Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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==Tipping points==

Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**~~ups and downs~~ may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**up may need to be swapped with down, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

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This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is ~~(P8, P5, b3/2) or~~ (P8, P5, vm3/2). It could also be called ~~(P8, P5, y3/3) or~~ (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.

Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the **DOL** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred.

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||= 4 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= same ||= same ||

||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||= same ||

||= 6 ||= (P8, P5, ~~vm3~~/2) ||= half-upminor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||

||= 6 ||= (P8, P5, ^m3/2) ||= half-upminor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||

||= 7 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||

||= 8 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, ^M6/2) ||= half-upmajor-6th ||

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There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.

==Notating Blackwood-like pergens*==

==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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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 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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.

A ~~blackwood~~-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits 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).

A Blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). 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 name ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ratio ||~ /1 ratio ||

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 ||

||= 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 ||= --- ||

||= 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\\... ||= (see below) ||= 81/80 ||

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

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

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

||= " ||= " ||= " ||= " ||= " ||= 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 ||

||= 17edo+yellow ||= (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 10, 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 13/12.

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

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

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

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

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 13/12.

~~The additional accidental has an equivalent ratio, found by adding the pergen's 3~~-~~limit comma onto~~ the ~~ratio~~. ~~Blackwood~~'s ~~comma~~ is 256/243, ~~and ^1 = 81~~/~~80 or equivalently~~, 16/15.

Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2). Blackwood-like pergens are a superset of rank-3 pergens, and are __very__ numerous.

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

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:

||~ temperament ||~ subgroup ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ ^1 ratio ||

||= large quintuple blue ||= 2.3.7 ||= (-14,0,0,5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= D E^^ Gv A^ Cvv D ||= C G D A E... ||= 49/48 ||

||= small quintuple red ||= 2.3.7 ||= (22,-5,0,-5) ||= " ||= " ||= " ||= " ||= " ||= 64/63 ||

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section).

~~//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 ^m6/2 or ^^A5/4.//~~

==Notating non-8ve and no-5ths pergens==

~~//When~~ the ~~edo implied by~~ the ~~octave split doesn't have a decent~~ 5th~~, as with P8/3, P8/4, P8/6, P8/8, etc.~~, and the ~~generator isn~~'t .~~..//~~

In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in "no-5ths~~" or "minus white"~~ pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. ~~The same is true of relative notation:~~ 75% of all intervals don't exist. There is no perfect 5th or major 2nd.

But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

~~Note~~-~~less names~~ can be ~~avoided~~ in ~~any pergen with~~ a ~~5th (or 4th or 11th, etc~~.~~). The full chain of 5ths must appear with neither ups~~/~~downs nor highs~~/~~lows in either the perchain or the genchain. For this reason~~, ~~Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same~~. ~~Thus the last row~~ of this ~~table~~ is ~~an invalid~~ notation.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.

~~||~ tuning ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~~~ notes ||

~~||= large quintuple blue~~

~~or small quintuple red ||= (-14,0,0,5)~~

~~or (22,-5,0,-5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E~~.~~.. ||=~~ a ~~valid~~ notation ||

~~||= Blackwood ||= 256/243 ||= (P8/5~~, ~~^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv..~~. ~~||= a valid notation ||~~

~~||= " ||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G,~~

~~D or A notes ||~~

~~== ==~~

~~edo-subset notations~~

~~P8/6, P8/8~~

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.

||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||

||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||

||= 1 ||= (P8, P5) ||= (P8, M3) ||= (P8, M2) ||= (P12, M6) ||= (P12, M3) ||= (WWM3, d5) ||

||~ half-splits ||~ ||~ ||~ ||~ ||~ ||~ ||

||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (WWM3/2, d5) ||

||= 3 ||= (P8, P4/2) ||= (P8, M3/2) ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M3/2) ||= (WWM3, d5/2) ||

||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, m7/2) ||= (P12, m7/2) ||= (P12, m10/2) ||= (WWM3, WA6/2) ||

||= 5 ||= (P8/2, P4/2) ||= (P8/2, M3/2) ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (WWM3/2, d5/2) ||

||~ third-splits ||~ ||~ ||~ ||~ ||~ ||~ ||

||= 6 ||= (P8/3, P5) ||= (P8/3, M3) ||= (P8/3, M2) ||= (P12/3, M6) ||= (P12/3, M3) ||= (WWM3/3, d5) ||

||= 7 ||= (P8, P4/3) ||= (P8, M3/3) ||= (P8, M2/3) ||= (P12, M6/3) ||= (P12, M3/3) ||= (WWM3, d5/3) ||

||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, m10/3) ||= (WWM3, WA6/3) ||

||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) ||

||= 10 ||= (P8/3, P4/2) ||= (P8/3, M3/2) ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||

||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, m7/2) ||= (P12/3, m7/2) ||= ||= ||

||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||= ||= ||

||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||= ||= ||

||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||= ||= ||

||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||= ||= ||

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

~~==Notating non~~-~~8ve~~ and no-~~5ths~~ pergens*==

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).

~~Just as all rank-2~~ pergens in ~~which 2 and 3 are present and independent can be numbered, so can~~ all ~~2.5 pergens~~, ~~all~~ 2.~~7 pergens, all~~ 3~~.5 pergens, etc~~. The ~~pergens are grouped into blocks~~ and ~~sections as before:~~

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but let's start with the standard 2.3 prime subgroup. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The complete rank-2 lattice is formed by tiling the squares.

~~||~ ||||||||||||~ __the first two independent primes in~~ the ~~prime subgroup__ ||~~

~~||~ pergen number ||~ 2~~.~~3 ||~ 2~~.~~5 ||~ 2~~.~~7 ||~ 3~~.~~5 ||~ 3~~.~~7 ||~ 5.7 ||~~

For (P8, P5), the pergen square has 4 notes:

~~||= 1 ||= (P8, P5) ||= (P8, y3) ||= (P8, r2) ||= (P12, y6) ||= (P12, r3) ||= (WWy3, bg5) ||~~

C2 -- G2

~~||~ half-splits ||~ ||~ ||~ ||~ ||~ ||~ ||~~

| |

~~||= 2 ||=~~ (P8/2, P5) ~~||= (P8/2~~, ~~y3) ||= (P8/2, r2) ||= ||= ||= ||~~

C1 -- G1

~~||= 3 ||= (P8, P4/2) ||= (P8, y3/2) ||= (P8, r2/2) ||= ||= ||= ||~~

~~||=~~ 4 ~~||= (P8, P5/2) ||= (P8, g6/2) ||= (P8, b7/2) ||= ||= ||= ||~~

Splitting the period or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes:

~~||= 5 ||= (P8/2, P4/2) ||= (P8/2, y3/2) ||= (P8/2, r2/2) ||= ||= ||= ||~~

C2 --- G2

~~||~ third~~-~~splits ||~ ||~ ||~ ||~ ||~ ||~ ||~~

F#v1 F#v2

~~||= 6 ||= (P8/3, P5) ||= (P8/3, y3) ||= ||= ||= ||=~~ ||

C1 --- G1

~~||= 7 ||= (P8, P4/3) ||= (P8, y3/3) ||= ||= ||= ||= ||~~

~~||= 8 ||= (P8, P5/3) ||= (P8, g6/3) ||= ||= ||= ||= ||~~

~~||= 9 ||= (P8, P11/3) ||= (P8, y10/3) ||= ||= ||= ||= ||~~

~~||= 10 ||=~~ (P8~~/3, P4~~/2~~) ||= (P8/3, y3/2) ||= ||= ||= ||= ||~~

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

~~||= 12 ||= (P8/2~~, ~~P4/3) ||= ||= ||= ||= ||= ||~~

~~||= 13 ||= (P8/2, P5/3) ||= ||= ||= ||= ||= ||~~

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

~~||= 15 ||= (P8/3, P4/3) ||= ||= ||= ||= ||= ||~~

~~Every rank-2 pergen~~ can be ~~identified~~ by the ~~first two independent primes and its pergen number~~. ~~Blackwood is~~ 2.5 ~~pergen #33. For prime~~ subgroup ~~p.q~~, ~~the unsplit pergen has period p~~/1. ~~The generator is found by dividing q by p until it's less than p~~/1, and ~~inverting if it's more than half of p~~/1.

The square can be generalized to any prime subgroup by representing the notes as dots. In the 2.5 subgroup, a horizontal side equals 5/4. In Bohlen-Peirce, horizontal = 5/3 and vertical = 3/1.

~~Every rank-3 pergen can be identified by its~~ first ~~__three__ independent primes and its pergen number. A similar table can be made for all~~ rank-3 pergens. ~~Each block is much larger~~. ~~The notation for 2~~.~~3.5 pergens and 2.3.7 pergens is often~~ the ~~same.~~

Here are the first 32 rank-2 pergens in a completely JI-agnostic format. True doubles are in red. Imperfect multigens are in green. These properties are independent of the prime subgroup.

~~||~ pergen number ||~ 2.3.5 ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5~~.~~7 ||~~

~~||= 1 ||= (P8, P5, ^1) ||= same ||= (P8, y3, r2) ||= (P12, y6, r3) ||~~

~~||~ half-splits ||~ ||~ ||~ ||~ ||~~

~~||= 2 ||= (P8/2, P5, ^1) ||= same ||= ||= ||~~

~~||= 3 ||= (P8, P4/2, ^1) ||= same ||= ||= ||~~

~~||= 4 ||= (P8, P5/2, ^1) ||= same ||= ||= ||~~

~~||= 5 ||= (P8/2, P4/2, ^1) ||= same ||= ||= ||~~

~~||= 6 ||= (P8, P5, vM3/2) ||= (P8, P5, ^M2/2) ||= ||= ||~~

~~||= 7 ||= (P8, P5, ^m6/2) ||= (P8, P5, vm7/2) ||= ||= ||~~

~~||= 8 ||= (P8/2, P5, vM3/2) ||= ||= ||= ||~~

~~||= 9 ||= (P8/2, P5, ^m6/2) ||= ||= ||= ||~~

~~||= 10 ||= (P8, P4/2, vM3/2) ||= ||= ||= ||~~

~~||= 11 ||= (P8, P4/2, ^m6/2) ||= ||= ||= ||~~

~~||= 12 ||= (P8, P5/2, vM3/2) ||= ||= ||= ||~~

~~||= 13 ||= (P8, P5/2, ^m6/2) ||= ||= ||= ||~~

~~||= 14 ||= (P8/2, P4/2, vM3/2) ||= ||= ||= ||~~

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to use a notation that isn't backwards compatible.

~~Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...~~

A similar chart could be made of all rank-3 pergen cubes.

A similar chart could be made for all rank-3 pergens, using pergen cubes.

==Notating tunings with an arbitrary generator==

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If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen.

~~Here are some example edo pairs.~~ The closer two edos are in the scale tree, the simpler the pergen they make:

The closer two edos are in the scale tree, the simpler the pergen they make. Examples::

||~ ||~ 12-edo ||~ 13b-edo ||~ 14-edo ||~ 15-edo ||~ 16-edo ||~ 17-edo ||~ 18b-edo ||~ 19-edo ||~ 20-edo ||

||~ 13b-edo ||= (P8, P5/7) ||= ||= ||= ||= ||= ||= ||= ||= ||

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needs more screenshots, including 12-edo's pergens and a page of the pdf

repost alt-pergenlister code

add pergens square pic

the half-step glitch

to do:

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__**Credits**__

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Pergen squares are Praveen's creation.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens">Notating rank-3 pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens*">Notating Blackwood-like pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens">Notating Blackwood-like pergens</a></div>

<div style="margin-left: 2em;"><a href="#~~toc18~~"> </a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating non-8ve and no-5ths pergens">Notating non-8ve and no-5ths pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating non-8ve and no-5ths pergens*">Notating non-8ve and no-5ths pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-~~Notating tunings with an arbitrary generator~~">~~Notating tunings with an arbitrary generator~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and ~~MOS scales~~">Pergens and ~~MOS scales~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-~~Pergens and EDOs~~">~~Pergens and EDOs~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: ~~2em~~;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-~~Notaion guide PDF~~">~~Notaion guide PDF~~</a></div>

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</a></div>

<div style="margin-left: ~~3em~~;"><a href="#Further Discussion-~~Supplemental materials*-pergenLister app~~">~~pergenLister app~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-~~Various proofs (unfinished)~~">~~Various proofs (unfinished)~~</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

<div ~~style="margin-left: 2em;"~~><~~a href="#Further Discussion~~-~~Miscellaneous Notes">Miscellaneous Notes</a></div>~~

</div>

~~</div>~~

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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<h2 id="toc4"><a name="Applications-Tipping points"></a>Tipping points</h2>

<br />

Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &quot;tipping point&quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>~~ups and downs~~ may need to be swapped, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br />

Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &quot;tipping point&quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>up may need to be swapped with down, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br />

<br />

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.<br />

Line 3,642:

Line 3,625:

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.<br />

<br />

Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is ~~(P8, P5, b3/2) or~~ (P8, P5, vm3/2). It could also be called ~~(P8, P5, y3/3) or~~ (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.<br />

Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.<br />

<br />

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the <strong>DOL</strong> (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.<br />

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</td>

</td>

<td style="text-align: center;">half-upminor-3rd<br />

Line 3,944:

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There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.<br />

<br />

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

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

<br />

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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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.~~<br />~~

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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.<br />

~~<br />~~

~~A blackwood-like pergen is~~ a ~~rank-3 pergen plus a 3-limit comma~~. ~~Adding this comma splits 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)~~.<br />

<br />

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:<br />

A Blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). 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:<br />

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<th>pergen<br />

</th>

<th>spoken<br />

<th>spoken name<br />

</th>

<th>enharmonics<br />

Line 3,967:

Line 3,948:

<th>genchain<br />

</th>

<th>^1 ratio<br />

</th>

<th>/1 ratio<br />

</th>

</tr>

Line 4,009:

Line 3,990:

</tr>

<tr>

<td style="text-align: center;">10edo+yellow<br />

</td>

Line 4,027:

Line 4,008:

</tr>

<tr>

</td>

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</tr>

<tr>

<td style="text-align: center;">17edo+yellow<br />

</td>

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</td>

</tr>

</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 10, 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 13/12.<br />

<br />

The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.<br />

<br />

Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2). Blackwood-like pergens are a superset of rank-3 pergens, and are <u>very</u> numerous.<br />

<br />

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:<br />

<tr>

<th>temperament<br />

</th>

<th>subgroup<br />

</th>

<th>comma<br />

</th>

<th>pergen<br />

</th>

<th>spoken name<br />

</th>

<th>enharmonic<br />

</th>

<th>perchain<br />

</th>

<th>genchain<br />

</th>

<th>^1 ratio<br />

</th>

</tr>

<tr>

<td style="text-align: center;">large quintuple blue<br />

</td>

</td>

</td>

</td>

<td style="text-align: center;">fifth-8ve<br />

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">small quintuple red<br />

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

</table>

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &quot;missing notes&quot; (see next section).<br />

<br />

<h2 id="toc18"><a name="Further Discussion-Notating non-8ve and no-5ths pergens"></a>Notating non-8ve and no-5ths pergens</h2>

<br />

In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3. <br />

<br />

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.<br />

<br />

But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.<br />

<br />

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.<br />

<br />

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.<br />

<tr>

<th><u>pergen number</u><br />

</th>

<th colspan="6"><u>prime subgroup</u><br />

</th>

</tr>

<tr>

<th>unsplit<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

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

~~</td>~~

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

</td>

</tr>

<tr>

<th>half-splits<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

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

~~</td>~~

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

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

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

~~</td>~~

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

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

~~</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 13/12.<br />

~~<br />~~

~~The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.<br />~~

~~<br />~~

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

~~<br />~~

<em>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 ^m6/2 or ^^A5/4.</em><br />

~~<br />~~

~~<em>When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...</em><br />~~

~~<br />~~

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.<br />

~~<br />~~

But in &quot;no-5ths&quot; or &quot;minus white&quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.<br />

~~<br />~~

Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.<br />

~~<table class="wiki_table">~~

<tr>

<th>~~tuning~~<br />

<th>third-splits<br />

</th>

<th>~~comma~~<br />

</th>

<th>~~pergen~~<br />

</th>

<th>~~spoken name~~<br />

</th>

<th>~~enharmonic~~<br />

</th>

<th>~~perchain~~<br />

</th>

<th>~~genchain<br />~~

~~</th>~~

~~<th>notes~~<br />

</th>

</tr>

<tr>

<td style="text-align: center;">~~large quintuple blue<br />~~

~~or small quintuple red~~<br />

</td>

<td style="text-align: center;">(~~-14,0,0,5)<br~~ /~~>~~

~~or (22,-5,0~~,-5)<br />

</td>

</td>

<td style="text-align: center;">~~fifth-8ve~~<br />

</td>

</td>

</td>

<td style="text-align: center;">~~C G D A E...<br~~ /~~>~~

~~</td>~~

~~<td style="text-align: center;">a valid notation~~<br />

</td>

</tr>

<tr>

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

</td>

</td>

</td>

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

~~</td>~~

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

</td>

</td>

</td>

<td style="text-align: center;">~~a valid notation~~<br />

</td>

</tr>

<tr>

</td>

</td>

</td>

<td style="text-align: center;">~~fifth-8ve major 3rd~~<br />

</td>

</td>

</td>

<td style="text-align: center;">~~C E G#...<br />~~

~~</td>~~

~~<td style="text-align: center;">invalid, no G~~,~~<br~~ /~~>~~

~~D or A notes~~<br />

</td>

</tr>

~~</table>~~

~~<h2 id="toc18"> </h2>~~

~~edo-subset notations<br />~~

~~P8/6, P8/8<br />~~

~~<br />~~

<h2 id="toc19"><a name="Further Discussion-Notating non-8ve and no-5ths pergens*"></a>Notating non-8ve and no-5ths pergens*</h2>

~~<br />~~

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:<br />

~~<table class="wiki_table">~~

<tr>

</th>

</td>

<~~th colspan~~="6"><u>~~the first two independent primes in the prime subgroup~~</u><br />

</th>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

~~<th>pergen number<br />~~

~~</th>~~

~~<th>2.3<br />~~

~~</th>~~

~~<th>2.5<br />~~

~~</th>~~

~~<th>2.7<br />~~

~~</th>~~

~~<th>3.5<br />~~

~~</th>~~

~~<th>3.7<br />~~

~~</th>~~

~~<th>5.7<br />~~

~~</th>~~

~~</tr>~~

~~<tr>~~

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<th>~~half-splits~~<br />

</th>

</td>

</th>

</td>

</th>

</td>

</th>

</td>

</th>

</td>

</th>

</td>

</th>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

Line 4,367:

Line 4,398:

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

Line 4,383:

Line 4,414:

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

Line 4,399:

Line 4,430:

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

Line 4,414:

Line 4,445:

</td>

</tr>

<~~tr>~~

</table>

~~<th>third-splits<br~~ />

~~</th>~~

For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.<br />

~~<th><br~~ /~~>~~

<br />

~~</th>~~

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).<br />

~~<th>~~<br />

<br />

</th>

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but let's start with the standard 2.3 prime subgroup. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The complete rank-2 lattice is formed by tiling the squares.<br />

~~<th><br />~~

<br />

~~</th>~~

For (P8, P5), the pergen square has 4 notes:<br />

~~<th><br />~~

C2 -- G2<br />

~~</th>~~

| |<br />

~~<th>&~~lt;br />

C1 -- G1<br />

</th>

<br />

~~<th><br />~~

Splitting the period or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes:<br />

~~</th>~~

C2 --- G2<br />

</tr>

F#v1 F#v2<br />

<tr>

C1 --- G1<br />

~~<td style="text-align~~: ~~center;">6~~<br />

<br />

</td>

The square can be generalized to any prime subgroup by representing the notes as dots. In the 2.5 subgroup, a horizontal side equals 5/4. In Bohlen-Peirce, horizontal = 5/3 and vertical = 3/1. <br />

<~~td style="text-align: center;"~~>~~(P8/3, P5)~~<br />

<br />

</td>

Here are the first 32 rank-2 pergens in a completely JI-agnostic format. True doubles are in red. Imperfect multigens are in green. These properties are independent of the prime subgroup.<br />

~~<td style="text-align: center;">~~(P8/3, y3)<br />

</td>

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

</td>

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

</td>

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">7<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P4/3)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, y3/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">8<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5/3)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, g6/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">9<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P11/3)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, y10/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">10<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/3, P4/2)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/3, y3/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">11<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/3, P5/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">12<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P4/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">13<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P5/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">14<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P11/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">15<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/3, P4/3)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~</table>~~

~~<br />~~

Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.<br />

~~<br />~~

Every rank-3 pergen can be identified by its first <u>three</u> independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.<br />

~~<table class="wiki_table">~~

~~<tr>~~

~~<th>pergen number<br />~~

~~</th>~~

~~<th>2.3.5<br />~~

~~</th>~~

~~<th>2.3.7<br />~~

~~</th>~~

~~<th>2.5.7<br />~~

~~</th>~~

~~<th>3.5.7<br />~~

~~</th>~~

~~</tr>~~

~~<tr>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~<td style="text-align: center;">(P8, y3, r2)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P12, y6, r3)<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<th>half-splits<br />~~

~~</th>~~

~~<th><br />~~

~~</th>~~

~~<th><br />~~

~~</th>~~

~~<th><br />~~

~~</th>~~

~~<th><br />~~

~~</th>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">2<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">3<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P4/2, ^1)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">4<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">5<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P4/2, ^1)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">6<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5, vM3/2)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5, ^M2/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">7<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5, ^m6/2)<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5, vm7/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">8<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P5, vM3/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">9<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P5, ^m6/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">10<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P4/2, vM3/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">11<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P4/2, ^m6/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">12<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5/2, vM3/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">13<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8, P5/2, ^m6/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">14<br />~~

~~</td>~~

~~<td style="text-align: center;">(P8/2, P4/2, vM3/2)<br />~~

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~</table>~~

<br />

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of note-less names. The composer may well want to use a notation that isn't backwards compatible.<br />

<br />

~~Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...<br />~~

<br />

A similar chart could be made of all rank-3 pergen cubes.<br />

A similar chart could be made for all rank-3 pergens, using pergen cubes.<br />

<br />

<h2 id="~~toc20~~"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<h2 id="toc19"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<br />

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br />

Line 5,145:

Line 4,816:

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

<br />

<h2 id="~~toc21~~"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<h2 id="toc20"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br />

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Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.<br />

<br />

<h2 id="~~toc22~~"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc21"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.<br />

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If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. <br />

<br />

~~Here are some example edo pairs.~~ The closer two edos are in the scale tree, the simpler the pergen they make:<br />

The closer two edos are in the scale tree, the simpler the pergen they make. Examples::<br />

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<br />

<h2 id="~~toc23~~"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<h2 id="toc22"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<br />

needs more screenshots, including 12-edo's pergens and a page of the pdf<br />

repost alt-pergenlister code<br />

add pergens square pic<br />

the half-step glitch<br />

<br />

to do:<br />

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make a glossary of all bolded terms?<br />

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br />

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

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

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

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

<br />

<h3 id="~~toc24~~"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<h3 id="toc23"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<br />

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.<br />

<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><br />

<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><br />

(screenshot)<br />

<br />

<h3 id="~~toc25~~"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

<h3 id="toc24"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

<br />

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.<br />

<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><br />

<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><br />

<br />

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br />

<br />

Screenshots of the first 38 pergens:<br />

<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;" /><br />

<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;" /><br />

<br />

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:<br />

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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br />

<br />

<h2 id="~~toc26~~"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<h2 id="toc25"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<br />

The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br />

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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br />

<br />

<h2 id="~~toc27~~"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<h2 id="toc26"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<br />

<u><strong>Staff notation</strong></u><br />

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<u><strong>Credits</strong></u><br />

<br />

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</body></html></pre></div>

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Pergen squares are Praveen's creation.</body></html></pre></div>

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 <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>2018-03-24 16:28:16 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-24 22:24:55 UTC</tt>.<br>
-: The original revision id was <tt>627980589</tt>.<br>
+: The original revision id was <tt>627984283</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 304: / Line 304: @@
 ==Tipping points==
-Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.
+Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**up may need to be swapped with down, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.
 Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.
@@ Line 602: / Line 602: @@
 This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.
-Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.
+Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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.
 There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the **DOL** (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
@@ Line 616: / Line 616: @@
 ||= 4 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= same ||= same ||
 ||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||= same ||
-||= 6 ||= (P8, P5, vm3/2) ||= half-upminor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||
+||= 6 ||= (P8, P5, ^m3/2) ||= half-upminor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||
 ||= 7 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||
 ||= 8 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, ^M6/2) ||= half-upmajor-6th ||
@@ Line 635: / Line 635: @@
 There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.
-==Notating Blackwood-like pergens*==
+==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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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 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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.
-A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits 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).
+A Blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). 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 name ||~ enharmonics ||~ perchain ||~ genchain ||~ ^1 ratio ||~ /1 ratio ||
-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 ||
 ||= 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 ||= --- ||
-||= 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\\... ||= (see below) ||= 81/80 ||
+||= 10edo+yellow ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= (see below) ||= 81/80 ||
-||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||
+||= 12edo+jade ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||
 ||= " ||= " ||= " ||= " ||= " ||= 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 ||
+||= 17edo+yellow ||= (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 10, 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 13/12.
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
+The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
-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 13/12.
-The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.
+Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2). Blackwood-like pergens are a superset of rank-3 pergens, and are __very__ numerous.
-(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)
+It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:
+||~ temperament ||~ subgroup ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ ^1 ratio ||
+||= large quintuple blue ||= 2.3.7 ||= (-14,0,0,5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= D E^^ Gv A^ Cvv D ||= C G D A E... ||= 49/48 ||
+||= small quintuple red ||= 2.3.7 ||= (22,-5,0,-5) ||= " ||= " ||= " ||= " ||= " ||= 64/63 ||
+Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause "missing notes" (see next section).
-//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 ^m6/2 or ^^A5/4.//
+==Notating non-8ve and no-5ths pergens==
-//When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...//
+In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.
 In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.
-But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.
+But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.
-Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.
-||~ tuning ||~ comma ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||
-||= large quintuple blue
-or small quintuple red ||= (-14,0,0,5)
-or (22,-5,0,-5) ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||
-||= Blackwood ||= 256/243 ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||
-||= " ||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G,
-D or A notes ||
-== ==
-edo-subset notations
-P8/6, P8/8
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.
+||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||
+||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||
+||= 1 ||= (P8, P5) ||= (P8, M3) ||= (P8, M2) ||= (P12, M6) ||= (P12, M3) ||= (WWM3, d5) ||
+||~ half-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (WWM3/2, d5) ||
+||= 3 ||= (P8, P4/2) ||= (P8, M3/2) ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M3/2) ||= (WWM3, d5/2) ||
+||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, m7/2) ||= (P12, m7/2) ||= (P12, m10/2) ||= (WWM3, WA6/2) ||
+||= 5 ||= (P8/2, P4/2) ||= (P8/2, M3/2) ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (WWM3/2, d5/2) ||
+||~ third-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+||= 6 ||= (P8/3, P5) ||= (P8/3, M3) ||= (P8/3, M2) ||= (P12/3, M6) ||= (P12/3, M3) ||= (WWM3/3, d5) ||
+||= 7 ||= (P8, P4/3) ||= (P8, M3/3) ||= (P8, M2/3) ||= (P12, M6/3) ||= (P12, M3/3) ||= (WWM3, d5/3) ||
+||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, m10/3) ||= (WWM3, WA6/3) ||
+||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) ||
+||= 10 ||= (P8/3, P4/2) ||= (P8/3, M3/2) ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||
+||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, m7/2) ||= (P12/3, m7/2) ||=   ||=   ||
+||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||=   ||=   ||
+||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||=   ||=   ||
+||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||=   ||=   ||
+||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||=   ||=   ||
+For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
-==Notating non-8ve and no-5ths pergens*==
+Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:
+Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but let's start with the standard 2.3 prime subgroup. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The complete rank-2 lattice is formed by tiling the squares.
-||~   ||||||||||||~ __the first two independent primes in the prime subgroup__ ||
-||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||
+For (P8, P5), the pergen square has 4 notes:
-||= 1 ||= (P8, P5) ||= (P8, y3) ||= (P8, r2) ||= (P12, y6) ||= (P12, r3) ||= (WWy3, bg5) ||
+C2 -- G2
-||~ half-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+| |
-||= 2 ||= (P8/2, P5) ||= (P8/2, y3) ||= (P8/2, r2) ||=   ||=   ||=   ||
+C1 -- G1
-||= 3 ||= (P8, P4/2) ||= (P8, y3/2) ||= (P8, r2/2) ||=   ||=   ||=   ||
-||= 4 ||= (P8, P5/2) ||= (P8, g6/2) ||= (P8, b7/2) ||=   ||=   ||=   ||
+Splitting the period or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes:
-||= 5 ||= (P8/2, P4/2) ||= (P8/2, y3/2) ||= (P8/2, r2/2) ||=   ||=   ||=   ||
+C2 --- G2
-||~ third-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+F#v1 F#v2
-||= 6 ||= (P8/3, P5) ||= (P8/3, y3) ||=   ||=   ||=   ||=   ||
+C1 --- G1
-||= 7 ||= (P8, P4/3) ||= (P8, y3/3) ||=   ||=   ||=   ||=   ||
-||= 8 ||= (P8, P5/3) ||= (P8, g6/3) ||=   ||=   ||=   ||=   ||
-||= 9 ||= (P8, P11/3) ||= (P8, y10/3) ||=   ||=   ||=   ||=   ||
-||= 10 ||= (P8/3, P4/2) ||= (P8/3, y3/2) ||=   ||=   ||=   ||=   ||
-||= 11 ||= (P8/3, P5/2) ||=   ||=   ||=   ||=   ||=   ||
-||= 12 ||= (P8/2, P4/3) ||=   ||=   ||=   ||=   ||=   ||
-||= 13 ||= (P8/2, P5/3) ||=   ||=   ||=   ||=   ||=   ||
-||= 14 ||= (P8/2, P11/3) ||=   ||=   ||=   ||=   ||=   ||
-||= 15 ||= (P8/3, P4/3) ||=   ||=   ||=   ||=   ||=   ||
-Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
+The square can be generalized to any prime subgroup by representing the notes as dots. In the 2.5 subgroup, a horizontal side equals 5/4. In Bohlen-Peirce, horizontal = 5/3 and vertical = 3/1.
-Every rank-3 pergen can be identified by its first __three__ independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.
+Here are the first 32 rank-2 pergens in a completely JI-agnostic format. True doubles are in red. Imperfect multigens are in green. These properties are independent of the prime subgroup.
-||~ pergen number ||~ 2.3.5 ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5.7 ||
-||= 1 ||= (P8, P5, ^1) ||= same ||= (P8, y3, r2) ||= (P12, y6, r3) ||
-||~ half-splits ||~   ||~   ||~   ||~   ||
-||= 2 ||= (P8/2, P5, ^1) ||= same ||=   ||=   ||
-||= 3 ||= (P8, P4/2, ^1) ||= same ||=   ||=   ||
-||= 4 ||= (P8, P5/2, ^1) ||= same ||=   ||=   ||
-||= 5 ||= (P8/2, P4/2, ^1) ||= same ||=   ||=   ||
-||= 6 ||= (P8, P5, vM3/2) ||= (P8, P5, ^M2/2) ||=   ||=   ||
-||= 7 ||= (P8, P5, ^m6/2) ||= (P8, P5, vm7/2) ||=   ||=   ||
-||= 8 ||= (P8/2, P5, vM3/2) ||=   ||=   ||=   ||
-||= 9 ||= (P8/2, P5, ^m6/2) ||=   ||=   ||=   ||
-||= 10 ||= (P8, P4/2, vM3/2) ||=   ||=   ||=   ||
-||= 11 ||= (P8, P4/2, ^m6/2) ||=   ||=   ||=   ||
-||= 12 ||= (P8, P5/2, vM3/2) ||=   ||=   ||=   ||
-||= 13 ||= (P8, P5/2, ^m6/2) ||=   ||=   ||=   ||
-||= 14 ||= (P8/2, P4/2, vM3/2) ||=   ||=   ||=   ||
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to use a notation that isn't backwards compatible.
-Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...
-A similar chart could be made of all rank-3 pergen cubes.
+A similar chart could be made for all rank-3 pergens, using pergen cubes.
 ==Notating tunings with an arbitrary generator==
@@ Line 934: / Line 916: @@
 If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen.
-Here are some example edo pairs. The closer two edos are in the scale tree, the simpler the pergen they make:
+The closer two edos are in the scale tree, the simpler the pergen they make. Examples::
 ||~   ||~ 12-edo ||~ 13b-edo ||~ 14-edo ||~ 15-edo ||~ 16-edo ||~ 17-edo ||~ 18b-edo ||~ 19-edo ||~ 20-edo ||
 ||~ 13b-edo ||= (P8, P5/7) ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 956: / Line 938: @@
 needs more screenshots, including 12-edo's pergens and a page of the pdf
 repost alt-pergenlister code
+add pergens square pic
+the half-step glitch
 to do:
@@ Line 1,180: / Line 1,164: @@
 __**Credits**__
-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Pergen squares are Praveen's creation.</pre></div>
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+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Naming very large intervals"&gt;Naming very large intervals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens*"&gt;Notating Blackwood-like pergens*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens"&gt;Notating Blackwood-like pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc18"&gt; &lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating non-8ve and no-5ths pergens"&gt;Notating non-8ve and no-5ths pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;Notating non-8ve and no-5ths pergens*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating tunings with an arbitrary generator"&gt;Notating tunings with an arbitrary generator&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating tunings with an arbitrary generator"&gt;Notating tunings with an arbitrary generator&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:142 --&gt;&lt;!-- ws:start:WikiTextTocRule:143: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:143 --&gt;&lt;!-- ws:start:WikiTextTocRule:144: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:144 --&gt;&lt;!-- ws:start:WikiTextTocRule:145: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:142 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
-&lt;!-- ws:end:WikiTextTocRule:145 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 &lt;br /&gt;
@@ Line 2,470: / Line 2,453: @@
 &lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Applications-Tipping points"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Tipping points&lt;/h2&gt;
   &lt;br /&gt;
-Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
+Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;up may need to be swapped with down, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
 &lt;br /&gt;
 Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.&lt;br /&gt;
@@ Line 3,642: / Line 3,625: @@
 This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.&lt;br /&gt;
 &lt;br /&gt;
-Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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;
+Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. 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^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. 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;
 There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from gen2 to make alternates. If gen2 can be expressed as a mapping comma, that is preferred. For demeter, any combination of vm3, double-8ves and double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the &lt;strong&gt;DOL&lt;/strong&gt; (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &amp;lt; 3, 5/4 is preferred.&lt;br /&gt;
@@ Line 3,739: / Line 3,722: @@
          &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5, vm3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, ^m3/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;half-upminor-3rd&lt;br /&gt;
@@ Line 3,944: / Line 3,927: @@
 There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:94:&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:94 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:94:&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:94 --&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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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;
+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. The 5th is not independent of the octave, thus it doesn't appear in the pergen. 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. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.&lt;br /&gt;
-&lt;br /&gt;
-A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits 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;
-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;
+A Blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma splits the octave and removes the middle term from the pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). 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 3,959: / Line 3,940: @@
          &lt;th&gt;pergen&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;spoken&lt;br /&gt;
+         &lt;th&gt;spoken name&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;enharmonics&lt;br /&gt;
@@ Line 3,967: / Line 3,948: @@
          &lt;th&gt;genchain&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;^1&lt;br /&gt;
+         &lt;th&gt;^1 ratio&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;/1&lt;br /&gt;
+         &lt;th&gt;/1 ratio&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
@@ Line 4,009: / Line 3,990: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;10edo+y&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;10edo+yellow&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/10, /1)&lt;br /&gt;
@@ Line 4,027: / Line 4,008: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;12edo+j&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;12edo+jade&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/12, ^1)&lt;br /&gt;
@@ Line 4,063: / Line 4,044: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;17edo+y&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;17edo+yellow&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/17, /1)&lt;br /&gt;
@@ Line 4,079: / Line 4,060: @@
          &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
 &lt;/td&gt;
+    &lt;/tr&gt;
+&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 10, 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 13/12.&lt;br /&gt;
+&lt;br /&gt;
+The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.&lt;br /&gt;
+&lt;br /&gt;
+Not all Blackwood-like pergens are of the form (P8/m, ^1). In the last section, we saw that demeter's pergen is (P8, P5, vm3/2). Tempering out 256/243 as well, the pergen becomes (P8/5, vm3/2). Blackwood-like pergens are a superset of rank-3 pergens, and are &lt;u&gt;very&lt;/u&gt; numerous.&lt;br /&gt;
+&lt;br /&gt;
+It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;temperament&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;subgroup&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;comma&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;pergen&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;spoken name&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;enharmonic&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;perchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;genchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;^1 ratio&lt;br /&gt;
+&lt;/th&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;large quintuple blue&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2.3.7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(-14,0,0,5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/5, P5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;fifth-8ve&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;D E^^ Gv A^ Cvv D&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C G D A E...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;49/48&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small quintuple red&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2.3.7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(22,-5,0,-5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;64/63&lt;br /&gt;
 &lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &amp;quot;missing notes&amp;quot; (see next section).&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:96:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:96 --&gt;Notating non-8ve and no-5ths pergens&lt;/h2&gt;
+ &lt;br /&gt;
+In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3. &lt;br /&gt;
+&lt;br /&gt;
+In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.&lt;br /&gt;
+&lt;br /&gt;
+But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.&lt;br /&gt;
+&lt;br /&gt;
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.&lt;br /&gt;
+&lt;br /&gt;
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;u&gt;pergen number&lt;/u&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="6"&gt;&lt;u&gt;prime subgroup&lt;/u&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;unsplit&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.3&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.5 (M3 = 5/4)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.7 (M2 = 8/7)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.5 (M6 = 5/3)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.7 (M3 = 9/7)&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5.7 (WWM3 = 5/1, d5 = 7/5)&lt;br /&gt;
+&lt;/th&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M6)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, d5)&lt;br /&gt;
-&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;&lt;br /&gt;
+        &lt;th&gt;half-splits&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5)&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;td style="text-align: center;"&gt;(P8/2, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/2, M6)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/2, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3/2, d5)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/2)&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;td style="text-align: center;"&gt;(P8, M3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M2/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, d5/2)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, m6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, m7/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, m7/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, m10/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, WA6/2)&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;5&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;(P8/2, M3/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, M2/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12/2, M6/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12/2, M3/2)&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3/2, d5/2)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
-&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 13/12.&lt;br /&gt;
-&lt;br /&gt;
-The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 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;
-&lt;br /&gt;
-&lt;em&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 ^m6/2 or ^^A5/4.&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.&lt;br /&gt;
-&lt;br /&gt;
-But in &amp;quot;no-5ths&amp;quot; or &amp;quot;minus white&amp;quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.&lt;br /&gt;
-&lt;br /&gt;
-Note-less names can be avoided in any pergen with a 5th (or 4th or 11th, etc.). The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th&gt;tuning&lt;br /&gt;
+         &lt;th&gt;third-splits&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;comma&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;pergen&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;spoken name&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;enharmonic&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;perchain&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;genchain&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;notes&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;large quintuple blue&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
-or small quintuple red&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(-14,0,0,5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P5)&lt;br /&gt;
-or (22,-5,0,-5)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/5, P5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;fifth-8ve&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, M6)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C D^^ Fv G^ Bbvv C&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C G D A E...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3/3, d5)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;a valid notation&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M3/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 5-edo&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M2/3)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;E = m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C D=Eb E=F G A=Bb B=C&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M6/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C Ev G#vv...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M3/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;a valid notation&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, d5/3)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/5, M3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, m6/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;fifth-8ve major 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, m7/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, m7/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C D^^ Fv G^ Bbvv C&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, m10/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C E G#...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, WA6/3)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;invalid, no G,&lt;br /&gt;
-D or A notes&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
-&lt;/table&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:96:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;!-- ws:end:WikiTextHeadingRule:96 --&gt; &lt;/h2&gt;
- edo-subset notations&lt;br /&gt;
-P8/6, P8/8&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:98:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Notating non-8ve and no-5ths pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:98 --&gt;Notating non-8ve and no-5ths pergens*&lt;/h2&gt;
- &lt;br /&gt;
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th colspan="6"&gt;&lt;u&gt;the first two independent primes in the prime subgroup&lt;/u&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P11/3)&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, M10/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, M9/3)&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;(P12, WWM3/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12, WM7/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(WWM3, WWm7/3)&lt;br /&gt;
+&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-        &lt;th&gt;pergen number&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.3&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.5&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.7&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;3.5&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;3.7&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;5.7&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P4/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, y3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, r2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M2/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12, y6)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, M6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12, r3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;etc.&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(WWy3, bg5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;etc.&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;th&gt;half-splits&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P5/2)&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3. m6/2)&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, m7/2)&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, m7/2)&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P4/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, y3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M3/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, r2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M2/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/2, M6/3)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 4,367: / Line 4,398: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, y3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, m6/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, r2/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, m7/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/2, m7/3)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 4,383: / Line 4,414: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P11/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, g6/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M10/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, b7/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M9/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/2, WWM3/3)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 4,399: / Line 4,430: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, P4/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P4/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, y3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M3/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, r2/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M2/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, M6/3)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 4,414: / Line 4,445: @@
 &lt;/td&gt;
      &lt;/tr&gt;
-    &lt;tr&gt;
+&lt;/table&gt;
-        &lt;th&gt;third-splits&lt;br /&gt;
-&lt;/th&gt;
+For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/th&gt;
+Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/th&gt;
+Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but let's start with the standard 2.3 prime subgroup. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The complete rank-2 lattice is formed by tiling the squares.&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/th&gt;
+For (P8, P5), the pergen square has 4 notes:&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+C2 -- G2&lt;br /&gt;
-&lt;/th&gt;
+| |&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+C1 -- G1&lt;br /&gt;
-&lt;/th&gt;
+&lt;br /&gt;
-        &lt;th&gt;&lt;br /&gt;
+Splitting the period or the multigen adds notes to the square. For (P8/2, P5), there are 6 notes:&lt;br /&gt;
-&lt;/th&gt;
+C2 --- G2&lt;br /&gt;
-    &lt;/tr&gt;
+F#v1 F#v2&lt;br /&gt;
-    &lt;tr&gt;
+C1 --- G1&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+The square can be generalized to any prime subgroup by representing the notes as dots. In the 2.5 subgroup, a horizontal side equals 5/4. In Bohlen-Peirce, horizontal = 5/3 and vertical = 3/1. &lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, P5)&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+Here are the first 32 rank-2 pergens in a completely JI-agnostic format. True doubles are in red. Imperfect multigens are in green. These properties are independent of the prime subgroup.&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, y3)&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;/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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P4/3)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, y3/3)&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;/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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5/3)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, g6/3)&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;/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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P11/3)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, y10/3)&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;/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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, P4/2)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, y3/2)&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;/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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, P5/2)&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;/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;/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;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P4/3)&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;/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;/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;13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P5/3)&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;/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;/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;14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P11/3)&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;/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;/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;15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/3, P4/3)&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;/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;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.&lt;br /&gt;
-&lt;br /&gt;
-Every rank-3 pergen can be identified by its first &lt;u&gt;three&lt;/u&gt; independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;th&gt;pergen number&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.3.5&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.3.7&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;2.5.7&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;3.5.7&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;1&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;same&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, y3, r2)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P12, y6, r3)&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;th&gt;half-splits&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P5, ^1)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;same&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P4/2, ^1)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;same&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;4&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;same&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P4/2, ^1)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;same&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5, vM3/2)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5, ^M2/2)&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5, ^m6/2)&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5, vm7/2)&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;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P5, vM3/2)&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;/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;9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P5, ^m6/2)&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;/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;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P4/2, vM3/2)&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;/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;11&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P4/2, ^m6/2)&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;/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;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5/2, vM3/2)&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;/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;13&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8, P5/2, ^m6/2)&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;/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;14&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;(P8/2, P4/2, vM3/2)&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;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of note-less names. The composer may well want to use a notation that isn't backwards compatible.&lt;br /&gt;
 &lt;br /&gt;
-Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all...&lt;br /&gt;
 &lt;br /&gt;
-A similar chart could be made of all rank-3 pergen cubes.&lt;br /&gt;
+A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:100:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:100 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:98:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:98 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
   &lt;br /&gt;
 Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;
@@ Line 5,145: / Line 4,816: @@
 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:102:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:102 --&gt;Pergens and MOS scales&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:100:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:100 --&gt;Pergens and MOS scales&lt;/h2&gt;
   &lt;br /&gt;
 Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
@@ Line 6,250: / Line 5,921: @@
 Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:104:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:104 --&gt;Pergens and EDOs&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:102:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:102 --&gt;Pergens and EDOs&lt;/h2&gt;
   &lt;br /&gt;
 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.&lt;br /&gt;
@@ Line 6,579: / Line 6,250: @@
 If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a Blackwood-like pergen. &lt;br /&gt;
 &lt;br /&gt;
-Here are some example edo pairs. The closer two edos are in the scale tree, the simpler the pergen they make:&lt;br /&gt;
+The closer two edos are in the scale tree, the simpler the pergen they make. Examples::&lt;br /&gt;
@@ Line 6,875: / Line 6,546: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:106:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:106 --&gt;Supplemental materials*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:104:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:104 --&gt;Supplemental materials*&lt;/h2&gt;
   &lt;br /&gt;
 needs more screenshots, including 12-edo's pergens and a page of the pdf&lt;br /&gt;
 repost alt-pergenlister code&lt;br /&gt;
+add pergens square pic&lt;br /&gt;
+the half-step glitch&lt;br /&gt;
 &lt;br /&gt;
 to do:&lt;br /&gt;
@@ Line 6,885: / Line 6,558: @@
 make a glossary of all bolded terms?&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:8598: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:8598 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8154: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:8154 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:8599: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:8599 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8155: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:8155 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:108:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:108 --&gt;Notaion guide PDF&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:106:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc23"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:106 --&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:8600: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:8600 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8156: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:8156 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:110:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc25"&gt;&lt;a name="Further Discussion-Supplemental materials*-pergenLister app"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:110 --&gt;pergenLister app&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:108:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Further Discussion-Supplemental materials*-pergenLister app"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:108 --&gt;pergenLister app&lt;/h3&gt;
   &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:8601: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:8601 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8157: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:8157 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;
 &lt;br /&gt;
 Screenshots of the first 38 pergens:&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextLocalImageRule:4917:&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:4917 --&gt;&lt;br /&gt;
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 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;
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 The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;
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 The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&lt;br /&gt;
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 Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;
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 &lt;u&gt;&lt;strong&gt;Staff notation&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
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 &lt;u&gt;&lt;strong&gt;Credits&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
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-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.&lt;/body&gt;&lt;/html&gt;</pre></div>
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Pergen squares are Praveen's creation.&lt;/body&gt;&lt;/html&gt;</pre></div>