Tour of regular temperaments: Difference between revisions

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==Why would I want to use a regular temperament?==

Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. ~~They~~ are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals.

Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.

==What do I need to know to understand all the numbers on the pages for individual regular temperaments?==

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===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)===

This family tempers out the limma, [8 -5 0> = 256/243, which implies 5-edo.

This family tempers out the limma, [8 -5 0> = 256/243, which implies [[5-edo]].

===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)===

This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies 7-edo.

This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies [[7-edo]].

===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)===

The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0>. This family includes the 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-edo, offset from one another justly tuned 5/4. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0>, which implies [[12-edo]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

== 2.3.5 Families ==

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===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)===

The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, 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]] temperament, where the intervals 50/49 and 64/63 are tempered out~~, of which~~ [[22edo]] is an excellent tuning.

The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. An obvious 7-limit interpretation of the period is 7/5, which makes [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out. [[22edo]] is an excellent pajara tuning.

===[[Bug family|Bug or Gugu family]] (P8, P4/2)===

This low-accuracy family of temperaments tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250¢, two of which make ~4/3. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore aka Zozo.

===[[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)===

This tempers out the immunity comma, [16 -13 2> (1638400/1594323).

This tempers out the immunity comma, [16 -13 2> (1638400/1594323). It has the same pergen as Bug/Gugu. Its generator is ~729/640 = ~247¢, two of which make ~4/3. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore aka Zozo.

~~===[[~~Bug ~~family|Bug or~~ Gugu ~~family]] (P8, P4~~/2)===

~~This tempers out [[27~~/~~25]],~~ the ~~large limma or bug comma. The~~ generator is ~~an approximate~~ 6~~/5 or 10/9~~.

===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)===

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). 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]] (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 a neutral-sized 3rd of ~350¢ that is 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]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.

===[[Augmented_family|Augmented or Trigu family]] (P8/3, P5)===

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===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)===

The porcupine family tempers out [1 -5 3> = [[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]].

The porcupine family tempers out [1 -5 3> = [[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]]. An important 7-limit extension also tempers out 64/63.

===[[Laconic family|Laconic or Latrigubi family]] (P8, P5/3)===

This tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others.

This low-accuracy family of temperaments tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. The generator is ~10/9 = ~230¢. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others. An obvious 7-limit interpretation of the generator is ~8/7, which leads to Gamelismic aka Latrizo.

===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)===

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===[[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)===

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

This tempers out the [[vulture comma]], [24 -21 4>. Its generator is ~320/243 = ~475¢, four of which make ~3/1. An obvious 7-limit interpretation of the generator is 21/16, which makes Saquadru.

===[[Comic family|Comic or Saquadyobi family]] (P8/2, M2/4)===

This tempers out the comic comma, [13 -14 4> = 5120000/4782969

This tempers out the comic comma, [13 -14 4> = 5120000/4782969. Its generator is ~81/80 = 55¢. An obvious 11-limit interpretation of the generator is 33/32, which makes Laquadlo.

===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)===

This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5>.

This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5>. The period is 59049/51200, and five periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

===[[Amity family|Amity or Saquinyo family]] (P8, P11/5)===

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

This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5>. The generator is 243/200 = ~339.5¢, five of which make ~8/3. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)===

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===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)===

This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938.

This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938. The period is ~4374/3125 = [1 7 -5>, two of which make an octave. The generator is ~27/25, five of which make ~3/2.

===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)===

This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10>.

This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10>. The period is ~524288/455625 = [19 -6 -4>, five of which equals an octave. The generator is ~16/15. A period plus a generator makes half a fifth. An obvious 7-limit interpretation of the period is 8/7.

===[[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)===

This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15>.

This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15>. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. An obvious 7-limit interpretation of the period is 8/7.

=== [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6) ===

This tempers out Ampersand's comma = 34171875/33554432 = [-25 7 6>. The generator is ~16/15, of which six make ~3/2. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the [[Miracle]] temperament.

===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)===

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===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)===

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

The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = [-21 3 7>, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 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|Orwell or Sepru]] temperament.

===[[Wesley family|Wesley or Lasepyobi family]] (P8, ccP4/7)===

This tempers out the wesley comma, [-13 -2 7> = 78125/73728. Seven generators equals a double-compound 4th of ~16/3.

This tempers out the wesley comma, [-13 -2 7> = 78125/73728. The generator is ~125/96 = ~412¢. Seven generators equals a double-compound 4th of ~16/3. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepru temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29-edo]].

===[[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)===

The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Seven generators equals a double-compound 5th of ~6/1.Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]].

The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Its generator is ~162/125 = ~443¢. Seven generators equals a double-compound 5th of ~6/1.Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.

===[[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)===

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===[[Mutt family|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)===

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

This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21>, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is not 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord.

===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)===

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===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)===

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. The generator is ~[-14 3 4> = ~55¢, and nine of them equal ~4/3. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritrilu temperament.

===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, ccP4/9)===

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===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)===

The sycamore family tempers out the sycamore comma, [-16 -6 11> = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. Eleven of these generators equals ~3/2.

===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c4P4/13)===

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators equals a quadruple-compound 4th.

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators of about 408¢ equals a quadruple-compound 4th. An obvious 3-limit interpretation of the generator is 81/64, which implies 53-edo, which is a good tuning for this high-accuracy family of temperaments.

===[[Luna family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)===

This tempers out the luna comma, [38 -2 -15> (274877906944/274658203125). Fifteen generators equals a double-compound 4th of ~16/3.

This tempers out the luna comma, [38 -2 -15> (274877906944/274658203125). The generator is ~{18 -1 -7> = ~193¢. Fifteen generators equals a double-compound 4th of ~16/3.

===[[Minortonic family|Minortonic or Trila-segu family]] (P8, ccP5/17)===

This tempers out the minortone comma, [-16 35 -17>. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th 6/1.

This tempers out the minortone comma, [-16 35 -17>. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th = ~6/1.

===[[Maja family|Maja or Saseyo family]] (P8, c6P4/17)===

This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. Seventeen generators equals a sextuple-compound 4th.

This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th.

===[[Maquila family|Maquila or Trisa-segu family]] (P8, c7P5/17)===

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17>. Seventeen generators equals a septuple-compound 5th.

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17>. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.

===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)===

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===[[Garischismic temperaments|Garischismic or Sasaru clan]] (P8, P5)===

This clan tempers out the garischisma, [25 -14 0 -1> = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048). This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.

This clan tempers out the garischisma, [25 -14 0 -1> = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048), and 7/4 to a double-diminished 8ve. This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.

===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) ===

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=== Sasa-zozo clan (P8, P5/2) ===

This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament.

This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)===

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=== Latriru clan (P8, P11/3) ===

This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = 81/56. ~~It includes as a strong extension~~ the [[Liese]] temperament, which is ~~in the~~ Meantone ~~family~~.

This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. An obvious 7-limit interpretation of the generator is 7/5, leading to the [[Liese]] temperament, which is a weak extension of Meantone.

===[[Stearnsmic temperaments|Stearnsmic or Latribiru clan]] (P8/2, P4/3)===

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649.

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6> = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternately one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. Equating this generator to ~10/9 creates a weak extension of the [[Porcupine|Porcupine or Triyo]] temperament, as does equating the period to ~7/5.

=== Laquadru clan (P8, P11/4) ===

This clan tempers out [-3 9 0 -4> = 42.3¢. Generator = 9/7. It includes as a strong extension the [[Squares]] temperament, which is ~~in the~~ Meantone ~~family~~.

This clan tempers out [-3 9 0 -4> = 42.3¢. Generator = ~9/7. It includes as a strong extension the [[Squares]] temperament, which is a weak extension of Meantone.

=== Saquadru clan (P8, P12/4) ===

This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = 21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.

This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = ~21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.

=== Saquinzo clan (P8, P12/5) ===

This clan tempers out [5 -12 0 5> = 20.7¢. ~~It includes as a strong extension~~ the [[Magic]] temperament, which is in the Magic family.

This clan tempers out [5 -12 0 5> = 20.7¢. Generator = ~243/196 = ~380¢. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[Magic]] temperament, which is in the Magic family.

=== Sepru clan (P8, P12/7) ===

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=== [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) ===

This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The ~~pergen's~~ M3 generator ~~equals~~ 5/4. The half-octave period ~~equals~~ 7/5.

This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The M3 generator is ~5/4. The half-octave period is ~7/5 or ~10/7.

===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M2)===

This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. Two M2 generators equals 5/4, and five of them equals 7/4.

This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. The M2 generator is ~28/25 = ~194¢. Two generators equals ~5/4, and five of them equals ~7/4.

===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, M2/2)===

This clan tempers out the quince, [-15 0 -2 7> = 823543/819200. Two generators equals 8/7 (a M2), and seven generators equals 8/5.

This clan tempers out the quince, [-15 0 -2 7> = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[Magic]] temperament, which is in the Magic family.

== 3.5.7 Clans ==

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===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)===

This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator = 9/7, and two generators equals 5/3.

This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator is ~9/7, and two generators equals ~5/3.

===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, cM7/4)===

This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. ~~Four~~ generators = a compound major 7th = 27/7.

This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5> = 16875/16807. The generator is ~7/5, and four generators equals a compound major 7th = ~27/7.

=Rank-3 temperaments=

@@ Line 9: / Line 9: @@
 ==Why would I want to use a regular temperament?==
-Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. They are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals.
+Regular temperaments are of most use to musicians who want their music to sound as much as possible like Just intonation, but without the difficulties normally associated with JI, such as wolf intervals, commas, and comma pumps. Specifically, if your chord progression pumps a comma, and you want to avoid pitch shifts, wolf intervals, and/or tonic drift, that comma must be tempered out. Temperaments are also of interest to musicians wishing to exploit the unique possibilities that arise when ratios that are distinct in JI become equated. For instance, 10/9 and 9/8 are equated in meantone. Equating distinct ratios through temperament allows for the construction of musical "puns", which are melodies or chord progressions that exploit the multiplicity of "meanings" of tempered intervals. Finally, some use temperaments solely for their sound. For example, one might like the sound of neutral 3rds, without caring much what ratio they are tuned to. Thus one might use Rastmic even though no commas are pumped.
 ==What do I need to know to understand all the numbers on the pages for individual regular temperaments?==
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 ===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)===
-This family tempers out the limma, [8 -5 0> = 256/243, which implies 5-edo.
+This family tempers out the limma, [8 -5 0> = 256/243, which implies [[5-edo]].
 ===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)===
-This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies 7-edo.
+This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies [[7-edo]].
 ===[[Pythagorean family|Pythagorean or Lalawa family]] (P8/12, ^1)===
-The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0&gt;. This family includes the 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-edo, offset from one another justly tuned 5/4. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.
+The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0&gt;, which implies [[12-edo]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.
 == 2.3.5 Families ==
@@ Line 61: / Line 61: @@
 ===[[Diaschismic family|Diaschismic or Sagugu family]] (P8/2, P5)===
-The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, 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]] temperament, where the intervals 50/49 and 64/63 are tempered out, of which [[22edo]] is an excellent tuning.
+The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. An obvious 7-limit interpretation of the period is 7/5, which makes [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out. [[22edo]] is an excellent pajara tuning.
+===[[Bug family|Bug or Gugu family]] (P8, P4/2)===
+This low-accuracy family of temperaments tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250¢, two of which make ~4/3. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore aka Zozo.
 ===[[Immunity family|Immunity or Sasa-yoyo family]] (P8, P4/2)===
-This tempers out the immunity comma, [16 -13 2> (1638400/1594323).
+This tempers out the immunity comma, [16 -13 2> (1638400/1594323). It has the same pergen as Bug/Gugu. Its generator is ~729/640 = ~247¢, two of which make ~4/3. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore aka Zozo.
-===[[Bug family|Bug or Gugu family]] (P8, P4/2)===
-This tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or 10/9.
 ===[[Dicot family|Dicot or Yoyo family]] (P8, P5/2)===
-The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). 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]] (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 a neutral-sized 3rd of ~350¢ that is 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]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.
 ===[[Augmented_family|Augmented or Trigu  family]] (P8/3, P5)===
@@ Line 76: / Line 76: @@
 ===[[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)===
-The porcupine family tempers out [1 -5 3> = [[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]].
+The porcupine family tempers out [1 -5 3> = [[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]]. An important 7-limit extension also tempers out 64/63.
 ===[[Laconic family|Laconic or Latrigubi family]] (P8, P5/3)===
-This tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others.
+This low-accuracy family of temperaments tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. The generator is ~10/9 = ~230¢. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others. An obvious 7-limit interpretation of the generator is ~8/7, which leads to Gamelismic aka Latrizo.
 ===[[Dimipent family|Dimipent or Quadgu family]] (P8/4, P5)===
@@ Line 88: / Line 88: @@
 ===[[Vulture family|Vulture or Sasa-quadyo family]] (P8, P12/4)===
-This tempers out the [[vulture comma]], [24 -21 4&gt;.
+This tempers out the [[vulture comma]], [24 -21 4&gt;. Its generator is ~320/243 = ~475¢, four of which make ~3/1. An obvious 7-limit interpretation of the generator is 21/16, which makes Saquadru.
 ===[[Comic family|Comic or Saquadyobi family]] (P8/2, M2/4)===
-This tempers out the comic comma, [13 -14 4> = 5120000/4782969
+This tempers out the comic comma, [13 -14 4> = 5120000/4782969. Its generator is ~81/80 = 55¢. An obvious 11-limit interpretation of the generator is 33/32, which makes Laquadlo.
 ===[[Pental family|Pental or Trila-quingu family]] (P8/5, P5)===
-This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5&gt;.
+This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5&gt;. The period is 59049/51200, and five periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.
 ===[[Amity family|Amity or Saquinyo family]] (P8, P11/5)===
-This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5&gt;.
+This tempers out the [[amity comma]], 1600000/1594323 = [9 -13 5&gt;. The generator is 243/200 = ~339.5¢, five of which make ~8/3. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.
 ===[[Magic family|Magic or Laquinyo family]] (P8, P12/5)===
@@ Line 103: / Line 103: @@
 ===[[Fifive family|Fifive or Saquinbiyo family]] (P8/2, P5/5)===
-This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938.
+This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938. The period is ~4374/3125 = [1 7 -5>, two of which make an octave. The generator is ~27/25, five of which make ~3/2.
 ===[[Qintosec family|Qintosec or Quadsa-quinbigu family]] (P8/5, P5/2)===
-This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10&gt;.
+This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10&gt;. The period is ~524288/455625 = [19 -6 -4>, five of which equals an octave. The generator is ~16/15. A period plus a generator makes half a fifth. An obvious 7-limit interpretation of the period is 8/7.
 ===[[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)===
-This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15&gt;.
+This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15&gt;. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. An obvious 7-limit interpretation of the period is 8/7.
+=== [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6) ===
+This tempers out Ampersand's comma = 34171875/33554432 = [-25 7 6>. The generator is ~16/15, of which six make ~3/2. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the [[Miracle]] temperament.
 ===[[Kleismic family|Kleismic or Tribiyo family]] (P8, P12/6)===
@@ Line 115: / Line 118: @@
 ===[[Semicomma_family|Orwell or Sepru, and the semicomma or Lasepyo family]] (P8, P12/7)===
-The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = [-21 3 7&gt;, is tempered out by the members of the semicomma family. It doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as ~7/6, leading to [[orwell]] temperament.
+The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = [-21 3 7&gt;, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 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|Orwell or Sepru]] temperament.
 ===[[Wesley family|Wesley or Lasepyobi family]] (P8, ccP4/7)===
-This tempers out the wesley comma, [-13 -2 7> = 78125/73728. Seven generators equals a double-compound 4th of ~16/3.
+This tempers out the wesley comma, [-13 -2 7> = 78125/73728. The generator is ~125/96 = ~412¢. Seven generators equals a double-compound 4th of ~16/3. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepru temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29-edo]].
 ===[[Sensipent family|Sensipent or Sepgu family]] (P8, ccP5/7)===
-The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Seven generators equals a double-compound 5th of ~6/1.Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]].
+The sensipent (sensi) family tempers out the [[sensipent comma]], [2 9 -7> (78732/78125), also known as the medium semicomma. Its generator is ~162/125 = ~443¢. Seven generators equals a double-compound 5th of ~6/1.Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.
 ===[[Vishnuzmic family|Vishnuzmic or Sasepbigu family]] (P8/2, P4/7)===
@@ Line 127: / Line 130: @@
 ===[[Mutt family|Mutt or Trila-septriyo family]] (P8/3, ccP4/7)===
-This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21&gt;, leading to some strange properties.
+This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21&gt;, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord.
 ===[[Würschmidt family|Würschmidt or Saquadbigu family]] (P8, ccP5/8)===
@@ Line 133: / Line 136: @@
 ===[[Escapade family|Escapade or Sasa-tritrigu family]] (P8, P4/9)===
-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. The generator is ~[-14 3 4> = ~55¢, and nine of them equal ~4/3. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritrilu temperament.
 ===[[Shibboleth family|Shibboleth or Tritriyo family]] (P8, ccP4/9)===
@@ Line 139: / Line 142: @@
 ===[[Sycamore family|Sycamore or Laleyo family]] (P8, P5/11)===
 The sycamore family tempers out the sycamore comma, [-16 -6 11&gt; = 48828125/47775744, which is the amount by which five stacked chromatic semitones, 25/24, exceed 6/5, and hence also the amount six exceeds 5/4. Eleven of these generators equals ~3/2.
 ===[[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c<sup>4</sup>P4/13)===
-This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators equals a quadruple-compound 4th.
+This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators of about 408¢ equals a quadruple-compound 4th. An obvious 3-limit interpretation of the generator is 81/64, which implies 53-edo, which is a good tuning for this high-accuracy family of temperaments.
 ===[[Luna family|Luna or Sasa-quintrigu family]] (P8, ccP4/15)===
-This tempers out the luna comma, [38 -2 -15&gt; (274877906944/274658203125). Fifteen generators equals a double-compound 4th of ~16/3.
+This tempers out the luna comma, [38 -2 -15&gt; (274877906944/274658203125). The generator is ~{18 -1 -7> = ~193¢. Fifteen generators equals a double-compound 4th of ~16/3.
 ===[[Minortonic family|Minortonic or Trila-segu family]] (P8, ccP5/17)===
-This tempers out the minortone comma, [-16 35 -17&gt;. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th 6/1.
+This tempers out the minortone comma, [-16 35 -17&gt;. The head of the family is minortonic temperament, with a generator of a minor tone (~10/9). Seventeen generators equals a double-compound 5th = ~6/1.
 ===[[Maja family|Maja or Saseyo family]] (P8, c<sup>6</sup>P4/17)===
-This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. Seventeen generators equals a sextuple-compound 4th.
+This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th.
 ===[[Maquila family|Maquila or Trisa-segu family]] (P8, c<sup>7</sup>P5/17)===
-This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17&gt;. Seventeen generators equals a septuple-compound 5th.
+This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17&gt;. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.
 ===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)===
@@ Line 172: / Line 175: @@
 ===[[Garischismic temperaments|Garischismic or Sasaru clan]] (P8, P5)===
-This clan tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048). This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.
+This clan tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048), and 7/4 to a double-diminished 8ve. This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.
 ===[[Trienstonic clan|Trienstonic or Zo clan]] (P8, P5) ===
@@ Line 181: / Line 184: @@
 === Sasa-zozo clan (P8, P5/2) ===
-This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament.
+This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.
 ===[[Gamelismic clan|Gamelismic or Latrizo clan]] (P8, P5/3)===
@@ Line 187: / Line 190: @@
 === Latriru clan (P8, P11/3) ===
-This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = 81/56. It includes as a strong extension the [[Liese]] temperament, which is in the Meantone family.
+This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. An obvious 7-limit interpretation of the generator is 7/5, leading to the [[Liese]] temperament, which is a weak extension of Meantone.
 ===[[Stearnsmic temperaments|Stearnsmic or Latribiru clan]] (P8/2, P4/3)===
-Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 118098/117649.
+Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternately one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. Equating this generator to ~10/9 creates a weak extension of the [[Porcupine|Porcupine or Triyo]] temperament, as does equating the period to ~7/5.
 === Laquadru clan (P8, P11/4) ===
-This clan tempers out  [-3 9 0 -4> = 42.3¢. Generator = 9/7. It includes as a strong extension the [[Squares]] temperament, which is in the Meantone family.
+This clan tempers out  [-3 9 0 -4> = 42.3¢. Generator = ~9/7. It includes as a strong extension the [[Squares]] temperament, which is a weak extension of Meantone.
 === Saquadru clan (P8, P12/4) ===
-This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = 21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.
+This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = ~21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.
 === Saquinzo clan (P8, P12/5) ===
-This clan tempers out [5 -12 0 5> = 20.7¢. It includes as a strong extension the [[Magic]] temperament, which is in the Magic family.
+This clan tempers out [5 -12 0 5> = 20.7¢. Generator = ~243/196 = ~380¢. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[Magic]] temperament, which is in the Magic family.
 === Sepru clan (P8, P12/7) ===
@@ Line 217: / Line 220: @@
 === [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3) ===
-This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The pergen's M3 generator equals 5/4. The half-octave period equals 7/5.
+This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The M3 generator is ~5/4. The half-octave period is ~7/5 or ~10/7.
 ===[[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M2)===
-This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. Two M2 generators equals 5/4, and five of them equals 7/4.
+This clan tempers out the hemimean comma, [6 0 -5 2> = 3136/3125. The M2 generator is ~28/25 = ~194¢. Two generators equals ~5/4, and five of them equals ~7/4.
 ===[[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, M2/2)===
-This clan tempers out the quince, [-15 0 -2 7&gt; = 823543/819200. Two generators equals 8/7 (a M2), and seven generators equals 8/5.
+This clan tempers out the quince, [-15 0 -2 7&gt; = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[Magic]] temperament, which is in the Magic family.
 == 3.5.7 Clans ==
@@ Line 229: / Line 232: @@
 ===[[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M3)===
-This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator = 9/7, and two generators equals 5/3.
+This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator is ~9/7, and two generators equals ~5/3.
 ===[[Mirkwai clan|Mirkwai or Quinru-aquadyo Noca clan]] (P12, cM7/4)===
-This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5&gt; = 16875/16807. Four generators = a compound major 7th = 27/7.
+This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5&gt; = 16875/16807. The generator is ~7/5, and four generators equals a compound major 7th = ~27/7.
 =Rank-3 temperaments=