Tour of regular temperaments: Difference between revisions

Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every strong extension of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but don't uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

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. 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. 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 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. An obvious 7-limit interpretation of the generator is 7/6, which makes Slendro aka Semaphore aka Zozo.

This tempers out the immunity comma, [16 -13 2> (1638400/1594323). It has the same pergen as Bug/Gugu. Its generator is ~729/640 = ~247¢, two of which make ~4/3. An obvious 7-limit interpretation of the generator is 7/6, which leads to Slendro aka Semaphore aka Zozo.

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. 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¢. 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]].

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]]. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]].

This tempers out the [[vulture comma]], [24 -21 4>. Its generator is ~320/243 = ~475¢, four of which make ~3/1. 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¢. 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>. The period is 59049/51200, and five 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. 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>. The generator is 243/200 = ~339.5¢, five of which make ~8/3. 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, [-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.

This tempers out the qintosec comma, 140737488355328/140126044921875 = [47 -15 -10>. 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. An obvious 7-limit interpretation of the period is 8/7.

This tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15>. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 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. 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. 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>, is tempered out by the members of the semicomma family. Its generator is ~75/64, seven of which equals ~3/1. It 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. 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.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>, or the amount by which seven chromatic semitones (25/24) fall short of a perfect fourth (4/3), or (4/3)/(25/24)^7.

This tempers out the [[escapade comma]], [32 -7 -9>, 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. 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.

The sycamore family tempers out the sycamore comma, [-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.  

This tempers out the ditonma, [-27 -2 13> = 1220703125/1207959552. Thirteen generators of about 408¢ equals a quadruple-compound 4th. 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> (274877906944/274658203125). The generator is ~{18 -1 -7> = ~193¢. Fifteen generators equals a double-compound 4th of ~16/3.  

This tempers out the minortone comma, [-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.

This tempers out the maja comma, [-3 -23 17> = 762939453125/753145430616. The generator is ~162/125 = ~453¢. Seventeen generators equals a sextuple-compound 4th.

This tempers out the maquila comma, 562949953421312/556182861328125 = [49 -6 -17>. The generator is ~512/375 = ~535¢. Seventeen generators equals a septuple-compound 5th. 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 septimal third-tone [[28/27]], a low-accuracy temperament that equates 7/6 with 9/8.

This clan tempers out the slendro diesis, [[49/48]]. Generator = 8/7 or 7/6. Its best downward extension is [[Godzilla]]. See also [[Semaphore]].

This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

This clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024, a no-fives comma. Three 8/7 generators equals a 5th. 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 [-9 11 0 -3> = 15.0¢. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. An obvious 7-limit 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> = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternately one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. Equating this 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  [-3 9 0 -4> = 42.3¢. Generator = ~9/7. It includes as a strong extension the [[Squares]] temperament, which is a weak extension of Meantone.

This clan tempers out [16 -3 0 -4> = 18.8¢. Generator = ~21/16. It includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.

This clan tempers out [5 -12 0 5> = 20.7¢. Generator = ~243/196 = ~380¢. 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 [7 8 0 -7> = 33.8¢. Generator = 7/6. It 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>. Generator = 11/9. It includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

This 2.3.13 clan tempers out 512/507 = [9 -1 0 0 0 -2>. Generator = 16/13. It includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The M3 generator is ~5/4. The half-octave period is ~7/5 or ~10/7.

Even less familiar than rank-2 temperaments are the [[Planar_Temperament|rank-3 temperaments]], generated by a set of three intervals. Since these temperaments may be mapped in many different ways, it is more common to identify rank-3 temperaments by the commas they temper out. They can also be identified by pergen. The 3rd element of a rank-3 pergen is usually a comma, but sometimes it's some fraction of a 5-limit or 7-limit interval.

These are the rank three temperaments tempering out the dischisma, [11 -4 -2> = 2048/2025. /1 = 64/63.

These are the rank three temperaments tempering out the porcupine comma or maximal diesis, [1 -5 3> = 250/243.  /1 = 64/63.

These are the rank three temperaments tempering out the kleisma, [-6 -5 6> = 15625/15552. /1 = 64/63.

Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6 and 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]].

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 [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. ^1 = 81/80.

These temper out [0 -5 1 2> = 245/243. ^1 = 64/63.

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

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

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

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

These temper out the nuwell comma, [1, 5, 1, -4> = 2430/2401. ^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. ^1 = 81/80.

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

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

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

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

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. ^1 = 81/80 or 64/63.

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

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

Cataharry temperaments temper out the cataharry comma, [-4 9 -2 -2&gt; = 19683/19600. Half of a 4th is ~81/70. ^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. Half a fith 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. The 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. The period = ~25/21.

These temper out the senga, [1 -3 -2 3> = 686/675. One generator = ~15/14, two = ~7/6 (a downminor 3rd), 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 an 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 an 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 a downmajor 3rd of ~5/4.

A wizmic temperament is one which tempers out the wizma, [ -6 -8 2 5 &gt; = 420175/419904. vm7 = 7/4.

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