Tour of regular temperaments: Difference between revisions
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Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 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 ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit: | Every 5-limit (color name: ya) comma defines a rank-3 family, thus every comma in the list of rank-2 2.3.5 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 ~7/1. This generator can be reduced to ~7/4, which can be reduced further to ~64/63. Hence in all the pergens below, the ^1 or /1 generator is ~64/63. An additional 7-limit or 11-limit comma creates an 11-limit (color name: yazala) temperament, and so forth. All these examples are 7-limit: | ||
; [[ | ; [[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. | : These are the rank-3 temperaments tempering out the didymus or meantone comma, 81/80. Its color name is Guti. | ||
; [[ | ; [[Diaschismic rank-3 family]] (P8/2, P5, /1) | ||
: These are the rank-3 temperaments tempering out the diaschisma, {{monzo| 11 -4 -2 }} (2048/2025). The half-octave period is ~45/32. Its color name is Saguguti. | : These are the rank-3 temperaments tempering out the diaschisma, {{monzo| 11 -4 -2 }} (2048/2025). The half-octave period is ~45/32. Its color name is Saguguti. | ||
; [[ | ; [[Porcupine rank-3 family]] (P8, P4/3, /1) | ||
: These are the rank-3 temperaments tempering out the porcupine comma a.k.a. maximal diesis, {{monzo| 1 -5 3 }} (250/243). In the pergen, P4/3 is ~10/9. Its color name is Triyoti. | : These are the rank-3 temperaments tempering out the porcupine comma a.k.a. maximal diesis, {{monzo| 1 -5 3 }} (250/243). In the pergen, P4/3 is ~10/9. Its color name is Triyoti. | ||
; [[ | ; [[Kleismic rank-3 family]] (P8, P12/6, /1) | ||
: These are the rank-3 temperaments tempering out the kleisma, {{monzo| -6 -5 6 }} (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti. | : These are the rank-3 temperaments tempering out the kleisma, {{monzo| -6 -5 6 }} (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti. | ||
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: 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-3 clan]] | ||
: 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. | : 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. | ||
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: 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-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. | : 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. | ||
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; [[Kalismic temperaments]] | ; [[Kalismic temperaments]] | ||
: These temper out the [[kalisma]], {{monzo| -3 4 -2 -2 2 }} (9801/9800). Its color name is Biloruguti. | : These temper out the [[kalisma]], {{monzo| -3 4 -2 -2 2 }} (9801/9800). Its color name is Biloruguti. | ||
== Rank-4 temperaments == | == Rank-4 temperaments == | ||