt;/a&gt; in a systematic way. This includes, in effect, certain linear temperaments.&lt;br /&gt;
-Vsrious theoretical solutions have been put forward for the vexing problem of how to indicate and define the tuning of Turkish &lt;a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian"&gt;maqam music&lt;/a&gt; in a systematic way. This includes, in effect, certain linear temperaments.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:142:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc71"&gt;&lt;a name="Temperaments for a given comma--Very low accuracy temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:142 --&gt;&lt;a class="wiki_link" href="/Very%20low%20accuracy%20temperaments"&gt;Very low accuracy temperaments&lt;/a&gt;&lt;/h3&gt;
@@ Line 644: / Line 640: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:158:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc79"&gt;&lt;a name="Rank 3 temperaments--Breed family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:158 --&gt;&lt;a class="wiki_link" href="/Breed%20family"&gt;Breed family&lt;/a&gt;&lt;/h3&gt;
-  Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749et will certainly do the trick. Breed has generators of 2, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.&lt;br /&gt;
+  Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749edo will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and your choice of 8/7 or 10/7.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:160:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc80"&gt;&lt;a name="Rank 3 temperaments--Ragisma family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:160 --&gt;&lt;a class="wiki_link" href="/Ragisma%20family"&gt;Ragisma family&lt;/a&gt;&lt;/h3&gt;
@@ Line 704: / Line 700: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:198:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc99"&gt;&lt;a name="Rank 3 temperaments--Porcupine rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:198 --&gt;&lt;a class="wiki_link" href="/Porcupine%20rank%20three%20family"&gt;Porcupine rank three family&lt;/a&gt;&lt;/h3&gt;
-  These are the rank three temperaments tempering out the porcupine comma or aximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.&lt;br /&gt;
+  These are the rank three temperaments tempering out the porcupine comma or maximal diesis, 250/243.If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:200:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc100"&gt;&lt;a name="Rank 3 temperaments--Archytas family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:200 --&gt;&lt;a class="wiki_link" href="/Archytas%20family"&gt;Archytas family&lt;/a&gt;&lt;/h3&gt;
@@ Line 746: / Line 742: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:224:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc112"&gt;&lt;a name="Commatic realms-Orgonia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:224 --&gt;&lt;a class="wiki_link" href="/Orgonia"&gt;Orgonia&lt;/a&gt;&lt;/h2&gt;
-  By &lt;em&gt;Orgonia&lt;/em&gt; is meant the commatic realm of the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; comma 65536/65219 = |16 0 0 -2 -3&amp;gt;, the orgonisma.&lt;br /&gt;
+  Orgonia is the commatic realm of the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; comma 65536/65219 = |16 0 0 -2 -3&amp;gt;, the orgonisma.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:226:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc113"&gt;&lt;a name="Commatic realms-The Biosphere"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:226 --&gt;&lt;a class="wiki_link" href="/The%20Biosphere"&gt;The Biosphere&lt;/a&gt;&lt;/h2&gt;