Tour of regular temperaments: Difference between revisions

This family tempers out the limma, [8 -5 0> = 256/243, which implies [[5-edo]].

This family tempers out the apotome, [-11 7 0> = 2187/2048, which implies [[7-edo]].

The Pythagorean family tempers out the [[Pythagorean comma]], 531441/524288 = [-19 12 0>, which implies [[12-edo]]. 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 12-edo, offset from one another by a justly tuned 5/4, or alternatively by a tempered 81/80. Several 12-edo instruments slightly detuned from each other provide an easy way to make music with these temperaments.

The schismatic family tempers out the schisma of [-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]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]].

The Sup'''<u>ra</u>'''pyth or Sayo family tempers out [12 -9 1> = 20480/19683, which equates 5/4 to a Pythagorean augmented 2nd. Being a fourthward comma, it tends to sharpen the 5th, hence it's "super-pythagorean". The best 7-limit extension adds the Archy or Ru comma to make the [[Superpyth|Sup'''<u>e</u>'''rpyth]] temperament.

This tempers out the pelogic comma, [-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]], [[16edo]], [[23edo]], and [[25edo]].

The diaschismic family tempers out the [[diaschisma]], [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]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. 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]] is an excellent pajara tuning.

This tempers out the immunity comma, [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 aka Zozo.

The augmented family tempers out the diesis of [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]], 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 [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]], [[22edo]], [[37edo]], and [[59edo]]. An important 7-limit extension also tempers out 64/63.

This low-accuracy family of temperaments tempers out the laconic comma, [-4 7 -3> (2187/2000), which is the difference between three 10/9's and one 3/2. The generator is ~10/9 = ~230¢. 5/4 is equated to 7 generators minus 1 octave. Laconic is supported by [[16edo]], [[21edo]], and [[37edo]] (using the 37b mapping), among others. An obvious 7-limit interpretation of the generator is ~8/7, which leads to Gamelismic aka Latrizo.

The dimipent (or diminished) family tempers out the major diesis or diminished comma, [3 4 -4> or [[648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo]]. 5/4 is equated to 1 fifth minus 1 period.

This tempers out the [[negri comma]], [-14 3 4&gt;. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.  

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by [5 -9 4> (20000/19683), the minimal diesis or [[tetracot comma]]. 5/4 is equated to 9 generators minus an octave. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]].

This tempers out the [[vulture comma]], [24 -21 4&gt;. 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 comic comma, [13 -14 4> = 5120000/4782969. Its generator is ~81/80 = 55¢. 5/4 is equated to 7 generators. An obvious 11-limit interpretation of the generator is 33/32, which makes Laquadlo.

This tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5&gt;. 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 [[amity comma]], 1600000/1594323 = [9 -13 5&gt;. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 5/4 is equated to 4 octaves minus 13 generators, or  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.

The magic family tempers out [-10 -1 5> (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.

This tempers out the fifive comma, [-1 -14 10> = 9765625/9565938. The period is ~4374/3125 = [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 qintosec comma, 140737488355328/140126044921875 = [47 -15 -10&gt;. The period is ~524288/455625 = [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 = [19 10 -15&gt;. 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 = [-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]] [-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]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.

The [[semicomma]] (also known as Fokker's comma), 2109375/2097152 = [-21 3 7&gt;, 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, [-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 [[29-edo]].  

The sensipent (sensi) family tempers out the [[sensipent comma]], [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]], [[19edo]], [[46edo]], and [[65edo]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.  

This tempers out the vishnuzma, [23 6 -14&gt;, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7. The period is ~[-11 -3 7> and the generator is ~25/24. 5/4 is equated to 1 period minus 3 generators.

This tempers out the [[mutt_comma|mutt comma]], [-44 -3 21&gt;, leading to some strange properties. Seven ~5/4 generators equals a double-compound 4th = ~16/3. The third-octave period is <u>not</u> 5/4, thus the generator is equivalently a period minus ~5/4, only about 14¢. The L/s ratio is very lopsided, and scales resemble a "fuzzy" augmented chord.

The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt comma]], 393216/390625 = [17 1 -8&gt;. 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]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.

This tempers out the [[escapade comma]], [32 -7 -9&gt;, which is the difference between nine just major thirds and seven just fourths. The generator is ~[-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, [-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.

The sycamore family tempers out the sycamore comma, [-16 -6 11&gt; = 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.   

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen ~[-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 53-edo, which is a good tuning for this high-accuracy family of temperaments.

This tempers out the luna comma, [38 -2 -15&gt; (274877906944/274658203125). The generator is ~{18 -1 -7> = ~193¢. Two generators equals ~5/4, and fifteen generators equals a double-compound 4th of ~16/3.  

This tempers out the minortone comma, [-16 35 -17&gt;. 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.

This tempers out the maja comma, [-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.

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17&gt;. 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.

The gammic family tempers out the gammic comma, [-29 -11 20&gt;. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~3/2. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

The tricot family tempers out the [[Tricot|tricot comma]], [39 -29 3&gt;. The generator is ~13/9 = ~634¢, or ~18/13 = ~566¢. Three generators of ~13/9 equals a compound 5th of ~3/1.

This clan tempers out the Laru comma [-13 10 0 -1> =  50.7¢. It equates 7/4 to an augmented 6th. Its best downward extension is [[Meantone family|Septimal Meantone]].

This clan tempers out the [[garischisma]], [25 -14 0 -1&gt; = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048), and 7/4 to a double-diminished 8ve [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 Laruru comma [-7 8 0 -2> = 78¢. 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 Diaschismatic or Sagugu temperament and the Jubalismic or Biruyo temperament.

This clan tempers out the Sasa-zozo comma [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament. 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 Triru comma, [-1 6 0 -3> = 105¢, 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|Augmented or Trigu]] temperament.

This clan tempers out the Trizo comma, [-2 -4 0 3> = 99¢, 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 gamelisma, [-10 1 0 3> = 1029/1024. Three ~8/7 generators equals a 5th. 7/4 is equated to an 8ve minus a generator. A particularly noteworthy member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps, thus it's a weak extension. Its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.

This clan tempers out the Latriru comma [-9 11 0 -3> = 15.0¢. 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.

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 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|Porcupine or Triyo]] temperament, as does equating the period to ~7/5.

This clan tempers out the Laquadru comma [-3 9 0 -4> = 42.3¢. 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 Saquadru comma [16 -3 0 -4> = 18.8¢. 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 Laquinzo comma [-14 0 0 5> = 44¢. Five ~8/7 periods equals an 8ve, and four periods equals ~7/4. 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 [3 7 0 -5> = 70¢. 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 [5 -12 0 5> = 20.7¢. 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 Sepru comma [7 8 0 -7> = 33.8¢. 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 2.3.11 clan tempers out 243/242 = [-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 2.3.11 clan tempers out the Laquadlo comma [-17 2 0 0 4>. Its half-ocave 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 as a strong extension the Comic aka Saquadyobi temperament, which is in the Comic family.

This 2.3.13 clan tempers out 512/507 = [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 Satritho comma 512/507 = [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 hemimean comma, [6 0 -5 2> = 3136/3125. The M2 generator is ~28/25 = ~194¢. Two generators equals ~5/4, and five of them equals ~7/4.

This clan tempers out the quince, [-15 0 -2 7&gt; = 823543/819200. The generator is ~343/320 = ~116¢. Two generators equals ~8/7 (a M2), and seven generators equals ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[Magic]] temperament, which is in the Magic family.

This 3.5.7 clan tempers out the sensamagic comma [0 -5 1 2> = 245/243. The M3 generator is ~9/7, and two generators equals ~5/3.

This 3.5.7 clan tempers out the mirkwai comma, [0 3 4 -5&gt; = 16875/16807. The generator is ~7/5, and four generators equals a compound major 7th = ~27/7.

These are the rank three temperaments tempering out the dischisma, [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, [1 -5 3> = 250/243. In the pergen, P4/3 is ~10/9.

These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552.  In the pergen, P12/6 is ~6/5.

A garischismic temperament is one which tempers out the garischisma, [25 -14 0 -1&gt; = 33554432/33480783.

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,  [-10 1 0 3> = 1029/1024. In the pergen, P5/3 is ~8/7.  

Stearnsmic temperaments temper out the stearnsma, [1 10 0 -6&gt; = 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 [-5 2 2 -1&gt; = [[225/224]]. It has a number of 11-limit children, including unidecimal marvel, prodigy, minerva and spectacle. Other family members include negri, sharp, mavila, wizard, tritonic, septimin, slender, triton, escapade and marvo. Considered elsewhere are meantone, miracle, magic, pajara, orwell, catakleismic, garibaldi, august and compton.

Starling tempers out the septimal semicomma or starling comma [1 2 -3 1> = [[126/125]], the difference between three 6/5s plus one 7/6, and an octave. Like marvel, it has the same generators as 5-limit JI. An excellent tuning for starling is [[77edo]], but 31, 46 or 58 also work nicely. Its family includes myna, sensi, valentine, casablanca and nusecond temperaments, not to mention meantone, keemun, muggles and opossum. In the pergen, ^1 = ~81/80.

These temper out [0 -5 1 2> = 245/243. In the pergen, ^1 = ~64/63.

These temper out the greenwoodma, [-3 4 1 -2&gt; = 405/392. In the pergen, ^1 = ~64/63.

These temper out the avicennma, [-9 1 2 1&gt; = 525/512, also known as Avicenna's enharmonic diesis. In the pergen, ^1 = ~81/80.

These temper out the keema [-5 -3 3 1> = 875/864. In the pergen, ^1 = ~81/80.

These temper out [6, 3, -1, -3> = 1728/1715. In the pergen, ^1 = ~64/63.

These temper out the nuwell comma, [1, 5, 1, -4> = 2430/2401. In the pergen, ^1 = ~64/63.

The 7-limit rank three microtemperament which tempers out the ragisma, [-1 -7 4 1&gt; = 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. Accurate ones include ennealimmal, supermajor, enneadecal, amity, mitonic, parakleismic, gamera and vishnu. Pontiac belongs on the list but falls under the schismatic family rubric. In the pergen, ^1 = ~81/80.

The hemifamity family of rank three temperaments tempers out the hemifamity comma, [10 -6 1 -1> = 5120/5103. In the pergen, ^1 = ~81/80.

The horwell family of rank three temperaments tempers out the horwell comma, [-16 1 5 1> = 65625/65536. In the pergen, ^1 = ~81/80.

The hemimage family of rank three temperaments tempers out the hemimage comma, [5 -7 -1 3> = 10976/10935. In the pergen, ^1 = ~64/63.

These temper out the tolerma, [10 -11 2 1> = 179200/177147. In the pergen, ^1 = ~81/80.

Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2&gt; = 19683/19600. In the pergen, half a 4th is ~81/70, and ^1 = ~81/80.

Breed is a 7-limit microtemperament which tempers out [-5 -1 -2 4&gt; = 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.

The mirwomo family of rank three temperaments tempers out the mirwomo comma, [-15 3 2 2> = 33075/32768. In the pergen, half a fifth is ~128/105, and ^1 = ~81/80.

The 7-limit rank three microtemperament which tempers out the lanscape comma, [-4 6 -6 3&gt; = 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 dimcomp family of rank three temperaments tempers out the dimcomp comma, [-1 -4 8 -4> = 390625/388962. In the pergen, the quarter-octave period is ~25/21, and ^1 = ~81/80.

These temper out the senga, [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, [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, [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, [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, [ -6 -8 2 5 &gt; = 420175/419904. Two ~324/245 generators equal the pergen's downminor 7th of ~7/4.

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

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

These temper out the valinorsma, [4 0 -2 -1 1> = 176/175.

These temper out the rastma, [-1 5 0 0 -2> = 243/242. As an ila (11-limit no-fives no-sevens) rank-2 temperament, it's (P8, P5/2).

These temper out the werckisma, [-3 2 -1 2 -1> = 441/440.

These temper out the swetisma, [2 3 1 -2 -1> = 540/539.

These temper out the lehmerisma, [-4 -3 2 -1 2> = 3025/3024.

These temper out the kalisma, [-3 4 -2 -2 2> = 9801/9800.

Orgonia is the commatic realm of the [[11-limit]] comma 65536/65219 = [16 0 0 -2 -3&gt;, the orgonisma.

The Archipelago is a name which has been given to the commatic realm of the [[13-limit]] comma [2 -3 -2 0 0 2> = 676/675.