Tour of regular temperaments: Difference between revisions

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=Regular temperaments=

Regular temperaments are non-Just tunings ~~wherein~~ the infinite number of intervals in [[Harmonic Limit|p-limit]] [[Just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[Comma|comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals ~~would be~~ located by a four-dimensional ~~set of~~ coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.

Regular temperaments are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|p-limit]] [[Just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[Comma|comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.

A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r [[Vals|vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.

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===[[Meantone family]]===

The meantone family tempers out 81/80, also called the ~~[[81_80|~~syntonic comma]]. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.

The meantone family tempers out [[81_80|81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.

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

The schismatic family tempers out the [[32805_32768|~~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]].

The schismatic family tempers out the schisma of [[32805_32768|32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. 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]]===

The kleismic family of temperaments tempers out the [[kleisma~~]] of~~ 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.

The kleismic family of temperaments tempers out the kleisma of [[kleisma|15625/15552]], which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. 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, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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 tempers out [[magic comma|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 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.

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

The diaschismic family tempers out 2048/2025, the [[diaschisma]], such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.

The diaschismic family tempers out [[diaschisma|2048/2025]], the diaschisma, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.

===[[Pelogic family]]===

This tempers out the [[135_128|~~pelogic comma]],~~ 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].

This tempers out the pelogic comma, [[135_128|135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].

===[[Porcupine family]]===

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or ~~[[250_243|~~porcupine comma]]. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].

The porcupine family tempers out [[250_243|250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].

===[[Würschmidt family]]===

The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt 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 MOSs 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 the Würschmidt comma, [[Würschmidt 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 MOSs 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 [[128_125|~~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" ([[3L 3s]]) 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]]" ([[3L 6s]]).

The augmented family tempers out the diesis of [[128_125|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" ([[3L 3s]]) 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]]" ([[3L 6s]]).

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

The dimipent family tempers out the ~~[[648_625|~~major diesis]] or diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].

The dimipent family tempers out the major diesis or diminished comma, [[648_625|648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].

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

The dicot family is a low-accuracy family of temperaments which temper out the [[25_24|~~chromatic semitone]],~~ 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25_24|25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].

===[[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 [[tetracot comma|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. 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), [[semicomma|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]]===

The Pythagorean family tempers out the [[Pythagorean comma]], <span style="background-color: #ffffff;">531441/524288 = </span>|-19 12 0>. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.

The Pythagorean family tempers out the Pythagorean comma, <span style="background-color: #ffffff;">[[Pythagorean comma|531441/524288]] = </span>|-19 12 0>. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.

===[[Apotome family]]===

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===[[Bug family]]===

This tempers out 27/25, the large limma or bug comma.

This tempers out [[27_25|27/25]], the large limma or bug comma.

===[[Father family]]===

This tempers out 16/15, the just diatonic semitone.

This tempers out [[16_15|16/15]], the just diatonic semitone.

===[[Sycamore family]]===

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===[[Escapade family]]===

This tempers out the escapade comma, |32 -7 -9>, which is the difference between nine just major thirds and seven just fourths.

This tempers out the [[escapade comma]], |32 -7 -9>, which is the difference between nine just major thirds and seven just fourths.

===[[Amity family]]===

This tempers out the amity comma, 1600000/1594323 = |9 -13 5>.

This tempers out the amity comma, [[amity comma|1600000/1594323]] = |9 -13 5>.

===[[Vulture family]]===

This tempers out the vulture comma, |24 -21 4>.

This tempers out the [[vulture comma]], |24 -21 4>.

===[[Vishnuzmic family]]===

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===[[Mutt family]]===

This tempers out the mutt comma, |-44 -3 21>, leading to some strange properties.

This tempers out the [[mutt comma]], |-44 -3 21>, leading to some strange properties.

==Clans==

If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just intonation subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of [[Normal lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.

===[[Gamelismic clan]]===

If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just intonation subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. ~~We can modify the definition of [[Normal lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.~~

Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps and its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.

Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.

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.

===[[Trienstonic clan]]===

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===[[Quince clan]]===

This tempers out quince, ~~the~~ no-threes comma |-15 0 -2 7> = 823543/819200.

This tempers out the quince, a no-threes comma |-15 0 -2 7> = 823543/819200.

=Temperaments for a given comma=

===[[Septisemi temperaments]]===

These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.

These are very low complexity temperaments tempering out the minor septimal semitone, [[21_20|21/20]] and hence equating 5/3 with 7/4.

===[[Mint temperaments]]===

These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.

These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36_35|36/35]].

===[[Greenwoodmic temperaments]]===

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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 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.

These temper out [[126_125|126/125]], the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave), and include</span> myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

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

These temper out |-5 2 2 -1> = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

These temper out |-5 2 2 -1> = [[225_224|225/224]], the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

===[[Orwellismic temperaments]]===

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===[[11-limit comma temperaments]]===

=Rank 3 temperaments=

=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.

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 family]]===

The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.

The head of the marvel family is marvel, which tempers out [[225_224|225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.

===[[Starling family]]===

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.

Starling tempers out [[126_125|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.

===[[Gamelismic family]]===

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

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.

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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These temper out the kalisma, 9801/9800.

=[[Rank four temperaments|Rank 4 temperaments]]=

=[[Rank four temperaments|Rank-4 temperaments]]=

Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tour of Regular Temperaments</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Regul%C3%A4re%20Temperaturen">Deutsch</a> - <a class="wiki_link" href="/%E3%83%AC%E3%82%AE%E3%83%A5%E3%83%A9%E3%83%BC%E3%83%86%E3%83%B3%E3%83%9A%E3%83%A9%E3%83%A1%E3%83%B3%E3%83%88%E3%81%A8%E3%83%A9%E3%83%B3%E3%82%AFr%E3%83%86%E3%83%B3%E3%83%9A%E3%83%A9%E3%83%A1%E3%83%B3%E3%83%88">日本語</a><br />

</span><br />

<a href="#Regular temperaments">Regular temperaments</a> | <a href="#Equal temperaments">Equal temperaments</a> | <a href="#Rank-2 (including linear) temperaments">Rank-2 (including linear) temperaments</a> | <a href="#Temperaments for a given comma">Temperaments for a given comma</a> | <a href="#Rank 3 temperaments">Rank 3 temperaments</a> | <a href="#Rank 4 temperaments">Rank 4 temperaments</a> | <a href="#Subgroup temperaments">Subgroup temperaments</a> | <a href="#Commatic realms">Commatic realms</a> | <a href="#Links">Links</a>

<a href="#Regular temperaments">Regular temperaments</a> | <a href="#Equal temperaments">Equal temperaments</a> | <a href="#Rank-2 (including linear) temperaments">Rank-2 (including linear) temperaments</a> | <a href="#Temperaments for a given comma">Temperaments for a given comma</a> | <a href="#Rank-3 temperaments">Rank-3 temperaments</a> | <a href="#Rank-4 temperaments">Rank-4 temperaments</a> | <a href="#Subgroup temperaments">Subgroup temperaments</a> | <a href="#Commatic realms">Commatic realms</a> | <a href="#Links">Links</a>

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<h1 id="toc0"><a name="Regular temperaments"></a>Regular temperaments</h1>

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Regular temperaments are non-Just tunings ~~wherein~~ the infinite number of intervals in <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> <a class="wiki_link" href="/Just%20intonation">Just intonation</a>, or any <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> thereof, are mapped to a smaller, though still infinite, set of <a class="wiki_link" href="/tempering%20out">tempered</a> intervals. This is done by deliberately mistuning some of the ratios such that a <a class="wiki_link" href="/Comma">comma</a> or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &quot;puns&quot; as commas are tempered out. Temperaments effectively reduce the &quot;dimensionality&quot; of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals ~~would be~~ located by a four-dimensional ~~set of~~ coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.<br />

Regular temperaments are non-Just tunings in which the infinite number of intervals in <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> <a class="wiki_link" href="/Just%20intonation">Just intonation</a>, or any <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> thereof, are mapped to a smaller, though still infinite, set of <a class="wiki_link" href="/tempering%20out">tempered</a> intervals. This is done by deliberately mistuning some of the ratios such that a <a class="wiki_link" href="/Comma">comma</a> or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &quot;puns&quot; as commas are tempered out. Temperaments effectively reduce the &quot;dimensionality&quot; of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.<br />

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A rank r <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow">regular temperament</a> in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r <a class="wiki_link" href="/Vals">vals</a>. An <a class="wiki_link" href="/abstract%20regular%20temperament">abstract regular temperament</a> can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the <a class="wiki_link" href="/comma%20pump%20examples">comma pumps</a> of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.<br />

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

The meantone family tempers out ~~81/80, also called the~~ <a class="wiki_link" href="/81_80">~~syntonic comma~~</a>. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/43edo">43edo</a>, <a class="wiki_link" href="/50edo">50edo</a>, <a class="wiki_link" href="/55edo">55edo</a> and <a class="wiki_link" href="/81edo">81edo</a>. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.<br />

The meantone family tempers out <a class="wiki_link" href="/81_80">81/80</a>, also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are <a class="wiki_link" href="/12edo">12edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, <a class="wiki_link" href="/43edo">43edo</a>, <a class="wiki_link" href="/50edo">50edo</a>, <a class="wiki_link" href="/55edo">55edo</a> and <a class="wiki_link" href="/81edo">81edo</a>. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.<br />

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

The schismatic family tempers out the <a class="wiki_link" href="/32805_32768">~~schisma~~</a> ~~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 />

The schismatic family tempers out the schisma of <a class="wiki_link" href="/32805_32768">32805/32768</a>, which is the amount by which the Pythagorean comma exceeds the syntonic comma. 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 />

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

The kleismic family of temperaments tempers out the <a class="wiki_link" href="/kleisma">~~kleisma~~</a> ~~of 15625/15552~~, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/34edo">34edo</a>, <a class="wiki_link" href="/49edo">49edo</a>, <a class="wiki_link" href="/53edo">53edo</a>, <a class="wiki_link" href="/72edo">72edo</a>, <a class="wiki_link" href="/87edo">87edo</a> and <a class="wiki_link" href="/140edo">140edo</a> among its possible tunings.<br />

The kleismic family of temperaments tempers out the kleisma of <a class="wiki_link" href="/kleisma">15625/15552</a>, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. 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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<h3 id="toc9"><a name="Rank-2 (including linear) 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~~ <a class="wiki_link" href="/magic%20comma">~~magic comma~~</a> or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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 tempers out <a class="wiki_link" href="/magic%20comma">3125/3072</a>, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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 />

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

The diaschismic family tempers out ~~2048/2025, the~~ <a class="wiki_link" href="/diaschisma">~~diaschisma~~</a>, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. 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>. A noted 7-limit extension to diaschismic is<a class="wiki_link" href="/pajara"> pajara</a> temperament, where the intervals 50/49 and 64/63 are tempered out, of which <a class="wiki_link" href="/22edo">22edo</a> is an excellent tuning.<br />

The diaschismic family tempers out <a class="wiki_link" href="/diaschisma">2048/2025</a>, the diaschisma, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. 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>. A noted 7-limit extension to diaschismic is<a class="wiki_link" href="/pajara"> pajara</a> temperament, where the intervals 50/49 and 64/63 are tempered out, of which <a class="wiki_link" href="/22edo">22edo</a> is an excellent tuning.<br />

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

This tempers out the <a class="wiki_link" href="/135_128">~~pelogic comma~~</a>~~, 135/128~~, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates <a class="wiki_link" href="/2L%205s">2L 5s</a> &quot;anti-diatonic&quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include <a class="wiki_link" href="/9edo">9edo</a>, <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/23edo">23edo</a>, and <a class="wiki_link" href="/25edo">25edo</a>.<br />

This tempers out the pelogic comma, <a class="wiki_link" href="/135_128">135/128</a>, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates <a class="wiki_link" href="/2L%205s">2L 5s</a> &quot;anti-diatonic&quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include <a class="wiki_link" href="/9edo">9edo</a>, <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/23edo">23edo</a>, and <a class="wiki_link" href="/25edo">25edo</a>.<br />

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

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or ~~<a class="wiki_link" href="/250_243">~~porcupine comma~~</a>~~. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/37edo">37edo</a>, and <a class="wiki_link" href="/59edo">59edo</a>.<br />

The porcupine family tempers out <a class="wiki_link" href="/250_243">250/243</a>, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include <a class="wiki_link" href="/15edo">15edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/37edo">37edo</a>, and <a class="wiki_link" href="/59edo">59edo</a>.<br />

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

The würschmidt (or wuerschmidt) family tempers out the <a class="wiki_link" href="/W%C3%BCrschmidt%20comma">~~Würschmidt comma~~</a>~~, 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 MOSs 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 the Würschmidt comma, <a class="wiki_link" href="/W%C3%BCrschmidt%20comma">393216/390625</a> = |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 MOSs 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 />

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

The augmented family tempers out the <a class="wiki_link" href="/128_125">~~diesis~~</a> ~~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; (<a class="wiki_link" href="/3L%203s">3L 3s</a>) 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; (<a class="wiki_link" href="/3L%206s">3L 6s</a>).<br />

The augmented family tempers out the diesis of <a class="wiki_link" href="/128_125">128/125</a>, 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; (<a class="wiki_link" href="/3L%203s">3L 3s</a>) 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; (<a class="wiki_link" href="/3L%206s">3L 6s</a>).<br />

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

The dimipent family tempers out the <a class="wiki_link" href="/648_625">~~major diesis~~</a> ~~or diminished comma, 648/625~~, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as <a class="wiki_link" href="/12edo">12edo</a>.<br />

The dimipent family tempers out the major diesis or diminished comma, <a class="wiki_link" href="/648_625">648/625</a>, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as <a class="wiki_link" href="/12edo">12edo</a>.<br />

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<h3 id="toc16"><a name="Rank-2 (including linear) temperaments-Families-Dicot family"></a><a class="wiki_link" href="/Dicot%20family">Dicot family</a></h3>

The dicot family is a low-accuracy family of temperaments which temper out the <a class="wiki_link" href="/25_24">~~chromatic semitone~~</a>~~, 25/24~~ (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. <a class="wiki_link" href="/7edo">7edo</a> makes for a &quot;good&quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &quot;neutral&quot; dicot 3rds span a 3/2. Tunings include <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/10edo">10edo</a>, and <a class="wiki_link" href="/17edo">17edo</a>.<br />

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, <a class="wiki_link" href="/25_24">25/24</a> (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. <a class="wiki_link" href="/7edo">7edo</a> makes for a &quot;good&quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &quot;neutral&quot; dicot 3rds span a 3/2. Tunings include <a class="wiki_link" href="/7edo">7edo</a>, <a class="wiki_link" href="/10edo">10edo</a>, and <a class="wiki_link" href="/17edo">17edo</a>.<br />

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<h3 id="toc17"><a name="Rank-2 (including linear) 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~~ <a class="wiki_link" href="/tetracot%20comma">~~tetracot comma~~</a>. <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 />

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 <a class="wiki_link" href="/tetracot%20comma">20000/19683</a>, 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 linear) temperaments-Families-Sensipent family"></a><a class="wiki_link" href="/Sensipent%20family">Sensipent family</a></h3>

This tempers out the <a class="wiki_link" href="/sensipent%20comma">~~sensipent comma~~</a>~~, 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 />

This tempers out the sensipent comma, <a class="wiki_link" href="/sensipent%20comma">78732/78125</a>, 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 linear) temperaments-Families-Orwell and the semicomma family"></a><a class="wiki_link" href="/Semicomma%20family">Orwell and the semicomma family</a></h3>

The <a class="wiki_link" href="/semicomma">~~semicomma~~</a> ~~(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 <a class="wiki_link" href="/orwell">orwell</a> temperament.<br />

The semicomma (also known as Fokker's comma), <a class="wiki_link" href="/semicomma">2109375/2097152</a> = |-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 linear) temperaments-Families-Pythagorean family"></a><a class="wiki_link" href="/Pythagorean%20family">Pythagorean family</a></h3>

The Pythagorean family tempers out the <a class="wiki_link" href="/Pythagorean%20comma">~~Pythagorean comma~~</a>~~, <span style="background-color: #ffffff;">531441/524288~~ = </span>|-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.<br />

The Pythagorean family tempers out the Pythagorean comma, <span style="background-color: #ffffff;"><a class="wiki_link" href="/Pythagorean%20comma">531441/524288</a> = </span>|-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.<br />

<br />

<h3 id="toc21"><a name="Rank-2 (including linear) temperaments-Families-Apotome family"></a><a class="wiki_link" href="/Apotome%20family">Apotome family</a></h3>

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<br />

<h3 id="toc24"><a name="Rank-2 (including linear) temperaments-Families-Bug family"></a><a class="wiki_link" href="/Bug%20family">Bug family</a></h3>

This tempers out 27/25, the large limma or bug comma.<br />

This tempers out <a class="wiki_link" href="/27_25">27/25</a>, the large limma or bug comma.<br />

<br />

<h3 id="toc25"><a name="Rank-2 (including linear) temperaments-Families-Father family"></a><a class="wiki_link" href="/Father%20family">Father family</a></h3>

This tempers out 16/15, the just diatonic semitone.<br />

This tempers out <a class="wiki_link" href="/16_15">16/15</a>, the just diatonic semitone.<br />

<br />

<h3 id="toc26"><a name="Rank-2 (including linear) temperaments-Families-Sycamore family"></a><a class="wiki_link" href="/Sycamore%20family">Sycamore family</a></h3>

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<br />

<h3 id="toc27"><a name="Rank-2 (including linear) temperaments-Families-Escapade family"></a><a class="wiki_link" href="/Escapade%20family">Escapade family</a></h3>

This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.<br />

This tempers out the <a class="wiki_link" href="/escapade%20comma">escapade comma</a>, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.<br />

<br />

<h3 id="toc28"><a name="Rank-2 (including linear) temperaments-Families-Amity family"></a><a class="wiki_link" href="/Amity%20family">Amity family</a></h3>

This tempers out the amity comma, 1600000/1594323 = |9 -13 5&gt;.<br />

This tempers out the amity comma, <a class="wiki_link" href="/amity%20comma">1600000/1594323</a> = |9 -13 5&gt;.<br />

<br />

<h3 id="toc29"><a name="Rank-2 (including linear) temperaments-Families-Vulture family"></a><a class="wiki_link" href="/Vulture%20family">Vulture family</a></h3>

This tempers out the vulture comma, |24 -21 4&gt;.<br />

This tempers out the <a class="wiki_link" href="/vulture%20comma">vulture comma</a>, |24 -21 4&gt;.<br />

<br />

<h3 id="toc30"><a name="Rank-2 (including linear) temperaments-Families-Vishnuzmic family"></a><a class="wiki_link" href="/Vishnuzmic%20family">Vishnuzmic family</a></h3>

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<br />

<h3 id="toc42"><a name="Rank-2 (including linear) temperaments-Families-Mutt family"></a><a class="wiki_link" href="/Mutt%20family">Mutt family</a></h3>

This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.<br />

This tempers out the <a class="wiki_link" href="/mutt%20comma">mutt comma</a>, |-44 -3 21&gt;, leading to some strange properties.<br />

<br />

<h2 id="toc43"><a name="Rank-2 (including linear) temperaments-Clans"></a>Clans</h2>

<br />

If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br />

<br />

<h3 id="toc44"><a name="Rank-2 (including linear) temperaments-Clans-Gamelismic clan"></a><a class="wiki_link" href="/Gamelismic%20clan">Gamelismic clan</a></h3>

If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another <a class="wiki_link" href="/Just%20intonation%20subgroups">subgroup</a> of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of <a class="wiki_link" href="/Normal%20lists">normal comma list</a> for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.<br />

Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps and its 21-note scale called &quot;blackjack&quot; and 31-note scale called &quot;canasta&quot; have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.<br />

~~<br />~~

Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.~~<br />~~

~~<br />~~

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 />

<br />

<h3 id="toc45"><a name="Rank-2 (including linear) temperaments-Clans-Trienstonic clan"></a><a class="wiki_link" href="/Trienstonic%20clan">Trienstonic clan</a></h3>

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<br />

<h3 id="toc52"><a name="Rank-2 (including linear) temperaments-Clans-Quince clan"></a><a class="wiki_link" href="/Quince%20clan">Quince clan</a></h3>

This tempers out quince, ~~the~~ no-threes comma |-15 0 -2 7&gt; = 823543/819200.<br />

This tempers out the quince, a no-threes comma |-15 0 -2 7&gt; = 823543/819200.<br />

<br />

<h1 id="toc53"><a name="Temperaments for a given comma"></a>Temperaments for a given comma</h1>

<br />

<h3 id="toc54"><a name="Temperaments for a given comma--Septisemi temperaments"></a><a class="wiki_link" href="/Septisemi%20temperaments">Septisemi temperaments</a></h3>

These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.<br />

These are very low complexity temperaments tempering out the minor septimal semitone, <a class="wiki_link" href="/21_20">21/20</a> and hence equating 5/3 with 7/4.<br />

<br />

<h3 id="toc55"><a name="Temperaments for a given comma--Mint temperaments"></a><a class="wiki_link" href="/Mint%20temperaments">Mint temperaments</a></h3>

These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.<br />

These are low complexity, high error temperaments tempering out the septimal quarter-tone, <a class="wiki_link" href="/36_35">36/35</a>.<br />

<br />

<h3 id="toc56"><a name="Temperaments for a given comma--Greenwoodmic temperaments"></a><a class="wiki_link" href="/Greenwoodmic%20temperaments">Greenwoodmic temperaments</a></h3>

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<h3 id="toc59"><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 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 />

These temper out <a class="wiki_link" href="/126_125">126/125</a>, the septimal semicomma or starling comma (<span class="commentBody">the difference between three 6/5s plus one 7/6, and an octave), and include</span> myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.<br />

<br />

<h3 id="toc60"><a name="Temperaments for a given comma--Marvel temperaments"></a><a class="wiki_link" href="/Marvel%20temperaments">Marvel temperaments</a></h3>

These temper out |-5 2 2 -1&gt; = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.<br />

These temper out |-5 2 2 -1&gt; = <a class="wiki_link" href="/225_224">225/224</a>, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.<br />

<br />

<h3 id="toc61"><a name="Temperaments for a given comma--Orwellismic temperaments"></a><a class="wiki_link" href="/Orwellismic%20temperaments">Orwellismic temperaments</a></h3>

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<h3 id="toc78"><a name="Temperaments for a given comma--11-limit comma temperaments"></a><a class="wiki_link" href="/11-limit%20comma%20temperaments">11-limit comma temperaments</a></h3>

<br />

<h1 id="toc79"><a name="Rank 3 temperaments"></a>Rank 3 temperaments</h1>

<h1 id="toc79"><a name="Rank-3 temperaments"></a>Rank-3 temperaments</h1>

<br />

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 />

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 />

<br />

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

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

The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.<br />

The head of the marvel family is marvel, which tempers out <a class="wiki_link" href="/225_224">225/224</a>. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.<br />

<br />

<h3 id="toc81"><a name="Rank 3 temperaments--Starling family"></a><a class="wiki_link" href="/Starling%20family">Starling family</a></h3>

<h3 id="toc81"><a name="Rank-3 temperaments--Starling family"></a><a class="wiki_link" href="/Starling%20family">Starling family</a></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 />

Starling tempers out <a class="wiki_link" href="/126_125">126/125</a>, 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 />

<br />

<h3 id="toc82"><a name="Rank 3 temperaments--Gamelismic family"></a><a class="wiki_link" href="/Gamelismic%20family">Gamelismic family</a></h3>

<h3 id="toc82"><a name="Rank-3 temperaments--Gamelismic family"></a><a class="wiki_link" href="/Gamelismic%20family">Gamelismic family</a></h3>

Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, 1029/1024.<br />

<br />

<h3 id="toc83"><a name="Rank 3 temperaments--Breed family"></a><a class="wiki_link" href="/Breed%20family">Breed family</a></h3>

<h3 id="toc83"><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, 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="toc84"><a name="Rank 3 temperaments--Ragisma family"></a><a class="wiki_link" href="/Ragisma%20family">Ragisma family</a></h3>

<h3 id="toc84"><a name="Rank-3 temperaments--Ragisma family"></a><a class="wiki_link" href="/Ragisma%20family">Ragisma family</a></h3>

The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.<br />

<br />

<h3 id="toc85"><a name="Rank 3 temperaments--Landscape family"></a><a class="wiki_link" href="/Landscape%20family">Landscape family</a></h3>

<h3 id="toc85"><a name="Rank-3 temperaments--Landscape family"></a><a class="wiki_link" href="/Landscape%20family">Landscape family</a></h3>

The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.<br />

<br />

<h3 id="toc86"><a name="Rank 3 temperaments--Hemifamity family"></a><a class="wiki_link" href="/Hemifamity%20family">Hemifamity family</a></h3>

<h3 id="toc86"><a name="Rank-3 temperaments--Hemifamity family"></a><a class="wiki_link" href="/Hemifamity%20family">Hemifamity family</a></h3>

The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.<br />

<br />

<h3 id="toc87"><a name="Rank 3 temperaments--Porwell family"></a><a class="wiki_link" href="/Porwell%20family">Porwell family</a></h3>

<h3 id="toc87"><a name="Rank-3 temperaments--Porwell family"></a><a class="wiki_link" href="/Porwell%20family">Porwell family</a></h3>

The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.<br />

<br />

<h3 id="toc88"><a name="Rank 3 temperaments--Horwell family"></a><a class="wiki_link" href="/Horwell%20family">Horwell family</a></h3>

<h3 id="toc88"><a name="Rank-3 temperaments--Horwell family"></a><a class="wiki_link" href="/Horwell%20family">Horwell family</a></h3>

The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.<br />

<br />

<h3 id="toc89"><a name="Rank 3 temperaments--Hemimage family"></a><a class="wiki_link" href="/Hemimage%20family">Hemimage family</a></h3>

<h3 id="toc89"><a name="Rank-3 temperaments--Hemimage family"></a><a class="wiki_link" href="/Hemimage%20family">Hemimage family</a></h3>

The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.<br />

<br />

<h3 id="toc90"><a name="Rank 3 temperaments--Sensamagic family"></a><a class="wiki_link" href="/Sensamagic%20family">Sensamagic family</a></h3>

<h3 id="toc90"><a name="Rank-3 temperaments--Sensamagic family"></a><a class="wiki_link" href="/Sensamagic%20family">Sensamagic family</a></h3>

These temper out 245/243.<br />

<br />

<h3 id="toc91"><a name="Rank 3 temperaments--Keemic family"></a><a class="wiki_link" href="/Keemic%20family">Keemic family</a></h3>

<h3 id="toc91"><a name="Rank-3 temperaments--Keemic family"></a><a class="wiki_link" href="/Keemic%20family">Keemic family</a></h3>

These temper out 875/864.<br />

<br />

<h3 id="toc92"><a name="Rank 3 temperaments--Sengic family"></a><a class="wiki_link" href="/Sengic%20family">Sengic family</a></h3>

<h3 id="toc92"><a name="Rank-3 temperaments--Sengic family"></a><a class="wiki_link" href="/Sengic%20family">Sengic family</a></h3>

These temper out the senga, 686/675.<br />

<br />

<h3 id="toc93"><a name="Rank 3 temperaments--Orwellismic family"></a><a class="wiki_link" href="/Orwellismic%20family">Orwellismic family</a></h3>

<h3 id="toc93"><a name="Rank-3 temperaments--Orwellismic family"></a><a class="wiki_link" href="/Orwellismic%20family">Orwellismic family</a></h3>

These temper out 1728/1715.<br />

<br />

<h3 id="toc94"><a name="Rank 3 temperaments--Nuwell family"></a><a class="wiki_link" href="/Nuwell%20family">Nuwell family</a></h3>

<h3 id="toc94"><a name="Rank-3 temperaments--Nuwell family"></a><a class="wiki_link" href="/Nuwell%20family">Nuwell family</a></h3>

These temper out the nuwell comma, 2430/2401.<br />

<br />

<h3 id="toc95"><a name="Rank 3 temperaments--Octagar family"></a><a class="wiki_link" href="/Octagar%20family">Octagar family</a></h3>

<h3 id="toc95"><a name="Rank-3 temperaments--Octagar family"></a><a class="wiki_link" href="/Octagar%20family">Octagar family</a></h3>

The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.<br />

<br />

<h3 id="toc96"><a name="Rank 3 temperaments--Mirkwai family"></a><a class="wiki_link" href="/Mirkwai%20family">Mirkwai family</a></h3>

<h3 id="toc96"><a name="Rank-3 temperaments--Mirkwai family"></a><a class="wiki_link" href="/Mirkwai%20family">Mirkwai family</a></h3>

The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.<br />

<br />

<h3 id="toc97"><a name="Rank 3 temperaments--Hemimean family"></a><a class="wiki_link" href="/Hemimean%20family">Hemimean family</a></h3>

<h3 id="toc97"><a name="Rank-3 temperaments--Hemimean family"></a><a class="wiki_link" href="/Hemimean%20family">Hemimean family</a></h3>

The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.<br />

<br />

<h3 id="toc98"><a name="Rank 3 temperaments--Mirwomo family"></a><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3>

<h3 id="toc98"><a name="Rank-3 temperaments--Mirwomo family"></a><a class="wiki_link" href="/Mirwomo%20family">Mirwomo family</a></h3>

The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.<br />

<br />

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

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

The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.<br />

<br />

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

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

These temper out the tolerma, 179200/177147.<br />

<br />

<h3 id="toc101"><a name="Rank 3 temperaments--Kleismic rank three family"></a><a class="wiki_link" href="/Kleismic%20rank%20three%20family">Kleismic rank three family</a></h3>

<h3 id="toc101"><a name="Rank-3 temperaments--Kleismic rank three family"></a><a class="wiki_link" href="/Kleismic%20rank%20three%20family">Kleismic rank three family</a></h3>

These are the rank three temperaments tempering out the kleisma, 15625/15552. 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="toc102"><a name="Rank 3 temperaments--Diaschismic rank three family"></a><a class="wiki_link" href="/Diaschismic%20rank%20three%20family">Diaschismic rank three family</a></h3>

<h3 id="toc102"><a name="Rank-3 temperaments--Diaschismic rank three family"></a><a class="wiki_link" href="/Diaschismic%20rank%20three%20family">Diaschismic rank three family</a></h3>

These are the rank three temperaments tempering out the dischisma, 2048/2025. 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="toc103"><a name="Rank 3 temperaments--Didymus rank three family"></a><a class="wiki_link" href="/Didymus%20rank%20three%20family">Didymus rank three family</a></h3>

<h3 id="toc103"><a name="Rank-3 temperaments--Didymus rank three family"></a><a class="wiki_link" href="/Didymus%20rank%20three%20family">Didymus rank three family</a></h3>

These are the rank three temperaments tempering out the didymus comma, 81/80. 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="toc104"><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>

<h3 id="toc104"><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 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 />

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="toc105"><a name="Rank 3 temperaments--Archytas family"></a><a class="wiki_link" href="/Archytas%20family">Archytas family</a></h3>

<h3 id="toc105"><a name="Rank-3 temperaments--Archytas family"></a><a class="wiki_link" href="/Archytas%20family">Archytas family</a></h3>

Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.<br />

<br />

<h3 id="toc106"><a name="Rank 3 temperaments--Jubilismic family"></a><a class="wiki_link" href="/Jubilismic%20family">Jubilismic family</a></h3>

<h3 id="toc106"><a name="Rank-3 temperaments--Jubilismic family"></a><a class="wiki_link" href="/Jubilismic%20family">Jubilismic family</a></h3>

Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.<br />

<br />

<h3 id="toc107"><a name="Rank 3 temperaments--Semiphore family"></a><a class="wiki_link" href="/Semiphore%20family">Semiphore family</a></h3>

<h3 id="toc107"><a name="Rank-3 temperaments--Semiphore family"></a><a class="wiki_link" href="/Semiphore%20family">Semiphore family</a></h3>

Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like &quot;semi-fourth&quot;.<br />

<br />

<h3 id="toc108"><a name="Rank 3 temperaments--Mint family"></a><a class="wiki_link" href="/Mint%20family">Mint family</a></h3>

<h3 id="toc108"><a name="Rank-3 temperaments--Mint family"></a><a class="wiki_link" href="/Mint%20family">Mint family</a></h3>

The mint temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.<br />

<br />

<h3 id="toc109"><a name="Rank 3 temperaments--Valinorismic temperaments"></a><a class="wiki_link" href="/Valinorismic%20temperaments">Valinorismic temperaments</a></h3>

<h3 id="toc109"><a name="Rank-3 temperaments--Valinorismic temperaments"></a><a class="wiki_link" href="/Valinorismic%20temperaments">Valinorismic temperaments</a></h3>

These temper out the valinorsma, 176/175.<br />

<br />

<h3 id="toc110"><a name="Rank 3 temperaments--Rastmic temperaments"></a><a class="wiki_link" href="/Rastmic%20temperaments">Rastmic temperaments</a></h3>

<h3 id="toc110"><a name="Rank-3 temperaments--Rastmic temperaments"></a><a class="wiki_link" href="/Rastmic%20temperaments">Rastmic temperaments</a></h3>

These temper out the rastma, 243/242.<br />

<br />

<h3 id="toc111"><a name="Rank 3 temperaments--Werckismic temperaments"></a><a class="wiki_link" href="/Werckismic%20temperaments">Werckismic temperaments</a></h3>

<h3 id="toc111"><a name="Rank-3 temperaments--Werckismic temperaments"></a><a class="wiki_link" href="/Werckismic%20temperaments">Werckismic temperaments</a></h3>

These temper out the werckisma, 441/440.<br />

<br />

<h3 id="toc112"><a name="Rank 3 temperaments--Swetismic temperaments"></a><a class="wiki_link" href="/Swetismic%20temperaments">Swetismic temperaments</a></h3>

<h3 id="toc112"><a name="Rank-3 temperaments--Swetismic temperaments"></a><a class="wiki_link" href="/Swetismic%20temperaments">Swetismic temperaments</a></h3>

These temper out the swetisma, 540/539.<br />

<br />

<h3 id="toc113"><a name="Rank 3 temperaments--Lehmerismic temperaments"></a><a class="wiki_link" href="/Lehmerismic%20temperaments">Lehmerismic temperaments</a></h3>

<h3 id="toc113"><a name="Rank-3 temperaments--Lehmerismic temperaments"></a><a class="wiki_link" href="/Lehmerismic%20temperaments">Lehmerismic temperaments</a></h3>

These temper out the lehmerisma, 3025/3024.<br />

<br />

<h3 id="toc114"><a name="Rank 3 temperaments--Kalismic temperaments"></a><a class="wiki_link" href="/Kalismic%20temperaments">Kalismic temperaments</a></h3>

<h3 id="toc114"><a name="Rank-3 temperaments--Kalismic temperaments"></a><a class="wiki_link" href="/Kalismic%20temperaments">Kalismic temperaments</a></h3>

These temper out the kalisma, 9801/9800.<br />

<br />

<h1 id="toc115"><a name="Rank 4 temperaments"></a><a class="wiki_link" href="/Rank%20four%20temperaments">Rank 4 temperaments</a></h1>

<h1 id="toc115"><a name="Rank-4 temperaments"></a><a class="wiki_link" href="/Rank%20four%20temperaments">Rank-4 temperaments</a></h1>

<br />

Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example <a class="wiki_link" href="/Hobbits">hobbit scales</a> can be constructed for them.<br />

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-: This revision was by author [[User:jake.huryn|jake.huryn]] and made on <tt>2017-05-15 15:19:56 UTC</tt>.<br>
+: This revision was by author [[User:jake.huryn|jake.huryn]] and made on <tt>2017-05-15 15:59:23 UTC</tt>.<br>
-: The original revision id was <tt>612911081</tt>.<br>
+: The original revision id was <tt>612914477</tt>.<br>
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 =Regular temperaments=
-Regular temperaments are non-Just tunings wherein the infinite number of intervals in [[Harmonic Limit|p-limit]] [[Just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[Comma|comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
+Regular temperaments are non-Just tunings in which the infinite number of intervals in [[Harmonic Limit|p-limit]] [[Just intonation]], or any [[Just intonation subgroups|subgroup]] thereof, are mapped to a smaller, though still infinite, set of [[tempering out|tempered]] intervals. This is done by deliberately mistuning some of the ratios such that a [[Comma|comma]] or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful "puns" as commas are tempered out. Temperaments effectively reduce the "dimensionality" of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.
 A rank r [[http://en.wikipedia.org/wiki/Regular_temperament|regular temperament]] in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r [[Vals|vals]]. An [[abstract regular temperament]] can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the [[comma pump examples|comma pumps]] of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.
@@ Line 41: / Line 41: @@
 ===[[Meantone family]]===
-The meantone family tempers out 81/80, also called the [[81_80|syntonic comma]]. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.
+The meantone family tempers out [[81_80|81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.
 ===[[Schismatic family]]===
-The schismatic family tempers out the [[32805_32768|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]].
+The schismatic family tempers out the schisma of [[32805_32768|32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. 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]]===
-The kleismic family of temperaments tempers out the [[kleisma]] of 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.
+The kleismic family of temperaments tempers out the kleisma of [[kleisma|15625/15552]], which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. 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, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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 tempers out [[magic comma|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 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.
 ===[[Diaschismic family]]===
-The diaschismic family tempers out 2048/2025, the [[diaschisma]], such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.
+The diaschismic family tempers out [[diaschisma|2048/2025]], the diaschisma, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.
 ===[[Pelogic family]]===
-This tempers out the [[135_128|pelogic comma]], 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].
+This tempers out the pelogic comma, [[135_128|135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].
 ===[[Porcupine family]]===
-The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or [[250_243|porcupine comma]]. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].
+The porcupine family tempers out [[250_243|250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].
 ===[[Würschmidt family]]===
-The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt 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 MOSs 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 the Würschmidt comma, [[Würschmidt 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 MOSs 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 [[128_125|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" ([[3L 3s]]) 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]]" ([[3L 6s]]).
+The augmented family tempers out the diesis of [[128_125|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" ([[3L 3s]]) 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]]" ([[3L 6s]]).
 ===[[Dimipent family]]===
-The dimipent family tempers out the [[648_625|major diesis]] or diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
+The dimipent family tempers out the major diesis or diminished comma, [[648_625|648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]].
 ===[[Dicot family]]===
-The dicot family is a low-accuracy family of temperaments which temper out the [[25_24|chromatic semitone]], 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].
+The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25_24|25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].
 ===[[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 [[tetracot comma|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. Tunings include [[xenharmonic/8edo|8edo]], [[xenharmonic/19edo|19edo]], [[xenharmonic/46edo|46edo]], and [[xenharmonic/65edo|65edo]].
+This tempers out the sensipent comma, [[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), [[semicomma|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]]===
-The Pythagorean family tempers out the [[Pythagorean comma]], &lt;span style="background-color: #ffffff;"&gt;531441/524288 = &lt;/span&gt;|-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.
+The Pythagorean family tempers out the Pythagorean comma, &lt;span style="background-color: #ffffff;"&gt;[[Pythagorean comma|531441/524288]] = &lt;/span&gt;|-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.
 ===[[Apotome family]]===
@@ Line 95: / Line 95: @@
 ===[[Bug family]]===
-This tempers out 27/25, the large limma or bug comma.
+This tempers out [[27_25|27/25]], the large limma or bug comma.
 ===[[Father family]]===
-This tempers out 16/15, the just diatonic semitone.
+This tempers out [[16_15|16/15]], the just diatonic semitone.
 ===[[Sycamore family]]===
@@ Line 104: / Line 104: @@
 ===[[Escapade family]]===
-This tempers out the escapade comma, |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.
+This tempers out the [[escapade comma]], |32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths.
 ===[[Amity family]]===
-This tempers out the amity comma, 1600000/1594323 = |9 -13 5&gt;.
+This tempers out the amity comma, [[amity comma|1600000/1594323]] = |9 -13 5&gt;.
 ===[[Vulture family]]===
-This tempers out the vulture comma, |24 -21 4&gt;.
+This tempers out the [[vulture comma]], |24 -21 4&gt;.
 ===[[Vishnuzmic family]]===
@@ Line 149: / Line 149: @@
 ===[[Mutt family]]===
-This tempers out the mutt comma, |-44 -3 21&gt;, leading to some strange properties.
+This tempers out the [[mutt comma]], |-44 -3 21&gt;, leading to some strange properties.
 ==Clans==
+If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just intonation subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of [[Normal lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.
 ===[[Gamelismic clan]]===
-If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another [[Just intonation subgroups|subgroup]] of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of [[Normal lists|normal comma list]] for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.
+Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps and its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.
-Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.
-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.
 ===[[Trienstonic clan]]===
@@ Line 182: / Line 180: @@
 ===[[Quince clan]]===
-This tempers out quince, the no-threes comma |-15 0 -2 7&gt; = 823543/819200.
+This tempers out the quince, a no-threes comma |-15 0 -2 7&gt; = 823543/819200.
 =Temperaments for a given comma=
 ===[[Septisemi temperaments]]===
-These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.
+These are very low complexity temperaments tempering out the minor septimal semitone, [[21_20|21/20]] and hence equating 5/3 with 7/4.
 ===[[Mint temperaments]]===
-These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.
+These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36_35|36/35]].
 ===[[Greenwoodmic temperaments]]===
@@ Line 202: / Line 200: @@
 ===[[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 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.
+These temper out [[126_125|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), and include&lt;/span&gt; myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.
 ===[[Marvel temperaments]]===
-These temper out |-5 2 2 -1&gt; = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.
+These temper out |-5 2 2 -1&gt; = [[225_224|225/224]], the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.
 ===[[Orwellismic temperaments]]===
@@ Line 260: / Line 258: @@
 ===[[11-limit comma temperaments]]===
-=Rank 3 temperaments=
+=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.
+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 family]]===
-The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.
+The head of the marvel family is marvel, which tempers out [[225_224|225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.
 ===[[Starling family]]===
-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.
+Starling tempers out [[126_125|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.
 ===[[Gamelismic family]]===
@@ Line 337: / Line 335: @@
 ===[[Porcupine rank three family]]===
-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.
+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: @@
 These temper out the kalisma, 9801/9800.
-=[[Rank four temperaments|Rank 4 temperaments]]=
+=[[Rank four temperaments|Rank-4 temperaments]]=
 Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.
@@ Line 399: / Line 397: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tour of Regular Temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/Regul%C3%A4re%20Temperaturen"&gt;Deutsch&lt;/a&gt; - &lt;a class="wiki_link" href="/%E3%83%AC%E3%82%AE%E3%83%A5%E3%83%A9%E3%83%BC%E3%83%86%E3%83%B3%E3%83%9A%E3%83%A9%E3%83%A1%E3%83%B3%E3%83%88%E3%81%A8%E3%83%A9%E3%83%B3%E3%82%AFr%E3%83%86%E3%83%B3%E3%83%9A%E3%83%A9%E3%83%A1%E3%83%B3%E3%83%88"&gt;日本語&lt;/a&gt;&lt;br /&gt;
 &lt;/span&gt;&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Regular temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Regular temperaments&lt;/h1&gt;
   &lt;br /&gt;
-Regular temperaments are non-Just tunings wherein the infinite number of intervals in &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20intonation"&gt;Just intonation&lt;/a&gt;, or any &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; thereof, are mapped to a smaller, though still infinite, set of &lt;a class="wiki_link" href="/tempering%20out"&gt;tempered&lt;/a&gt; intervals. This is done by deliberately mistuning some of the ratios such that a &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &amp;quot;puns&amp;quot; as commas are tempered out. Temperaments effectively reduce the &amp;quot;dimensionality&amp;quot; of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals would be located by a four-dimensional set of coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.&lt;br /&gt;
+Regular temperaments are non-Just tunings in which the infinite number of intervals in &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Just%20intonation"&gt;Just intonation&lt;/a&gt;, or any &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; thereof, are mapped to a smaller, though still infinite, set of &lt;a class="wiki_link" href="/tempering%20out"&gt;tempered&lt;/a&gt; intervals. This is done by deliberately mistuning some of the ratios such that a &lt;a class="wiki_link" href="/Comma"&gt;comma&lt;/a&gt; or set of commas vanishes by becoming a unison. The utility of regular temperament is partly to produce scales that are simpler and have more consonances than strict JI, while maintaining a high level of concordance (or similarity to JI), and partly to introduce useful &amp;quot;puns&amp;quot; as commas are tempered out. Temperaments effectively reduce the &amp;quot;dimensionality&amp;quot; of JI, thereby simplifying the pitch relationships. For instance, the pitch relationships in 7-limit JI can be thought of as 4-dimensional, with each prime up to 7 (2, 3, 5, and 7) representing an axis, and all intervals located by four-dimensional coordinates. In a 7-limit regular temperament, however, the dimensionality is reduced in some way, depending on which and how many commas are tempered out. In this way, intervals can be located with a set of one-, two-, or three-dimensional coordinates depending on the number of commas that have been tempered out. The dimensionality is the rank of the temperament.&lt;br /&gt;
 &lt;br /&gt;
 A rank r &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Regular_temperament" rel="nofollow"&gt;regular temperament&lt;/a&gt; in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r &lt;a class="wiki_link" href="/Vals"&gt;vals&lt;/a&gt;. An &lt;a class="wiki_link" href="/abstract%20regular%20temperament"&gt;abstract regular temperament&lt;/a&gt; can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament. A characteristic feature of any temperament tempering out a comma are the &lt;a class="wiki_link" href="/comma%20pump%20examples"&gt;comma pumps&lt;/a&gt; of the comma, which are sequences of harmonically related notes or chords which return to their starting point when tempered, but which would not do so in just intonation. An example is the pump I-vii-IV-ii-V-I of meantone temperament.&lt;br /&gt;
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-  The meantone family tempers out 81/80, also called the &lt;a class="wiki_link" href="/81_80"&gt;syntonic comma&lt;/a&gt;. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43edo&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;, &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; and &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.&lt;br /&gt;
+  The meantone family tempers out &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, &lt;a class="wiki_link" href="/43edo"&gt;43edo&lt;/a&gt;, &lt;a class="wiki_link" href="/50edo"&gt;50edo&lt;/a&gt;, &lt;a class="wiki_link" href="/55edo"&gt;55edo&lt;/a&gt; and &lt;a class="wiki_link" href="/81edo"&gt;81edo&lt;/a&gt;. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc7"&gt;&lt;a name="Rank-2 (including linear) 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 &lt;a class="wiki_link" href="/32805_32768"&gt;schisma&lt;/a&gt; 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;
+  The schismatic family tempers out the schisma of &lt;a class="wiki_link" href="/32805_32768"&gt;32805/32768&lt;/a&gt;, which is the amount by which the Pythagorean comma exceeds the syntonic comma. 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 linear) 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;
-  The kleismic family of temperaments tempers out the &lt;a class="wiki_link" href="/kleisma"&gt;kleisma&lt;/a&gt; of 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt; among its possible tunings.&lt;br /&gt;
+  The kleismic family of temperaments tempers out the kleisma of &lt;a class="wiki_link" href="/kleisma"&gt;15625/15552&lt;/a&gt;, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp. The kleismic family includes &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, &lt;a class="wiki_link" href="/72edo"&gt;72edo&lt;/a&gt;, &lt;a class="wiki_link" href="/87edo"&gt;87edo&lt;/a&gt; and &lt;a class="wiki_link" href="/140edo"&gt;140edo&lt;/a&gt; among its possible tunings.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Rank-2 (including linear) 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 &lt;a class="wiki_link" href="/magic%20comma"&gt;magic comma&lt;/a&gt; or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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 tempers out &lt;a class="wiki_link" href="/magic%20comma"&gt;3125/3072&lt;/a&gt;, known as the magic comma or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/2. The 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;
 &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 linear) 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;
-  The diaschismic family tempers out 2048/2025, the &lt;a class="wiki_link" href="/diaschisma"&gt;diaschisma&lt;/a&gt;, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. A noted 7-limit extension to diaschismic is&lt;a class="wiki_link" href="/pajara"&gt; pajara&lt;/a&gt; temperament, where the intervals 50/49 and 64/63 are tempered out, of which &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is an excellent tuning.&lt;br /&gt;
+  The diaschismic family tempers out &lt;a class="wiki_link" href="/diaschisma"&gt;2048/2025&lt;/a&gt;, the diaschisma, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. A noted 7-limit extension to diaschismic is&lt;a class="wiki_link" href="/pajara"&gt; pajara&lt;/a&gt; temperament, where the intervals 50/49 and 64/63 are tempered out, of which &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is an excellent tuning.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Pelogic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;a class="wiki_link" href="/Pelogic%20family"&gt;Pelogic family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the &lt;a class="wiki_link" href="/135_128"&gt;pelogic comma&lt;/a&gt;, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates &lt;a class="wiki_link" href="/2L%205s"&gt;2L 5s&lt;/a&gt; &amp;quot;anti-diatonic&amp;quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;.&lt;br /&gt;
+  This tempers out the pelogic comma, &lt;a class="wiki_link" href="/135_128"&gt;135/128&lt;/a&gt;, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates &lt;a class="wiki_link" href="/2L%205s"&gt;2L 5s&lt;/a&gt; &amp;quot;anti-diatonic&amp;quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Porcupine family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;a class="wiki_link" href="/Porcupine%20family"&gt;Porcupine family&lt;/a&gt;&lt;/h3&gt;
-  The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or &lt;a class="wiki_link" href="/250_243"&gt;porcupine comma&lt;/a&gt;. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/59edo"&gt;59edo&lt;/a&gt;.&lt;br /&gt;
+  The porcupine family tempers out &lt;a class="wiki_link" href="/250_243"&gt;250/243&lt;/a&gt;, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/59edo"&gt;59edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Rank-2 (including linear) 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 würschmidt (or wuerschmidt) family tempers out the &lt;a class="wiki_link" href="/W%C3%BCrschmidt%20comma"&gt;Würschmidt comma&lt;/a&gt;, 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 MOSs 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 the Würschmidt comma, &lt;a class="wiki_link" href="/W%C3%BCrschmidt%20comma"&gt;393216/390625&lt;/a&gt; = |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 MOSs 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 linear) 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 &lt;a class="wiki_link" href="/128_125"&gt;diesis&lt;/a&gt; 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; (&lt;a class="wiki_link" href="/3L%203s"&gt;3L 3s&lt;/a&gt;) 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; (&lt;a class="wiki_link" href="/3L%206s"&gt;3L 6s&lt;/a&gt;).&lt;br /&gt;
+  The augmented family tempers out the diesis of &lt;a class="wiki_link" href="/128_125"&gt;128/125&lt;/a&gt;, 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; (&lt;a class="wiki_link" href="/3L%203s"&gt;3L 3s&lt;/a&gt;) 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; (&lt;a class="wiki_link" href="/3L%206s"&gt;3L 6s&lt;/a&gt;).&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 linear) 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;
-  The dimipent family tempers out the &lt;a class="wiki_link" href="/648_625"&gt;major diesis&lt;/a&gt; or diminished comma, 648/625, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;.&lt;br /&gt;
+  The dimipent family tempers out the major diesis or diminished comma, &lt;a class="wiki_link" href="/648_625"&gt;648/625&lt;/a&gt;, the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Dicot family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;a class="wiki_link" href="/Dicot%20family"&gt;Dicot family&lt;/a&gt;&lt;/h3&gt;
-  The dicot family is a low-accuracy family of temperaments which temper out the &lt;a class="wiki_link" href="/25_24"&gt;chromatic semitone&lt;/a&gt;, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a &amp;quot;good&amp;quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &amp;quot;neutral&amp;quot; dicot 3rds span a 3/2. Tunings include &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;.&lt;br /&gt;
+  The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, &lt;a class="wiki_link" href="/25_24"&gt;25/24&lt;/a&gt; (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a &amp;quot;good&amp;quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &amp;quot;neutral&amp;quot; dicot 3rds span a 3/2. Tunings include &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;.&lt;br /&gt;
 &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 linear) 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 &lt;a class="wiki_link" href="/tetracot%20comma"&gt;tetracot comma&lt;/a&gt;. &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;
+  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 &lt;a class="wiki_link" href="/tetracot%20comma"&gt;20000/19683&lt;/a&gt;, 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 linear) 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 &lt;a class="wiki_link" href="/sensipent%20comma"&gt;sensipent comma&lt;/a&gt;, 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;
+  This tempers out the sensipent comma, &lt;a class="wiki_link" href="/sensipent%20comma"&gt;78732/78125&lt;/a&gt;, 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 linear) 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 &lt;a class="wiki_link" href="/semicomma"&gt;semicomma&lt;/a&gt; (also known as Fokker's comma), 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;
+  The semicomma (also known as Fokker's comma), &lt;a class="wiki_link" href="/semicomma"&gt;2109375/2097152&lt;/a&gt; = |-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 linear) 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;
-  The Pythagorean family tempers out the &lt;a class="wiki_link" href="/Pythagorean%20comma"&gt;Pythagorean comma&lt;/a&gt;, &lt;span style="background-color: #ffffff;"&gt;531441/524288 = &lt;/span&gt;|-19 12 0&amp;gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.&lt;br /&gt;
+  The Pythagorean family tempers out the Pythagorean comma, &lt;span style="background-color: #ffffff;"&gt;&lt;a class="wiki_link" href="/Pythagorean%20comma"&gt;531441/524288&lt;/a&gt; = &lt;/span&gt;|-19 12 0&amp;gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate chains of 12-equal, offset from one another justly tuned 5/4.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc21"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Apotome family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;a class="wiki_link" href="/Apotome%20family"&gt;Apotome family&lt;/a&gt;&lt;/h3&gt;
@@ Line 486: / Line 484: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Bug family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;&lt;a class="wiki_link" href="/Bug%20family"&gt;Bug family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out 27/25, the large limma or bug comma.&lt;br /&gt;
+  This tempers out &lt;a class="wiki_link" href="/27_25"&gt;27/25&lt;/a&gt;, the large limma or bug comma.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc25"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Father family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;a class="wiki_link" href="/Father%20family"&gt;Father family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out 16/15, the just diatonic semitone.&lt;br /&gt;
+  This tempers out &lt;a class="wiki_link" href="/16_15"&gt;16/15&lt;/a&gt;, the just diatonic semitone.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc26"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Sycamore family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;&lt;a class="wiki_link" href="/Sycamore%20family"&gt;Sycamore family&lt;/a&gt;&lt;/h3&gt;
@@ Line 495: / Line 493: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc27"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Escapade family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;a class="wiki_link" href="/Escapade%20family"&gt;Escapade family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the escapade comma, |32 -7 -9&amp;gt;, which is the difference between nine just major thirds and seven just fourths.&lt;br /&gt;
+  This tempers out the &lt;a class="wiki_link" href="/escapade%20comma"&gt;escapade comma&lt;/a&gt;, |32 -7 -9&amp;gt;, which is the difference between nine just major thirds and seven just fourths.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc28"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Amity family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;a class="wiki_link" href="/Amity%20family"&gt;Amity family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the amity comma, 1600000/1594323 = |9 -13 5&amp;gt;.&lt;br /&gt;
+  This tempers out the amity comma, &lt;a class="wiki_link" href="/amity%20comma"&gt;1600000/1594323&lt;/a&gt; = |9 -13 5&amp;gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc29"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Vulture family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;a class="wiki_link" href="/Vulture%20family"&gt;Vulture family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the vulture comma, |24 -21 4&amp;gt;.&lt;br /&gt;
+  This tempers out the &lt;a class="wiki_link" href="/vulture%20comma"&gt;vulture comma&lt;/a&gt;, |24 -21 4&amp;gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc30"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Vishnuzmic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;a class="wiki_link" href="/Vishnuzmic%20family"&gt;Vishnuzmic family&lt;/a&gt;&lt;/h3&gt;
@@ Line 540: / Line 538: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc42"&gt;&lt;a name="Rank-2 (including linear) temperaments-Families-Mutt family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;&lt;a class="wiki_link" href="/Mutt%20family"&gt;Mutt family&lt;/a&gt;&lt;/h3&gt;
-  This tempers out the mutt comma, |-44 -3 21&amp;gt;, leading to some strange properties.&lt;br /&gt;
+  This tempers out the &lt;a class="wiki_link" href="/mutt%20comma"&gt;mutt comma&lt;/a&gt;, |-44 -3 21&amp;gt;, leading to some strange properties.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc43"&gt;&lt;a name="Rank-2 (including linear) temperaments-Clans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;Clans&lt;/h2&gt;
   &lt;br /&gt;
+If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. We can modify the definition of &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.&lt;br /&gt;
+&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc44"&gt;&lt;a name="Rank-2 (including linear) temperaments-Clans-Gamelismic clan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;&lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;Gamelismic clan&lt;/a&gt;&lt;/h3&gt;
-  If a 5-limit comma defines a family of rank two temperaments, then we might say a comma belonging to another &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;subgroup&lt;/a&gt; of the 7-limit can define a clan. In particular we might say a triprime comma (one with exactly three primes in the factorization) can define a clan. Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. We can modify the definition of &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; for clans by changing the ordering of prime numbers, and using this to sort out clan relationships.&lt;br /&gt;
+  Notable among such clans are the temperaments which temper out the gamelisma, 1029/1024. Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps and its 21-note scale called &amp;quot;blackjack&amp;quot; and 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.&lt;br /&gt;
-&lt;br /&gt;
-Particularly noteworthy as member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds.&lt;br /&gt;
-&lt;br /&gt;
-Miracle temperament divides the fifth into 6 equal steps. A 21-note scale called &amp;quot;blackjack&amp;quot; and a 31-note scale called &amp;quot;canasta&amp;quot; have some useful properties. It's the most efficient 11-limit temperament for many purposes with a tuning close to 72-EDO.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc45"&gt;&lt;a name="Rank-2 (including linear) temperaments-Clans-Trienstonic clan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;&lt;a class="wiki_link" href="/Trienstonic%20clan"&gt;Trienstonic clan&lt;/a&gt;&lt;/h3&gt;
@@ Line 573: / Line 569: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:104:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc52"&gt;&lt;a name="Rank-2 (including linear) temperaments-Clans-Quince clan"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:104 --&gt;&lt;a class="wiki_link" href="/Quince%20clan"&gt;Quince clan&lt;/a&gt;&lt;/h3&gt;
-  This tempers out quince, the no-threes comma |-15 0 -2 7&amp;gt; = 823543/819200.&lt;br /&gt;
+  This tempers out the quince, a no-threes comma |-15 0 -2 7&amp;gt; = 823543/819200.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:106:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc53"&gt;&lt;a name="Temperaments for a given comma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:106 --&gt;Temperaments for a given comma&lt;/h1&gt;
   &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:108:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc54"&gt;&lt;a name="Temperaments for a given comma--Septisemi temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:108 --&gt;&lt;a class="wiki_link" href="/Septisemi%20temperaments"&gt;Septisemi temperaments&lt;/a&gt;&lt;/h3&gt;
-  These are very low complexity temperaments tempering out the minor septimal semitone, 21/20 and hence equating 5/3 with 7/4.&lt;br /&gt;
+  These are very low complexity temperaments tempering out the minor septimal semitone, &lt;a class="wiki_link" href="/21_20"&gt;21/20&lt;/a&gt; and hence equating 5/3 with 7/4.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:110:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc55"&gt;&lt;a name="Temperaments for a given comma--Mint temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:110 --&gt;&lt;a class="wiki_link" href="/Mint%20temperaments"&gt;Mint temperaments&lt;/a&gt;&lt;/h3&gt;
-  These are low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35.&lt;br /&gt;
+  These are low complexity, high error temperaments tempering out the septimal quarter-tone, &lt;a class="wiki_link" href="/36_35"&gt;36/35&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:112:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc56"&gt;&lt;a name="Temperaments for a given comma--Greenwoodmic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:112 --&gt;&lt;a class="wiki_link" href="/Greenwoodmic%20temperaments"&gt;Greenwoodmic temperaments&lt;/a&gt;&lt;/h3&gt;
@@ Line 593: / Line 589: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:118:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc59"&gt;&lt;a name="Temperaments for a given comma--Starling temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:118 --&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 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;
+  These temper out &lt;a class="wiki_link" href="/126_125"&gt;126/125&lt;/a&gt;, the septimal semicomma or starling comma (&lt;span class="commentBody"&gt;the difference between three 6/5s plus one 7/6, and an octave), and include&lt;/span&gt; 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:120:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc60"&gt;&lt;a name="Temperaments for a given comma--Marvel temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:120 --&gt;&lt;a class="wiki_link" href="/Marvel%20temperaments"&gt;Marvel temperaments&lt;/a&gt;&lt;/h3&gt;
-  These temper out |-5 2 2 -1&amp;gt; = 225/224, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.&lt;br /&gt;
+  These temper out |-5 2 2 -1&amp;gt; = &lt;a class="wiki_link" href="/225_224"&gt;225/224&lt;/a&gt;, the marvel comma, and include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:122:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc61"&gt;&lt;a name="Temperaments for a given comma--Orwellismic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:122 --&gt;&lt;a class="wiki_link" href="/Orwellismic%20temperaments"&gt;Orwellismic temperaments&lt;/a&gt;&lt;/h3&gt;
@@ Line 651: / Line 647: @@
 &lt;!-- ws:start:WikiTextHeadingRule:156:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc78"&gt;&lt;a name="Temperaments for a given comma--11-limit comma temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:156 --&gt;&lt;a class="wiki_link" href="/11-limit%20comma%20temperaments"&gt;11-limit comma temperaments&lt;/a&gt;&lt;/h3&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:158:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc79"&gt;&lt;a name="Rank 3 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:158 --&gt;Rank 3 temperaments&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:158:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc79"&gt;&lt;a name="Rank-3 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:158 --&gt;Rank-3 temperaments&lt;/h1&gt;
   &lt;br /&gt;
-Even less familiar than rank 2 temperaments are the &lt;a class="wiki_link" href="/Planar%20Temperament"&gt;rank 3 temperaments&lt;/a&gt;, 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.&lt;br /&gt;
+Even less familiar than rank-2 temperaments are the &lt;a class="wiki_link" href="/Planar%20Temperament"&gt;rank-3 temperaments&lt;/a&gt;, 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.&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--Marvel family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:160 --&gt;&lt;a class="wiki_link" href="/Marvel%20family"&gt;Marvel family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:160:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc80"&gt;&lt;a name="Rank-3 temperaments--Marvel family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:160 --&gt;&lt;a class="wiki_link" href="/Marvel%20family"&gt;Marvel family&lt;/a&gt;&lt;/h3&gt;
-  The head of the marvel family is marvel, which tempers out 225/224. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.&lt;br /&gt;
+  The head of the marvel family is marvel, which tempers out &lt;a class="wiki_link" href="/225_224"&gt;225/224&lt;/a&gt;. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:162:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc81"&gt;&lt;a name="Rank 3 temperaments--Starling family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:162 --&gt;&lt;a class="wiki_link" href="/Starling%20family"&gt;Starling family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:162:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc81"&gt;&lt;a name="Rank-3 temperaments--Starling family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:162 --&gt;&lt;a class="wiki_link" href="/Starling%20family"&gt;Starling family&lt;/a&gt;&lt;/h3&gt;
-  Starling tempers out 126/125, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;, but 31, 46 or 58 also work nicely.&lt;br /&gt;
+  Starling tempers out &lt;a class="wiki_link" href="/126_125"&gt;126/125&lt;/a&gt;, and like marvel it has the same generators as the 5-limit. An excellent tuning for starling is &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;, but 31, 46 or 58 also work nicely.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:164:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc82"&gt;&lt;a name="Rank 3 temperaments--Gamelismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:164 --&gt;&lt;a class="wiki_link" href="/Gamelismic%20family"&gt;Gamelismic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:164:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc82"&gt;&lt;a name="Rank-3 temperaments--Gamelismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:164 --&gt;&lt;a class="wiki_link" href="/Gamelismic%20family"&gt;Gamelismic family&lt;/a&gt;&lt;/h3&gt;
   Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma, 1029/1024.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:166:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc83"&gt;&lt;a name="Rank 3 temperaments--Breed family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:166 --&gt;&lt;a class="wiki_link" href="/Breed%20family"&gt;Breed family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:166:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc83"&gt;&lt;a name="Rank-3 temperaments--Breed family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:166 --&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, 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:168:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc84"&gt;&lt;a name="Rank 3 temperaments--Ragisma family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:168 --&gt;&lt;a class="wiki_link" href="/Ragisma%20family"&gt;Ragisma family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:168:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc84"&gt;&lt;a name="Rank-3 temperaments--Ragisma family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:168 --&gt;&lt;a class="wiki_link" href="/Ragisma%20family"&gt;Ragisma family&lt;/a&gt;&lt;/h3&gt;
   The 7-limit rank three microtemperament which tempers out the ragisma, 4375/4374, extends to various higher limit rank three temperaments such as thor.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:170:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc85"&gt;&lt;a name="Rank 3 temperaments--Landscape family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:170 --&gt;&lt;a class="wiki_link" href="/Landscape%20family"&gt;Landscape family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:170:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc85"&gt;&lt;a name="Rank-3 temperaments--Landscape family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:170 --&gt;&lt;a class="wiki_link" href="/Landscape%20family"&gt;Landscape family&lt;/a&gt;&lt;/h3&gt;
   The 7-limit rank three microtemperament which tempers out the lanscape comma, 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:172:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc86"&gt;&lt;a name="Rank 3 temperaments--Hemifamity family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:172 --&gt;&lt;a class="wiki_link" href="/Hemifamity%20family"&gt;Hemifamity family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:172:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc86"&gt;&lt;a name="Rank-3 temperaments--Hemifamity family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:172 --&gt;&lt;a class="wiki_link" href="/Hemifamity%20family"&gt;Hemifamity family&lt;/a&gt;&lt;/h3&gt;
   The hemifamity family of rank three temperaments tempers out the hemifamity comma, 5120/5103.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:174:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc87"&gt;&lt;a name="Rank 3 temperaments--Porwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:174 --&gt;&lt;a class="wiki_link" href="/Porwell%20family"&gt;Porwell family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:174:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc87"&gt;&lt;a name="Rank-3 temperaments--Porwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:174 --&gt;&lt;a class="wiki_link" href="/Porwell%20family"&gt;Porwell family&lt;/a&gt;&lt;/h3&gt;
   The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:176:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc88"&gt;&lt;a name="Rank 3 temperaments--Horwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:176 --&gt;&lt;a class="wiki_link" href="/Horwell%20family"&gt;Horwell family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:176:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc88"&gt;&lt;a name="Rank-3 temperaments--Horwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:176 --&gt;&lt;a class="wiki_link" href="/Horwell%20family"&gt;Horwell family&lt;/a&gt;&lt;/h3&gt;
   The horwell family of rank three temperaments tempers out the horwell comma, 65625/65536.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:178:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc89"&gt;&lt;a name="Rank 3 temperaments--Hemimage family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:178 --&gt;&lt;a class="wiki_link" href="/Hemimage%20family"&gt;Hemimage family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:178:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc89"&gt;&lt;a name="Rank-3 temperaments--Hemimage family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:178 --&gt;&lt;a class="wiki_link" href="/Hemimage%20family"&gt;Hemimage family&lt;/a&gt;&lt;/h3&gt;
   The hemimage family of rank three temperaments tempers out the hemimage comma, 10976/10935.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:180:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc90"&gt;&lt;a name="Rank 3 temperaments--Sensamagic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:180 --&gt;&lt;a class="wiki_link" href="/Sensamagic%20family"&gt;Sensamagic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:180:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc90"&gt;&lt;a name="Rank-3 temperaments--Sensamagic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:180 --&gt;&lt;a class="wiki_link" href="/Sensamagic%20family"&gt;Sensamagic family&lt;/a&gt;&lt;/h3&gt;
   These temper out 245/243.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:182:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc91"&gt;&lt;a name="Rank 3 temperaments--Keemic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:182 --&gt;&lt;a class="wiki_link" href="/Keemic%20family"&gt;Keemic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:182:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc91"&gt;&lt;a name="Rank-3 temperaments--Keemic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:182 --&gt;&lt;a class="wiki_link" href="/Keemic%20family"&gt;Keemic family&lt;/a&gt;&lt;/h3&gt;
   These temper out 875/864.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:184:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc92"&gt;&lt;a name="Rank 3 temperaments--Sengic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:184 --&gt;&lt;a class="wiki_link" href="/Sengic%20family"&gt;Sengic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:184:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc92"&gt;&lt;a name="Rank-3 temperaments--Sengic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:184 --&gt;&lt;a class="wiki_link" href="/Sengic%20family"&gt;Sengic family&lt;/a&gt;&lt;/h3&gt;
   These temper out the senga, 686/675.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:186:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc93"&gt;&lt;a name="Rank 3 temperaments--Orwellismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:186 --&gt;&lt;a class="wiki_link" href="/Orwellismic%20family"&gt;Orwellismic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:186:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc93"&gt;&lt;a name="Rank-3 temperaments--Orwellismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:186 --&gt;&lt;a class="wiki_link" href="/Orwellismic%20family"&gt;Orwellismic family&lt;/a&gt;&lt;/h3&gt;
   These temper out 1728/1715.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:188:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc94"&gt;&lt;a name="Rank 3 temperaments--Nuwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:188 --&gt;&lt;a class="wiki_link" href="/Nuwell%20family"&gt;Nuwell family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:188:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc94"&gt;&lt;a name="Rank-3 temperaments--Nuwell family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:188 --&gt;&lt;a class="wiki_link" href="/Nuwell%20family"&gt;Nuwell family&lt;/a&gt;&lt;/h3&gt;
   These temper out the nuwell comma, 2430/2401.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:190:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc95"&gt;&lt;a name="Rank 3 temperaments--Octagar family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:190 --&gt;&lt;a class="wiki_link" href="/Octagar%20family"&gt;Octagar family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:190:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc95"&gt;&lt;a name="Rank-3 temperaments--Octagar family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:190 --&gt;&lt;a class="wiki_link" href="/Octagar%20family"&gt;Octagar family&lt;/a&gt;&lt;/h3&gt;
   The octagar family of rank three temperaments tempers out the octagar comma, 4000/3969.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:192:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc96"&gt;&lt;a name="Rank 3 temperaments--Mirkwai family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:192 --&gt;&lt;a class="wiki_link" href="/Mirkwai%20family"&gt;Mirkwai family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:192:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc96"&gt;&lt;a name="Rank-3 temperaments--Mirkwai family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:192 --&gt;&lt;a class="wiki_link" href="/Mirkwai%20family"&gt;Mirkwai family&lt;/a&gt;&lt;/h3&gt;
   The mirkwai family of rank three temperaments tempers out the mirkwai comma, 16875/16807.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:194:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc97"&gt;&lt;a name="Rank 3 temperaments--Hemimean family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:194 --&gt;&lt;a class="wiki_link" href="/Hemimean%20family"&gt;Hemimean family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:194:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc97"&gt;&lt;a name="Rank-3 temperaments--Hemimean family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:194 --&gt;&lt;a class="wiki_link" href="/Hemimean%20family"&gt;Hemimean family&lt;/a&gt;&lt;/h3&gt;
   The hemimean family of rank three temperaments tempers out the hemimean comma, 3136/3125.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:196:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc98"&gt;&lt;a name="Rank 3 temperaments--Mirwomo family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:196 --&gt;&lt;a class="wiki_link" href="/Mirwomo%20family"&gt;Mirwomo family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:196:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc98"&gt;&lt;a name="Rank-3 temperaments--Mirwomo family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:196 --&gt;&lt;a class="wiki_link" href="/Mirwomo%20family"&gt;Mirwomo family&lt;/a&gt;&lt;/h3&gt;
   The mirwomo family of rank three temperaments tempers out the mirwomo comma, 33075/32768.&lt;br /&gt;
 &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--Dimcomp family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:198 --&gt;&lt;a class="wiki_link" href="/Dimcomp%20family"&gt;Dimcomp family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:198:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc99"&gt;&lt;a name="Rank-3 temperaments--Dimcomp family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:198 --&gt;&lt;a class="wiki_link" href="/Dimcomp%20family"&gt;Dimcomp family&lt;/a&gt;&lt;/h3&gt;
   The dimcomp family of rank three temperaments tempers out the dimcomp comma, 390625/388962.&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--Tolermic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:200 --&gt;&lt;a class="wiki_link" href="/Tolermic%20family"&gt;Tolermic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:200:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc100"&gt;&lt;a name="Rank-3 temperaments--Tolermic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:200 --&gt;&lt;a class="wiki_link" href="/Tolermic%20family"&gt;Tolermic family&lt;/a&gt;&lt;/h3&gt;
   These temper out the tolerma, 179200/177147.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:202:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc101"&gt;&lt;a name="Rank 3 temperaments--Kleismic rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:202 --&gt;&lt;a class="wiki_link" href="/Kleismic%20rank%20three%20family"&gt;Kleismic rank three family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:202:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc101"&gt;&lt;a name="Rank-3 temperaments--Kleismic rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:202 --&gt;&lt;a class="wiki_link" href="/Kleismic%20rank%20three%20family"&gt;Kleismic rank three family&lt;/a&gt;&lt;/h3&gt;
   These are the rank three temperaments tempering out the kleisma, 15625/15552. 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:204:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc102"&gt;&lt;a name="Rank 3 temperaments--Diaschismic rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:204 --&gt;&lt;a class="wiki_link" href="/Diaschismic%20rank%20three%20family"&gt;Diaschismic rank three family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:204:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc102"&gt;&lt;a name="Rank-3 temperaments--Diaschismic rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:204 --&gt;&lt;a class="wiki_link" href="/Diaschismic%20rank%20three%20family"&gt;Diaschismic rank three family&lt;/a&gt;&lt;/h3&gt;
   These are the rank three temperaments tempering out the dischisma, 2048/2025. 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:206:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc103"&gt;&lt;a name="Rank 3 temperaments--Didymus rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:206 --&gt;&lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&gt;Didymus rank three family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:206:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc103"&gt;&lt;a name="Rank-3 temperaments--Didymus rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:206 --&gt;&lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&gt;Didymus rank three family&lt;/a&gt;&lt;/h3&gt;
   These are the rank three temperaments tempering out the didymus comma, 81/80. 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:208:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc104"&gt;&lt;a name="Rank 3 temperaments--Porcupine rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:208 --&gt;&lt;a class="wiki_link" href="/Porcupine%20rank%20three%20family"&gt;Porcupine rank three family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:208:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc104"&gt;&lt;a name="Rank-3 temperaments--Porcupine rank three family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:208 --&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 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;
+  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:210:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc105"&gt;&lt;a name="Rank 3 temperaments--Archytas family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:210 --&gt;&lt;a class="wiki_link" href="/Archytas%20family"&gt;Archytas family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:210:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc105"&gt;&lt;a name="Rank-3 temperaments--Archytas family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:210 --&gt;&lt;a class="wiki_link" href="/Archytas%20family"&gt;Archytas family&lt;/a&gt;&lt;/h3&gt;
   Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:212:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc106"&gt;&lt;a name="Rank 3 temperaments--Jubilismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:212 --&gt;&lt;a class="wiki_link" href="/Jubilismic%20family"&gt;Jubilismic family&lt;/a&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:212:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc106"&gt;&lt;a name="Rank-3 temperaments--Jubilismic family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:212 --&gt;&lt;a class="wiki_link" href="/Jubilismic%20family"&gt;Jubilismic family&lt;/a&gt;&lt;/h3&gt;
   Jubilismic temperament tempers out 50/49 and thereby identifies the two septimal tritones, 7/5 and 10/7.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:214:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc107"&gt;&lt;a name="Rank 3 temperaments--Semiphore family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:214 --&gt;&lt;a class="wiki_link" href="/Semiphore%20family"&gt;Semiphore family&lt;/a&gt;&lt;/h3&gt;
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   Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like &amp;quot;semi-fourth&amp;quot;.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:216:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc108"&gt;&lt;a name="Rank-3 temperaments--Mint family"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:216 --&gt;&lt;a class="wiki_link" href="/Mint%20family"&gt;Mint family&lt;/a&gt;&lt;/h3&gt;
   The mint temperament tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.&lt;br /&gt;
 &lt;br /&gt;
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   These temper out the valinorsma, 176/175.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:220:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc110"&gt;&lt;a name="Rank-3 temperaments--Rastmic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:220 --&gt;&lt;a class="wiki_link" href="/Rastmic%20temperaments"&gt;Rastmic temperaments&lt;/a&gt;&lt;/h3&gt;
   These temper out the rastma, 243/242.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:222:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc111"&gt;&lt;a name="Rank-3 temperaments--Werckismic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:222 --&gt;&lt;a class="wiki_link" href="/Werckismic%20temperaments"&gt;Werckismic temperaments&lt;/a&gt;&lt;/h3&gt;
   These temper out the werckisma, 441/440.&lt;br /&gt;
 &lt;br /&gt;
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   These temper out the swetisma, 540/539.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:226:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc113"&gt;&lt;a name="Rank-3 temperaments--Lehmerismic temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:226 --&gt;&lt;a class="wiki_link" href="/Lehmerismic%20temperaments"&gt;Lehmerismic temperaments&lt;/a&gt;&lt;/h3&gt;
   These temper out the lehmerisma, 3025/3024.&lt;br /&gt;
 &lt;br /&gt;
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   These temper out the kalisma, 9801/9800.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:230:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc115"&gt;&lt;a name="Rank-4 temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:230 --&gt;&lt;a class="wiki_link" href="/Rank%20four%20temperaments"&gt;Rank-4 temperaments&lt;/a&gt;&lt;/h1&gt;
   &lt;br /&gt;
 Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit scales&lt;/a&gt; can be constructed for them.&lt;br /&gt;