Tour of regular temperaments: Difference between revisions

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; [[Bug 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 {{nowrap| 10/9 {{=}} ~~~250¢~~ }}, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

: This low-accuracy family of temperaments tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or {{nowrap| 10/9 {{=}} ~250{{c}} }}, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

; [[Immunity family]] (P8, P4/2)

: This tempers out the immunity comma, {{monzo| 16 -13 2 }} (1638400/1594323). Its generator is {{nowrap| ~729/640 {{=}} ~~~247¢~~ }}, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

: This tempers out the immunity comma, {{monzo| 16 -13 2 }} (1638400/1594323). Its generator is {{nowrap| ~729/640 {{=}} ~247{{c}} }}, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.

; [[Dicot family]] (P8, P5/2)

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

: This tempers out the luna comma, {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is ~~{{nowrap|~~ ~{{monzo| 18 -1 -7 }} ~~{{=}}~~ ~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 4th of ~16/3. Its color name is Sasa-quintriguti.

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

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=== Temperaments defined by an 11-limit comma ===

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

: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Luyoyoti.

: These temper out the [[ptolemisma]], {{monzo| 2 -2 2 0 -1 }} (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Luyoyoti.

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

: These temper out the biyatisma, {{monzo| -3 -1 -1 0 2 }} (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.

: These temper out the [[biyatisma]], {{monzo| -3 -1 -1 0 2 }} (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.

; [[Valinorsmic clan]]

: These temper out the valinorsma, {{monzo| 4 0 -2 -1 1 }} (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.

: These temper out the [[valinorsma]], {{monzo| 4 0 -2 -1 1 }} (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.

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

: These temper out the rastma, ~~{{nowrap|~~{{monzo| 1 5 0 0 -2 }} ~~{{=}}~~ 243/242}}. In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2). Its color name is Luluti.

: These temper out the [[rastma]], {{monzo| 1 5 0 0 -2 }} (243/242). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2). Its color name is Luluti.

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

: These temper out the pentacircle comma, {{monzo| 7 -4 0 1 -1 }} (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.

: These temper out the [[pentacircle comma]], {{monzo| 7 -4 0 1 -1 }} (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.

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

: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} (14641/14580). 5/4 is equated to a 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Quadlo-aguti.

: These temper out the [[semicanousma]], {{monzo| -2 -6 -1 0 4 }} (14641/14580). 5/4 is equated to a 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Quadlo-aguti.

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

: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} (16384/16335). 5/4 is equated to an 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Saluluguti.

: These temper out the [[semiporwellisma]], {{monzo| 14 -3 -1 0 -2 }} (16384/16335). 5/4 is equated to an 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Saluluguti.

; [[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.

: 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 three clan|Alphaxenic rank-3 clan]]

: These temper out the Alpharabian comma, ~~{{nowrap|~~{{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.

: 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.

== Rank-4 temperaments ==

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.

@@ Line 45: / Line 45: @@
 ; [[Bug 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 {{nowrap| 10/9 {{=}} ~250¢ }}, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
+: This low-accuracy family of temperaments tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or {{nowrap| 10/9 {{=}} ~250{{c}} }}, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. Its color name is Guguti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
 ; [[Immunity family]] (P8, P4/2)
-: This tempers out the immunity comma, {{monzo| 16 -13 2 }} (1638400/1594323). Its generator is {{nowrap| ~729/640 {{=}} ~247¢ }}, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
+: This tempers out the immunity comma, {{monzo| 16 -13 2 }} (1638400/1594323). Its generator is {{nowrap| ~729/640 {{=}} ~247{{c}} }}, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. Its color name is Sasa-yoyoti. An obvious 7-limit interpretation of the generator is 7/6~8/7, which leads to semaphore or Zozoti.
 ; [[Dicot family]] (P8, P5/2)
@@ Line 156: / Line 156: @@
 ; [[Luna family]] (P8, ccP4/15)
-: This tempers out the luna comma, {{monzo| 38 -2 -15 }} (274877906944/274658203125). The generator is {{nowrap| ~{{monzo| 18 -1 -7 }} {{=}} ~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 4th of ~16/3. Its color name is Sasa-quintriguti.
 ; [[Vavoom family]] (P8, P12/17)
@@ Line 599: / Line 599: @@
 === Temperaments defined by an 11-limit comma ===
 ; [[Ptolemismic clan]] (P8, P5, ^1)
-: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Luyoyoti.
+: These temper out the [[ptolemisma]], {{monzo| 2 -2 2 0 -1 }} (100/99). 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, {{nowrap| ^1 {{=}} ~81/80 }}. Its color name is Luyoyoti.
 ; [[Biyatismic clan]] (P8, P5, ^1)
-: These temper out the biyatisma, {{monzo| -3 -1 -1 0 2 }} (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.
+: These temper out the [[biyatisma]], {{monzo| -3 -1 -1 0 2 }} (121/120). 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Lologuti.
 ; [[Valinorsmic clan]]
-: These temper out the valinorsma, {{monzo| 4 0 -2 -1 1 }} (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.
+: These temper out the [[valinorsma]], {{monzo| 4 0 -2 -1 1 }} (176/175). To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen. Its color name is Loruguguti.
 ; [[Rastmic rank three clan|Rastmic rank-3 clan]]
-: These temper out the rastma, {{nowrap|{{monzo| 1 5 0 0 -2 }} {{=}} 243/242}}. In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2). Its color name is Luluti.
+: These temper out the [[rastma]], {{monzo| 1 5 0 0 -2 }} (243/242). In the corresponding [[#Clans defined by a 2.3.11 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2). Its color name is Luluti.
 ; [[Pentacircle clan]] (P8, P5, ^1)
-: These temper out the pentacircle comma, {{monzo| 7 -4 0 1 -1 }} (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.
+: These temper out the [[pentacircle comma]], {{monzo| 7 -4 0 1 -1 }} (896/891). The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704. Its color name is Saluzoti.
 ; [[Semicanousmic clan]] (P8, P5, ^1)
-: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} (14641/14580). 5/4 is equated to a 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Quadlo-aguti.
+: These temper out the [[semicanousma]], {{monzo| -2 -6 -1 0 4 }} (14641/14580). 5/4 is equated to a 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Quadlo-aguti.
 ; [[Semiporwellismic clan]] (P8, P5, ^1)
-: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} (16384/16335). 5/4 is equated to an 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Saluluguti.
+: These temper out the [[semiporwellisma]], {{monzo| 14 -3 -1 0 -2 }} (16384/16335). 5/4 is equated to an 2.3.11 (color name: ila) interval, thus every 2.3.5.11 interval is equated to a 2.3.11 interval, and both the pergen and the lattice are identical to that of 2.3.11-subgroup JI. In the pergen, ^1 is either ~33/32 or ~729/704. Its color name is Saluluguti.
 ; [[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.
+: 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 three clan|Alphaxenic rank-3 clan]]
-: These temper out the Alpharabian comma, {{nowrap|{{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.
+: 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.
 == Rank-4 temperaments ==
-{{Main| Rank-4 temperament }}
+{{Main| Catalog of rank-4 temperaments }}
 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.