Kite's thoughts on pergens: Difference between revisions

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A '''pergen''' (pronounced "peer-gen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

A '''pergen''' (pronounced "peer-jen", from '''per'''iod and '''gen'''erator) is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka ~~Triyo~~]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka ~~Trigu~~ tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to ~~imply inflections of~~ the generator rather than the fifth.)

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Triguti tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. ~~The~~ Dicot aka ~~Yoyo temperament~~ (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka ~~Triyo~~ is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka ~~Zozo~~, a pun on "semi-fourth", is of course half-fourth.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka ~~Sagugu~~ and Injera aka Gu & ~~Biruyo~~ sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.

Line 35:

| | 81/80

| | Meantone

| | Gu

| | Guti

| | gT

|-

Line 42:

| | 64/63

| | Archy

| | Ru

| | Ruti

| | rT

|-

Line 49:

| | (-14,8,1)

| | Schismic

| | ~~Layo~~

| | Layoti

| | LyT

|-

Line 56:

| | (11, -4, -2)

| | Srutal

| | ~~Sagugu~~

| | Saguguti

| | sggT

|-

Line 63:

| | 81/80, 50/49

| | Injera

| | Gu & ~~Biruyo~~

| | Gu & Biruyoti

| | g&rryyT

|-

Line 70:

| | 25/24

| | Dicot

| | ~~Yoyo~~

| | Yoyoti

| | yyT

|-

Line 77:

| | (-1,5,0,0,-2)

| | Mohajira

| | ~~Lulu~~

| | Luluti

| | 1uuT

|-

Line 84:

| | 49/48

| | Semaphore

| | ~~Zozo~~

| | Zozoti

| | zzT

|-

Line 91:

| | 25/24, 49/48

| | Decimal

| | Yoyo & ~~Zozo~~

| | Yoyo & Zozoti

| | yy&zzT

|-

Line 98:

| | 250/243

| | Porcupine

| | ~~Triyo~~

| | Triyoti

| | y3T

|-

Line 105:

| | (12,-1,0,0,-3)

| | Satrilu

| | ~~Satrilu~~

| | Satriluti

| | s1u3T

|-

Line 112:

| | (3,4,-4)

| | Diminished

| | ~~Quadgu~~

| | Quadguti

| | g4T

|-

Line 119:

| | (-17,2,0,0,4)

| | Laquadlo

| | ~~Laquadlo~~

| | Laquadloti

| | L1o4T

|-

Line 126:

| | (-10,-1,5)

| | Magic

| | ~~Laquinyo~~

| | Laquinyoti

| | Ly5T

|}

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Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

For example, Marvel aka ~~Ruyoyo~~ (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: ~~Trizogu~~ (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. ~~Biruyo~~ (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, ~~Biruyo~~ Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

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In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.

For example, consider Sawa & ~~Ruyoyo~~ (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

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A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, Porcupine aka ~~Triyo~~ (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic aka ~~Bizozogu~~ is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

{| class="wikitable" style="text-align:center;"

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Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is ~~Triyo~~, and the second one is Triyo & Ru. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.

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The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[~~Ups_and_Downs_Notation~~|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka ~~Lulu~~ and Dicot aka ~~Yoyo~~ are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka ~~Layo~~ is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Line 386:

| | P8/2 = vA4 = ^d5

| | C - vF#=^Gb - C

| | Srutal aka ~~Sagugu~~

| | Srutal aka Saguguti

^1 = 81/80

|-

Line 397:

| | P8/2 = ^A4 = vd5

| | C - ^F#=vGb - C

| | Injera aka Gu & ~~Biruyo~~

| | Injera aka Gu & Biruyoti

^1 = 64/63

Line 407:

| | P8/2 = ^4 = v5

| | C - ^F=vG - C

| | ~~Thotho~~, if 13/8 = M6

| | Thothoti, if 13/8 = M6

^1 = 27/26

Line 419:

| | P4/2 = ^M2 = vm3

| | C - ^D=vEb - F

| | Semaphore aka ~~Zozo~~

| | Semaphore aka Zozoti

^1 = 64/63

Line 429:

| | P4/2 = vA2 = ^d3

| | C - vD#=^Ebb - F

| | Lala-~~yoyo~~

| | Lala-yoyoti

^1 = 81/80

Line 441:

| | P5/2 = ^m3 = vM3

| | C - ^Eb=vE - G

| | Mohajira aka ~~Lulu~~

| | Mohajira aka Luluti

^1 = 33/32

Line 477:

C - ^/F=v\G - C

| | Zozo & ~~Lulu~~

| | Zozo & Luluti

^1 = 33/32

Line 505:

C - ^/Eb=v\E - G

| | Sagugu & ~~Zozo~~

| | Sagugu & Zozoti

^1 = 81/80

Line 533:

C - v/D=^\Eb - F

| | Sagugu & ~~Lulu~~

| | Sagugu & Luluti

^1 = 81/80

Line 555:

| | P8/3 = vM3 = ^^d4

| | C - vE - ^Ab - C

| | Augmented aka ~~Trigu~~

| | Augmented aka Triguti

^1 = 81/80

Line 567:

| | P4/3 = vM2 = ^^m2

| | C - vD - ^Eb - F

| | Porcupine aka ~~Triyo~~

| | Porcupine aka Triyoti

^1 = 81/80

Line 579:

| | P5/3 = ^M2 = vvm3

| | C - ^D - vF - G

| | Slendric aka ~~Latrizo~~

| | Slendric aka Latrizoti

^1 = 64/63

Line 591:

| | P11/3 = vA4 = ^^dd5

| | C - vF# - ^Cb - F

| | ~~Satrilu~~, if 11/8 = A4

| | Satriluti, if 11/8 = A4

^1 = 729/704

Line 601:

| | P11/3 = ^4 = vv5

| | C - ^F - vC - F

| | ~~Satrilu~~, if 11/8 = P4

| | Satriluti, if 11/8 = P4

^1 = 33/32

Line 617:

C - ^3Db=v3E - F

| | ~~Tribilo~~, if 11/8 = P4

| | Tribiloti, if 11/8 = P4

^1 = 33/32

Line 635:

C - /D=\Eb - F

| | Triforce aka Trigu & ~~Zozo~~

| | Triforce aka Trigu & Zozoti

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

Line 655:

C - /Eb=\E - G

| | ~~Satribizo~~

| | Satribizoti

^1 = 49/48, /1 = 343/324

Line 675:

C - \D - /Eb - F

| | ~~Latribiru~~

| | Latribiruti

^1 = 1029/1024, /1 = 49/48

Line 691:

C - vvD# - ^^Fb - G

| | ~~Lartribiyo~~

| | Latribiyoti

^1 = 81/80

Line 709:

C - /D - \F - G

| | Lemba aka Latrizo & ~~Biruyo~~

| | Lemba aka Latrizo & Biruyoti

^1 = (10,-6,1,-1), /1 = 64/63

Line 725:

C - ^^F - vvC - F

| | ~~Latribilo~~, if 11/8 = P4

| | Latribiloti, if 11/8 = P4

^1 = 33/32

Line 751:

C - v/D - ^\F - G

| | Triyo & ~~Triru~~

| | Triyo & Triguti

^1 = 64/63

Line 775:

C - v\D - ^/Eb - F

| | Trigu & ~~Latrizo~~

| | Trigu & Latrizoti

^1 = 81/80

Line 799:

C - v/E - ^\Ab - C

| | Triyo & ~~Latrizo~~

| | Triyo & Latrizoti

^1 = 81/80

Line 819:

| | P8/4 = vm3 = ^3A2

| | C vEb vvGb=^^F# ^A C

| | Diminished aka ~~Quadgu~~

| | Diminished aka Quadguti

|-

| | 17

Line 827:

| | P4/4 = ^m2 = v3AA1

| | C ^Db ^^Ebb=vvD# vE F

| | Negri aka ~~Laquadyo~~

| | Negri aka Laquadyoti

|-

| | 18

Line 835:

| | P5/4 = vM2 = ^3m2

| | C vD vvE=^^Eb ^F G

| | Tetracot aka ~~Saquadyo~~

| | Tetracot aka Saquadyoti

|-

| | 19

Line 843:

| | P11/4 = ^M3 = v3dd5

| | C ^E ^^G# vDb F

| | Squares aka ~~Laquadru~~

| | Squares aka Laquadruti

|-

| | 20

Line 851:

| | P12/4 = v4 = ^3M3

| | C vF vvBb=^^A ^D G

| | Vulture aka Sasa-~~quadyo~~

| | Vulture aka Sasa-quadyoti

|-

| |

Line 1,000:

==Secondary splits==

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka ~~Triyo~~) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - vD - ^Eb - F

Line 1,098:

| | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24

|-

| | (~~P83~~, P4/3)

| | (P8/3, P4/3)

| | third-everything

| | every 3-limit interval is split three times as much as before

Line 1,115:

==Finding an example temperament==

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, the ~~diminished~~ temperament. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, the ~~Quadru~~ temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4⋅P and P8. If P is 6/5, the comma is 4⋅P - P8 = (6/5)4 ÷ (2/1) = 648/625, making the Diminished temperament aka Quadguti. If P is 7/6, the comma is P8 - 4⋅P = (2/1) · (7/6)-4, making the Quadruti temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), ~~Saseplo~~.

Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n⋅x gens = n⋅I = x⋅M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5⋅(7/6) = 2⋅P5. Thus 2⋅P5 - 5⋅(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7⋅(11/9) = 2⋅P8, and the comma is (-2, -14, 0, 0, 7), making Saseploti.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Line 1,141:

==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma ~~= Ly-2~~ = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x · 3y · P±1, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & ~~Biruyo~~, where ^1 equals 64/63 minus 81/80.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.

Line 1,161:

* quarter-comma Meantone: # = 76¢

* fifth-comma Meantone: # = 84¢

* third-comma Archy aka Ru: # = 177¢

* third-comma Archy aka Ruti: # = 177¢

* eighth-comma Porcupine aka ~~Triyo~~: # = 157¢, ^ = 52¢ (^ = third-sharp)

* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)

* seventh-comma Srutal aka Sagugu & ~~Zoquadyo~~: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)

* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)

* third-comma Injera aka Gu & ~~Biruyo~~: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)

* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)

* eighth-comma Hedgehog aka Triyo & ~~Biruyo~~: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)

* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)

The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.

Line 1,220:

It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & ~~Zozo~~), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

Sixth-4th with single-pair notation has an awkward ^6d64 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v3A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v3\\M2 and ^3\\d2.

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4

Line 1,249:

For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v3m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d32. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^3d32. Add the multi-EU ^3[-2,1] to the multigen P5 = [7,4] to get ^3[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^3[-2,1] = v3[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d32 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d32 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d32 = -200¢ - 26·c, ^ = (-d32) / 3 = 67¢ + 8.67·c, about a third-tone.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka ~~Zozo~~ (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-~~yoyo~~, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is > 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.

For example, ~~Satrilu~~ tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v3M2, but if 11/8 is notated as a vA4, the EU is ^3dd2.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & ~~Trizogu~~ (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v3dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.

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==Chord names and staff notation==

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[~~Ups_and_Downs_Notation~~|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka ~~Sagugu~~ & Ru (2.3.5.7 with ~~2048~~/~~2025~~ and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & ~~Biruyo~~ (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka ~~Triyo~~ (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v3A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

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The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a very unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka ~~Triyo~~'s 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka ~~Yoyo~~'s 25/24 comma is also an A1, and has the same tipping point. Semaphore aka ~~Zozo~~'s 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.

Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.

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

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

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

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

==Notating unsplit pergens==

Line 1,319:

! | cents

|-

| | Meantone aka Gu

| | Meantone aka Guti

| | 81/80 = P1

| | c = -3¢ to -5¢

Line 1,330:

| | ---

|-

| | Mavila aka ~~Layobi~~

| | Mavila aka Layobiti

| | 135/128 = A1

| | c = -21¢ to -22¢

Line 1,341:

| | -100¢ - 7c = 47¢-54¢

|-

| | ~~Lagu~~

| | Laguti

| | (-15,11,-1) = A1

| | c = -10¢ to -12¢

Line 1,352:

| | 100¢ + 7c = 26¢-30¢

|-

| | Schismic aka ~~Layo~~

| | Schismic aka Layoti

| | (-15,8,1) = -d2

| | c = 1.7¢ to 2.0¢

Line 1,363:

| | 12c = 20¢-24¢

|-

| | ~~Lalagu~~

| | Lalaguti

| | (-23,16,-1) = -d2

| | c = -0.9¢ to -1.2¢

Line 1,374:

| | -12c = 10¢-15¢

|-

| | Father aka ~~Gubi~~

| | Father aka Gubiti

| | 16/15 = m2

| | c = 56¢ to 58¢

Line 1,385:

| | -100¢ + 5c = 180-190¢

|-

| | Superpyth aka ~~Sasayo~~

| | Superpyth aka Sasayoti

| | (12,-9,1) = m2

| | c = 9¢ to 10¢

Line 1,400:

For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka ~~Archy~~ (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.

==Notating rank-3 pergens==

Line 1,456:

| | ---

|-

| | Meantone aka Gu

| | Meantone aka Guti

| | (P8, P5)

| | rank-2

Line 1,464:

| | ---

|-

| | Diaschismic aka ~~Sagugu~~

| | Diaschismic aka Saguguti

| | (P8/2, P5)

| | rank-2

Line 1,472:

| | EU = ^^d2

|-

| | Semaphore aka ~~Zozo~~

| | Semaphore aka Zozoti

| | (P8, P4/2)

| | rank-2

Line 1,480:

| | EU = vvm2

|-

| | Decimal aka Yoyo & ~~Zozo~~

| | Decimal aka Yoyo & Zozoti

| | (P8/2, P4/2)

| | rank-2

Line 1,496:

| | ---

|-

| | Marvel aka ~~Ruyoyo~~

| | Marvel aka Ruyoyoti

| | (P8, P5, ^1)

| | rank-3

Line 1,504:

| | ---

|-

| | Breedsmic aka ~~Bizozogu~~

| | Breedsmic aka Bizozoguti

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

| | rank-3

Line 1,542:

! | EU

|-

| | Marvel aka ~~Ruyoyo~~

| | Marvel aka Ruyoyoti

| | 225/224

| | (P8, P5, ^1)

Line 1,562:

| | ^^\d2

|-

| | ~~Biruyo~~

| | Biruyoti

| | 50/49

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

Line 1,572:

| | ^^\\d2

|-

| | ~~Trizogu~~

| | Trizoguti

| | 1029/1000

| | (P8, P11/3, ^1)

Line 1,582:

| | ^^^\\\dd3

|-

| | Breedsmic aka ~~Bizozogu~~

| | Breedsmic aka Bizozoguti

| | 2401/2400

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

Line 1,592:

| | ^^\4dd3

|-

| | Demeter aka Trizo-~~agugu~~

| | Demeter aka Trizo-aguguti

| | 686/675

| | (P8, P5, \m3/2)

Line 1,604:

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

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, ~~Biruyo~~ 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 EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. 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 vE G \Bb.

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyoti 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 EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. 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 vE G \Bb.

With this standardization, the EU can be derived directly 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). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.

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 ~~Biruyo~~, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore ~~Biruyo~~ doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

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 Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.

Unlike the previous examples, Demeter aka Trizo-~~agugu~~'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 EU. 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, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

Unlike the previous examples, Demeter aka Trizo-aguguti'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 EU. 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, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops 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 \m3, 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 there will always be a 3-limit comma small enough to 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''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred.

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 \m3, 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 there will always be a 3-limit comma small enough to 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''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.

If ^1 = 81/80, possible half-split gen2'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.

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:

Line 1,689:

| | 10

| | (P8/2, P5, ^m3/2)

| | half-8ve half-upminor 3rd

| | half-8ve half-upminor-3rd

| | (P8/2, P5, ^M2/2)

| | half-8ve half-upmajor-2nd

Line 1,695:

| | 11

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

| | half-8ve half-downmajor 3rd

| | half-8ve half-downmajor-3rd

| | (P8/2, P5, vm3/2)

| | ~~etc.~~

| | half-8ve half-downminor-3rd

|-

| | 12

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

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

| | half-~~8ve~~ half-~~upminor 6th~~

| | half-4th half-downmajor-3rd

| | (P8/2~~, P5~~, ^M6/2)

| | (P8, P4/2, ^M2/2)

| |

| | half-4th half-upmajor-2nd

|-

| | 13

~~| | (P8/2, P5, vM6/2)~~

~~| | half-8ve half-downmajor 6th~~

~~| | (P8/2, P5, vm7/2)~~

~~| |~~

|-

~~| | 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)

| |

| | half-4th half-upminor-6th

~~| | (P8, P4/2, ^M6/2)~~

~~| |~~

|-

~~| | 17~~

~~| | (P8, P4/2, vM6/2)~~

~~| |~~

| | (P8, P4/2, vm7/2)

| |

| | half-4th half-downminor-7th

|-

| | 18

| | 14

| | (P8, P5/2, ~~^m3~~/2)

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

| |

| | half-5th half-downmajor-3rd

| | (P8, P5/2, ^M2/2)

| |

| | half-5th half-upmajor-2nd

|-

| | 19

| | 15

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

~~| |~~

~~| | (P8, P5/2, vm3/2)~~

~~| |~~

|-

~~| | 20~~

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

| |

| | half-5th half-upminor-6th

~~| | (P8, P5/2, ^M6/2)~~

~~| |~~

|-

~~| | 21~~

~~| | (P8, P5/2, vM6/2)~~

~~| |~~

| | (P8, P5/2, vm7/2)

| |

| | half-5th half-downminor-7th

|-

| | 22

| | 16

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

| | half-everything half-downmajor-3rd

| | (P8/2, P4/2, ^M2/2)

| |

| | half-everything half-upmajor-2nd

|}

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.

Line 1,771:

Line 1,735:

A multi-EDO pergen 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 multi-EDO 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 aka ~~Sawa~~ plus Ya 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:

A multi-EDO 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 aka Sawati plus Ya 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:

{| class="wikitable" style="text-align:center;"

Line 1,784:

Line 1,748:

! | /1 ratio

|-

| | Blackwood aka ~~Sawa~~+ya

| | Blackwood aka Sawati+ya

| | (P8/5, ^1)

| | rank-2 5-edo

Line 1,793:

Line 1,757:

| | ---

|-

| | Whitewood aka ~~Lawa~~+ya

| | Whitewood aka Lawati+ya

| | (P8/7, ^1)

| | rank-2 7-edo

Line 1,858:

Line 1,822:

! | ^1 ratio

|-

| | ~~Laquinzo~~

| | Laquinzoti

| | 2.3.7

| | (-14,0,0,5)

Line 1,868:

Line 1,832:

| | 49/48

|-

| | ~~Saquinru~~

| | Saquinruti

| | 2.3.7

| | (22,-5,0,-5)

Line 1,886:

Line 1,850:

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 EUs, 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 non-8ve and no-5ths pergens, not every name has a note. For example, ~~Biruyo~~ Nowa (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 - vF# - 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.

But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (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 - vF# - 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.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths 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. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Line 2,825:

Line 2,789:

|-

| | 5L 5s

| | (P8/2, P5) [10]

| | (P8/5, P5) [10]

| | ~~half~~-8ve decatonic

| | fifth-8ve decatonic

| | (lopsided unless 5th is quite flat)

|-

Line 2,850:

Line 2,814:

|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & ~~Ruyoyo~~ generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyoti generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka ~~Zozoquingu~~ Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquinguti Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

==Pergens and EDOs==

Line 4,201:

Line 4,165:

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. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.

Screenshots of the first 69 pergens:

Screenshots of the first 170 pergens:

[[File:alt-pergenLister_1.png|~~alt=alt-pergenLister 1.png|800x427px~~|alt-pergenLister 1.png]]

[[File:alt-pergenLister_1.png|852x852px|alt-pergenLister 1.png]]

[[File:~~alt-pergenLister_2.png|alt=alt~~-pergenLister 2.png|~~800x455px~~|~~alt-pergenLister 2.png~~]]

[[File:Alt-pergenLister 2a.png|frameless|852x852px]]

~~The first 29 pergens supported by 12edo~~:

[[File:Alt-pergenLister 3.png|frameless|854x854px]]

~~[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]~~

~~Some of the~~ pergens supported by ~~15edo. A red asterisk means partial support.~~

The first 39 pergens supported by 12edo:

[[File:alt-~~pergenLister_15edo~~.png|~~alt=~~alt-pergenLister 15edo.png|~~800x493px~~|alt-pergenLister 15edo.png]]

[[File:alt-pergenLister_12edo.png|857x857px|alt-pergenLister 12edo.png]]

Some of the pergens supported by 15edo. A red asterisk means partial support, e.g. (P8, P5) only uses a 5edo subset of 15edo.

[[File:alt-pergenLister_15edo.png|854x854px|alt-pergenLister 15edo.png]]

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.

[[File:alt-pergenLister_19edo.png|~~alt=~~alt-pergenLister 19edo.png|~~800x459px~~|~~alt-pergenLister 19edo.png~~]]

[[File:alt-pergenLister_19edo.png|857x857px|alt-pergenLister 19edo.png]]

The first 54 imperfect pergens:

[[File:Imperfect pergens.png|frameless|863x863px]]

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:

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== Addenda (late 2023) ==

=== New terminology===

All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/~~Triyo~~ has pergen (P8, P4/3) and is triple-chain. Diaschismatic/~~Sagugu~~ has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.

===The EU(s) define the pergen===

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The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.

Consider ~~Triyo~~/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For ~~Triyo~~/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v3A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v3A1, the vAC is v(81/80) or [-4 4 -1 -1].

===The three commas ===

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The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.

In our ~~Triyo~~ example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

In our Triyoti example, 250/243 plus 3 downed syntonic commas = v3A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.

Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v3AA1 and v3d44 respectively), making a very awkward notation.

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, ~~Sagugu~~/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Saguguti/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).

More examples: ~~Laquinyo~~/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. ~~Gugu~~/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^5dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. ~~Zozo~~/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozoti/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}

== Addenda (late 2024) ==

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|12.24%

|}

== Addenda (Spring 2026) ==

=== Initial commas ===

As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.

The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.

{| class="wikitable sortable"

|+Initial comma of each pergen

!#

!pergen

!interval

!cents

!genspan

!notes per octave

|-

!1

!(P8, P5)

|±d2

|abs(12c)

|±12G

|12

|-

!2

!(P8/2, P5)

|±d2/2

|abs(6c)

|±6G

|12

|-

!3

!(P8, P4/2)

|m2/2

|50¢ - 2.5c

|5G

|5

|-

!4

!(P8, P5/2)

|A1/2

|50¢ + 3.5c

|7G

|7

|-

!5

!(P8/2, P4/2)

| colspan="2" | ''same as #3 (P8, P4/2)''

|5G

|10

|-

!6

!(P8/3, P5)

|±d2/3

|abs(4c)

|±4G

|12

|-

!7

!(P8, P4/3)

|A1/3

|33.3¢ + 2.33c

| -7G

|7

|-

!8

!(P8, P5/3)

|m2/3

|33.3¢ - 1.67c

| -5G

|5

|-

!9

!(P8, P11/3)

|M2/3

|66.7¢ + 0.67c

|2G

|2 (or >= 14)

|-

!10

!(P8/3, P4/2)

|A2/6

|50¢ + 1.5c

|3G

|9

|-

!11

!(P8/3, P5/2)

|m3/6

|50¢ - 0.5c

|1G

|3

|-

!12

!(P8/2, P4/3)

| colspan="2" | ''same as #7 (P8, P4/3)''

| -7G

|14

|-

!13

!(P8/2, P5/3)

| colspan="2" | ''same as #8 (P8, P5/3)''

| -5G

|10

|-

!14

!(P8/2, P11/3)

|M2/6

|33.3¢ + 0.33c

|1G

|2

|-

!15

!(P8/3, P4/3)

| colspan="2" | ''same as #7 (P8, P4/3)''

| -7G

|21

|-

!16

!(P8/4, P5)

|±d2/4

|abs(3c)

|±3G

|12

|-

!17

!(P8, P4/4)

| colspan="2" | ''same as #3 (P8, P4/2)''

|10G

|10

|-

!18

!(P8, P5/4)

|A1/4

|25¢ + 1.75c

|7G

|7

|-

!19

!(P8, P11/4)

|dd3/4

|25¢ - 4.25c

| -17G

|17

|-

!20

!(P8, P12/4)

|m2/4

|25¢ - 1.25c

| -5G

|5

|-

!21

!(P8/4, P4/2)

|M2/4

|50¢ + c/2

|G

|4

|-

!22

!(P8/2, M2/4)

|M2/4

|50¢ + c/2

|G

|2

|-

!23

!(P8/2, P4/4)

|m2/4

|25¢ - 1.25c

|5G

|10

|-

!24

!(P8/2, P5/4)

| colspan="2" |''same as #18 (P8, P5/4)''

|7G

|14

|-

!25

!(P8/4, P4/3)

|d4/12

|33.3¢ - 0.67c

|2G

|8

|}

The initial comma of #9 (P8, P11/3) is about 67¢, which is not too small to be a scale step. But if there are more than 2 notes per 8ve, the L/s ratio becomes enormous. The ratio only becomes reasonable (roughly 3) when there are at least 14 notes per octave.

Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.

This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.

For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v12d4, and ^12C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^12d94. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.

Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v3m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. Fortunately, the next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister.

True doubles require double-pair notation and thus require finding two commas.

[[Category:Regular temperament theory]]

[[Category:Notation]]

[[Category:Pages with proofs]]

@@ Line 1: / Line 1: @@
-A '''pergen''' (pronounced "peer-gen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
+A '''pergen''' (pronounced "peer-jen", from '''per'''iod and '''gen'''erator) is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
-Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to imply inflections of the generator rather than the fifth.)
+Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyoti]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Triguti tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name. (Note these uses of comma fractions are not convention universally; temperament pages tend to use comma fractions to indicate the damage to the generator rather than the fifth. This is analogous to indicating the amount of stretching of an edo by the damage to the edostep rather than to the octave. But the latter approach is much more common, because the damage to the octave is much more audible and thus much more musically relevant than the damage to an edostep, which often doesn't correspond to a simple ratio. Likewise for pergens: the damage to Triyoti/Porcupine's generator, which is both 10/9 and 11/10, is less relevant than the damage to Triyoti's 4th or 5th.)
 Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.
@@ Line 10: / Line 10: @@
 If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.
-For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The Dicot aka Yoyo temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyo is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozo, a pun on "semi-fourth", is of course half-fourth.
+For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. Dicot aka Yoyoti (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyoti is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozoti, a pun on "semi-fourth", is of course half-fourth.
-Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Sagugu and Injera aka Gu & Biruyo sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.
+Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Saguguti and Injera aka Gu & Biruyoti sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.
 The largest category contains all single-comma rank-2 temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime &gt; 3 (a '''higher prime'''), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called '''unsplit'''.
@@ Line 35: / Line 35: @@
 | | 81/80
 | | Meantone
-| | Gu
+| | Guti
 | | gT
 |-
@@ Line 42: / Line 42: @@
 | | 64/63
 | | Archy
-| | Ru
+| | Ruti
 | | rT
 |-
@@ Line 49: / Line 49: @@
 | | (-14,8,1)
 | | Schismic
-| | Layo
+| | Layoti
 | | LyT
 |-
@@ Line 56: / Line 56: @@
 | | (11, -4, -2)
 | | Srutal
-| | Sagugu
+| | Saguguti
 | | sggT
 |-
@@ Line 63: / Line 63: @@
 | | 81/80, 50/49
 | | Injera
-| | Gu & Biruyo
+| | Gu & Biruyoti
 | | g&rryyT
 |-
@@ Line 70: / Line 70: @@
 | | 25/24
 | | Dicot
-| | Yoyo
+| | Yoyoti
 | | yyT
 |-
@@ Line 77: / Line 77: @@
 | | (-1,5,0,0,-2)
 | | Mohajira
-| | Lulu
+| | Luluti
 | | 1uuT
 |-
@@ Line 84: / Line 84: @@
 | | 49/48
 | | Semaphore
-| | Zozo
+| | Zozoti
 | | zzT
 |-
@@ Line 91: / Line 91: @@
 | | 25/24, 49/48
 | | Decimal
-| | Yoyo & Zozo
+| | Yoyo & Zozoti
 | | yy&amp;zzT
 |-
@@ Line 98: / Line 98: @@
 | | 250/243
 | | Porcupine
-| | Triyo
+| | Triyoti
 | | y<span style="vertical-align: super;">3</span>T
 |-
@@ Line 105: / Line 105: @@
 | | (12,-1,0,0,-3)
 | | Satrilu
-| | Satrilu
+| | Satriluti
 | | s1u<span style="vertical-align: super;">3</span>T
 |-
@@ Line 112: / Line 112: @@
 | | (3,4,-4)
 | | Diminished
-| | Quadgu
+| | Quadguti
 | | g<span style="vertical-align: super;">4</span>T
 |-
@@ Line 119: / Line 119: @@
 | | (-17,2,0,0,4)
 | | Laquadlo
-| | Laquadlo
+| | Laquadloti
 | | L1o<span style="vertical-align: super;">4</span>T
 |-
@@ Line 126: / Line 126: @@
 | | (-10,-1,5)
 | | Magic
-| | Laquinyo
+| | Laquinyoti
 | | Ly<span style="vertical-align: super;">5</span>T
 |}
@@ Line 136: / Line 136: @@
 Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.
-For example, Marvel aka Ruyoyo (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.
+For example, Marvel aka Ruyoyoti (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.
-More examples: Trizogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.
+More examples: Trizoguti (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyoti (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyoti Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.
 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.
@@ Line 150: / Line 150: @@
 In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is '''dependent''' on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two '''independent''' primes.
-For example, consider Sawa & Ruyoyo (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).
+For example, consider Sawa & Ruyoyoti (2.3.5.7 with commas 256/243 and 225/224). The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Notating multi-EDO pergens|multi-EDO pergens]] below).
 To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [http://x31eq.com/temper/uv.html x31eq.com/temper/uv.html] that will find such a matrix, it's the reduced mapping. Next make a '''square mapping''' by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.
@@ Line 168: / Line 168: @@
 A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.
-For example, Porcupine aka Triyo (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 &lt;= i &lt;= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
+For example, Porcupine aka Triyoti (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 &lt;= i &lt;= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
-Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozogu is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:
+Rank-3 pergens are trickier to find. For example, Breedsmic aka Bizozoguti is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:
 {| class="wikitable" style="text-align:center;"
@@ Line 323: / Line 323: @@
 Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.
-Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyo, and the second one is Triyo & Ru. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
+Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyoti, and the second one is Triyo & Ruti. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
 Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further Discussion-Chord names and scale names|Chord names and scale names]] below.
@@ Line 329: / Line 329: @@
 The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.
-All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups_and_Downs_Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.
+All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Luluti and Dicot aka Yoyoti are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u implies Mohajira, but using ^ and v implies neither, and is a more general notation.
 One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, which is octave-equivalent, fifth-generated and heptatonic. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and all notation used here is backwards compatible.
-Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layo is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.
+Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, Schismic aka Layoti is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
@@ Line 386: / Line 386: @@
 | | P8/2 = vA4 = ^d5
 | | C - vF#=^Gb - C
-| | Srutal aka Sagugu
+| | Srutal aka Saguguti
 ^1 = 81/80
 |-
@@ Line 397: / Line 397: @@
 | | P8/2 = ^A4 = vd5
 | | C - ^F#=vGb - C
-| | Injera aka Gu & Biruyo
+| | Injera aka Gu & Biruyoti
 ^1 = 64/63
@@ Line 407: / Line 407: @@
 | | P8/2 = ^4 = v5
 | | C - ^F=vG - C
-| | Thotho, if 13/8 = M6
+| | Thothoti, if 13/8 = M6
 ^1 = 27/26
@@ Line 419: / Line 419: @@
 | | P4/2 = ^M2 = vm3
 | | C - ^D=vEb - F
-| | Semaphore aka Zozo
+| | Semaphore aka Zozoti
 ^1 = 64/63
@@ Line 429: / Line 429: @@
 | | P4/2 = vA2 = ^d3
 | | C - vD#=^Ebb - F
-| | Lala-yoyo
+| | Lala-yoyoti
 ^1 = 81/80
@@ Line 441: / Line 441: @@
 | | P5/2 = ^m3 = vM3
 | | C - ^Eb=vE - G
-| | Mohajira aka Lulu
+| | Mohajira aka Luluti
 ^1 = 33/32
@@ Line 477: / Line 477: @@
 C - ^/F=v\G - C
-| | Zozo &amp; Lulu
+| | Zozo &amp; Luluti
 ^1 = 33/32
@@ Line 505: / Line 505: @@
 C - ^/Eb=v\E - G
-| | Sagugu &amp; Zozo
+| | Sagugu &amp; Zozoti
 ^1 = 81/80
@@ Line 533: / Line 533: @@
 C - v/D=^\Eb - F
-| | Sagugu & Lulu
+| | Sagugu & Luluti
 ^1 = 81/80
@@ Line 555: / Line 555: @@
 | | P8/3 = vM3 = ^^d4
 | | C - vE - ^Ab - C
-| | Augmented aka Trigu
+| | Augmented aka Triguti
 ^1 = 81/80
@@ Line 567: / Line 567: @@
 | | P4/3 = vM2 = ^^m2
 | | C - vD - ^Eb - F
-| | Porcupine aka Triyo
+| | Porcupine aka Triyoti
 ^1 = 81/80
@@ Line 579: / Line 579: @@
 | | P5/3 = ^M2 = vvm3
 | | C - ^D - vF - G
-| | Slendric aka Latrizo
+| | Slendric aka Latrizoti
 ^1 = 64/63
@@ Line 591: / Line 591: @@
 | | P11/3 = vA4 = ^^dd5
 | | C - vF# - ^Cb - F
-| | Satrilu, if 11/8 = A4
+| | Satriluti, if 11/8 = A4
 ^1 = 729/704
@@ Line 601: / Line 601: @@
 | | P11/3 = ^4 = vv5
 | | C - ^F - vC - F
-| | Satrilu, if 11/8 = P4
+| | Satriluti, if 11/8 = P4
 ^1 = 33/32
@@ Line 617: / Line 617: @@
 C - ^<span style="vertical-align: super;">3</span>Db=v<span style="vertical-align: super;">3</span>E - F
-| | Tribilo, if 11/8 = P4
+| | Tribiloti, if 11/8 = P4
 ^1 = 33/32
@@ Line 635: / Line 635: @@
 C - /D=\Eb - F
-| | Triforce aka Trigu & Zozo
+| | Triforce aka Trigu & Zozoti
 ^1 = 81/80, /1 = 64/63
@@ Line 655: / Line 655: @@
 C - /Eb=\E - G
-| | Satribizo
+| | Satribizoti
 ^1 = 49/48, /1 = 343/324
@@ Line 675: / Line 675: @@
 C - \D - /Eb - F
-| | Latribiru
+| | Latribiruti
 ^1 = 1029/1024, /1 = 49/48
@@ Line 691: / Line 691: @@
 C - vvD# - ^^Fb - G
-| | Lartribiyo
+| | Latribiyoti
 ^1 = 81/80
@@ Line 709: / Line 709: @@
 C - /D - \F - G
-| | Lemba aka Latrizo & Biruyo
+| | Lemba aka Latrizo & Biruyoti
 ^1 = (10,-6,1,-1), /1 = 64/63
@@ Line 725: / Line 725: @@
 C - ^^F - vvC - F
-| | Latribilo, if 11/8 = P4
+| | Latribiloti, if 11/8 = P4
 ^1 = 33/32
@@ Line 751: / Line 751: @@
 C - v/D - ^\F - G
-| | Triyo &amp; Triru
+| | Triyo &amp; Triguti
 ^1 = 64/63
@@ Line 775: / Line 775: @@
 C - v\D - ^/Eb - F
-| | Trigu &amp; Latrizo
+| | Trigu &amp; Latrizoti
 ^1 = 81/80
@@ Line 799: / Line 799: @@
 C - v/E - ^\Ab - C
-| | Triyo &amp; Latrizo
+| | Triyo &amp; Latrizoti
 ^1 = 81/80
@@ Line 819: / Line 819: @@
 | | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2
 | | C vEb vvGb=^^F# ^A C
-| | Diminished aka Quadgu
+| | Diminished aka Quadguti
 |-
 | | 17
@@ Line 827: / Line 827: @@
 | | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1
 | | C ^Db ^^Ebb=vvD# vE F
-| | Negri aka Laquadyo
+| | Negri aka Laquadyoti
 |-
 | | 18
@@ Line 835: / Line 835: @@
 | | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2
 | | C vD vvE=^^Eb ^F G
-| | Tetracot aka Saquadyo
+| | Tetracot aka Saquadyoti
 |-
 | | 19
@@ Line 843: / Line 843: @@
 | | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5
 | | C ^E ^^G# vDb F
-| | Squares aka Laquadru
+| | Squares aka Laquadruti
 |-
 | | 20
@@ Line 851: / Line 851: @@
 | | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3
 | | C vF vvBb=^^A ^D G
-| | Vulture aka Sasa-quadyo
+| | Vulture aka Sasa-quadyoti
 |-
 | |
@@ Line 1,000: / Line 1,000: @@
 ==Secondary splits==
-Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyo) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:
+Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. Porcupine aka Triyoti) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:
 P4/3: C - vD - ^Eb - F
@@ Line 1,098: / Line 1,098: @@
 | | half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24
 |-
-| | (P83, P4/3)
+| | (P8/3, P4/3)
 | | third-everything
 | | every 3-limit interval is split three times as much as before
@@ Line 1,115: / Line 1,115: @@
 ==Finding an example temperament==
-To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span style="">⋅</span>P and P8. If P is 6/5, the comma is 4<span style="">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4<span style="">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>, the Quadru temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.
+To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span style="">⋅</span>P and P8. If P is 6/5, the comma is 4<span style="">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625, making the Diminished temperament aka Quadguti. If P is 7/6, the comma is P8 - 4<span style="">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>, making the Quadruti temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.
-Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n<span style="">⋅x</span> gens = n<span style="">⋅</span>I = x<span style="">⋅</span>M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5<span style="">⋅</span>(7/6) = 2<span style="">⋅P5. Thus </span>2<span style="">⋅P</span>5 - 5<span style="">⋅</span>(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7<span style="">⋅</span>(11/9) = 2<span style="">⋅</span>P8, and the comma is (-2, -14, 0, 0, 7), Saseplo.
+Another method: if the generator's cents are known, look on the genchain for an interval that approximates a ratio of color depth ±1. Let the interval be I, and the genspan of this interval be x. Then n<span style="">⋅x</span> gens = n<span style="">⋅</span>I = x<span style="">⋅</span>M, where M is the multigen and M/n is the generator. The comma can be found from this equation, if n and x are coprime. For example, suppose (P8, P5/5) has G = 140¢. The genchain is all multiples of 140¢. Looking at the cents, 280¢ is about 7/6. Thus 2G = 7/6, and 10G = 5<span style="">⋅</span>(7/6) = 2<span style="">⋅P5. Thus </span>2<span style="">⋅P</span>5 - 5<span style="">⋅</span>(7/6) = 0G = 0¢, and the comma is (3, 7, 0, -5). If the period is split and the generator isn't, use the perchain instead of the genchain. For example, (P8/7, P5) has a period of 141¢. 2 gens = 343¢, about 11/9. Thus 7<span style="">⋅</span>(11/9) = 2<span style="">⋅</span>P8, and the comma is (-2, -14, 0, 0, 7), making Saseploti.
 If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
@@ Line 1,141: / Line 1,141: @@
 ==Ratio and cents of the accidentals==
-The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.
+The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup> · 3<sup>y</sup> · P<sup>±1</sup>, where P is a higher prime. They are called mapping commas because they equate or map the ratio P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for prime 5 include 81/80, 135/128, and the schisma aka Layoma = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 (besides 1/1 of course) are the currently employed commas and combinations of them.
-If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyo, where ^1 equals 64/63 minus 81/80.
+If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in Lemba aka Latrizo & Biruyoti, where ^1 equals 64/63 minus 81/80.
 Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also [[Pergen#Notating_multi-EDO_pergens|multi-EDO pergens]] pergens below.
@@ Line 1,161: / Line 1,161: @@
 * quarter-comma Meantone: # = 76¢
 * fifth-comma Meantone: # = 84¢
-* third-comma Archy aka Ru: # = 177¢
+* third-comma Archy aka Ruti: # = 177¢
-* eighth-comma Porcupine aka Triyo: # = 157¢, ^ = 52¢ (^ = third-sharp)
+* eighth-comma Porcupine aka Triyoti: # = 157¢, ^ = 52¢ (^ = third-sharp)
-* seventh-comma Srutal aka Sagugu & Zoquadyo: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
+* seventh-comma Srutal aka Sagugu & Zoquadyoti: # = 139¢, ^ = 33¢ (no fixed relationship between ^ and #, seventh-comma means 1/7 of 2048/2025)
-* third-comma Injera aka Gu & Biruyo: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
+* third-comma Injera aka Gu & Biruyoti: # = 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma means 1/3 of 81/80)
-* eighth-comma Hedgehog aka Triyo & Biruyo: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
+* eighth-comma Hedgehog aka Triyo & Biruyoti: # = 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma means 1/8 of 250/243)
 The last five examples are a generalization of the practice of naming meantone tunings as a fraction of a comma, e.g. quarter-comma. The 5th is sharpened or flattened by some fraction of the first comma in the color name. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both Srutal and Injera are half-8ve, but their optimal tunings are very different.
@@ Line 1,220: / Line 1,220: @@
 It's not yet known if every pergen can avoid large EUs (those of a 3rd or more) with double-pair notation. One situation in which very large EUs occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the EU's stepspan to equal the multigen's stepspan.
-Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore  commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Third-4th has EU = v<span style="vertical-align: super;">3</span>A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v<sup>3</sup>\\M2 and ^<sup>3</sup>\\d2.
+Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 EU. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore  commas, aka Triyo & Zozoti), and its double-pair notation can also combine both. Third-4th has EU = v<span style="vertical-align: super;">3</span>A1 and G'= vM2 = ^^m2. Half-4th has EU' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent EUs are v<sup>3</sup>\\M2 and ^<sup>3</sup>\\d2.
 <span style="display: block; text-align: center;">P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
@@ Line 1,249: / Line 1,249: @@
 For example, (P8, P5/3) has n = 3, G = ^M2, and EU = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-EU is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-EU [-2,1]. This can't be simplified, so the new EU is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, EU needs to be upped, so EU = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-EU ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·EU to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- vD# -- ^Fb -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone.
-Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is &gt; 1. For example, consider Semaphore aka Zozo (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyo, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
+Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one EU (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain EU. Specifically, the comma tempered out should map to the EU, or the multi-EU if the count is &gt; 1. For example, consider Semaphore aka Zozoti (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 EU makes sense. G = ^M2 and the genchain is C -- ^D=vEb -- F. But consider Lala-yoyoti, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus EU = ^^dd2, G = vA2, and the genchain is C -- vD#=^Ebb -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
 Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 EU is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of EU.
-For example, Satrilu tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the EU is ^<span style="vertical-align: super;">3</span>dd2.
+For example, Paralimmal aka Satriluti tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the EU is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the EU is ^<span style="vertical-align: super;">3</span>dd2.
-Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizogu (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.
+Sometimes the temperament implies an EU that isn't even a 2nd. For example, Liese aka Gu & Trizoguti (2.3.5.7 with 81/80 and 1029/1000) is (P8, P11/3), with G = 7/5 = vd5, and EU = 3·vd5 - P11 = v<span style="vertical-align: super;">3</span>dd3. The genchain is P1 -- vd5 -- ^M7 -- P11, or C -- vGb -- ^B -- F.
 This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an EU, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different EUs.
@@ Line 1,261: / Line 1,261: @@
 ==Chord names and staff notation==
-Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups_and_Downs_Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
+Using pergens, all rank-2 chords can be named using ups and downs, and if needed lifts and drops as well. See the [[Ups and downs notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G ^A and C E G vBb are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
-Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Sagugu & Ru (2.3.5.7 with 2048/2025 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.
+Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara aka Ru & Biruyoti (2.3.5.7 with 64/63 and 50/49) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and EU = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C vE G Bb = Cv,7.
-A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyo (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.
+A different temperament may result in the same pergen with the same EU, but may still produce a different name for the same chord. For example, Injera aka Gu & Biruyoti (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 EU is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and EU = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G vBb = C,v7.
 Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes. This is in fact a requirement for MOS scales.
-Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyo (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.
+Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. Iv -- vIIIv -- vVI^m -- Iv. A Porcupine aka Triyoti (third-4th) comma pump can be written out like so: Cv -- vA^m -- vDv -- [vvB=^Bb]^m -- ^Ebv -- G^m -- Gv -- Cv. Brackets are used to show that vvB and ^Bb are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. vvB is written first to show that this root is a vM6 above the previous root, vD. ^Bb is second to show the P4 relationship to the next root, ^Eb. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either vvB^m or ^Bb^m, or possibly ^BbvvM = ^Bb vD ^F.
 Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with EU = v<sup>3</sup>A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.
@@ Line 1,285: / Line 1,285: @@
 The tipping point depends on the choice of EU. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an EU of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, the EU usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of EU is a given. However, the mapping of primes 11 and 13 is not agreed on.
-The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyo's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyo's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozo's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.
+The notation's tipping point is determined by the uninflected EU, which is implied by the vanishing comma. For example, Porcupine aka Triyoti's 250/243 comma is an A1 = (-11,7), which implies an uninflected EU of A1, which implies 7-edo, and a 685.7¢ tipping point. Dicot aka Yoyoti's 25/24 comma is also an A1, and has the same tipping point. Semaphore aka Zozoti's 49/48 comma is a minor 2nd = (8,-5), implying 5-edo and a 720¢ tipping point. See [[pergen#Applications-Tipping points|tipping points]] above for a more complete list.
 Double-pair notation has two EUs, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two EUs can be any two of A1, m2 or d2. One can choose to use whichever two EUs best avoid tipping.
-An example of a temperament that tips easily is Negri aka Laquadyo, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.
+An example of a temperament that tips easily is Negri aka Laquadyoti, 2.3.5 and (-14,3,4). Because Negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with Negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and the EU could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's perhaps best to avoid inverting the ratio. 81/80 implies an EU of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma Negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.
-Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyo comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.
+Another "tippy" temperament is found by adding the mapping comma 81/80 to the Negri comma and getting the Latriyoma comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.
 ==Notating unsplit pergens==
@@ Line 1,319: / Line 1,319: @@
 ! | cents
 |-
-| | Meantone aka Gu
+| | Meantone aka Guti
 | | 81/80 = P1
 | | c = -3¢ to -5¢
@@ Line 1,330: / Line 1,330: @@
 | | ---
 |-
-| | Mavila aka Layobi
+| | Mavila aka Layobiti
 | | 135/128 = A1
 | | c = -21¢ to -22¢
@@ Line 1,341: / Line 1,341: @@
 | | -100¢ - 7c = 47¢-54¢
 |-
-| | Lagu
+| | Laguti
 | | (-15,11,-1) = A1
 | | c = -10¢ to -12¢
@@ Line 1,352: / Line 1,352: @@
 | | 100¢ + 7c = 26¢-30¢
 |-
-| | Schismic aka Layo
+| | Schismic aka Layoti
 | | (-15,8,1) = -d2
 | | c = 1.7¢ to 2.0¢
@@ Line 1,363: / Line 1,363: @@
 | | 12c = 20¢-24¢
 |-
-| | Lalagu
+| | Lalaguti
 | | (-23,16,-1) = -d2
 | | c = -0.9¢ to -1.2¢
@@ Line 1,374: / Line 1,374: @@
 | | -12c = 10¢-15¢
 |-
-| | Father aka Gubi
+| | Father aka Gubiti
 | | 16/15 = m2
 | | c = 56¢ to 58¢
@@ Line 1,385: / Line 1,385: @@
 | | -100¢ + 5c = 180-190¢
 |-
-| | Superpyth aka Sasayo
+| | Superpyth aka Sasayoti
 | | (12,-9,1) = m2
 | | c = 9¢ to 10¢
@@ Line 1,400: / Line 1,400: @@
 For unsplit pergens only, the up symbol's ratio can be expressed as a 3-limit comma, which is the sum or difference of the vanishing comma and the mapping comma. This 3-limit comma is tempered, as is every ratio. It can also be found directly from the 3-limit mapping of the vanishing comma. For example, the Schismic comma is a descending d2, and d2 = (19,-12), therefore the 3-limit comma is the pythagorean comma (-19,12).
-A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Ru aka Archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.
+A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for Archy aka Ruti (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit Schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C vE G vBb.
 ==Notating rank-3 pergens==
@@ Line 1,456: / Line 1,456: @@
 | | ---
 |-
-| | Meantone aka Gu
+| | Meantone aka Guti
 | | (P8, P5)
 | | rank-2
@@ Line 1,464: / Line 1,464: @@
 | | ---
 |-
-| | Diaschismic aka Sagugu
+| | Diaschismic aka Saguguti
 | | (P8/2, P5)
 | | rank-2
@@ Line 1,472: / Line 1,472: @@
 | | EU = ^^d2
 |-
-| | Semaphore aka Zozo
+| | Semaphore aka Zozoti
 | | (P8, P4/2)
 | | rank-2
@@ Line 1,480: / Line 1,480: @@
 | | EU = vvm2
 |-
-| | Decimal aka Yoyo & Zozo
+| | Decimal aka Yoyo & Zozoti
 | | (P8/2, P4/2)
 | | rank-2
@@ Line 1,496: / Line 1,496: @@
 | | ---
 |-
-| | Marvel aka Ruyoyo
+| | Marvel aka Ruyoyoti
 | | (P8, P5, ^1)
 | | rank-3
@@ Line 1,504: / Line 1,504: @@
 | | ---
 |-
-| | Breedsmic aka Bizozogu
+| | Breedsmic aka Bizozoguti
 | | (P8, P5/2, ^1)
 | | rank-3
@@ Line 1,542: / Line 1,542: @@
 ! | EU
 |-
-| | Marvel aka Ruyoyo
+| | Marvel aka Ruyoyoti
 | | 225/224
 | | (P8, P5, ^1)
@@ Line 1,562: / Line 1,562: @@
 | | ^^\d2
 |-
-| | Biruyo
+| | Biruyoti
 | | 50/49
 | | (P8/2, P5, ^1)
@@ Line 1,572: / Line 1,572: @@
 | | ^^\\d2
 |-
-| | Trizogu
+| | Trizoguti
 | | 1029/1000
 | | (P8, P11/3, ^1)
@@ Line 1,582: / Line 1,582: @@
 | | ^^^\\\dd3
 |-
-| | Breedsmic aka Bizozogu
+| | Breedsmic aka Bizozoguti
 | | 2401/2400
 | | (P8, P5/2, ^1)
@@ Line 1,592: / Line 1,592: @@
 | | ^^\<span style="vertical-align: super;">4</span>dd3
 |-
-| | Demeter aka Trizo-agugu
+| | Demeter aka Trizo-aguguti
 | | 686/675
 | | (P8, P5, \m3/2)
@@ Line 1,604: / Line 1,604: @@
 If using single-pair notation, Marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - vE - G - vvA#. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - vE - G - \Bb.
-There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyo 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 EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. 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 vE G \Bb.
+There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, Biruyoti 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 EU = //d2. The ratio for /1 is (-10,6,-1,1), a descending interval. 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled awkwardly as C vE G v/Bb. 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 vE G \Bb.
 With this standardization, the EU can be derived directly 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)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the EU never is, therefore the EU = ^^\\d2. The period is found by adding/subtracting the EU from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both EU and P become more complex, but the ratio for /1 becomes simpler.
-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 Biruyo, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyo doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.
+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 Biruyoti, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore Biruyoti doesn't tip. Single-pair rank-3 notation has no EU, and thus no tipping point. Double-pair rank-3 notation has 1 EU, but two mapping commas. Rank-3 notations rarely tip.
-Unlike the previous examples, Demeter aka Trizo-agugu'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 EU. 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, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.
+Unlike the previous examples, Demeter aka Trizo-aguguti'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 EU. 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, EU = ^^\\\dd3, and ^^\\\C = A##. Genchain2 is C -- v/C# -- \Eb -- vE -- \\Gb -- v\G -- vvG#=\\\Bbb -- v\\Bb... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own EU, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own EU, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops 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 \m3, 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 there will always be a 3-limit comma small enough to 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''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred.
+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 \m3, 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 there will always be a 3-limit comma small enough to 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''' ([[Odd limit|double odd limit]]) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 &lt; 3, 5/4 is preferred. And as a tie-breaker, in case of two ratios with the same DOL (such as 5/3 and 6/5), to minimize the size in cents of the multigen2. Since 5/3 = 884.4¢ and 6/5 = 315.6¢, 6/5 is preferred.
 If ^1 = 81/80, possible half-split gen2'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.
 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:
@@ Line 1,689: / Line 1,689: @@
 | | 10
 | | (P8/2, P5, ^m3/2)
-| | half-8ve half-upminor 3rd
+| | half-8ve half-upminor-3rd
 | | (P8/2, P5, ^M2/2)
 | | half-8ve half-upmajor-2nd
@@ Line 1,695: / Line 1,695: @@
 | | 11
 | | (P8/2, P5, vM3/2)
-| | half-8ve half-downmajor 3rd
+| | half-8ve half-downmajor-3rd
 | | (P8/2, P5, vm3/2)
-| | etc.
+| | half-8ve half-downminor-3rd
 |-
 | | 12
-| | (P8/2, P5, ^m6/2)
+| | (P8, P4/2, vM3/2)
-| | half-8ve half-upminor 6th
+| | half-4th half-downmajor-3rd
-| | (P8/2, P5, ^M6/2)
+| | (P8, P4/2, ^M2/2)
-| |
+| | half-4th half-upmajor-2nd
 |-
 | | 13
-| | (P8/2, P5, vM6/2)
-| | half-8ve half-downmajor 6th
-| | (P8/2, P5, vm7/2)
-| |
-|-
-| | 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)
-| |
+| | half-4th half-upminor-6th
-| | (P8, P4/2, ^M6/2)
-| |
-|-
-| | 17
-| | (P8, P4/2, vM6/2)
-| |
 | | (P8, P4/2, vm7/2)
-| |
+| | half-4th half-downminor-7th
 |-
-| | 18
+| | 14
-| | (P8, P5/2, ^m3/2)
+| | (P8, P5/2, vM3/2)
-| |
+| | half-5th half-downmajor-3rd
 | | (P8, P5/2, ^M2/2)
-| |
+| | half-5th half-upmajor-2nd
 |-
-| | 19
+| | 15
-| | (P8, P5/2, vM3/2)
-| |
-| | (P8, P5/2, vm3/2)
-| |
-|-
-| | 20
 | | (P8, P5/2, ^m6/2)
-| |
+| | half-5th half-upminor-6th
-| | (P8, P5/2, ^M6/2)
-| |
-|-
-| | 21
-| | (P8, P5/2, vM6/2)
-| |
 | | (P8, P5/2, vm7/2)
-| |
+| | half-5th half-downminor-7th
 |-
-| | 22
+| | 16
 | | (P8/2, P4/2, vM3/2)
 | | half-everything half-downmajor-3rd
 | | (P8/2, P4/2, ^M2/2)
-| |
+| | half-everything half-upmajor-2nd
 |}
 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.
@@ Line 1,771: / Line 1,735: @@
 A multi-EDO pergen 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 multi-EDO 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 aka Sawa plus Ya 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:
+A multi-EDO 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 aka Sawati plus Ya 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:
 {| class="wikitable" style="text-align:center;"
@@ Line 1,784: / Line 1,748: @@
 ! | /1 ratio
 |-
-| | Blackwood aka Sawa+ya
+| | Blackwood aka Sawati+ya
 | | (P8/5, ^1)
 | | rank-2 5-edo
@@ Line 1,793: / Line 1,757: @@
 | | ---
 |-
-| | Whitewood aka Lawa+ya
+| | Whitewood aka Lawati+ya
 | | (P8/7, ^1)
 | | rank-2 7-edo
@@ Line 1,858: / Line 1,822: @@
 ! | ^1 ratio
 |-
-| | Laquinzo
+| | Laquinzoti
 | | 2.3.7
 | | (-14,0,0,5)
@@ Line 1,868: / Line 1,832: @@
 | | 49/48
 |-
-| | Saquinru
+| | Saquinruti
 | | 2.3.7
 | | (22,-5,0,-5)
@@ Line 1,886: / Line 1,850: @@
 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 EUs, 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 non-8ve and no-5ths pergens, not every name has a note. For example, Biruyo Nowa (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 - vF# - 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.
+But in non-8ve and no-5ths pergens, not every name has a note. For example, Biruyoti Nowa (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 - vF# - 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.
 Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths 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. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
@@ Line 2,825: / Line 2,789: @@
 |-
 | | 5L 5s
-| | (P8/2, P5) [10]
+| | (P8/5, P5) [10]
-| | half-8ve decatonic
+| | fifth-8ve decatonic
 | | (lopsided unless 5th is quite flat)
 |-
@@ Line 2,850: / Line 2,814: @@
 |}
-The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyo generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
+The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the Sensei aka Sepgu & Ruyoyoti generator would be (P8, ccP5/7) [5]. The rationale would be that two Sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
-Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquingu Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.
+Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be Roulette [7] aka Zozoquinguti Nowa [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.
 ==Pergens and EDOs==
@@ Line 4,201: / Line 4,165: @@
 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. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. EUs of a 3rd or more are in red.
-Screenshots of the first 69 pergens:
+Screenshots of the first 170 pergens:
-[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]
+[[File:alt-pergenLister_1.png|852x852px|alt-pergenLister 1.png]]
-[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]
+[[File:Alt-pergenLister 2a.png|frameless|852x852px]]
-The first 29 pergens supported by 12edo:
+[[File:Alt-pergenLister 3.png|frameless|854x854px]]
-[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]
-Some of the pergens supported by 15edo. A red asterisk means partial support.
+The first 39 pergens supported by 12edo:
-[[File:alt-pergenLister_15edo.png|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]
+[[File:alt-pergenLister_12edo.png|857x857px|alt-pergenLister 12edo.png]]
+Some of the pergens supported by 15edo. A red asterisk means partial support, e.g. (P8, P5) only uses a 5edo subset of 15edo.
+[[File:alt-pergenLister_15edo.png|854x854px|alt-pergenLister 15edo.png]]
 Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.
-[[File:alt-pergenLister_19edo.png|alt=alt-pergenLister 19edo.png|800x459px|alt-pergenLister 19edo.png]]
+[[File:alt-pergenLister_19edo.png|857x857px|alt-pergenLister 19edo.png]]
+The first 54 imperfect pergens:
+[[File:Imperfect pergens.png|frameless|863x863px]]
 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:
@@ Line 4,424: / Line 4,396: @@
 == Addenda (late 2023) ==
 === New terminology===
-All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyo has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Sagugu has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.
+All temperaments have a '''chain number''', which is the number of fifthchains in the temperament's lattice. Any (P8, P5) temperament has a chain number of 1, and is '''single-chain'''. All other pergens are '''multi-chain'''. For example, Porcupine/Triyoti has pergen (P8, P4/3) and is triple-chain. Diaschismatic/Saguguti has pergen (P8/2, P5) and is double-chain. A pergen (P8/m, M/n) has chain number m * n / f, where M is the multigen and f is the absolute value of M's [[fifthspan]]. For example (P8/2, M2/4) is quadruple-chain.
 ===The EU(s) define the pergen===
@@ Line 4,510: / Line 4,482: @@
 The '''[[arrow]] comma''' is the ratio that equals the up [[arrow]]. This term overlaps a lot with the term mapping comma, but it isn't quite identical to it. For 5-limit temperaments the arrow comma is almost always 81/80, and for 7-limit it's almost always 64/63. But other commas can occur.
-Consider Triyo/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v<sup>3</sup>A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyo/Porcupine with an EU of v<sup>3</sup>A1, the vAC is v(81/80) or [-4 4 -1 -1].
+Consider Triyoti/Porcupine which is (P8, P4/3). The vanishing comma or '''VC''' is 250/243, which has a 2.3.5 monzo of [2 -5 3]. The EU is v<sup>3</sup>A1, which has a 2.3.^ monzo of [-11 7 -3]. The arrow comma or '''AC''' equals an up, therefore it vanishes when downed. The downed AC (or '''vAC''') can be expressed as a 2.3.5.^ monzo. For Triyoti/Porcupine with an EU of v<sup>3</sup>A1, the vAC is v(81/80) or [-4 4 -1 -1].
 ===The three commas ===
@@ Line 4,519: / Line 4,491: @@
 The EU always equals the VC (possibly inverted) plus or minus some number of vAC's. That number is whatever is needed to eliminate the higher prime. The VC must be inverted if the resulting EU would otherwise have a negative stepspan, or is a diminished unison.
-In our Triyo example, 250/243 plus 3 downed syntonic commas = v<sup>3</sup>A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.
+In our Triyoti example, 250/243 plus 3 downed syntonic commas = v<sup>3</sup>A1. As 2.3.5.^ monzos, we have [1 -5 3 0] + 3·[-4 4 -1 -1] = [-11 7 0 -3]. Note the zeros. The VC always has a zero arrow-count and the EU always has a zero prime-5-count.
 Visualizing the three commas in the lattice, one starts at the VC and heads towards the row of 3-limit intervals via the vAC. Wherever one lands is the uninflected EU. One can try other AC's besides 81/80. The AC's prime-5-count must be ±1, so [[135/128|Layobi]] (135/128) or [[Schisma|Layo]] (the schisma) are possibilities. But either of these would lead to a very remote EU (v<sup>3</sup>AA1 and v<sup>3</sup>d<sup>4</sup>4 respectively), making a very awkward notation.
-Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Sagugu/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).
+Next let's specify the AC, experiment with the VC and see what the EU turns out to be. For 5-limit temperaments, one can require that the AC always be 81/80, and derive the EU (and thus the notation) from the VC. For example, Saguguti/Diaschismic has VC = [11 -4 -2 0]. Subtracting two vAC's makes [19 -12 0 2] = ^^d2. This is in fact the recommended EU for (P8/2, P5).
-More examples: Laquinyo/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^<sup>5</sup>dd2. Gugu/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.
+More examples: Laquinyoti/Magic is (P8, P12/5) and has VC = [-10 -1 5 0]. Adding five vAC's makes [-30 19 0 -5]. This appears to be a AA7 but is actually a negative 2nd. We invert to get [30 -19 0 5] = ^<sup>5</sup>dd2. Guguti/Bug has VC = 27/25 = [0 3 -2 0]. Subtracting two vAC's makes [8 -5 0 2] = ^^m2. Again, this is the recommended EU.
-Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozo/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}
+Let's try a 7-limit temperament with the obvious vAC of 64/63 = [6 -2 -1 -1]. Zozoti/Semaphore has VC = 49/48 = [-4 -1 2 0]. Adding two vAC's makes [8 -5 0 -2] = vvm2.{{todo|complete section|comment=apply to multi-comma temperaments|inline=1}}
 == Addenda (late 2024) ==
@@ Line 4,819: / Line 4,791: @@
 |12.24%
 |}
+== Addenda (Spring 2026) ==
+=== Initial commas ===
+As one stacks generators and octave-reduces, at some point one overshoots or undershoots the octave by an interval of about a quartertone or less. This small interval is the pergen's initial comma. For example, (P8, P5)'s initial comma is the pythagorean comma, its next comma is Mercator's comma, etc. Each comma has a certain genspan, here 12 and 53. The genspan of the initial comma limits the size of a scale one can construct, assuming one wants to avoid overly-small steps. Thus one can have a pythagorean scale of up to 12 notes, but a 13-note scale will have a very small step. Note that the genspan gives the maximum notes per period, not per octave. Assuming one also wants to avoid extreme L/s step ratios also limits the maximum notes per octave.
+The table below lists the initial comma of various pergens. "±" indicates a tippy pergen. "c" is the difference between the fifth and 7\12. "abs(6c)" means the absolute value of 6c. The dim 2nd is a pythagorean comma.
+{| class="wikitable sortable"
+|+Initial comma of each pergen
+!#
+!pergen
+!interval
+!cents
+!genspan
+!notes per octave
+|-
+!1
+!(P8, P5)
+|±d2
+|abs(12c)
+|±12G
+|12
+|-
+!2
+!(P8/2, P5)
+|±d2/2
+|abs(6c)
+|±6G
+|12
+|-
+!3
+!(P8, P4/2)
+|m2/2
+|50¢ - 2.5c
+|5G
+|5
+|-
+!4
+!(P8, P5/2)
+|A1/2
+|50¢ + 3.5c
+|7G
+|7
+|-
+!5
+!(P8/2, P4/2)
+| colspan="2" |   ''same as #3 (P8, P4/2)''
+|5G
+|10
+|-
+!6
+!(P8/3, P5)
+|±d2/3
+|abs(4c)
+|±4G
+|12
+|-
+!7
+!(P8, P4/3)
+|A1/3
+|33.3¢ + 2.33c
+| -7G
+|7
+|-
+!8
+!(P8, P5/3)
+|m2/3
+|33.3¢ - 1.67c
+| -5G
+|5
+|-
+!9
+!(P8, P11/3)
+|M2/3
+|66.7¢ + 0.67c
+|2G
+|2 (or >= 14)
+|-
+!10
+!(P8/3, P4/2)
+|A2/6
+|50¢ + 1.5c
+|3G
+|9
+|-
+!11
+!(P8/3, P5/2)
+|m3/6
+|50¢ - 0.5c
+|1G
+|3
+|-
+!12
+!(P8/2, P4/3)
+| colspan="2" |   ''same as #7 (P8, P4/3)''
+| -7G
+|14
+|-
+!13
+!(P8/2, P5/3)
+| colspan="2" |   ''same as #8 (P8, P5/3)''
+| -5G
+|10
+|-
+!14
+!(P8/2, P11/3)
+|M2/6
+|33.3¢ + 0.33c
+|1G
+|2
+|-
+!15
+!(P8/3, P4/3)
+| colspan="2" |   ''same as #7 (P8, P4/3)''
+| -7G
+|21
+|-
+!16
+!(P8/4, P5)
+|±d2/4
+|abs(3c)
+|±3G
+|12
+|-
+!17
+!(P8, P4/4)
+| colspan="2" |   ''same as #3 (P8, P4/2)''
+|10G
+|10
+|-
+!18
+!(P8, P5/4)
+|A1/4
+|25¢ + 1.75c
+|7G
+|7
+|-
+!19
+!(P8, P11/4)
+|dd3/4
+|25¢ - 4.25c
+| -17G
+|17
+|-
+!20
+!(P8, P12/4)
+|m2/4
+|25¢ - 1.25c
+| -5G
+|5
+|-
+!21
+!(P8/4, P4/2)
+|M2/4
+|50¢ + c/2
+|G
+|4
+|-
+!22
+!(P8/2, M2/4)
+|M2/4
+|50¢ + c/2
+|G
+|2
+|-
+!23
+!(P8/2, P4/4)
+|m2/4
+|25¢ - 1.25c
+|5G
+|10
+|-
+!24
+!(P8/2, P5/4)
+| colspan="2" |''same as #18 (P8, P5/4)''
+|7G
+|14
+|-
+!25
+!(P8/4, P4/3)
+|d4/12
+|33.3¢ - 0.67c
+|2G
+|8
+|}
+The initial comma of #9 (P8, P11/3) is about 67¢, which is not too small to be a scale step. But if there are more than 2 notes per 8ve, the L/s ratio becomes enormous. The ratio only becomes reasonable (roughly 3) when there are at least 14 notes per octave.
+Note the initial comma is often equivalent to the uninflected EU divided by the height. For example, (P8, P4/2) has comma m2/2 and EU vvm2. In other words, the up-arrow is often the initial comma.
+This suggests a new algorithm for finding a good EU for a pergen. Search the cents table (in the Notation Guide For rank-2 Pergens pdf, the first table of each pergen) for a small step. The search can be easily done by computer. Then derive the EU from that small step.
+For example, pergen #25 (P8/4, P4/3) has a 33¢ step at 2G - P. Thus ^1 = 2G - P. Multiplying 2G by 3 gives us unsplit 4ths, and multiplying P by 4 gives us unsplit octaves. Thus we must multiply the up-arrow by 12 to get an unsplit 3-limit interval. 12 ups = 24G - 12P = 8P4 - 3P8 = dim 4th. Thus the EU is v<sup>12</sup>d4, and ^<sup>12</sup>C = Fb. But the pergenLister program lists the single-pair notation for this pergen as having an EU of ^<sup>12</sup>d<sup>9</sup>4. Thus the pergenLister algorithm missed a much simpler EU, and hence a much simpler notation.
+Not all initial commas imply a valid notation. For pergen #66 (P8, P5/6), the initial comma is P - 10G = m2/3 = 33.3¢ - 1.67c. The implied EU is v<sup>3</sup>m2. But this notation is incomplete because it only notates 3 of the 6 fifthchains. There must be 6 arrows in the EU, not 3. Or else there must be a second EU with 2 arrows. Fortunately, the next comma on the genchain is 21G - 2P = A1/2 = 50¢ - 3.5c, which implies a double-pair notation with EU' = \\A1. This is the notation found by pergenLister.
+True doubles require double-pair notation and thus require finding two commas.
 [[Category:Regular temperament theory]]
 [[Category:Notation]]
 [[Category:Pages with proofs]]