Tour of regular temperaments: Difference between revisions

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; [[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 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.

; [[Parakleismic family]] (P8, c3P4/13)

: This tempers out the [[parakleisma]], {{monzo| 8 14 -13 }} (1224440064/1220703125). The generator is a slightly sharpened ~5/3 major sixth. Thirteen generators minus three octaves equal ~4/3, and fourteen generators minus three octaves equal ~8/5.

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

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

: This tempers out the [[minortone comma]], {{monzo| -16 35 -17 }}. The head of the family is minortone, 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 fourth. 5/4 is equated to 9 octaves minus 23 generators. Its color name is Saseyoti.

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

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: This clan tempers out [[Harrison's comma]], {{monzo| -13 10 0 -1 }} (59049/57344). It equates 7/4 to an augmented sixth. Its color name is Laruti. Its best downward extension is [[septimal meantone]].

; [[~~Garischismic~~ clan]] (P8, P5)

; [[Septischismic clan]] (P8, P5)

: This clan tempers out the [[~~garischisma~~]], {{monzo| 25 -14 0 -1 }} (33554432/33480783). It equates 8/7 to two apotomes ({{monzo| -22 14 }}, (2187/2048)2) and 7/4 to a double-diminished octave {{monzo| 23 -14 }}. This clan includes [[schismatic family #Garibaldi|garibaldi]], [[~~garischismic~~ clan #Newt|newt]], [[~~garischismic~~ clan #Satin|satin]], [[vulture family #Vulture|vulture]], and [[~~garischismic~~ clan #Sextile|sextile]]. Its color name is Sasaruti.

: This clan tempers out the [[septischisma]], {{monzo| 25 -14 0 -1 }} (33554432/33480783). It equates 8/7 to two apotomes ({{monzo| -22 14 }}, (2187/2048)2) and 7/4 to a double-diminished octave {{monzo| 23 -14 }}. This clan includes [[schismatic family #Garibaldi|garibaldi]], [[septichismic clan #Newt|newt]], [[septischismic clan #Satin|satin]], [[vulture family #Vulture|vulture]], and [[septischismic clan #Sextile|sextile]]. Its color name is Sasaruti.

; Sasazoti clan (P8, P5)

: This clan tempers out the [[leapfrog comma]], {{monzo| 21 -15 0 1 }} (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes [[~~aberschismic~~ temperaments #Leapday|leapday]], [[sensamagic clan #Leapweek|leapweek]] and [[diaschismic family #Srutal|srutal]].

: This clan tempers out the [[leapfrog comma]], {{monzo| 21 -15 0 1 }} (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes [[sengic temperaments #Leapday|leapday]], [[sensamagic clan #Leapweek|leapweek]] and [[diaschismic family #Srutal|srutal]].

; Laruruti clan (P8/2, P5)

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: These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.

; Sengic temperaments

; [[Sengic temperaments]]

: Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.

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: Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti.

; Nuwell temperaments

; [[Nuwell temperaments]]

: Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.

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; [[Didymus rank-3 family]] (P8, P5, ^1)

: These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.

; [[Schismic rank-3 family]] (P8, P5, /1)

: These are the rank-3 temperaments tempering out the schisma, {{monzo| -15 8 1 }} (32805/32768).

; [[Diaschismic rank-3 family]] (P8/2, P5, /1)

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=== Families defined by a 2.3.7 comma ===

Every no-5's 7-limit (color name: za) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, {{nowrap| ^1 {{=}} ~81/80 }}. An additional 5-limit or 11-limit comma creates a 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:

Every no-5's 7-limit (color name: za) comma defines a rank-3 family; thus every comma in the list of rank-2 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, {{nowrap| ^1 {{=}} ~81/80 }}. An additional 5-limit or 11-limit comma creates a 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:

; [[Archytas family]] (P8, P5, ^1)

: ~~Archytas~~ temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.

: The archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.

; [[~~Garischismic~~ family]] (P8, P5, ^1)

; [[Septischismic family]] (P8, P5, ^1)

: ~~A garischismic~~ temperament ~~is one which~~ tempers out the ~~garischisma~~, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.

: The septischismic temperament tempers out the septischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.

; Laruruti clan (P8/2, P5)

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

: ~~Semaphoresmic~~ temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.

: The semaphoresmic temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.

; [[Gamelismic family]] (P8, P5/3, ^1)

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; [[Wizardharry clan]] (P8, P4/3, ^1)

: These temper out the [[4000/3993|wizardharry comma]], {{monzo| 5 -1 3 0 -3 }} (4000/3993), and split the fourth in three. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Triluyoti.

; [[Vishdelismic clan]] (P8, P5, ^1)

: These temper out the [[vishdelisma]], {{monzo| 9 -2 -4 0 1 }} (5632/5625). 11/8 is equated to 5632/4096, the tetraptolemaic double-augmented third.

; [[Semicathartismic clan]] (P8, P5, ^1)

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; [[Olympic clan]] (P8, P5, ^1)

: These temper out the [[olympia]], {{monzo| 17 -5 0 -2 -1 }} (131072/130977). 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Salururuti.

~~; [[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]]

@@ Line 154: / Line 154: @@
 ; [[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 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.
+; [[Parakleismic family]] (P8, c3P4/13)
+: This tempers out the [[parakleisma]], {{monzo| 8 14 -13 }} (1224440064/1220703125). The generator is a slightly sharpened ~5/3 major sixth. Thirteen generators minus three octaves equal ~4/3, and fourteen generators minus three octaves equal ~8/5.
 ; [[Luna family]] (P8, ccP4/15)
@@ Line 163: / Line 166: @@
 ; [[Minortone family]] (P8, ccP5/17)
 : This tempers out the [[minortone comma]], {{monzo| -16 35 -17 }}. The head of the family is minortone, 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 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)
@@ Line 187: / Line 187: @@
 : This clan tempers out [[Harrison's comma]], {{monzo| -13 10 0 -1 }} (59049/57344). It equates 7/4 to an augmented sixth. Its color name is Laruti. Its best downward extension is [[septimal meantone]].
-; [[Garischismic clan]] (P8, P5)
+; [[Septischismic clan]] (P8, P5)
-: This clan tempers out the [[garischisma]], {{monzo| 25 -14 0 -1 }} (33554432/33480783). It equates 8/7 to two apotomes ({{monzo| -22 14 }}, (2187/2048)<sup>2</sup>) and 7/4 to a double-diminished octave {{monzo| 23 -14 }}. This clan includes [[schismatic family #Garibaldi|garibaldi]], [[garischismic clan #Newt|newt]], [[garischismic clan #Satin|satin]], [[vulture family #Vulture|vulture]], and [[garischismic clan #Sextile|sextile]]. Its color name is Sasaruti.
+: This clan tempers out the [[septischisma]], {{monzo| 25 -14 0 -1 }} (33554432/33480783). It equates 8/7 to two apotomes ({{monzo| -22 14 }}, (2187/2048)<sup>2</sup>) and 7/4 to a double-diminished octave {{monzo| 23 -14 }}. This clan includes [[schismatic family #Garibaldi|garibaldi]], [[septichismic clan #Newt|newt]], [[septischismic clan #Satin|satin]], [[vulture family #Vulture|vulture]], and [[septischismic clan #Sextile|sextile]]. Its color name is Sasaruti.
 ; Sasazoti clan (P8, P5)
-: This clan tempers out the [[leapfrog comma]], {{monzo| 21 -15 0 1 }} (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes [[aberschismic temperaments #Leapday|leapday]], [[sensamagic clan #Leapweek|leapweek]] and [[diaschismic family #Srutal|srutal]].
+: This clan tempers out the [[leapfrog comma]], {{monzo| 21 -15 0 1 }} (14680064/14348907). It equates 7/6 to two apotomes and 7/4 to double augmented fifth. This clan includes [[sengic temperaments #Leapday|leapday]], [[sensamagic clan #Leapweek|leapweek]] and [[diaschismic family #Srutal|srutal]].
 ; Laruruti clan (P8/2, P5)
@@ Line 341: / Line 341: @@
 : These temper out the [[avicennma]], {{monzo| -9 1 2 1 }} (525/512), also known as Avicenna's enharmonic diesis. Its color name is Zoyoyoti.
-; Sengic temperaments
+; [[Sengic temperaments]]
 : Sengic rank-2 temperaments temper out the [[senga]], {{monzo| 1 -3 -2 3 }} (686/675). Its color name is Trizo-aguguti.
@@ Line 350: / Line 350: @@
 : Secanticorn rank-2 temperaments temper out the [[secanticornisma]], {{monzo| -3 11 -5 -1 }} (177147/175000). Its color name is Laruquinguti.
-; Nuwell temperaments
+; [[Nuwell temperaments]]
 : Nuwell rank-2 temperaments temper out the [[nuwell comma]], {{monzo| 1 5 1 -4 }} (2430/2401). Its color name is Quadru-ayoti.
@@ Line 487: / Line 487: @@
 ; [[Didymus rank-3 family]] (P8, P5, ^1)
 : These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti.
+; [[Schismic rank-3 family]] (P8, P5, /1)
+: These are the rank-3 temperaments tempering out the schisma, {{monzo| -15 8 1 }} (32805/32768).
 ; [[Diaschismic rank-3 family]] (P8/2, P5, /1)
@@ Line 498: / Line 501: @@
 === Families defined by a 2.3.7 comma ===
-Every no-5's 7-limit (color name: za) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, {{nowrap| ^1 {{=}} ~81/80 }}. An additional 5-limit or 11-limit comma creates a 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:
+Every no-5's 7-limit (color name: za) comma defines a rank-3 family; thus every comma in the list of rank-2 2.3.7 families could be included here. If nothing else is tempered out, the prime subgroup is assumed to be 2.3.5.7, and we have a 7-limit (color name: yaza) temperament in which one of the generators is ~5/1. This generator can be reduced to ~5/4, which may be reduced further to ~81/80. Hence in all the pergens below, {{nowrap| ^1 {{=}} ~81/80 }}. An additional 5-limit or 11-limit comma creates a 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit:
 ; [[Archytas family]] (P8, P5, ^1)
-: Archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.
+: The archytas temperament tempers out 64/63. This comma equates every 7-limit interval to some 3-limit interval. If 81/80 were tempered out too, the otonal tetrad 4:5:6:7 would be identified with the dominant seventh chord. Its color name is Ruti.
-; [[Garischismic family]] (P8, P5, ^1)
+; [[Septischismic family]] (P8, P5, ^1)
-: A garischismic temperament is one which tempers out the garischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.
+: The septischismic temperament tempers out the septischisma, {{monzo| 25 -14 0 -1 }} (33554432/33480783). Its color name is Sasaruti.
 ; Laruruti clan (P8/2, P5)
@@ Line 510: / Line 513: @@
 ; [[Semaphoresmic family]] (P8, P4/2, ^1)
-: Semaphoresmic temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.
+: The semaphoresmic temperament tempers out 49/48 and thereby identifies the septimal minor third 7/6 with the septimal whole tone 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like ''semi-fourth''. See also [[semaphore]]. Its color name is Zozoti.
 ; [[Gamelismic family]] (P8, P5/3, ^1)
@@ Line 624: / Line 627: @@
 ; [[Wizardharry clan]] (P8, P4/3, ^1)
 : These temper out the [[4000/3993|wizardharry comma]], {{monzo| 5 -1 3 0 -3 }} (4000/3993), and split the fourth in three. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Triluyoti.
+; [[Vishdelismic clan]] (P8, P5, ^1)
+: These temper out the [[vishdelisma]], {{monzo| 9 -2 -4 0 1 }} (5632/5625). 11/8 is equated to 5632/4096, the tetraptolemaic double-augmented third.
 ; [[Semicathartismic clan]] (P8, P5, ^1)
@@ Line 633: / Line 639: @@
 ; [[Olympic clan]] (P8, P5, ^1)
 : These temper out the [[olympia]], {{monzo| 17 -5 0 -2 -1 }} (131072/130977). 11/8 is equated with a 2.3.7 interval, and thus every 2.3.7.11 interval is equated with a 2.3.7 interval. In the pergen, {{nowrap| ^1 {{=}} ~64/63 }}. Its color name is Salururuti.
-; [[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]]