Tour of regular temperaments: Difference between revisions

These are families defined by a wa or 3-limit comma. If only primes 2 and 3 are part of the [[Just intonation subgroup|subgroup]], the comma creates a rank-1 temperament, an [[EDO|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.

: This family tempers out the [[limma]], {{nowrap|{{monzo| 8 -5 }} {{=}} 256/243}}. It equates 5 fifths with 3 octaves, which creates multiple copies of [[5edo]]. The fifth is ~720¢, quite sharp. This family includes 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, to match the sharp 5th.

: This family equates 7 fifths with 4 octaves, which implies [[7edo]]. It tempers out the apotome, {{nowrap|{{monzo| -11 7 }} {{=}} 2187/2048}}.

: This tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo| -19 12 }}, which implies [[12edo]]. This family includes the compton and catler temperaments. Temperaments in this family tend to have a period of 1/12th octave The 5-limit compton temperament can be thought of as multiple rings of 12edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

: The schismatic family tempers out the schisma of {{Monzo|-15 8 1}} = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], and [[118edo|118]] EDOs.

: This tempers out the pelogic comma, {{Monzo|-7 3 1}} = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo|9]], [[16edo|16]], [[23edo|23]], and [[25edo|25]] EDOs.

: The diaschismic family tempers out the [[diaschisma]], {{Monzo|11 -4 -2}} or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. 5/4 is equated to 3 periods minus 2 fifths. The major 2nd ~9/8 is divided in half, with each half equated to ~16/15. Diaschismic tunings include [[12edo|12]], [[22edo|22]], [[34edo|34]], [[46edo|46]], [[56edo|56]], [[58edo|58]] and [[80edo|80]] EDOs. An obvious 7-limit interpretation of the period is 7/5, which makes [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out. [[22edo|22EDO]] is an excellent pajara tuning.

: This low-accuracy family of temperaments tempers out [[27/25]], the large limma or bug comma. The generator is an approximate 6/5 or 10/9 = ~250¢, two of which make ~4/3. 5/4 is equated to 1 octave minus 3 generators. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore or Zozo.

: This tempers out the immunity comma, {{Monzo|16 -13 2}} (1638400/1594323). Its generator is ~729/640 = ~247¢, two of which make ~4/3. 5/4 is equated to 3 octaves minus 13 generators. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore or Zozo.

: The augmented family tempers out the diesis of {{Monzo|7 0 -3}} = [[128/125]], the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo|12EDO]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]).

: The porcupine family tempers out {{Monzo|1 -5 3}} = [[250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include [[15edo|15]], [[22edo|22]], [[37edo|37]], and [[59edo|59]] EDOs. An important 7-limit extension also tempers out 64/63.

: The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = {{monzo|-13 10 -1}} = 633¢, or its octave inverse ~81920/59049 = 567¢. Three of the latter generators equals a compound 4th of ~8/3. 5/4 is equated to 14 octaves minus 29 of these generators. An obvious 7-limit interpretation of the generator is 81/56 = 639¢, a much simpler ratio which leads to the [[Tour of Regular Temperaments#Latriru clan (P8, P11/3)|Latriru clan]]. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the [[Tour of Regular Temperaments #Satritho clan (P8, P11/3)|Satritho clan]].

: This tempers out the symbolic comma, 2048/1875 = {{Monzo|11 -1 -4}}. Its generator is ~5/4 = ~421¢, four of which make ~8/3.

: This tempers out the [[vulture comma]], {{Monzo|24 -21 4}}. Its generator is ~320/243 = ~475¢, four of which make ~3/1. 5/4 is equated to 21 generators minus 8 octaves. An obvious 7-limit interpretation of the generator is 21/16, which makes Saquadru.

: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}. The period is ~59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio.  5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

: This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]], and five of them with [[16/9]]. The generator is [[27/25]], two of which equals 10/9, three of which equals [[6/5]], and five of which equals [[4/3]]. 5/4 is equated to an octave minus 8 generators. As one might expect, [[12edo|12EDO]] is about as accurate as it can be.

: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of four [[16/15]]'s with [[5/4]], and five of them with [[4/3]].

: This tempers out the [[amity comma]], 1600000/1594323 = {{Monzo|9 -13 5}}. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or 2 fifths minus 3 generators. An obvious 11-limit interpretation of the generator is 11/9, which makes Saquinlo. An obvious 13-limit interpretation of the generator is 39/32, which makes Lala-quintho.

: This tempers out the fifive comma, {{Monzo|-1 -14 10}} = 9765625/9565938. The period is ~4374/3125 = {{Monzo|1 7 -5}}, two of which make an octave. The generator is ~27/25, five of which make ~3/2.  5/4 is equated to 7 generators minus 1 period.  

: This tempers out the quintosec comma, 140737488355328/140126044921875 = {{monzo| 47 -15 -10 }}. The period is ~524288/455625 = {{monzo| 19 -6 -4 }}, five of which equals an octave. The generator is ~16/15. A period plus a generator makes half a fifth. 5/4 is equated to 3 periods minus 3 generators. An obvious 7-limit interpretation of the period is 8/7.

: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}}. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.  

: This tempers out Ampersand's comma = 34171875/33554432 = {{Monzo|-25 7 6}}. The generator is ~16/15, of which six make ~3/2. 5/4 is equated to 1 octave minus 7 generators. If the generator is also equated to ~15/14, and three generators to ~11/9, one gets the [[Miracle]] temperament.

: The kleismic family of temperaments tempers out the [[kleisma]] {{Monzo|-6 -5 6}} = 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp.  5/4 is equated to 5 generators minus 1 octave. The kleismic family includes [[15edo|15]], [[19edo|19]], [[34edo|34]], [[49edo|49]], [[53edo|53]], [[72edo|72]], [[87edo|87]] and [[140edo|140]] EDOs among its possible tunings.

: The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = {{Monzo|-21 3 7}}, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. 5/4 is equated to 1 octave minus 3 generators. This temperament doesn't have much independent existence as a 5-limit temperament, since its generator has a natural interpretation as ~7/6, leading to [[orwell|Orwell or Sepru]] temperament.

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

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

: This tempers out the unicorn comma, 1594323/1562500 = {{Monzo|-2 13 -8}}. The generator is ~250/243 = ~62¢ and eight of them equal ~4/3.

: The würschmidt (or wuerschmidt) 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, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOSs as [[Magic_family|magic temperament]], but is tuned slightly more accurately. Both [[31edo|31EDO]] and [[34edo|34EDO]] can be used as würschmidt tunings, as can [[65edo|65EDO]], which is quite accurate.

: This tempers out the [[escapade comma]], {{Monzo|32 -7 -9}}, which is the difference between nine just major thirds and seven just fourths. The generator is ~{{Monzo|-14 3 4}} = ~55¢ and nine of them equal ~4/3. Seven of them equal ~5/4, thus two of them equal ~16/15. An obvious 11-limit interpretation of the generator is 33/32, leading to the Trisa-tritrilu temperament.   

: This tempers out the shibboleth comma, {{Monzo|-5 -10 9}} = 1953125/1889568. Nine generators of ~6/5 equal a double compound 4th of ~16/3.  5/4 is equated to 3 octaves minus 10 generators.

; [[Mabila family|Mabila or Sasa-quinbigu family]] (P8, c⁴P4/10)

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

: 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.

: The quartonic family tempers out the quartonic comma, {{Monzo|3 -18 11}} = 390625000/387420489. The generator is ~250/243 = ~45¢, seven generators equals ~6/5, and eleven generators equals ~4/3. An obvious 7-limit interpretation of the generator is ~36/35.

: This tempers out the lafa comma, {{Monzo|77 -31 -12}}. The generator is ~4982259375/4294967296 = ~258.6¢. Twelve generators equals a twelfth = ~3/1. 5/4 is equated to 7 octaves minus 31 generators.

; [[Ditonmic family|Ditonmic or Lala-theyo family]] (P8, c⁴P4/13)

: This tempers out the ditonma, {{Monzo|-27 -2 13}} = 1220703125/1207959552. Thirteen ~{{Monzo|-12 -1 6}} generators of about 407¢ 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.

: This tempers out the luna comma, {{Monzo|38 -2 -15}} = 274877906944/274658203125. The generator is ~{{Monzo|18 -1 -7}} = ~193¢. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3.  

: This tempers out the vavoom comma, {{Monzo|-68 18 17}}. The generator is ~16/15 = ~111.9¢. Seventeen generators equals a twelfth = ~3/1. 5/4 is equated to two octaves minus 18 generators.

: 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.

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

: This tempers out the maja comma, {{Monzo|-3 -23 17}} = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th. 5/4 is equated to 9 octaves minus 23 generators.

; [[Maquila family|Maquila or Trisa-segu family]] (P8, c⁷P5/17)

: This tempers out the maquila comma, 562949953421312/556182861328125 = {{Monzo|49 -6 -17}}. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th.  5/4 is equated to 3 octaves minus 6 generators. An obvious 11-limit interpretation of the generator is 11/8, leading to the Lala-selo temperament. However, Lala-selo isn't nearly as accurate as Trisa-segu.

: This clan tempers out the Laru comma, {{Monzo|-13 10 0 -1}} = 59049/57344. It equates 7/4 to an augmented 6th. Its best downward extension is [[Meantone family|septimal meantone]].

: This clan tempers out the [[garischisma]], {{Monzo|25 -14 0 -1}} = 33554432/33480783. It equates 8/7 to two apotomes ({{Monzo|-11 7}} = 2187/2048) and 7/4 to a double-diminished 8ve {{Monzo|23 -14}}. This clan includes [[Vulture family #Vulture|vulture]], [[Breedsmic temperaments #Newt|newt]], [[Schismatic family #Garibaldi|garibaldi]], [[Landscape microtemperaments #Sextile|sextile]], and [[Canousmic temperaments #Satin|satin]].

: This clan tempers out the Sasazo 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 [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]] and [[Diaschismic family #Srutal|srutal]].

: This clan tempers out the Laruru comma, {{Monzo|-7 8 0 -2}} = 6561/6272. Two ~81/56 periods equal an 8ve. The generator is ~3/2, and four generators minus three periods equals ~7/4. The major 2nd ~9/8 is divided in half, with each half equated to ~28/27. See also the Diaschismic or Sagugu temperament and the Jubilismic or Biruyo temperament.

: This clan tempers out the parahemif comma, {{Monzo| 15 -13 0 2 }} = 1605632/1594323, and includes the [[hemif]] temperament and its strong extension [[hemififths]]. 7/4 is equated to 13 generators minus 3 octaves. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

: This clan tempers out the gamelisma, {{Monzo|-10 1 0 3}} = 1029/1024. Three ~8/7 generators equals a 5th. 7/4 is equated to an 8ve minus a generator. Five generators is slightly flat of 2/1, making this a [[cluster temperament]]. See also Sawa and Lasepzo.

: This clan tempers out the Trizo comma, {{Monzo|-2 -4 0 3}} = 343/324, a low-accuracy temperament. Three ~7/6 generators equals a 5th, and four equal ~7/4. An obvious interpretation of the ~234¢ generator is 8/7, leading to the much more accurate Gamelismic or Latrizo temperament.

: This clan tempers out the Triru comma, {{Monzo|-1 6 0 -3}} = 729/686, a low-accuracy temperament. Three ~9/7 periods equals an 8ve. The generator is ~3/2, and two generators minus a period equals ~7/4. An obvious 5-limit interpretation of the ~400¢ period is 5/4, leading to the [[augmented]] temperament.

: This clan tempers out the Latriru comma, {{Monzo|-9 11 0 -3}} = 177147/175616. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the [[liese]] temperament, which is a weak extension of Meantone.

: This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5.

: This clan tempers out the ''buzzardisma'', {{Monzo|16 -3 0 -4}} = 65536/64827. Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. This clan includes as a strong extension the [[Vulture family|vulture]] temperament, which is in the vulture family.

: This clan tempers out the Laquadru comma, {{Monzo|-3 9 0 -4}} = 19683/19208. its generator is ~9/7. Four generators equals ~8/3. 7/4 is equated to 4 octaves minus 9 generators. This clan includes as a strong extension the [[squares]] temperament, which is a weak extension of meantone.

: This clan tempers out the [[cloudy comma]], {{Monzo|-14 0 0 5}} = 16807/16384. It has a period of 1/5 octave, which represents ~8/7. The generator is ~3/2. Unlike the Blackwood or Sawa family, ~3/2 is not equated with three-fifths of an octave, resulting in very small intervals.

: This clan tempers out the Quinru comma, {{Monzo|3 7 0 -5}} = 17496/16807. The ~54/49 generator is about 139¢. Two of them equal ~7/6, three equal ~9/7, five equal ~3/2, and seven equal ~7/4.  

: This clan tempers out the Saquinzo comma, {{Monzo|5 -12 0 5}} = 537824/531441. Its generator is ~243/196 = ~380¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[magic]] temperament, which is in the Magic family.

: This clan tempers out the Lasepzo comma {{Monzo|-18 -1 0 7}} = 823543/786432. Its generator is ~8/7. Six generators equals ~7/3, and seven generators equals ~8/3. Three generators is ~30¢ sharp of 3/2, and five generators is ~15¢ sharp of 2/1, making this a [[cluster temperament]]. See also Sawa and Latrizo.

: This clan tempers out the ''septiness'' comma {{Monzo|26 -4 0 -7}} = 67108864/66706983. Its generator is ~147/128, four of them gives ~7/4, and seven of them gives ~8/3. Five generators is ~12.5¢ sharp of 2/1, making this a [[cluster temperament]].

: This clan tempers out the sepru comma, {{Monzo|7 8 0 -7}} = 839808/823543. Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the [[orwell]] temperament, which is in the Semicomma family.

: This clan tempers out the ''[[Septimal ennealimma|septiennealimma]]'' (tritrizo comma), {{Monzo|-11 -9 0 9}} = 40353607/40310784. It has a period of 1/9 octave, which represents ~7/6. The generator is ~3/2. This clan includes a number of regular temperaments including [[enneaportent]], [[ennealimmal]], and [[Kleismic family #Novemkleismic|novemkleismic]].

: This 2.3.11 clan tempers out [[243/242]] = {{monzo| -1 5 0 0 -2 }}. Its generator is ~11/9. Two generators equals ~3/2. 11/8 is equated to 5 generators minus an octave. This clan includes as a strong extension the [[dicot]] temperament, which is in the dicot family.

: This 2.3.13 clan tempers out [[512/507]] = {{monzo| 9 -1 0 0 0 -2 }}. Its generator is ~16/13. Two generators equals ~3/2. 13/8 is equated to 1 octave minus 1 generator. This clan includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

: This 2.3.13 clan tempers out the threedie, [[2197/2187]] = {{monzo| 0 -7 0 0 0 3 }}. Its generator is ~18/13. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan.

: This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd = ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.

: This clan tempers out the bapbo comma, [[256/245]]. The genarator is ~8/7 = ~202¢ and two of them equals ~5/4.

: This clan tempers out the hemimean comma, {{monzo| 6 0 -5 2 }} = 3136/3125. The generator is ~28/25 = ~194¢. Two generators equals the nowa major 3rd = ~5/4, three of them equals ~7/5, and five of them equals ~7/4.

: This clan tempers out the mabilisma, {{monzo| -20 0 5 3 }} = 1071875/1048576. The generator is ~175/128 = ~527¢. Three generators equals ~5/2 and five of them equals ~32/7.

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

: This clan tempers out the [[rainy comma]], {{monzo| -21 0 3 5 }} = 2100875/2097152. The rurugu generator is ~256/245 = ~77¢. Three generators equals ~8/7 and five of them equals the nowa major 3rd = ~5/4.

: This clan tempers out the [[llywelynsma]], {{monzo| 22 0 -1 -7 }} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.

: This clan tempers out the quince, {{monzo| -15 0 -2 7 }} = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan.

: This clan tempers out the slither comma, {{monzo| 16 0 4 -9 }} = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.

: This 3.5.7 clan tempers out the Arcturus comma {{Monzo|0 -7 6 -1}} = 15625/15309. The generator is the noca major 6th = ~5/3, and six generators equals ~21/1.

: This 3.5.7 clan tempers out the sensamagic comma {{Monzo|0 -5 1 2}} = 245/243. The generator is ~9/7, and two generators equals the noca major 6th = ~5/3.

: This 3.5.7 clan tempers out the gariboh comma {{Monzo|0 -2 5 -3}} = 3125/3087. The generator is ~25/21, two generators equals ~7/5, and three generators equals the noca major 6th = ~5/3.

: This 3.5.7 clan tempers out the mirkwai comma, {{Monzo|0 3 4 -5}} = 16875/16807. The generator is ~7/5, four generators equals ~27/7, and five generators equals the noca compound minor 7th = ~27/5.

: This 3.5.7 clan tempers out the Procyon comma {{Monzo|0 -8 -3 7}} = 823543/820125. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the noca minor seventh = ~9/5.

: This 3.5.7 clan tempers out the Betelgeuse comma {{Monzo|0 -13 -2 9}} = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th = ~45/1.

: This 3.5.7 clan tempers out the Izar comma (also known as bapbo schismina), {{Monzo|0 -11 -7 12}} = 13841287201/13839609375. The generator is ~16807/10125, five generators equals ~63/5, seven equals ~243/7, and twelve equals ~2187/5.

: These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392.

: Keegic rank-two temperaments temper out the keega, {{Monzo|-3 1 -3 3}} = 1029/1000.

: These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512, also known as Avicenna's enharmonic diesis.

: Sengic rank-two temperaments temper out the senga, {{Monzo|1 -3 -2 3}} = 686/675.

: Keemic rank-two temperaments temper out the keema, {{Monzo|-5 -3 3 1}} = 875/864.

: Secanticorn rank-two temperaments temper out the ''secanticornisma'', {{monzo|-3 11 -5 -1}} = 177147/175000.

: Nuwell rank-two temperaments temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401.

: Mermismic rank-two temperaments temper out the ''mermisma'', {{Monzo|5 -1 7 -7}} = 2500000/2470629.

: Negricorn rank-two temperaments temper out the ''negricorn'' comma, {{Monzo|6 -5 -4 4}} = 153664/151875.

: These temper out the tolerma, {{Monzo|10 -11 2 1}} = 179200/177147.

: Valenwuer rank-two temperaments temper out the ''valenwuer'' comma, {{Monzo|12 3 -6 -1}} = 110592/109375.

: Mirwomo rank-two temperaments temper out the mirwomo comma, {{Monzo|-15 3 2 2}} = 33075/32768.

: Catasyc rank-two temperaments temper out the ''catasyc'' comma, {{Monzo|-11 -3 8 -1}} = 390625/387072.

: Compass rank-two temperaments temper out the compass comma, {{Monzo|-6 -2 10 -5}} = 9765625/9680832.

: The trimyna rank-two temperaments temper out the trimyna comma, {{Monzo|-4 1 -5 5}} = 50421/50000.

: Starling rank-two temperaments temper out the septimal semicomma or starling comma {{Monzo|1 2 -3 1}} = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. It includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum.

: Octagar rank-two temperaments temper out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969.

: Orwellismic rank-two temperaments temper out orwellisma, {{Monzo|6 3 -1 -3}} = 1728/1715.

: Mynaslendric rank-two temperaments temper out the ''mynaslender'' comma, {{Monzo|11 4 1 -7}} = 829440/823543.

: Mistismic rank-two temperaments temper out the ''mistisma'', {{Monzo|16 -6 -4 1}} = 458752/455625.

: Varunismic rank-two temperaments temper out the varunisma, {{monzo|-9 8 -4 2}} = 321489/320000.

: Marvel rank-two temperaments temper out {{Monzo|-5 2 2 -1}} = [[225/224]]. It includes negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

: Dimcomp rank-two temperaments temper out the dimcomp comma, {{Monzo|-1 -4 8 -4}} = 390625/388962.

: Cataharry rank-two temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600.

: Canousmic rank-two temperaments temper out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969.

: Triwellismic rank-two temperaments temper out the ''triwellisma'', {{Monzo|1 -1 -7 6}} = 235298/234375.

: Hemimage rank-two temperaments temper out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935.

: Hemifamity rank-two temperaments temper out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103.

: Parkleiness rank-two temperaments temper out the ''parkleiness'' comma, {{Monzo|7 7 -9 1}} = 1959552/1953125.

: Porwell rank-two temperaments temper out the porwell comma, {{Monzo|11 1 -3 -2}} = 6144/6125.

: Cartoonismic temperaments temper out the cartoonisma, {{monzo| 12 -3 -14 9 }} = 165288374272/164794921875.

: Hemfiness rank-two temperaments temper out the ''hemfiness'' comma, {{Monzo|15 -5 3 -5}} = 4096000/4084101.

: Hewuermera rank-two temperaments temper out the ''hewuermera'' comma, {{Monzo|16 2 -1 -6}} = 589824/588245.

: Lokismic rank-two temperaments temper out the ''lokisma'', {{Monzo|21 -8 -6 2}} = 102760448/102515625.

: Decovulture rank-two temperaments temper out the ''decovulture'' comma, {{Monzo|26 -7 -4 -2}} = 67108864/66976875.

: Pontiqak rank-two temperaments temper out the ''pontiqak'' comma, {{Monzo|-17 -6 9 2}} = 95703125/95551488.

: Mitonismic rank-two temperaments temper out the ''mitonisma'', {{Monzo|-20 7 -1 4}} = 5250987/5242880.

: Horwell rank-two temperaments temper out the horwell comma, {{Monzo|-16 1 5 1}} = 65625/65536.

: Neptunismic rank-two temperaments temper out the ''neptunisma'', {{Monzo|-12 -5 11 -2}} = 48828125/48771072.

: Metric rank-two temperaments temper out the meter comma, {{Monzo|-11 2 7 -3}} = 703125/702464.

: Wizmic rank-two temperaments temper out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904.

: Supermatertismic rank-two temperaments temper out the ''supermatertisma'', {{Monzo|-6 3 9 -7}} = 52734375/52706752.

: Breedsmic rank-two temperaments temper out the breedsma, {{Monzo|-5 -1 -2 4}} = 2401/2400.

: Supermasesquartismic rank-two temperaments temper out the ''supermasesquartisma'', {{Monzo|-5 10 5 -8}} = 184528125/184473632.

: Ragismic rank-two temperaments temper out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374.

: Landscape rank-two 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.

: These are the rank three temperaments tempering out the dischisma, {{Monzo|11 -4 -2}} = 2048/2025.  The half-octave period is ~45/32.

: These are the rank three temperaments tempering out the porcupine comma or maximal diesis, {{Monzo|1 -5 3}} = 250/243. In the pergen, P4/3 is ~10/9.

: These are the rank three temperaments tempering out the kleisma, {{Monzo|-6 -5 6}} = 15625/15552.  In the pergen, P12/6 is ~6/5.

Every za or 7-limit-no-fives comma defines a rank-3 family, thus every comma in the list of rank-two 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 yaza or 7-limit 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, ^1 = ~81/80. An additional 5-limit or 11-limit comma creates a yazala or 11-limit temperament, and so forth. All these examples are yaza:

: A garischismic temperament is one which tempers out the garischisma, {{Monzo|25 -14 0 -1}} = 33554432/33480783.

: Semiphore 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|Sem'''<u>a</u>'''phore]] and [[Slendro clan|Slendro]].

: Not to be confused with the gamelismic clan of rank two temperaments, the gamelismic family are those rank three temperaments which temper out the gamelisma,  {{Monzo|-10 1 0 3}} = 1029/1024. In the pergen, P5/3 is ~8/7.  

: Stearnsmic temperaments temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. In the pergen, P8/2 is ~343/243 and P4/3 is ~54/49.

: The head of the marvel family is marvel, which tempers out {{Monzo|-5 2 2 -1}} = [[225/224]]. It divides 8/7 into two 16/15s, or equivalently, two 15/14s. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle.

The marvel comma equates every 7-limit interval to some 5-limit interval, therefore the generators are the same as for 5-limit JI: 2/1, 3/1 and 5/1. These may be reduced to 2/1, 3/2 and 5/4, and 5/4 may be reduced further to 81/80. Hence in the pergen, ^1 = ~81/80.  

: Starling tempers out the septimal semicomma or starling comma {{Monzo|1 2 -3 1}} = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. It divides 10/7 into two 6/5s. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo|77EDO]], but 31, 46 or 58 also work nicely. In the pergen, ^1 = ~81/80.

: These temper out {{Monzo|0 -5 1 2}} = 245/243, which divides 16/15 into two 28/27s. In the pergen, ^1 = ~64/63.

: These temper out the greenwoodma, {{Monzo|-3 4 1 -2}} = 405/392, which divides 10/9 into two 15/14s. In the pergen, ^1 = ~64/63.

: These temper out the avicennma, {{Monzo|-9 1 2 1}} = 525/512, which divides 7/6 into two 16/15s. In the pergen, ^1 = ~81/80.

: These temper out the keema {{Monzo|-5 -3 3 1}} = 875/864, which divides 15/14 into two 25/24s. In the pergen, ^1 = ~81/80.

: These temper out {{Monzo|6 3 -1 -3}} = 1728/1715. In the pergen, ^1 = ~64/63.

: These temper out the nuwell comma, {{Monzo|1 5 1 -4}} = 2430/2401. In the pergen, ^1 = ~64/63.

: The 7-limit rank three microtemperament which tempers out the ragisma, {{Monzo|-1 -7 4 1}} = 4375/4374, extends to various higher limit rank three temperaments such as thor. These are not by any means all microtemperaments, but those which are not highly accurate are probably best discussed under another heading. In the pergen, ^1 = ~81/80.

: The hemifamity family of rank three temperaments tempers out the hemifamity comma, {{Monzo|10 -6 1 -1}} = 5120/5103, which divides 10/7 into three 9/8s. In the pergen, ^1 = ~81/80.

: The horwell family of rank three temperaments tempers out the horwell comma, {{Monzo|-16 1 5 1}} = 65625/65536. In the pergen, ^1 = ~81/80.

: The hemimage family of rank three temperaments tempers out the hemimage comma, {{Monzo|5 -7 -1 3}} = 10976/10935, which divides 10/9 into three 28/27s. In the pergen, ^1 = ~64/63.

: The mint temperament is a low complexity, high error temperament, tempering out the septimal quarter-tone 36/35, equating 7/6 with 6/5, and 5/4 with 9/7. In the pergen, ^1 = ~81/80 or ~64/63.

: These are very low accuracy temperaments tempering out the minor septimal semitone, [[21/20]] and hence equating 5/3 with 7/4. In the pergen, ^1 = ~81/80.

: Jubilismic temperament tempers out 50/49 and thereby equates the two septimal tritones, 7/5 and 10/7. This is the half-octave period. In the pergen, ^1 = ~81/80.

: Cataharry temperaments temper out the cataharry comma, {{Monzo|-4 9 -2 -2}} = 19683/19600. In the pergen, half a 4th is ~81/70, and ^1 = ~81/80.

: Breed is a 7-limit microtemperament which tempers out {{Monzo|-5 -1 -2 4}} = 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2749EDO will certainly do the trick. Breed has generators of 2/1, a 49/40-cum-60/49 neutral third, and ~64/63.

: These temper out the senga, {{Monzo|1 -3 -2 3}} = 686/675. One generator = ~15/14, two = ~7/6 (the downminor 3rd in the pergen), and three = ~6/5.

: The porwell family of rank three temperaments tempers out the porwell comma, {{Monzo|11 1 -3 -2}} = 6144/6125. Two ~35/32 generators equal the pergen's upminor 3rd of ~6/5.

: The octagar family of rank three temperaments tempers out the octagar comma, {{Monzo|5 -4 3 -2}} = 4000/3969. Two ~80/63 generators equal the pergen's upminor 6th of ~8/5.

: The hemimean family of rank three temperaments tempers out the hemimean comma, {{Monzo|6 0 -5 2}} = 3136/3125.  Two ~28/25 generators equal the pergen's downmajor 3rd of ~5/4.

: A wizmic temperament is one which tempers out the wizma, {{Monzo|-6 -8 2 5}} = 420175/419904. Two ~324/245 generators equal the pergen's downminor 7th of ~7/4.

: The 7-limit rank three microtemperament which tempers out the lanscape comma, {{Monzo|-4 6 -6 3}} = 250047/250000, extends to various higher limit rank three temperaments such as tyr and odin. In the pergen, the third-octave period is ~63/50, and ^1 = ~81/80.

: The gariboh family of rank three temperaments tempers out the gariboh comma, {{monzo| 0 -2 5 -3 }} = 3125/3087. Three ~25/21 generators equal the pergen's downmajor 6th of ~5/3.  

: The canou family of rank three temperaments tempers out the canousma, {{Monzo|4 -14 3 4}} = 4802000/4782969. Three ~81/70 generators equal the pergen's downminor 6th of ~14/9.  

: The dimcomp family of rank three temperaments tempers out the dimcomp comma, {{Monzo|-1 -4 8 -4}} = 390625/388962. In the pergen, the quarter-octave period is ~25/21, and ^1 = ~81/80.

: The mirkwai family of rank three temperaments tempers out the mirkwai comma, {{Monzo|0 3 4 -5}} = 16875/16807. Four ~7/5 generators equal the pergen's compound upmajor 7th of  ~27/7.

: 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, ^1 = ~81/80.

: 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.

: 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.

: These temper out the rastma, {{monzo| 1 5 0 0 -2 }} = 243/242. In the corresponding [[Tour of regular temperaments#Clans defined by a 2.3.11 .28ila.29 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2).

: 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.

: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (2.3.11) interval, thus 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.

: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (2.3.11) interval, thus 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.

: 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, ^1 = ~64/63.

: These temper out the Alpharabian comma, {{monzo| -17 2 0 0 4 }} = 131769/131072. In the corresponding [[Tour of regular temperaments#Clans defined by a 2.3.11 .28ila.29 comma|2.3.11 rank-2 temperament]], the pergen is (P8/2, M2/4).

: 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 isn't), ~33/32 or ~729/704.

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

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

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

: 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 isn't), ~33/32 or ~729/704.

: The Archipelago is a name which has been given to the commatic realm of the 13-limit comma 676/675 = {{monzo| 2 -3 -2 0 0 2 }}, the [[island comma]].

: This is the commatic realm of the 11-limit comma 65536/65219 = {{monzo| 16 0 0 -2 -3 }}, the [[orgonisma]].

: This is the commatic realm of the 11-limit comma 1771561/1769472 = {{monzo| -16 -3 0 0 6 }}, the [[nexus comma]].  

: This is the commatic realm of the 11-limit comma 117440512/117406179 = {{monzo| 24 -6 0 1 -5 }}, the [[quartisma]].

[[Category:Lists of temperaments| ]] <!-- main article -->