Kite's thoughts on pergens: Difference between revisions

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

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

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

||= decimal ||= (P8/2, P4/2) ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 = ^^\\A1 ||

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

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

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

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

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

When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.

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

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)

All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.

Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ notation ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic ||

||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||

||= " ||= " ||= " ||= " ||= double-pair ||= " ||= " ||= " ||= ^^\d2 ||

||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= double-pair ||= v/A4 = 10/7 ||= P5 ||= ^1 = 81/80 ||= ^^\\d2 ||

||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= rank-3 third-11th ||= double-pair ||= P8 ||= ^\d5 = 7/5 ||= ^1 = 81/80 ||= ^^^\\\dd3 ||

||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\4dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.

With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as ~~50/49 =~~ vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)-2 · (64/63)2 · (19,-12)-1. This can be rewritten as vv1 + ``//``1 - d2 = vv``//``-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

The 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 ~~is tempered to 0¢~~. However, a rank-3 temperament has two mapping commas, and neither is forced to ~~be 0¢~~. 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.

The 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.

~~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~~.

Demeter is unusual in that its 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^^. 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.

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

There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to or subtracted from the 2nd generator 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.

~~Demeter is unusual in that its 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~~^^~~. Standard double-pair notation is better. Gen2 = v~~/A1, ~~E =~~ ^^~~\\\dd3~~, and C^~~^\\\ = A##~~. ~~The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C~~ -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with ~~no other accidentals in it~~, ~~it always adds~~ up ~~to something simpler eventually. If it doesn't have its own enharmonic, it~~'s ~~infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals~~.

If ^1 = 81/80, possible half-split gens2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

~~The next table lists all~~ rank-3 pergens ~~up to half-splits. Each block~~ is much ~~larger. The notation~~ for 2~~.3.5~~ pergens ~~and 2~~.~~3.7~~ pergens ~~is often the same.~~

All possible rank-3 pergens can be listed, but the table is much longer than for rank-2 pergens. Here are all the half-split pergens:

||~ unsplit ||~ ^1 = 81/80 ||~ spoken name ||~ ^1 = 64/63 ||~ spoken name ||

||~ unsplit ||~ ~~2.3.5~~ ||~ ||~ ~~2.3.7~~ ||~ ||

||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= same ||

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

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

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

||= 7 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, vm7/2) ||= half-downminor-7th ||

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

||= 8 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||= ||= ||

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

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

||= 9 ||= (P8, P5, vM6/2) ||= half-downmajor-6th ||= (P8, P5, vm7/2) ||= half-downminor-7th ||

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

||= 10 ||= (P8/2, P5, ^m3/2) ||= half-8ve half-upminor 3rd ||= (P8/2, P5, ^M2/2) ||= half-8ve half-upmajor-2nd ||

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

||= 11 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||= (P8/2, P5, vm3/2) ||= etc. ||

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

||= 12 ||= (P8/2, P5, ^m6/2) ||= half-8ve half-upminor 6th ||= (P8/2, P5, ^M6/2) ||= ||

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

||= 13 ||= (P8/2, P5, vM6/2) ||= half-8ve half-downmajor 6th ||= (P8/2, P5, vm7/2) ||= ||

||= 14 ||= (P8/2, P4/2, vM3/2) ||= half-everything half-downmajor-3rd ||= ||= ||

||= 14 ||= (P8, P4/2, ^m3/2) ||= half-4th half-upminor 3rd ||= (P8, P4/2, ^M2/2) ||= ||

||= 15 ||= (P8, P4/2, vM3/2) ||= etc. ||= (P8, P4/2, vm3/2) ||= ||

||= 16 ||= (P8, P4/2, ^m6/2) ||= ||= (P8, P4/2, ^M6/2) ||= ||

||= 17 ||= (P8, P4/2, vM6/2) ||= ||= (P8, P4/2, vm7/2) ||= ||

||= 18 ||= (P8, P5/2, ^m3/2) ||= ||= (P8, P5/2, ^M2/2) ||= ||

||= 19 ||= (P8, P5/2, vM3/2) ||= ||= (P8, P5/2, vm3/2) ||= ||

||= 20 ||= (P8, P5/2, ^m6/2) ||= ||= (P8, P5/2, ^M6/2) ||= ||

||= 21 ||= (P8, P5/2, vM6/2) ||= ||= (P8, P5/2, vm7/2) ||= ||

||= 22 ||= (P8/2, P4/2, vM3/2) ||= half-everything half-downmajor-3rd ||= (P8/2, P4/2, ^M2/2) ||= ||

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*==

~~Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.~~

~~another example might work?~~

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. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

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

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

Such a pergen is in effect multiple copies of an edo. Its 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:

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

//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.//

//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 ...//

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==Notating non-8ve and ~~non~~-~~5th~~ pergens*==

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

Just as all rank-2 pergens in which 2 and 3 are 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:

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:

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

||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||

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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.

||~ pergen number ||~ 2.3.5 ~~||~~~ ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5.7 ||

||~ pergen number ||~ 2.3.5 ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5.7 ||

||= 1 ||= (P8, P5, ^1) ~~||= rank-3 unsplit~~ ||= same ||= (P8, y3, r2) ||= (P12, y6, r3) ||

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

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

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

||= 2 ||= (P8/2, P5, ^1) ~~||= rank-3 half-8ve~~ ||= same ||= ||= ||

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

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

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

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

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

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

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

||= 6 ||= (P8, P5, vM3/2) ~~||= half-downmajor-3rd~~ ||= (P8, P5, ^M2/2) ||= ||= ||

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

||= 7 ||= (P8, P5, ^m6/2) ~~||= half-upminor-6th~~ ||= (P8, P5, vm7/2) ||= ||= ||

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

||= 8 ||= (P8/2, P5, vM3/2) ~~||= half-8ve half-downmajor 3rd~~ ||= ||= ||= ||

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

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

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

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

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

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

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

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

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

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

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

||= 14 ||= (P8/2, P4/2, vM3/2) ~~||= rank-3 half-everything~~ ||= ||= ||= ||

||= 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 ~~devise~~ a ~~personal~~ notation that isn't backwards compatible.

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.

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<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 ~~non~~-~~5th~~ pergens*">Notating non-8ve and ~~non~~-~~5th~~ 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-Pergens and MOS scales">Pergens and MOS scales</a></div>

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

</td>

<td style="text-align: center;">E = vvd2, E' = \\m2

<td style="text-align: center;">E = vvd2, E' = \\m2 = ^^\\A1

</td>

</tr>

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</table>

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

When there is more than one enharmonic, the first one can be combined with the 2nd to make an equivalent 2nd enharmonic.

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

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.)

All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one might simply use colors.

Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.

Some examples of 7-limit rank-3 temperaments:

Line 3,491:

Line 3,505:

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