Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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||~ ||~ pergen ||~ enharmonic

interval(s) ||~ ~~equiva-~~

interval(s) ||~ equivalence(s) ||~ split

~~lence~~(s) ||~ split

interval(s) ||~ perchain(s) and/or

interval(s) ||~ perchain(s) and

genchains(s) ||~ examples ||

||= 1 ||= (P8, P5)

unsplit ||= none ||= none ||= none ||= C - G ||= meantone,

unsplit ||= none ||= none ||= none ||= C - G ||= pythagorean, meantone, dominant,

schismic ||

schismic, mavila, archy, etc. ||

||~ ||~ halves ||~ ||~ ||~ ||~ ||~ ||

||= 2 ||= (P8/2, P5)

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^1 = 81/80 ||

||= ||= " ||= vvd2 (if 5th

< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= ~~large deep red~~

< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= injera

^1 = 64/63 ||

||= ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= ~~169~~/~~162~~

||= ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= deep emerald, if 13/8 = M6

^1 = 27/26 ||

||= 3 ||= (P8, P4/2)

half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore

^1 = 64/63 ||

||= ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= ~~(-22,-11,2)~~ ||

||= ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= double large deep yellow

^1 = 81/80 ||

||= 4 ||= (P8, P5/2)

half-5th ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira

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C - Eb^=Ev - G,

C - F#v/=Gb^\ - C,

C - F^/=Gv\ - C ||= ~~49/48~~ & ~~243/242~~

C - F^/=Gv\ - C ||= deep blue & deep amber

^1 = 33/32

/1 = 64/63 ||

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P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C,

C - D/=Eb\ - F,

C - Eb^/=Ev\ - G ||= ~~2048/2025~~ & ~~49/48~~

C - Eb^/=Ev\ - G ||= small deep green & deep blue

^1 = 81/80

/1 = 64/63 ||

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P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C,

C - Eb/=E\ - G,

C - Dv/=Eb^\ - F ||= ~~2048/2025 & 243/242~~

C - Dv/=Eb^\ - F ||= small deep green and deep amber

^1 = 81/80

/1 = 33/32 ||

||~ ||~ thirds ||~ ||~ ||~ ||~ ||~ ||

||= 6 ||= (P8/3, P5)

third-8ve ||= ^3d2 ||= C^3 ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||

third-8ve ||= ^3d2 ||= C^3 ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented

^1 = 81/80 ||

||= 7 ||= (P8, P4/3)

third-4th ||= v3A1 ||= C^3 ~~``=``~~ C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||

third-4th ||= v3A1 ||= C^3 ``=`` C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine

^1 = 81/80 ||

||= 8 ||= (P8, P5/3)

third-5th ||= v3m2 ||= C^3 ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||

third-5th ||= v3m2 ||= C^3 ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric

^1 = 64/63 ||

||= 9 ||= (P8, P11/3)

third-11th ||= ^3dd2 ||= C^3 ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,

third-11th ||= ^3dd2 ||= C^3 ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber, if 11/8 = A4

~~with~~ 11/8 = A4 ||

^1 = 729/704 ||

||= ||= " ||= v3M2 ||= C^3 ``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= ~~same~~, ~~with~~ 11/8 = P4 ||

||= ||= " ||= v3M2 ||= C^3 ``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= small triple amber, if 11/8 = P4

^1 = 33/32 ||

||= 10 ||= (P8/3, P4/2)

third-8ve, half-4th ||= v6A2 ||= C^6 ``=`` D# ||= P8/3 = ^^m3 = v4A4

P4/2 = ^3m2 = v3M3 ||= C - Eb^^ - Avv - C

C - Db^3=Ev3 - F ||= sixfold jade ||

C - Db^3=Ev3 - F ||= sixfold jade, if 11/8 = P4

^1 = 33/32 ||

||= ||= " ||= ^3d2,

\\m2 ||= C^3 ``=`` B#

C``//`` = Db ||= P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C

C - D/=Eb\ - F ||= 128/125 & 49/48 ||

C - D/=Eb\ - F ||= triforce (128/125 & 49/48)

^1 = 81/80, /1 = 64/63 ||

||= 11 ||= (P8/3, P5/2)

third-8ve, half-5th ||= ^3d2

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C``///`` = Db ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C

C - /D - \F - G ||= 50/49 & 1029/1024 ||

C - /D - \F - G ||= lemba (50/49 & 1029/1024) ||

||= 14 ||= (P8/2, P11/3)

half-8ve, third-11th ||= v6M2 ||= C^6 ``=`` D ||= P8/2 = ^34 = v35

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==Ratio and cents of the accidentals==

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.

We can assign cents to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.

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In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:

15-edo: # ``=`` 240¢, ^ ``=`` 80¢ (^ = 1/3 #)

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eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ (^ = 1/3 #)

sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ (no fixed relationship between ^ and #)

third-comma injera: # ``=`` 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80)

third-comma injera: # ``=`` 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)

eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243)

eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)

Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.

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An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32.

Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

==Notating unsplit pergens==

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==Notating rank-3 pergens==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

||~ tuning ||~ tuning's rank ||~ notation ||~ notation's rank ||~ ``#`` of enharmonics needed ||~ enharmonics ||

||~ tuning ||~ pergen ||~ tuning's rank ||~ notation ||~ notation's rank ||~ ``#`` of enharmonics needed ||~ enharmonics ||

||= 12-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||

||= 12-edo ||= (P8/12) ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||

||= 19-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = dd2 ||

||= 19-edo ||= (P8/19) ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = dd2 ||

||= 15-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v3A1 = v3M2 ||

||= 15-edo ||= (P8/15) ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v3A1 = v3M2 ||

||= 24-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 ||

||= 24-edo ||= (P8/24) ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 ||

||= pythagorean ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= pythagorean ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= meantone ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= meantone ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||

||= semaphore ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||

||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = d2 ||

||= decimal ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 ||

||= semaphore ||= (P8, P4/2) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||

||= 5-limit JI ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= decimal ||= (P8/2, P4/2) ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 ||

||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= 5-limit JI ||= (P8, P5, ^1) ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||

||= marvel ||= (P8, P5, ^1) ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||

||= breedsmic ||= (P8, P5/2, ^1) ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||

||= 7-limit JI ||= (P8, P5, ^1, /1) ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||

When there is more than one enharmonic, they combine to make new enharmonics. Decimal's 2nd enharmonic could be written as ^^\\A1, but combining two accidentals in one enharmonic is avoided.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.

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If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.

Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A convention that colors are used for infinitely stackable accidentals and ups/downs/highs/lows for the other kind of accidentals ~~is arguably a good thing~~.

Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.

==Notating Blackwood-like pergens*==

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a ~~near-~~just ~~5th~~, like P8/5, P8/7, P8/10 and especially P8/12.

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).

The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.

Such a pergen is in effect multiple copies of an edo. Its notation ~~can be~~ based on the edo's notation, expanded with an additional microtonal accidental pair~~. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator~~. Examples:

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

||~ temperament ~~||~~~ ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 ||

||~ temperament ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 ||

||= Blackwood ~~||= 5edo+y~~ ||= (P8/5, ^1) ||= ~~fifth~~-~~8ve no~~-~~threes~~ ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 ||= --- ||

||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 = 16/15 ||= --- ||

||= ||= ||= ||= ||= ||= ||= ||= ||= ||

||= Whitewood ||= (P8/7, ^1) ||= rank-2 7-edo ||= E = A1 ||= D E F G A B C D ||= D F^ A^^... ||= 80/81 = 135/128 ||= --- ||

||= ||= 12edo+j ||= ||= ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||

||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ??? ||= 81/80 ||

||= ||= ~~10edo+y~~ ||= ||= ||= ||= ||= ||= ||= ||

||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||

~~||=~~ ||= 17edo+y ||= ||= ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\... ||= 256/243 ||= 81/80 ||

||= ||= ||= ||= ||= ||= D G#v=Abv Dvv... ||= 729/704 = ||= ||

~~||=~~ ||= ||= ||= ||= ||= ||= ||= ||= ||

||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 ||

~~||=~~ ||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||

~~||=~~ ||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||

~~||=~~ ||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||

~~||=~~ ||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||= ||

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or perhaps 13/12. The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.

(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)

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Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.

See also the [[Map of rank-2 temperaments|map of rank-2 temperaments]].

==Pergens and MOS scales==

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See the screenshots in the next section for examples of which pergens are supported by a specific edo.

Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the ~~bezout pair~~ of (N~~/m,~~N'~~/m) is (g,g'),~~ G = |g'|\N= |g|\N'. ~~The bezout pair~~ is ~~also either ancestor of N/N' in the scale tree. Alternate~~ generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.

Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo & N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.

||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen ||

||= 7 & 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= (P8, WWP5/6) ||

Line 801:

Line 815:

||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= **1\22= 1\24**, 10\22 = 11\24 ||= 13\22 = 14\24 ||= half-8ve quarter-tone ||= ||

A specific pergen can be converted to an edo pair by looking up its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines ~~half-5th~~.

A specific pergen can be converted to an edo pair by looking up its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).

==Supplemental materials*==

Line 808:

Line 822:

needs pergen squares picture

fill in the 2nd pergens column above

to do:

add a mapping commas section somewhere?

finish proofs

link from: ups and downs page, Kite Giedraitis page, MOS scale names page, ~~is there a~~ rank-2 ~~page?~~

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments

http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments

===Notaion guide PDF===

Line 1,601:

Line 1,619:

interval(s)

</th>

<th>~~equiva- ~~

<th>equivalence(s)

~~lence~~(s)

</th>

<th>split

interval(s)

</th>

<th>perchain(s) and

<th>perchain(s) and/or

genchains(s)

</th>

Line 1,627:

Line 1,644:

<td style="text-align: center;">C - G

</td>

<td style="text-align: center;">meantone,

<td style="text-align: center;">pythagorean, meantone, dominant,

schismic

schismic, mavila, archy, etc.

</td>

</tr>

Line 1,680:

Line 1,697:

<td style="text-align: center;">C - F#^=Gbv - C

</td>

<td style="text-align: center;">~~large deep red~~

<td style="text-align: center;">injera

^1 = 64/63

</td>

Line 1,697:

Line 1,714:

<td style="text-align: center;">C - F^=Gv - C

</td>

<td style="text-align: center;">~~169~~/~~162~~

<td style="text-align: center;">deep emerald, if 13/8 = M6

^1 = 27/26

</td>

Line 1,732:

Line 1,749:

<td style="text-align: center;">C - D#v=Ebb^ - F

</td>

<td style="text-align: center;">~~(-22,-11,2)~~

<td style="text-align: center;">double large deep yellow

^1 = 81/80

</td>

</tr>

Line 1,779:

Line 1,797:

C - F^/=Gv\ - C

</td>

<td style="text-align: center;">~~49/48~~ &amp; ~~243/242~~

<td style="text-align: center;">deep blue &amp; deep amber

^1 = 33/32

/1 = 64/63

Line 1,805:

Line 1,823:

C - Eb^/=Ev\ - G

</td>

<td style="text-align: center;">~~2048/2025~~ &amp; ~~49/48~~

<td style="text-align: center;">small deep green &amp; deep blue

^1 = 81/80

/1 = 64/63

Line 1,831:

Line 1,849:

C - Dv/=Eb^\ - F

</td>

<td style="text-align: center;">~~2048/2025 &amp; 243/242~~

<td style="text-align: center;">small deep green and deep amber

^1 = 81/80

/1 = 33/32

Line 1,867:

Line 1,885:

</td>

<td style="text-align: center;">augmented

^1 = 81/80

</td>

</tr>

Line 1,877:

Line 1,896:

<td style="text-align: center;">v3A1

</td>

<td style="text-align: center;">C^3 =<!-- ws:end:WikiTextRawRule:09 --~~> </span~~>C#

<td style="text-align: center;">C^3 = C#

</td>

<td style="text-align: center;">P4/3 = vM2 = ^^m2

Line 1,884:

Line 1,903:

</td>

<td style="text-align: center;">porcupine

^1 = 81/80

</td>

</tr>

Line 1,901:

Line 1,921:

</td>

<td style="text-align: center;">slendric

^1 = 64/63

</td>

</tr>

Line 1,917:

Line 1,938:

<td style="text-align: center;">C - F#v - Cb^ - F

</td>

<td style="text-align: center;">small triple amber,

<td style="text-align: center;">small triple amber, if 11/8 = A4

~~with 11~~/~~8 = A4~~

^1 = 729/704

</td>

</tr>

Line 1,934:

Line 1,955:

<td style="text-align: center;">C - F^ - Cv - F

</td>

<td style="text-align: center;">~~same~~, ~~with~~ 11/8 = P4

<td style="text-align: center;">small triple amber, if 11/8 = P4

^1 = 33/32

</td>

</tr>

Line 1,953:

Line 1,975:

C - Db^3=Ev3 - F

</td>

<td style="text-align: center;">sixfold jade

<td style="text-align: center;">sixfold jade, if 11/8 = P4

^1 = 33/32

</td>

</tr>

Line 1,973:

Line 1,996:

C - D/=Eb\ - F

</td>

<td style="text-align: center;">128/125 &amp; 49/48

<td style="text-align: center;">triforce (128/125 &amp; 49/48)

^1 = 81/80, /1 = 64/63

</td>

</tr>

Line 2,054:

Line 2,078:

C - /D - \F - G

</td>

<td style="text-align: center;">50/49 &amp; 1029/1024

<td style="text-align: center;">lemba (50/49 &amp; 1029/1024)

</td>

</tr>

Line 2,719:

Line 2,743:

<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.

We can assign cents to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.

Line 2,725:

Line 2,749:

In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:

15-edo: # = 240¢, ^ = 80¢ (^ = 1/3 #)

Line 2,740:

Line 2,764:

eighth-comma porcupine: # = 157¢, ^ = 52¢ (^ = 1/3 #)

sixth-comma srutal: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #)

third-comma injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80)

third-comma injera: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)

eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243)

eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)

Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.

Line 2,834:

Line 2,858:

An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^4dd2 or v4dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^4dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.

Another &quot;tippy&quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32.

Another &quot;tippy&quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.

<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

Line 3,044:

Line 3,068:

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a>Notating rank-3 pergens</h2>

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

Line 3,050:

Line 3,074:

<tr>

<th>tuning

</th>

<th>pergen

</th>

<th>tuning's rank

Line 3,064:

Line 3,090:

<tr>

<td style="text-align: center;">12-edo

</td>

<td style="text-align: center;">(P8/12)

</td>

<td style="text-align: center;">rank-1

Line 3,078:

Line 3,106:

<tr>

<td style="text-align: center;">19-edo

</td>

<td style="text-align: center;">(P8/19)

</td>

<td style="text-align: center;">rank-1

Line 3,092:

Line 3,122:

<tr>

<td style="text-align: center;">15-edo

</td>

<td style="text-align: center;">(P8/15)

</td>

<td style="text-align: center;">rank-1

Line 3,106:

Line 3,138:

<tr>

<td style="text-align: center;">24-edo

</td>

<td style="text-align: center;">(P8/24)

</td>

<td style="text-align: center;">rank-1

Line 3,120:

Line 3,154:

<tr>

<td style="text-align: center;">pythagorean

</td>

<td style="text-align: center;">(P8, P5)

</td>

<td style="text-align: center;">rank-2

Line 3,134:

Line 3,170:

<tr>

<td style="text-align: center;">meantone

</td>

<td style="text-align: center;">(P8, P5)

</td>

<td style="text-align: center;">rank-2

Line 3,144:

Line 3,182:

</td>

<td style="text-align: center;">---

</td>

</tr>

<tr>

<td style="text-align: center;">srutal

</td>

<td style="text-align: center;">(P8/2, P5)

</td>

<td style="text-align: center;">rank-2

</td>

<td style="text-align: center;">single-pair

</td>

<td style="text-align: center;">rank-3

</td>

<td style="text-align: center;">1

</td>

<td style="text-align: center;">E = d2

</td>

</tr>

<tr>

<td style="text-align: center;">semaphore

</td>

<td style="text-align: center;">(P8, P4/2)

</td>

<td style="text-align: center;">rank-2

Line 3,162:

Line 3,218:

<tr>

<td style="text-align: center;">decimal

</td>

<td style="text-align: center;">(P8/2, P4/2)

</td>

<td style="text-align: center;">rank-2

Line 3,176:

Line 3,234:

<tr>

<td style="text-align: center;">5-limit JI

</td>

<td style="text-align: center;">(P8, P5, ^1)

</td>

<td style="text-align: center;">rank-3

Line 3,190:

Line 3,250:

<tr>

<td style="text-align: center;">marvel

</td>

<td style="text-align: center;">(P8, P5, ^1)

</td>

<td style="text-align: center;">rank-3

Line 3,204:

Line 3,266:

<tr>

<td style="text-align: center;">breedsmic

</td>

<td style="text-align: center;">(P8, P5/2, ^1)

</td>

<td style="text-align: center;">rank-3

Line 3,218:

Line 3,282:

<tr>

<td style="text-align: center;">7-limit JI

</td>

<td style="text-align: center;">(P8, P5, ^1, /1)

</td>

<td style="text-align: center;">rank-4

Line 3,232:

Line 3,298:

</table>

When there is more than one enharmonic, they combine to make new enharmonics. Decimal's 2nd enharmonic could be written as ^^\\A1, but combining two accidentals in one enharmonic is avoided.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.

Line 3,371:

Line 3,438:

If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.

Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A convention that colors are used for infinitely stackable accidentals and ups/downs/highs/lows for the other kind of accidentals ~~is arguably a good thing~~.

Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.

<h2 id="toc17"><a name="Further Discussion-Notating Blackwood-like pergens*"></a>Notating Blackwood-like pergens*</h2>

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a ~~near-~~just ~~5th~~, like P8/5, P8/7, P8/10 and especially P8/12.

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).

The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.

Such a pergen is in effect multiple copies of an edo. Its notation ~~can be~~ based on the edo's notation, expanded with an additional microtonal accidental pair~~. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator~~. Examples:

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:

Line 3,385:

Line 3,454:

<tr>

<th>temperament

~~</th>~~

~~<th> ~~

</th>

<th>pergen

Line 3,405:

Line 3,472:

<tr>

<td style="text-align: center;">Blackwood

~~</td>~~

~~<td style="text-align: center;">5edo+y ~~

</td>

<td style="text-align: center;">(P8/5, ^1)

</td>

<td style="text-align: center;">~~fifth~~-~~8ve no~~-~~threes~~

<td style="text-align: center;">rank-2 5-edo

</td>

<td style="text-align: center;">E = m2

Line 3,418:

Line 3,483:

<td style="text-align: center;">D F#v=Gv Bvv...

</td>

<td style="text-align: center;">81/80

<td style="text-align: center;">81/80 = 16/15

</td>

<td style="text-align: center;">---

Line 3,424:

Line 3,489:

</tr>

<tr>

<td style="text-align: center;">Whitewood

</td>

<td style="text-align: center;">(P8/7, ^1)

</td>

<td style="text-align: center;">rank-2 7-edo

</td>

<td style="text-align: center;">E = A1

</td>

<td style="text-align: center;">D E F G A B C D

</td>

<td style="text-align: center;">D F^ A^^...

</td>

<td style="text-align: center;">80/81 = 135/128

</td>

<td style="text-align: center;">---

</td>

</tr>

<tr>

<td style="text-align: center;">10edo+y

</td>

<td style="text-align: center;">(P8/10, /1)

</td>

<td style="text-align: center;">rank-2 10-edo

</td>

<td style="text-align: center;">E = m2, E' = vvA1 = vvM2

</td>

<td style="text-align: center;">D D^=Ev E=F F^=Gv G...

</td>

<td style="text-align: center;">D F#\=G\ B\\...

</td>

<td style="text-align: center;">???

</td>

<td style="text-align: center;">81/80

</td>

</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

<td style="text-align: center;">12edo+j

</td>

<td style="text-align: center;">(P8/12, ^1)

</td>

<td style="text-align: center;">rank-2 12-edo

</td>

<td style="text-align: center;">E = d2

Line 3,465:

Line 3,544:

<tr>

~~</td>~~

~~<td style="text-align: center;">10edo+y ~~

</td>

Line 3,476:

Line 3,553:

</td>

<td style="text-align: center;">D G#v=Abv Dvv...

</td>

<td style="text-align: center;">729/704 =

</td>

Line 3,484:

Line 3,561:

</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

<td style="text-align: center;">17edo+y

</td>

<td style="text-align: center;">(P8/17, /1)

</td>

<td style="text-align: center;">rank-2 17-edo

</td>

<td style="text-align: center;">E = dd3, E' = vm2 = vvA1

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<td style="text-align: center;">D D^=Eb D#=Ev E F...

</td>

<td style="text-align: center;">D F#\ A#\\...

<td style="text-align: center;">D F#\ A#\\=Bv\\...

</td>

<td style="text-align: center;">256/243

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</tr>

<tr>

~~<td style="text-align: center;"> ~~

~~</td>~~

</td>

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</table>

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or perhaps 13/12. The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.

(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)

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Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.

See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.

<h2 id="toc19"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

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See the screenshots in the next section for examples of which pergens are supported by a specific edo.

Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the ~~bezout pair~~ of (N~~/m,~~N'~~/m) is (g,g'),~~ G = |g'|\N= |g|\N'. ~~The bezout pair~~ is ~~also either ancestor of N/N' in the scale tree. Alternate~~ generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.

Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.

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A specific pergen can be converted to an edo pair by looking up its generator cents in the <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator">arbitrary generator</a> table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines ~~half-5th~~.

A specific pergen can be converted to an edo pair by looking up its generator cents in the <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator">arbitrary generator</a> table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).

<h2 id="toc21"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

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needs pergen squares picture

fill in the 2nd pergens column above

to do:

add a mapping commas section somewhere?

finish proofs

link from: ups and downs page, Kite Giedraitis page, MOS scale names page, ~~is there~~ a rank-~~2 page?~~

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a>

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a>

(screenshot)

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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a>

Screenshot of the first 38 pergens:

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" />

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-22 14:58:53 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-25 19:22:43 UTC</tt>.<br>
-: The original revision id was <tt>626778259</tt>.<br>
+: The original revision id was <tt>626876991</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 142: / Line 142: @@
 ||~   ||~ pergen ||~ enharmonic
-interval(s) ||~ equiva-
+interval(s) ||~ equivalence(s) ||~ split
-lence(s) ||~ split
+interval(s) ||~ perchain(s) and/or
-interval(s) ||~ perchain(s) and
 genchains(s) ||~ examples ||
 ||= 1 ||= (P8, P5)
-unsplit ||= none ||= none ||= none ||= C - G ||= meantone,
+unsplit ||= none ||= none ||= none ||= C - G ||= pythagorean, meantone, dominant,
-schismic ||
+schismic, mavila, archy, etc. ||
 ||~   ||~ halves ||~   ||~   ||~   ||~   ||~   ||
 ||= 2 ||= (P8/2, P5)
@@ Line 155: / Line 154: @@
 ^1 = 81/80 ||
 ||=   ||= " ||= vvd2 (if 5th
-&lt; 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red
+&lt; 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= injera
 ^1 = 64/63 ||
-||=   ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 169/162
+||=   ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= deep emerald, if 13/8 = M6
 ^1 = 27/26 ||
 ||= 3 ||= (P8, P4/2)
 half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore
 ^1 = 64/63 ||
-||=   ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||
+||=   ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= double large deep yellow
+^1 = 81/80 ||
 ||= 4 ||= (P8, P5/2)
 half-5th ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira
@@ Line 179: / Line 179: @@
 C - Eb^=Ev - G,
 C - F#v/=Gb^\ - C,
-C - F^/=Gv\ - C ||= 49/48 &amp; 243/242
+C - F^/=Gv\ - C ||= deep blue &amp; deep amber
 ^1 = 33/32
 /1 = 64/63 ||
@@ Line 190: / Line 190: @@
 P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C,
 C - D/=Eb\ - F,
-C - Eb^/=Ev\ - G ||= 2048/2025 &amp; 49/48
+C - Eb^/=Ev\ - G ||= small deep green &amp; deep blue
 ^1 = 81/80
 /1 = 64/63 ||
@@ Line 201: / Line 201: @@
 P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C,
 C - Eb/=E\ - G,
-C - Dv/=Eb^\ - F ||= 2048/2025 &amp; 243/242
+C - Dv/=Eb^\ - F ||= small deep green and deep amber
 ^1 = 81/80
 /1 = 33/32 ||
 ||~   ||~ thirds ||~   ||~   ||~   ||~   ||~   ||
 ||= 6 ||= (P8/3, P5)
-third-8ve ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||
+third-8ve ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented
+^1 = 81/80 ||
 ||= 7 ||= (P8, P4/3)
-third-4th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3 ``=`` &lt;/span&gt;C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||
+third-4th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine
+^1 = 81/80 ||
 ||= 8 ||= (P8, P5/3)
-third-5th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
+third-5th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric
+^1 = 64/63 ||
 ||= 9 ||= (P8, P11/3)
-third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,
+third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber, if 11/8 = A4
-with 11/8 = A4 ||
+^1 = 729/704 ||
-||=   ||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||
+||=   ||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= small triple amber, if 11/8 = P4
+^1 = 33/32 ||
 ||= 10 ||= (P8/3, P4/2)
 third-8ve, half-4th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;A2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D# ||= P8/3 = ^^m3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A4
 P4/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= C - Eb^^ - Avv - C
-C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||= sixfold jade ||
+C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||= sixfold jade, if 11/8 = P4
+^1 = 33/32 ||
 ||=   ||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
 \\m2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B#
 C``//`` = Db ||= P8/3 = vM3 = ^^d4
 P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C
-C - D/=Eb\ - F ||= 128/125 &amp; 49/48 ||
+C - D/=Eb\ - F ||= triforce (128/125 &amp; 49/48)
+^1 = 81/80, /1 = 64/63 ||
 ||= 11 ||= (P8/3, P5/2)
 third-8ve, half-5th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2
@@ Line 244: / Line 250: @@
 C``///`` = Db ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
-C - /D - \F - G ||= 50/49 &amp; 1029/1024 ||
+C - /D - \F - G ||= lemba (50/49 &amp; 1029/1024) ||
 ||= 14 ||= (P8/2, P11/3)
 half-8ve, third-11th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D ||= P8/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5
@@ Line 393: / Line 399: @@
 ==Ratio and cents of the accidentals==
-In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.
 We can assign cents to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.
@@ Line 399: / Line 405: @@
 In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.
-This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:
+This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:
 -edo: # ``=`` 240¢, ^ ``=`` 80¢ (^ = 1/3 #)
@@ Line 414: / Line 420: @@
 eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ (^ = 1/3 #)
 sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ (no fixed relationship between ^ and #)
-third-comma injera: # ``=`` 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80)
+third-comma injera: # ``=`` 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
-eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243)
+eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
 Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.
@@ Line 517: / Line 523: @@
 An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.
-Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32.
+Another "tippy" temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.
 ==Notating unsplit pergens==
@@ Line 541: / Line 547: @@
 ==Notating rank-3 pergens==
-Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:
+Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:
-||~ tuning ||~ tuning's rank ||~ notation ||~ notation's rank ||~ ``#`` of enharmonics needed ||~ enharmonics ||
+||~ tuning ||~ pergen ||~ tuning's rank ||~ notation ||~ notation's rank ||~ ``#`` of enharmonics needed ||~ enharmonics ||
-||= 12-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||
+||= 12-edo ||= (P8/12) ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||
-||= 19-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = dd2 ||
+||= 19-edo ||= (P8/19) ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = dd2 ||
-||= 15-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||
+||= 15-edo ||= (P8/15) ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||
-||= 24-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 ||
+||= 24-edo ||= (P8/24) ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 ||
-||= pythagorean ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
+||= pythagorean ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
-||= meantone ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
+||= meantone ||= (P8, P5) ||= rank-2 ||= conventional ||= rank-2 ||= 0 ||= --- ||
-||= semaphore ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||
+||= srutal ||= (P8/2, P5) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = d2 ||
-||= decimal ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 ||
+||= semaphore ||= (P8, P4/2) ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||
-||= 5-limit JI ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||
+||= decimal ||= (P8/2, P4/2) ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 ||
-||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||
+||= 5-limit JI ||= (P8, P5, ^1) ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||
-||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||
+||= marvel ||= (P8, P5, ^1) ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||
-||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||
+||= breedsmic ||= (P8, P5/2, ^1) ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||
+||= 7-limit JI ||= (P8, P5, ^1, /1) ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||
+When there is more than one enharmonic, they combine to make new enharmonics. Decimal's 2nd enharmonic could be written as ^^\\A1, but combining two accidentals in one enharmonic is avoided.
 A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.
@@ Line 575: / Line 583: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.
-Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A convention that colors are used for infinitely stackable accidentals and ups/downs/highs/lows for the other kind of accidentals is arguably a good thing.
+Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.
 ==Notating Blackwood-like pergens*==
-A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a near-just 5th, like P8/5, P8/7, P8/10 and especially P8/12.
+A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
+A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).
 The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.
-Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples:
+Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:
-||~ temperament ||~   ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 ||
+||~ temperament ||~ pergen ||~ spoken ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ||~ /1 ||
-||= Blackwood ||= 5edo+y ||= (P8/5, ^1) ||= fifth-8ve no-threes ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 ||= --- ||
+||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 = 16/15 ||= --- ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||= Whitewood ||= (P8/7, ^1) ||= rank-2 7-edo ||= E = A1 ||= D E F G A B C D ||= D F^ A^^... ||= 80/81 = 135/128 ||= --- ||
-||=   ||= 12edo+j ||=   ||=   ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||
+||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ??? ||= 81/80 ||
-||=   ||= 10edo+y ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||
-||=   ||= 17edo+y ||=   ||=   ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\... ||= 256/243 ||= 81/80 ||
+||=   ||=   ||=   ||=   ||=   ||= D G#v=Abv Dvv... ||= 729/704 = ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or perhaps 13/12. The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.
 (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)
@@ Line 638: / Line 650: @@
 Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.
+See also the [[Map of rank-2 temperaments|map of rank-2 temperaments]].
 ==Pergens and MOS scales==
@@ Line 786: / Line 800: @@
 See the screenshots in the next section for examples of which pergens are supported by a specific edo.
-Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. The bezout pair is also either ancestor of N/N' in the scale tree. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.
+Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.
 ||~ edos ||~ octave split ||~ period ||~ generator(s) ||~ 5th's keyspans ||~ pergen ||~ 2nd pergen ||
 ||= 7 &amp; 12 ||= 1 ||= 7\7 = 12\12 ||= 3\7 = 5\12 ||= 4\7 = 7\12 ||= unsplit ||= (P8, WWP5/6) ||
@@ Line 801: / Line 815: @@
 ||= 22 &amp; 24 ||= 2 ||= 11\22 = 12\24 ||= **1\22= 1\24**, 10\22 = 11\24 ||= 13\22 = 14\24 ||= half-8ve quarter-tone ||=   ||
-A specific pergen can be converted to an edo pair by looking up its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th.
+A specific pergen can be converted to an edo pair by looking up its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).
 ==Supplemental materials*==
@@ Line 808: / Line 822: @@
 needs pergen squares picture
 fill in the 2nd pergens column above
+to do:
 add a mapping commas section somewhere?
 finish proofs
-link from: ups and downs page, Kite Giedraitis page, MOS scale names page, is there a rank-2 page?
+link from: ups and downs page, Kite Giedraitis page, MOS scale names page,
+http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments
+http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments
 ===Notaion guide PDF===
@@ Line 1,601: / Line 1,619: @@
 interval(s)&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;equiva-&lt;br /&gt;
+         &lt;th&gt;equivalence(s)&lt;br /&gt;
-lence(s)&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;split&lt;br /&gt;
 interval(s)&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;perchain(s) and&lt;br /&gt;
+         &lt;th&gt;perchain(s) and/or&lt;br /&gt;
 genchains(s)&lt;br /&gt;
 &lt;/th&gt;
@@ Line 1,627: / Line 1,644: @@
          &lt;td style="text-align: center;"&gt;C - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;meantone,&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;pythagorean, meantone, dominant,&lt;br /&gt;
-schismic&lt;br /&gt;
+schismic, mavila, archy, etc.&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,680: / Line 1,697: @@
          &lt;td style="text-align: center;"&gt;C - F#^=Gbv - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;large deep red&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;injera&lt;br /&gt;
 ^1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,697: / Line 1,714: @@
          &lt;td style="text-align: center;"&gt;C - F^=Gv - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;169/162&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;deep emerald, if 13/8 = M6&lt;br /&gt;
 ^1 = 27/26&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,732: / Line 1,749: @@
          &lt;td style="text-align: center;"&gt;C - D#v=Ebb^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(-22,-11,2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;double large deep yellow&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,779: / Line 1,797: @@
 C - F^/=Gv\ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;49/48 &amp;amp; 243/242&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;deep blue &amp;amp; deep amber&lt;br /&gt;
 ^1 = 33/32&lt;br /&gt;
 /1 = 64/63&lt;br /&gt;
@@ Line 1,805: / Line 1,823: @@
 C - Eb^/=Ev\ - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;2048/2025 &amp;amp; 49/48&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small deep green &amp;amp; deep blue&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
 /1 = 64/63&lt;br /&gt;
@@ Line 1,831: / Line 1,849: @@
 C - Dv/=Eb^\ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;2048/2025 &amp;amp; 243/242&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small deep green and deep amber&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
 /1 = 33/32&lt;br /&gt;
@@ Line 1,867: / Line 1,885: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;augmented&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,877: / Line 1,896: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;!-- ws:start:WikiTextRawRule:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; &lt;/span&gt;C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; &lt;!-- ws:start:WikiTextRawRule:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P4/3 = vM2 = ^^m2&lt;br /&gt;
@@ Line 1,884: / Line 1,903: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;porcupine&lt;br /&gt;
+^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,901: / Line 1,921: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;slendric&lt;br /&gt;
+^1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,917: / Line 1,938: @@
          &lt;td style="text-align: center;"&gt;C - F#v - Cb^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;small triple amber,&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small triple amber, if 11/8 = A4&lt;br /&gt;
-with 11/8 = A4&lt;br /&gt;
+^1 = 729/704&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,934: / Line 1,955: @@
          &lt;td style="text-align: center;"&gt;C - F^ - Cv - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;same, with 11/8 = P4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small triple amber, if 11/8 = P4&lt;br /&gt;
+^1 = 33/32&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,953: / Line 1,975: @@
 C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;sixfold jade&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;sixfold jade, if 11/8 = P4&lt;br /&gt;
+^1 = 33/32&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,973: / Line 1,996: @@
 C - D/=Eb\ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;128/125 &amp;amp; 49/48&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;triforce (128/125 &amp;amp; 49/48)&lt;br /&gt;
+^1 = 81/80, /1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,054: / Line 2,078: @@
 C - /D - \F - G&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;50/49 &amp;amp; 1029/1024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;lemba (50/49 &amp;amp; 1029/1024)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,719: / Line 2,743: @@
 &lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
-In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition. See also blackwood-like pergens below.&lt;br /&gt;
 &lt;br /&gt;
 We can assign cents to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.&lt;br /&gt;
@@ Line 2,725: / Line 2,749: @@
 In certain edos, the up symbol's cents can be directly related to the sharp's cents. For example, in 15edo, ^ is 1/3 the cents of #. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.&lt;br /&gt;
 &lt;br /&gt;
-This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, information for playing the score. Examples:&lt;br /&gt;
+This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, perhaps the cents of the 5th, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, basic information for playing the score. Examples:&lt;br /&gt;
 &lt;br /&gt;
 -edo: # &lt;!-- ws:start:WikiTextRawRule:036:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:036 --&gt; 240¢, ^ &lt;!-- ws:start:WikiTextRawRule:037:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:037 --&gt; 80¢ (^ = 1/3 #)&lt;br /&gt;
@@ Line 2,740: / Line 2,764: @@
 eighth-comma porcupine: # &lt;!-- ws:start:WikiTextRawRule:043:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:043 --&gt; 157¢, ^ = 52¢ (^ = 1/3 #)&lt;br /&gt;
 sixth-comma srutal: # &lt;!-- ws:start:WikiTextRawRule:044:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:044 --&gt; 139¢, ^ = 33¢ (no fixed relationship between ^ and #)&lt;br /&gt;
-third-comma injera: # &lt;!-- ws:start:WikiTextRawRule:045:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:045 --&gt; 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80)&lt;br /&gt;
+third-comma injera: # &lt;!-- ws:start:WikiTextRawRule:045:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:045 --&gt; 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)&lt;br /&gt;
-eighth-comma hedgehog: # &lt;!-- ws:start:WikiTextRawRule:046:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:046 --&gt; 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243)&lt;br /&gt;
+eighth-comma hedgehog: # &lt;!-- ws:start:WikiTextRawRule:046:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:046 --&gt; 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)&lt;br /&gt;
 &lt;br /&gt;
 Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&lt;br /&gt;
@@ Line 2,834: / Line 2,858: @@
 An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an E of ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.&lt;br /&gt;
 &lt;br /&gt;
-Another &amp;quot;tippy&amp;quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32.&lt;br /&gt;
+Another &amp;quot;tippy&amp;quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Notating unsplit pergens&lt;/h2&gt;
@@ Line 3,044: / Line 3,068: @@
 &lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;Notating rank-3 pergens&lt;/h2&gt;
   &lt;br /&gt;
-Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:&lt;br /&gt;
+Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:&lt;br /&gt;
@@ Line 3,050: / Line 3,074: @@
      &lt;tr&gt;
          &lt;th&gt;tuning&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;pergen&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;tuning's rank&lt;br /&gt;
@@ Line 3,064: / Line 3,090: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;12-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/12)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-1&lt;br /&gt;
@@ Line 3,078: / Line 3,106: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;19-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/19)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-1&lt;br /&gt;
@@ Line 3,092: / Line 3,122: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;15-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/15)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-1&lt;br /&gt;
@@ Line 3,106: / Line 3,138: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;24-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/24)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-1&lt;br /&gt;
@@ Line 3,120: / Line 3,154: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;pythagorean&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-2&lt;br /&gt;
@@ Line 3,134: / Line 3,170: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;meantone&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-2&lt;br /&gt;
@@ Line 3,144: / Line 3,182: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;srutal&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;single-pair&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = d2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;semaphore&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-2&lt;br /&gt;
@@ Line 3,162: / Line 3,218: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;decimal&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P4/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-2&lt;br /&gt;
@@ Line 3,176: / Line 3,234: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;5-limit JI&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-3&lt;br /&gt;
@@ Line 3,190: / Line 3,250: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;marvel&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-3&lt;br /&gt;
@@ Line 3,204: / Line 3,266: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;breedsmic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2, ^1)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-3&lt;br /&gt;
@@ Line 3,218: / Line 3,282: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;7-limit JI&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, ^1, /1)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;rank-4&lt;br /&gt;
@@ Line 3,232: / Line 3,298: @@
 &lt;/table&gt;
+When there is more than one enharmonic, they combine to make new enharmonics. Decimal's 2nd enharmonic could be written as ^^\\A1, but combining two accidentals in one enharmonic is avoided.&lt;br /&gt;
 &lt;br /&gt;
 A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.&lt;br /&gt;
@@ Line 3,371: / Line 3,438: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.&lt;br /&gt;
 &lt;br /&gt;
-Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A convention that colors are used for infinitely stackable accidentals and ups/downs/highs/lows for the other kind of accidentals is arguably a good thing.&lt;br /&gt;
+Demeter is unusual in that its gen2 isn't a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^. Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
   &lt;br /&gt;
-A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a near-just 5th, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
+A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
+&lt;br /&gt;
+A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).&lt;br /&gt;
 &lt;br /&gt;
 The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.&lt;br /&gt;
 &lt;br /&gt;
-Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair. Removing the 3-limit comma makes a rank-3 pergen, and blackwood-like pergens are like rank-3 pergens minus the first generator. Examples:&lt;br /&gt;
+Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:&lt;br /&gt;
@@ Line 3,385: / Line 3,454: @@
      &lt;tr&gt;
          &lt;th&gt;temperament&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;pergen&lt;br /&gt;
@@ Line 3,405: / Line 3,472: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;5edo+y&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;fifth-8ve no-threes&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-2 5-edo&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E = m2&lt;br /&gt;
@@ Line 3,418: / Line 3,483: @@
          &lt;td style="text-align: center;"&gt;D F#v=Gv Bvv...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;81/80 = 16/15&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
@@ Line 3,424: / Line 3,489: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Whitewood&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/7, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 7-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = A1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D E F G A B C D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F^ A^^...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;80/81 = 135/128&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;10edo+y&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/10, /1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-2 10-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;E = m2, E' = vvA1 = vvM2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;D D^=Ev E=F F^=Gv G...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;D F#\=G\ B\\...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;???&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12edo+j&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/12, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-2 12-edo&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E = d2&lt;br /&gt;
@@ Line 3,465: / Line 3,544: @@
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10edo+y&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,476: / Line 3,553: @@
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;D G#v=Abv Dvv...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;729/704 =&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,484: / Line 3,561: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;17edo+y&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/17, /1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-2 17-edo&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;E = dd3, E' = vm2 = vvA1&lt;br /&gt;
@@ Line 3,496: / Line 3,571: @@
          &lt;td style="text-align: center;"&gt;D D^=Eb D#=Ev E F...&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D F#\ A#\\...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;D F#\ A#\\=Bv\\...&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
@@ Line 3,504: / Line 3,579: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,524: / Line 3,597: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,544: / Line 3,615: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,564: / Line 3,633: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,584: / Line 3,651: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,604: / Line 3,669: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,625: / Line 3,688: @@
 &lt;/table&gt;
+If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or perhaps 13/12. The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.&lt;br /&gt;
 &lt;br /&gt;
 (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)&lt;br /&gt;
@@ Line 3,976: / Line 4,040: @@
 &lt;br /&gt;
 Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.&lt;br /&gt;
+&lt;br /&gt;
+See also the &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;map of rank-2 temperaments&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:95:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Pergens and MOS scales&lt;/h2&gt;
@@ Line 5,391: / Line 5,457: @@
 See the screenshots in the next section for examples of which pergens are supported by a specific edo.&lt;br /&gt;
 &lt;br /&gt;
-Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp;amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. If the bezout pair of (N/m,N'/m) is (g,g'), G = |g'|\N= |g|\N'. The bezout pair is also either ancestor of N/N' in the scale tree. Alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking sometimes creates a 2nd pergen.&lt;br /&gt;
+Just as a pair of edos can specify a rank-2 temperament, a pair of edos can specify a rank-2 pergen. For N-edo &amp;amp; N'-edo, m = GCD (N,N'). P = (N/m)\N = (N'/m)\N'. Let g\g' be the smaller-numbered ancestor of N\N' in the scale tree. G = g\N = g'\N'. If the octave is split, alternate generators are found by adding periods, or by subtracting periods and inverting. One generator is chosen (bolded in the table) and stacked until it totals some voicing of the 4th or 5th of both edos, or if the octave is split, some appropriate imperfect multigen. Further stacking, or stacking an alternate generator, sometimes creates a 2nd pergen.&lt;br /&gt;
@@ Line 5,591: / Line 5,657: @@
 &lt;br /&gt;
-A specific pergen can be converted to an edo pair by looking up its generator cents in the &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator"&gt;arbitrary generator&lt;/a&gt; table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines half-5th.&lt;br /&gt;
+A specific pergen can be converted to an edo pair by looking up its generator cents in the &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator"&gt;arbitrary generator&lt;/a&gt; table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&gt;Supplemental materials*&lt;/h2&gt;
@@ Line 5,598: / Line 5,664: @@
 needs pergen squares picture&lt;br /&gt;
 fill in the 2nd pergens column above&lt;br /&gt;
+&lt;br /&gt;
+to do:&lt;br /&gt;
 add a mapping commas section somewhere?&lt;br /&gt;
 finish proofs&lt;br /&gt;
-link from: ups and downs page, Kite Giedraitis page, MOS scale names page, is there a rank-2 page?&lt;br /&gt;
+link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7095:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7095 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7096:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7096 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Notaion guide PDF&lt;/h3&gt;
   &lt;br /&gt;
 This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7026:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7026 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7097:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7097 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,615: / Line 5,685: @@
   &lt;br /&gt;
 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7027:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7027 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7098:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7098 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshot of the first 38 pergens:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:4216:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4216 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:4252:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4252 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;