Tour of regular temperaments: Difference between revisions

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; Blackwood family (P8/5, ^1)

: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is the [[blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp ~~5th~~. Its color name is Sawati.

: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is the [[blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp fifth. Its color name is Sawati.

; [[Whitewood family]] (P8/7, ^1)

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: The alphatricot family tempers out the [[alphatricot comma]], {{monzo| 39 -29 3 }}. The generator is {{nowrap| ~59049/40960 ({{monzo| -13 10 -1 }}) {{=}} 633{{c}} }}, or its octave inverse {{nowrap| ~81920/59049 {{=}} 567{{c}} }}. Three of the former generators equals the third harmonic, ~3/1. 5/4 is equated to 29 of these generators octave-reduced. Its color name is Quadsa-triyoti. An obvious 7-limit interpretation of the generator is {{nowrap| 81/56 {{=}} 639{{c}} }}, a much simpler ratio which leads to the [[Tour of Regular Temperaments #Latriruti clan (P8, P11/3)|Latriruti clan]]. An obvious 13-limit interpretation is {{nowrap| 13/9 {{=}} 637¢ }}, an even simpler ratio implying the [[Tour of Regular Temperaments #Satrithoti clan (P8, P11/3)|Satrithoti clan]].

; [[~~Dimipent family|Dimipent or diminished~~ family]] (P8/4, P5)

; [[Diminished family]] (P8/4, P5)

: The ~~dimipent or~~ diminished family tempers out the major diesis or diminished comma, {{monzo| 3 4 -4 }} or [[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]]. 5/4 is equated to 1 fifth minus 1 period. Its color name is Quadguti.

: The diminished family tempers out the major diesis or diminished comma, {{monzo| 3 4 -4 }} or [[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]]. 5/4 is equated to 1 fifth minus 1 period. Its color name is Quadguti.

; [[Undim family]] (P8/4, P5)

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; [[Wesley family]] (P8, ccP4/7)

: This tempers out the [[wesley comma]], 78125/73728 ({{monzo| -13 -2 7 }}). The generator is {{nowrap| ~125/96 {{=}} ~412{{c}} }}. Seven generators equals a double-compound ~~4th~~ of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29edo]].

: This tempers out the [[wesley comma]], 78125/73728 ({{monzo| -13 -2 7 }}). The generator is {{nowrap| ~125/96 {{=}} ~412{{c}} }}. Seven generators equals a double-compound fourth of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29edo]].

; [[Sensipent family]] (P8, ccP5/7)

: The sensipent family tempers out the [[sensipent comma]], 78732/78125 ({{monzo| 2 9 -7 }}), also known as the medium semicomma. Its generator is {{nowrap| ~162/125 {{=}} ~443{{c}} }}. Seven generators equals a double-compound ~~5th~~ of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.

: The sensipent family tempers out the [[sensipent comma]], 78732/78125 ({{monzo| 2 9 -7 }}), also known as the medium semicomma. Its generator is {{nowrap| ~162/125 {{=}} ~443{{c}} }}. Seven generators equals a double-compound fifth of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.

; [[Vishnuzmic family]] (P8/2, P4/7)

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; [[Würschmidt family]] (P8, ccP5/8)

: The würschmidt family tempers out the [[würschmidt comma]], 393216/390625 ({{monzo| 17 1 -8 }}). Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect ~~5th~~); that is, {{nowrap| (5/4)8⋅(393216/390625) {{=}} 6 }}. It tends to generate the same mos scales as the [[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. Its color name is Saquadbiguti.

: The würschmidt family tempers out the [[würschmidt comma]], 393216/390625 ({{monzo| 17 1 -8 }}). Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect fifth); that is, {{nowrap| (5/4)8⋅(393216/390625) {{=}} 6 }}. It tends to generate the same mos scales as the [[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. Its color name is Saquadbiguti.

; [[Escapade family]] (P8, P4/9)

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; [[Mabila family]] (P8, c4P4/10)

: The mabila family tempers out the mabila comma, {{monzo| 28 -3 -10 }} (268435456/263671875). The generator is {{nowrap| ~512/375 {{=}} ~530{{c}} }}, three generators equals ~5/2 and ten of them equals a quadruple-compound ~~4th~~ of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.

: The mabila family tempers out the [[mabila comma]], {{monzo| 28 -3 -10 }} (268435456/263671875). The generator is {{nowrap| ~512/375 {{=}} ~530{{c}} }}, three generators equals ~5/2 and ten of them equals a quadruple-compound fourth of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.

; [[Sycamore family]] (P8, P5/11)

: The sycamore family tempers out the sycamore comma, {{monzo| -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. Its color name is Laleyoti.

: The sycamore family tempers out the [[sycamore comma]], {{monzo| -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. Its color name is Laleyoti.

; [[Quartonic family]] (P8, P4/11)

: The quartonic family tempers out the quartonic comma, {{monzo| 3 -18 11 }} (390625000/387420489). The generator is {{nowrap| ~250/243 {{=}} ~45{{c}} }}, seven generators equals ~6/5, and eleven generators equals ~4/3. Its color name is Saleyoti. An obvious 7-limit interpretation of the generator is ~36/35.

: The quartonic family tempers out the [[quartonic comma]], {{monzo| 3 -18 11 }} (390625000/387420489). The generator is {{nowrap| ~250/243 {{=}} ~45{{c}} }}, seven generators equals ~6/5, and eleven generators equals ~4/3. Its color name is Saleyoti. An obvious 7-limit interpretation of the generator is ~36/35.

; [[Lafa family]] (P8, P12/12)

: This tempers out the lafa comma, {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.

: This tempers out the [[lafa comma]], {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.

; [[Ditonmic family]] (P8, c4P4/13)

: This tempers out the ditonma, {{monzo| -27 -2 13 }} (1220703125/1207959552). Thirteen ~{{monzo| -12 -1 6 }} generators of about 407{{c}} equals a quadruple-compound ~~4th~~. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53edo, which is a good tuning for this high-accuracy family of temperaments. Its color name is Lala-theyoti.

: This tempers out the [[ditonma]], {{monzo| -27 -2 13 }} (1220703125/1207959552). Thirteen ~{{monzo| -12 -1 6 }} generators of about 407{{c}} equals a quadruple-compound fourth. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53edo, which is a good tuning for this high-accuracy family of temperaments. Its color name is Lala-theyoti.

; [[Luna family]] (P8, ccP4/15)

: This tempers out the luna comma, {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. Two generators equals ~5/4, and fifteen generators equals a double-compound ~~4th~~ of ~16/3. Its color name is Sasa-quintriguti.

: This tempers out the [[luna comma]], {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. Two generators equals ~5/4, and fifteen generators equals a double-compound fourth of ~16/3. Its color name is Sasa-quintriguti.

; [[Vavoom family]] (P8, P12/17)

: This tempers out the vavoom comma, {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti.

: This tempers out the [[vavoom comma]], {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti.

; [[Minortonic family]] (P8, ccP5/17)

: This tempers out the minortone comma, {{monzo| -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). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.

: This tempers out the [[minortone comma]], {{monzo| -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 fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.

; [[Maja family]] (P8, c6P4/17)

: This tempers out the maja comma, {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. Seventeen generators equals a sextuple-compound ~~4th~~. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.

: This tempers out the [[maja comma]], {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. Seventeen generators equals a sextuple-compound fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.

; [[Maquila family]] (P8, c7P5/17)

: This tempers out the maquila comma, {{monzo| 49 -6 -17 }} (562949953421312/556182861328125). The generator is {{nowrap| ~512/375 {{=}} ~535{{c}} }}. Seventeen generators equals a septuple-compound ~~5th~~. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.

: This tempers out the [[maquila comma]], {{monzo| 49 -6 -17 }} (562949953421312/556182861328125). The generator is {{nowrap| ~512/375 {{=}} ~535{{c}} }}. Seventeen generators equals a septuple-compound fifth. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.

; [[Gammic family]] (P8, P5/20)

: The gammic family tempers out the gammic comma, {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.

: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.

=== Clans defined by a 2.3.7 comma ===

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; [[Landscape microtemperaments]]

: Landscape rank-2 temperaments temper out the [[landscape comma]], {{monzo| -4 6 -6 3 }} (250047/250000). These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals. Its color name is Trizoguguti.

== Rank-3 temperaments ==

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; [[Cataharry family]] (P8, P4/2, ^1)

: Cataharry temperaments temper out the cataharry comma, {{monzo| -4 9 -2 -2 }} (19683/19600). In the pergen, half a ~~4th~~ is ~81/70, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Labiruguti.

: Cataharry temperaments temper out the cataharry comma, {{monzo| -4 9 -2 -2 }} (19683/19600). In the pergen, half a fourth is ~81/70, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Labiruguti.

; [[Breed family]] (P8, P5/2, ^1)

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; [[Alphaxenic rank three clan|Alphaxenic rank-3 clan]]

: These temper out the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} (131769/131072). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8/2, M2/4). Its color name is Laquadloti.

; [[Keenanismic temperaments]]

: These temper out the [[keenanisma]], {{monzo| -7 -1 1 1 1 }} (385/384). Its color name is Lozoyoti.

; [[Werckismic temperaments]]

: These temper out the [[werckisma]], {{monzo| -3 2 -1 2 -1 }} (441/440). Its color name is Luzozoguti.

; [[Swetismic temperaments]]

: These temper out the [[swetisma]], {{monzo| 2 3 1 -2 -1 }} (540/539). Its color name is Lururuyoti.

; [[Lehmerismic temperaments]]

: These temper out the [[lehmerisma]], {{monzo| -4 -3 2 -1 2 }} (3025/3024). Its color name is Loloruyoyoti.

; [[Kalismic temperaments]]

: These temper out the [[kalisma]], {{monzo| -3 4 -2 -2 2 }} (9801/9800). Its color name is Biloruguti.

== Rank-4 temperaments ==

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Even less explored than rank-3 temperaments are rank-4 temperaments. In fact, unless one counts 7-limit JI they do not seem to have been explored at all. However, they could be used; for example [[hobbit]] scales can be constructed for them.

; [[Keenanismic ~~temperaments~~]] (P8, P5, ^1, /1)

; [[Keenanismic family]] (P8, P5, ^1, /1)

: These temper out the keenanisma, {{monzo| -7 -1 1 1 1 }} (385/384). In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Lozoyoti.

; [[Werckismic ~~temperaments]]~~ (P8, P5, ^1, /1)

; Werckismic family (P8, P5, ^1, /1)

: These temper out the werckisma, {{monzo| -3 2 -1 2 -1 }} (441/440). 11/8 is equated to {{monzo| -6 2 -1 2 }} and 5/4 is equated to {{monzo| -5 2 0 2 -1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704. Its color name is Luzozoguti.

; [[Swetismic ~~temperaments]]~~ (P8, P5, ^1, /1)

; Swetismic family (P8, P5, ^1, /1)

: These temper out the swetisma, {{monzo| 2 3 1 -2 -1 }} (540/539). 11/8 is equated to {{monzo| -1 3 1 -2 }} (135/98) and 5/4 is equated to {{monzo| -4 -3 0 2 1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704. Its color name is Lururuyoti.

; [[Lehmerismic ~~temperaments]]~~ (P8, P5, ^1, /1)

; Lehmerismic family (P8, P5, ^1, /1)

: These temper out the lehmerisma, {{monzo| -4 -3 2 -1 2 }} (3025/3024). Since 7/4 is equated to a no-7's 11-limit (color name: yala) interval, both the pergen and the lattice are identical to that of no-7's 11-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }} and /1 is either ~33/32 or ~729/704. Its color name is Loloruyoyoti.

; [[Kalismic ~~temperaments]]~~ (P8/2, P5, ^1, /1)

; Kalismic family (P8/2, P5, ^1, /1)

: These temper out the kalisma, {{monzo| -3 4 -2 -2 2 }} (9801/9800). The octave is split into two ~99/70 periods. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Biloruguti.

@@ Line 12: / Line 12: @@
 ; Blackwood family (P8/5, ^1)
-: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is the [[blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp 5th. Its color name is Sawati.
+: This family tempers out the [[limma]], {{monzo| 8 -5 }} (256/243). It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. The only member of this family is the [[blackwood]] temperament, which is 5-limit. Blackwood's edo copies are offset from one another by 5/4, or alternatively by 81/80. 5/4 is usually tempered sharp, perhaps ~400¢, to match the sharp fifth. Its color name is Sawati.
 ; [[Whitewood family]] (P8/7, ^1)
@@ Line 65: / Line 65: @@
 : The alphatricot family tempers out the [[alphatricot comma]], {{monzo| 39 -29 3 }}. The generator is {{nowrap| ~59049/40960 ({{monzo| -13 10 -1 }}) {{=}} 633{{c}} }}, or its octave inverse {{nowrap| ~81920/59049 {{=}} 567{{c}} }}. Three of the former generators equals the third harmonic, ~3/1. 5/4 is equated to 29 of these generators octave-reduced. Its color name is Quadsa-triyoti. An obvious 7-limit interpretation of the generator is {{nowrap| 81/56 {{=}} 639{{c}} }}, a much simpler ratio which leads to the [[Tour of Regular Temperaments #Latriruti clan (P8, P11/3)|Latriruti clan]]. An obvious 13-limit interpretation is {{nowrap| 13/9 {{=}} 637¢ }}, an even simpler ratio implying the [[Tour of Regular Temperaments #Satrithoti clan (P8, P11/3)|Satrithoti clan]].
-; [[Dimipent family|Dimipent or diminished family]] (P8/4, P5)
+; [[Diminished family]] (P8/4, P5)
-: The dimipent or diminished family tempers out the major diesis or diminished comma, {{monzo| 3 4 -4 }} or [[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]]. 5/4 is equated to 1 fifth minus 1 period. Its color name is Quadguti.
+: The diminished family tempers out the major diesis or diminished comma, {{monzo| 3 4 -4 }} or [[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]]. 5/4 is equated to 1 fifth minus 1 period. Its color name is Quadguti.
 ; [[Undim family]] (P8/4, P5)
@@ Line 123: / Line 123: @@
 ; [[Wesley family]] (P8, ccP4/7)
-: This tempers out the [[wesley comma]], 78125/73728 ({{monzo| -13 -2 7 }}). The generator is {{nowrap| ~125/96 {{=}} ~412{{c}} }}. Seven generators equals a double-compound 4th of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29edo]].
+: This tempers out the [[wesley comma]], 78125/73728 ({{monzo| -13 -2 7 }}). The generator is {{nowrap| ~125/96 {{=}} ~412{{c}} }}. Seven generators equals a double-compound fourth of ~16/3. 5/4 is equated to 1 octave minus 2 generators. Its color name is Lasepyobiti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepruti temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29edo]].
 ; [[Sensipent family]] (P8, ccP5/7)
-: The sensipent family tempers out the [[sensipent comma]], 78732/78125 ({{monzo| 2 9 -7 }}), also known as the medium semicomma. Its generator is {{nowrap| ~162/125 {{=}} ~443{{c}} }}. Seven generators equals a double-compound 5th of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.
+: The sensipent family tempers out the [[sensipent comma]], 78732/78125 ({{monzo| 2 9 -7 }}), also known as the medium semicomma. Its generator is {{nowrap| ~162/125 {{=}} ~443{{c}} }}. Seven generators equals a double-compound fifth of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. Its color name is Sepguti. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzoti temperament.
 ; [[Vishnuzmic family]] (P8/2, P4/7)
@@ Line 135: / Line 135: @@
 ; [[Würschmidt family]] (P8, ccP5/8)
-: The würschmidt family tempers out the [[würschmidt comma]], 393216/390625 ({{monzo| 17 1 -8 }}). Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect 5th); that is, {{nowrap| (5/4)<sup>8</sup>⋅(393216/390625) {{=}} 6 }}. It tends to generate the same mos scales as the [[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. Its color name is Saquadbiguti.
+: The würschmidt family tempers out the [[würschmidt comma]], 393216/390625 ({{monzo| 17 1 -8 }}). Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect fifth); that is, {{nowrap| (5/4)<sup>8</sup>⋅(393216/390625) {{=}} 6 }}. It tends to generate the same mos scales as the [[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. Its color name is Saquadbiguti.
 ; [[Escapade family]] (P8, P4/9)
@@ Line 141: / Line 141: @@
 ; [[Mabila family]] (P8, c4P4/10)
-: The mabila family tempers out the mabila comma, {{monzo| 28 -3 -10 }} (268435456/263671875). The generator is {{nowrap| ~512/375 {{=}} ~530{{c}} }}, three generators equals ~5/2 and ten of them equals a quadruple-compound 4th of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.
+: The mabila family tempers out the [[mabila comma]], {{monzo| 28 -3 -10 }} (268435456/263671875). The generator is {{nowrap| ~512/375 {{=}} ~530{{c}} }}, three generators equals ~5/2 and ten of them equals a quadruple-compound fourth of ~64/3. Its color name is Sasa-quinbiguti. An obvious 11-limit interpretation of the generator is ~15/11.
 ; [[Sycamore family]] (P8, P5/11)
-: The sycamore family tempers out the sycamore comma, {{monzo| -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. Its color name is Laleyoti.
+: The sycamore family tempers out the [[sycamore comma]], {{monzo| -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. Its color name is Laleyoti.
 ; [[Quartonic family]] (P8, P4/11)
-: The quartonic family tempers out the quartonic comma, {{monzo| 3 -18 11 }} (390625000/387420489). The generator is {{nowrap| ~250/243 {{=}} ~45{{c}} }}, seven generators equals ~6/5, and eleven generators equals ~4/3. Its color name is Saleyoti. An obvious 7-limit interpretation of the generator is ~36/35.
+: The quartonic family tempers out the [[quartonic comma]], {{monzo| 3 -18 11 }} (390625000/387420489). The generator is {{nowrap| ~250/243 {{=}} ~45{{c}} }}, seven generators equals ~6/5, and eleven generators equals ~4/3. Its color name is Saleyoti. An obvious 7-limit interpretation of the generator is ~36/35.
 ; [[Lafa family]] (P8, P12/12)
-: This tempers out the lafa comma, {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.
+: This tempers out the [[lafa comma]], {{monzo| 77 -31 -12 }}. The generator is {{nowrap| ~4982259375/4294967296 {{=}} ~258.6{{c}} }}. Twelve generators equals a twelfth (~3/1). 5/4 is equated to 7 octaves minus 31 generators. Its color name is Tribisa-quadtriguti.
 ; [[Ditonmic family]] (P8, c4P4/13)
-: This tempers out the ditonma, {{monzo| -27 -2 13 }} (1220703125/1207959552). Thirteen ~{{monzo| -12 -1 6 }} generators of about 407{{c}} equals a quadruple-compound 4th. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53edo, which is a good tuning for this high-accuracy family of temperaments. Its color name is Lala-theyoti.
+: This tempers out the [[ditonma]], {{monzo| -27 -2 13 }} (1220703125/1207959552). Thirteen ~{{monzo| -12 -1 6 }} generators of about 407{{c}} equals a quadruple-compound fourth. 5/4 is equated to 1 octave minus 2 generators. An obvious 3-limit interpretation of the generator is 81/64, which implies 53edo, which is a good tuning for this high-accuracy family of temperaments. Its color name is Lala-theyoti.
 ; [[Luna family]] (P8, ccP4/15)
-: This tempers out the luna comma, {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3. Its color name is Sasa-quintriguti.
+: This tempers out the [[luna comma]], {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~{{monzo| 18 -1 -7 }} at ~193{{c}}. Two generators equals ~5/4, and fifteen generators equals a double-compound fourth of ~16/3. Its color name is Sasa-quintriguti.
 ; [[Vavoom family]] (P8, P12/17)
-: This tempers out the vavoom comma, {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti.
+: This tempers out the [[vavoom comma]], {{monzo| -68 18 17 }}. The generator is {{nowrap| ~16/15 {{=}} ~111.9{{c}} }}. Seventeen generators equals a twelfth (~3/1). 5/4 is equated to two octaves minus 18 generators. Its color name is Quinla-seyoti.
 ; [[Minortonic family]] (P8, ccP5/17)
-: This tempers out the minortone comma, {{monzo| -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). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.
+: This tempers out the [[minortone comma]], {{monzo| -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 fifth (~6/1). 5/4 is equated to 35 generators minus 5 octaves. Its color name is Trila-seguti.
 ; [[Maja family]] (P8, c<sup>6</sup>P4/17)
-: This tempers out the maja comma, {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.
+: This tempers out the [[maja comma]], {{monzo| -3 -23 17 }} (762939453125/753145430616). The generator is {{nowrap| ~162/125 {{=}} ~453{{c}} }}. Seventeen generators equals a sextuple-compound fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.
 ; [[Maquila family]] (P8, c<sup>7</sup>P5/17)
-: This tempers out the maquila comma, {{monzo| 49 -6 -17 }} (562949953421312/556182861328125). The generator is {{nowrap| ~512/375 {{=}} ~535{{c}} }}. Seventeen generators equals a septuple-compound 5th. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.
+: This tempers out the [[maquila comma]], {{monzo| 49 -6 -17 }} (562949953421312/556182861328125). The generator is {{nowrap| ~512/375 {{=}} ~535{{c}} }}. Seventeen generators equals a septuple-compound fifth. 5/4 is equated to 3 octaves minus 6 generators. Its color name is Trisa-seguti. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-seloti temperament. However, Lala-seloti is not nearly as accurate as Trisa-seguti.
 ; [[Gammic family]] (P8, P5/20)
-: The gammic family tempers out the gammic comma, {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.
+: The gammic family tempers out the [[gammic comma]], {{monzo| -29 -11 20 }}. Nine generators of about 35{{c}} equals ~6/5, eleven equals ~5/4 and twenty equals ~3/2. 34edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is the [[neptune]] temperament. Its color name is Laquinquadyoti.
 === Clans defined by a 2.3.7 comma ===
@@ Line 471: / Line 471: @@
 ; [[Landscape microtemperaments]]
 : Landscape rank-2 temperaments temper out the [[landscape comma]], {{monzo| -4 6 -6 3 }} (250047/250000). These have a period of 1/3 octave, but ~5/4 is not equated with a period, resulting in small intervals. Its color name is Trizoguguti.
 == Rank-3 temperaments ==
@@ Line 562: / Line 562: @@
 ; [[Cataharry family]] (P8, P4/2, ^1)
-: Cataharry temperaments temper out the cataharry comma, {{monzo| -4 9 -2 -2 }} (19683/19600). In the pergen, half a 4th is ~81/70, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Labiruguti.
+: Cataharry temperaments temper out the cataharry comma, {{monzo| -4 9 -2 -2 }} (19683/19600). In the pergen, half a fourth is ~81/70, and {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Labiruguti.
 ; [[Breed family]] (P8, P5/2, ^1)
@@ Line 624: / Line 624: @@
 ; [[Alphaxenic rank three clan|Alphaxenic rank-3 clan]]
 : These temper out the [[Alpharabian comma]], {{monzo| -17 2 0 0 4 }} (131769/131072). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8/2, M2/4). Its color name is Laquadloti.
+; [[Keenanismic temperaments]]
+: These temper out the [[keenanisma]], {{monzo| -7 -1 1 1 1 }} (385/384). Its color name is Lozoyoti.
+; [[Werckismic temperaments]]
+: These temper out the [[werckisma]], {{monzo| -3 2 -1 2 -1 }} (441/440). Its color name is Luzozoguti.
+; [[Swetismic temperaments]]
+: These temper out the [[swetisma]], {{monzo| 2 3 1 -2 -1 }} (540/539). Its color name is Lururuyoti.
+; [[Lehmerismic temperaments]]
+: These temper out the [[lehmerisma]], {{monzo| -4 -3 2 -1 2 }} (3025/3024). Its color name is Loloruyoyoti.
+; [[Kalismic temperaments]]
+: These temper out the [[kalisma]], {{monzo| -3 4 -2 -2 2 }} (9801/9800). Its color name is Biloruguti.
 == Rank-4 temperaments ==
@@ Line 630: / Line 645: @@
 Even less explored than rank-3 temperaments are rank-4 temperaments. In fact, unless one counts 7-limit JI they do not seem to have been explored at all. However, they could be used; for example [[hobbit]] scales can be constructed for them.
-; [[Keenanismic temperaments]] (P8, P5, ^1, /1)
+; [[Keenanismic family]] (P8, P5, ^1, /1)
 : These temper out the keenanisma, {{monzo| -7 -1 1 1 1 }} (385/384). In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Lozoyoti.
-; [[Werckismic temperaments]] (P8, P5, ^1, /1)
+; Werckismic family (P8, P5, ^1, /1)
 : These temper out the werckisma, {{monzo| -3 2 -1 2 -1 }} (441/440). 11/8 is equated to {{monzo| -6 2 -1 2 }} and 5/4 is equated to {{monzo| -5 2 0 2 -1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704. Its color name is Luzozoguti.
-; [[Swetismic temperaments]] (P8, P5, ^1, /1)
+; Swetismic family (P8, P5, ^1, /1)
 : These temper out the swetisma, {{monzo| 2 3 1 -2 -1 }} (540/539). 11/8 is equated to {{monzo| -1 3 1 -2 }} (135/98) and 5/4 is equated to {{monzo| -4 -3 0 2 1 }}, thus the lattice can be thought of as either 7-limit JI or no-5's 11-limit JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704. Its color name is Lururuyoti.
-; [[Lehmerismic temperaments]] (P8, P5, ^1, /1)
+; Lehmerismic family (P8, P5, ^1, /1)
 : These temper out the lehmerisma, {{monzo| -4 -3 2 -1 2 }} (3025/3024). Since 7/4 is equated to a no-7's 11-limit (color name: yala) interval, both the pergen and the lattice are identical to that of no-7's 11-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }} and /1 is either ~33/32 or ~729/704. Its color name is Loloruyoyoti.
-; [[Kalismic temperaments]] (P8/2, P5, ^1, /1)
+; Kalismic family (P8/2, P5, ^1, /1)
 : These temper out the kalisma, {{monzo| -3 4 -2 -2 2 }} (9801/9800). The octave is split into two ~99/70 periods. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 is not), ~33/32 or ~729/704. Its color name is Biloruguti.