Tour of regular temperaments

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==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. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. 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.

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

[[Paul Erlich]] has given us a [[Catalog of five-limit rank two temperaments|catalog of 5-limit rank two temperaments]]. As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the [[Normal lists|normal comma list]] of the various temperaments, where a **comma** is a small interval, not a square or cube or other power, which is tempered out by the temperament.

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

==[[Meantone 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.)

===Injera[[#injera]]=== 

Injera has a half-octave period and a small step-sized generator, which is the difference between a half-octave and a perfect fifth. It differs from pajara temperament in having a slightly smaller generator, and a different mapping of the fifth and seventh harmonics.

===[[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.

===[[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. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.

===Miracle[[#miracle]]=== 
Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called "blackjack" and a 31-note scale called "canasta" have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.

===Orwell[[#orwell]]=== 

So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.

===[[Diaschismic family]]=== 
The diaschismic family tempers out 2048/2025, the diaschisma. It has a period of half an octave and a generator of a fifth, usually a sharpened one. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. Using [[22edo]] as a tuning is associated with pajara temperament, where the intervals 50/49 and 64/63 are tempered out.

===[[Porcupine family]]=== 
The porcupine family tempers out 250/243, known as the maximal diesis or porcupine comma. It has a generator of a minor whole tone (10/9), three of which make up a fourth.

===[[Schismatic family]]=== 
The schismatic family tempers out the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymas comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity.

==Rank 3 temperaments== 

Even less familiar than rank 2 temperaments are the [[Planar Temperament|rank 3 temperaments]], based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.

===Marvel[[#marvel]]=== 
Tempers out 225/224. An excellent tuning for marvel is [[240edo]]. It extends in a natural way to unidecimal marvel, which tempers out 385/384, for which the slightly less accurate tuning of [[72edo]] works well. Marvel is generated by 2, 3, and 5, the same generators as the [[Harmonic Limit|5-limit]], which means a scale can be converted to marvel or unidecimal marvel simply by tempering it.

===Starling[[#starling]]=== 
Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely.

==Breed[[#breed]]==
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, 2578et will certainly do the trick. Breed has generators of a 49/40 neutral third, and 10/7.

===Jove, aka Wonder[[#wonder]]=== 
Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been depreciated as a name due to conflict with another temperament also given that name. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72 and 130 are good edos for jove, and if that doesn't suit there's 476edo.
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<html><head><title>Tour of Regular Temperaments</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Equal temperaments</h2>
 <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. An EDO is simply a division of the octave into equal steps (specifically, steps of equal size in cents). An ET, on the other hand, is a temperament, an altered representation of some subset of the intervals of just intonation. 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.<br />
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-Rank 2 (including &quot;linear&quot;) temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank 2 (including &quot;linear&quot;) temperaments<!-- ws:start:WikiTextAnchorRule:32:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@lineartemperaments&quot; title=&quot;Anchor: lineartemperaments&quot;/&gt; --><a name="lineartemperaments"></a><!-- ws:end:WikiTextAnchorRule:32 --></h2>
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<a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> has given us a <a class="wiki_link" href="/Catalog%20of%20five-limit%20rank%20two%20temperaments">catalog of 5-limit rank two temperaments</a>. As we go up from rank two, the various 5-limit temperaments often break up as families of related temperaments, depending on how higher primes are mapped (or equivalently, on which higher limit commas are introduced.) Members of families and their relationships can be classified by the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> of the various temperaments, where a <strong>comma</strong> is a small interval, not a square or cube or other power, which is tempered out by the temperament.<br />
<br />
Meantone is a familar historical temperament based on a chain of fifths (or fourths), but it is only one of many possibilities for temperaments based on a chain of generating intervals. These are referred to as &quot;rank 2&quot; temperaments, since they are based on a set of two linearly independent intervals. One of these intervals (typically an octave or fraction of an octave) can be selected as the &quot;period&quot;, and another interval, usually chosen to be smaller than the period, is referred to as the &quot;generator&quot;.<br />
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Meantone family"></a><!-- ws:end:WikiTextHeadingRule:4 --><a class="wiki_link" href="/Meantone%20family">Meantone family</a></h2>
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 />
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<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="x-Meantone family-Injera"></a><!-- ws:end:WikiTextHeadingRule:6 -->Injera<!-- ws:start:WikiTextAnchorRule:33:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@injera&quot; title=&quot;Anchor: injera&quot;/&gt; --><a name="injera"></a><!-- ws:end:WikiTextAnchorRule:33 --></h3>
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Injera has a half-octave period and a small step-sized generator, which is the difference between a half-octave and a perfect fifth. It differs from pajara temperament in having a slightly smaller generator, and a different mapping of the fifth and seventh harmonics.<br />
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Meantone family-Kleismic family"></a><!-- ws:end:WikiTextHeadingRule:8 --><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 />
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Meantone family-Magic family"></a><!-- ws:end:WikiTextHeadingRule:10 --><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. It's optimal, in a sense, for 9-limit harmony with a tuning close to 41-EDO. It's more accurate than meantone and simpler than schismatic. It works with 19 and 22 note scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. It's a little tricky to work with because 3/2 fifths are a relatively complex interval and it doesn't naturally work with scales of around 7 notes to the octave.<br />
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Meantone family-Miracle"></a><!-- ws:end:WikiTextHeadingRule:12 -->Miracle<!-- ws:start:WikiTextAnchorRule:34:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@miracle&quot; title=&quot;Anchor: miracle&quot;/&gt; --><a name="miracle"></a><!-- ws:end:WikiTextAnchorRule:34 --></h3>
 Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &quot;blackjack&quot; and a 31-note scale called &quot;canasta&quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.<br />
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Meantone family-Orwell"></a><!-- ws:end:WikiTextHeadingRule:14 -->Orwell<!-- ws:start:WikiTextAnchorRule:35:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@orwell&quot; title=&quot;Anchor: orwell&quot;/&gt; --><a name="orwell"></a><!-- ws:end:WikiTextAnchorRule:35 --></h3>
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So called because 19/84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with 22, 31 and 53-EDO. It's reasonable in the 7-limit and naturally extends into the 11-limit.<br />
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Meantone family-Diaschismic family"></a><!-- ws:end:WikiTextHeadingRule:16 --><a class="wiki_link" href="/Diaschismic%20family">Diaschismic family</a></h3>
 The diaschismic family tempers out 2048/2025, the diaschisma. It has a period of half an octave and a generator of a fifth, usually a sharpened one. Diaschismic tunings include <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/56edo">56edo</a>, <a class="wiki_link" href="/58edo">58edo</a> and <a class="wiki_link" href="/80edo">80edo</a>. Using <a class="wiki_link" href="/22edo">22edo</a> as a tuning is associated with pajara temperament, where the intervals 50/49 and 64/63 are tempered out.<br />
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Meantone family-Porcupine family"></a><!-- ws:end:WikiTextHeadingRule:18 --><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 />
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<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="x-Meantone family-Schismatic family"></a><!-- ws:end:WikiTextHeadingRule:20 --><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="/microtemperament">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 />
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x-Rank 3 temperaments"></a><!-- ws:end:WikiTextHeadingRule:22 -->Rank 3 temperaments</h2>
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Even less familiar than rank 2 temperaments are the <a class="wiki_link" href="/Planar%20Temperament">rank 3 temperaments</a>, based on a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank 3 temperaments by the commas they temper out.<br />
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Rank 3 temperaments-Marvel"></a><!-- ws:end:WikiTextHeadingRule:24 -->Marvel<!-- ws:start:WikiTextAnchorRule:36:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@marvel&quot; title=&quot;Anchor: marvel&quot;/&gt; --><a name="marvel"></a><!-- ws:end:WikiTextAnchorRule:36 --></h3>
 Tempers out 225/224. An excellent tuning for marvel is <a class="wiki_link" href="/240edo">240edo</a>. It extends in a natural way to unidecimal marvel, which tempers out 385/384, for which the slightly less accurate tuning of <a class="wiki_link" href="/72edo">72edo</a> works well. Marvel is generated by 2, 3, and 5, the same generators as the <a class="wiki_link" href="/Harmonic%20Limit">5-limit</a>, which means a scale can be converted to marvel or unidecimal marvel simply by tempering it.<br />
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Rank 3 temperaments-Starling"></a><!-- ws:end:WikiTextHeadingRule:26 -->Starling<!-- ws:start:WikiTextAnchorRule:37:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@starling&quot; title=&quot;Anchor: starling&quot;/&gt; --><a name="starling"></a><!-- ws:end:WikiTextAnchorRule:37 --></h3>
 Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is <a class="wiki_link" href="/77edo">77edo</a>, but 31, 46 or 58 also work nicely.<br />
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<!-- ws:start:WikiTextHeadingRule:28:&lt;h2&gt; --><h2 id="toc14"><a name="x-Breed"></a><!-- ws:end:WikiTextHeadingRule:28 -->Breed<!-- ws:start:WikiTextAnchorRule:38:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@breed&quot; title=&quot;Anchor: breed&quot;/&gt; --><a name="breed"></a><!-- ws:end:WikiTextAnchorRule:38 --></h2>
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, 2578et will certainly do the trick. Breed has generators of a 49/40 neutral third, and 10/7.<br />
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<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="x-Breed-Jove, aka Wonder"></a><!-- ws:end:WikiTextHeadingRule:30 -->Jove, aka Wonder<!-- ws:start:WikiTextAnchorRule:39:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@wonder&quot; title=&quot;Anchor: wonder&quot;/&gt; --><a name="wonder"></a><!-- ws:end:WikiTextAnchorRule:39 --></h3>
 Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been depreciated as a name due to conflict with another temperament also given that name. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72 and 130 are good edos for jove, and if that doesn't suit there's 476edo.<br />
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