Tour of regular temperaments: Difference between revisions

These are families defined by a 3-limit (color name: wa) comma. If only primes 2 and 3 are part of the [[subgroup]], the comma creates a rank-1 temperament, an [[edo]]. But if another prime such as 5 is present, the comma creates a rank-2 temperament. Since edos are discussed elsewhere, this section assumes the presence of at least one additional prime. The rank-2 temperament created consists of multiple "copies" of an edo. The edo copies can be thought of as being offset from one another by a small comma. This small comma is represented in the [[pergen]] by ^1.  

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

; [[Mercator family]] (P8/53, ^1)

=== Families defined by a 2.3.5 comma ===

: 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]].

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

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

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

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

; [[Würschmidt family]] (P8, ccP5/8)

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

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

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

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

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

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

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

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

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

; [[Maja family]] (P8, c⁶P4/17)

; [[Maquila family]] (P8, c⁷P5/17)

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

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

; Triruti clan (P8/3, P5)

; [[Gamelismic clan]] (P8, P5/3)

: This clan tempers out the [[bleu comma]], {{monzo| 3 7 0 -5 }} (17496/16807). The ~54/49 generator is about 139{{c}}. Two of them equal ~7/6, three equal ~9/7, five equal ~3/2, and seven equal ~7/4.  

; Lasepzoti clan (P8, P11/7)

; [[Septiennealimmal clan]] (P8/9, P5)

=== Clans defined by a 2.3.11 comma ===

: This 2.3.11 clan tempers out the [[nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its 1/3-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3. Its color name is Tribiloti.  

: This 2.3.11 clan tempers out the [[Alpharabian comma]] {{monzo| -17 2 0 0 4 }}. Its half-octave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes a strong extension to the comic or Saquadyobiti temperament, which is in the jubilismic clan. Its color name is Laquadloti.  

: This clan tempers out the [[vorwell comma]] (named for being tempered in [[septimal vulture]] and [[orwell]]), {{monzo| 27 0 -8 -3 }} (134217728/133984375). The generator is {{nowrap| ~1024/875 {{=}} ~272{{c}} }}. Three generators equals ~8/5 and eight of them equals ~7/2. Its color name is Sasatriru-aquadbiguti Nowa.  

; [[Llywelynsmic clan]] (P8, cM3/7)

; [[Quince clan]] (P8, m6/7)

: This clan tempers out the [[quince comma]], {{monzo| -15 0 -2 7 }} (823543/819200). The generator is {{nowrap| ~343/320 {{=}} ~116{{c}} }}. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the classical minor sixth ~8/5. An obvious 5-limit interpretation of the generator is ~16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan. Its color name is Lasepzo-aguguti Nowa.  

; Slither clan (P8, ccm6/9)

; Compass temperaments

: Compass rank-2 temperaments temper out the [[compass comma]], {{monzo| -6 -2 10 -5 }} (9765625/9680832). Its color name is Quinruyoyoti.  

; Trimyna temperaments

; [[Mistismic temperaments]]

: Mistismic rank-2 temperaments temper out the [[mistisma]], {{monzo| 16 -6 -4 1 }} (458752/455625). Its color name is Sazoquadguti.  

; [[Varunismic temperaments]]

; [[Landscape microtemperaments]]

== Rank-3 temperaments ==

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:

: 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 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 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 kleisma, {{monzo| -6 -5 6 }} (15625/15552). In the pergen, P12/6 is ~6/5. Its color name is Tribiyoti.  

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

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

: 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 [[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.  

; [[Semicanousmic 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 [[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.  

; [[Kalismic temperaments]]

== Rank-4 temperaments ==

; [[Miscellaneous 5-limit temperaments]]

: High in badness, but worth cataloging for one reason or another.

; [[Low harmonic entropy linear temperaments]]

; Middle Path tables

: Tables of temperaments where {{nowrap| complexity/7.65 + damage/10 < 1 }}. Useful for beginners looking for a list of a manageable number of temperaments, which approximate harmonious intervals accurately with a manageable number of notes.

== Maps of temperaments ==

* [[Map of rank-2 temperaments]], sorted by generator size

* [[Rank-2 temperaments by mapping of 3]]

* [[Temperaments for MOS shapes]]

== Temperament nomenclature ==