Kite's thoughts on pergens: Difference between revisions

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A '''pergen''' (pronounced "peer-jen") 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 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 ~~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 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.)

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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==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 1,456:

| | ---

|-

| | 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 1,604:

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.

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|}

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 4,165:

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|852x852px|alt-pergenLister 1.png]]

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

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

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

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

The first 29 pergens supported by 12edo:

The first 39 pergens supported by 12edo:

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

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

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

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|~~alt=alt-pergenLister 15edo.png|800x493px~~|alt-pergenLister 15edo.png]]

[[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 4,487:

Line 4,495:

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: 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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== Addenda (Spring 2026) ==

=== Initial commas ===

~~WORK IN PROGRESS~~

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.

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.

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.

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!cents

!genspan

!~~max~~

!notes per octave

~~!min~~

~~!comments~~

|-

!1

Line 4,810:

Line 4,813:

|±12G

|12

|

|-

!2

Line 4,819:

Line 4,820:

|±6G

|12

|

|-

!3

Line 4,828:

Line 4,827:

|5G

|5

|

|-

!4

Line 4,837:

Line 4,834:

|7G

|7

|

|-

!5

Line 4,845:

Line 4,840:

|5G

|10

|

|-

!6

Line 4,854:

Line 4,847:

|±4G

|12

|

|-

!7

Line 4,863:

Line 4,854:

| -7G

|7

|

|-

!8

Line 4,872:

Line 4,861:

| -5G

|5

|

|-

!9

Line 4,880:

Line 4,867:

|66.7¢ + 0.67c

|2G

|2

|2 (or >= 14)

~~|15~~

|

|-

!10

Line 4,890:

Line 4,875:

|3G

|9

|

|-

!11

Line 4,899:

Line 4,882:

|1G

|3

|

|-

!12

Line 4,907:

Line 4,888:

| -7G

|14

|

|-

!13

Line 4,915:

Line 4,894:

| -5G

|10

|

|-

!14

Line 4,924:

Line 4,901:

|1G

|2

~~|30~~

|

|-

!15

Line 4,932:

Line 4,907:

| -7G

|21

|

|-

!16

Line 4,941:

Line 4,914:

|±3G

|12

|

|-

!17

Line 4,949:

Line 4,920:

|10G

|10

|

|-

!18

Line 4,958:

Line 4,927:

|7G

|7

|

|-

|

|-

!19

!(P8, P11/4)

Line 4,967:

Line 4,934:

| -17G

|17

|

|-

!20

Line 4,976:

Line 4,941:

| -5G

|5

|

|-

!21

!(P8/4, P4/2)

|

|M2/4

|

|50¢ + c/2

|

|G

|

|4

|

|-

!22

Line 4,994:

Line 4,955:

|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

|

|-

~~!26~~

~~!(P8/4, P5/3)~~

|

|-

~~!27~~

~~!(P8/4, P11/3)~~

|

|-

~~!28~~

~~!(P8/3, P4~~/4)

|

|-

~~!29~~

~~!(P8/3, P5/4)~~

|

|}

Note ~~the similarity of~~ the initial comma to the EU divided by the height.

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.

~~The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous!~~

~~Note the unusability of certain pergens such as (P8/2, P11/3).~~

[[Category:Regular temperament theory]]

[[Category:Notation]]

[[Category:Pages with proofs]]

@@ Line 1: / Line 1: @@
-A '''pergen''' (pronounced "peer-jen") 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 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 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 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.)
+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 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 1,456: / 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 1,604: / 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.
@@ Line 2,814: / 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 4,165: / 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|852x852px|alt-pergenLister 1.png]]
+[[File:Alt-pergenLister 2a.png|frameless|852x852px]]
-[[File:alt-pergenLister_1.png|alt=alt-pergenLister 1.png|800x427px|alt-pergenLister 1.png]]
+[[File:Alt-pergenLister 3.png|frameless|854x854px]]
-[[File:alt-pergenLister_2.png|alt=alt-pergenLister 2.png|800x455px|alt-pergenLister 2.png]]
-The first 29 pergens supported by 12edo:
+The first 39 pergens supported by 12edo:
-[[File:alt-pergenLister_12edo.png|alt=alt-pergenLister 12edo.png|800x449px|alt-pergenLister 12edo.png]]
+[[File:alt-pergenLister_12edo.png|857x857px|alt-pergenLister 12edo.png]]
-Some of the pergens supported by 15edo. A red asterisk means partial support.
+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|alt=alt-pergenLister 15edo.png|800x493px|alt-pergenLister 15edo.png]]
+[[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 4,487: / Line 4,495: @@
 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: 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 4,786: / Line 4,794: @@
 == Addenda (Spring 2026) ==
+=== Initial commas ===
-WORK IN PROGRESS
+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.
-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.
 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.
@@ Line 4,800: / Line 4,805: @@
 !cents
 !genspan
-!max
+!notes per octave
-!min
-!comments
 |-
 !1
@@ Line 4,810: / Line 4,813: @@
 |±12G
 |12
-|
-|
 |-
 !2
@@ Line 4,819: / Line 4,820: @@
 |±6G
 |12
-|
-|
 |-
 !3
@@ Line 4,828: / Line 4,827: @@
 |5G
 |5
-|
-|
 |-
 !4
@@ Line 4,837: / Line 4,834: @@
 |7G
 |7
-|
-|
 |-
 !5
@@ Line 4,845: / Line 4,840: @@
 |5G
 |10
-|
-|
 |-
 !6
@@ Line 4,854: / Line 4,847: @@
 |±4G
 |12
-|
-|
 |-
 !7
@@ Line 4,863: / Line 4,854: @@
 | -7G
 |7
-|
-|
 |-
 !8
@@ Line 4,872: / Line 4,861: @@
 | -5G
 |5
-|
-|
 |-
 !9
@@ Line 4,880: / Line 4,867: @@
 |66.7¢ + 0.67c
 |2G
-|2
+|2 (or >= 14)
-|15
-|
 |-
 !10
@@ Line 4,890: / Line 4,875: @@
 |3G
 |9
-|
-|
 |-
 !11
@@ Line 4,899: / Line 4,882: @@
 |1G
 |3
-|
-|
 |-
 !12
@@ Line 4,907: / Line 4,888: @@
 | -7G
 |14
-|
-|
 |-
 !13
@@ Line 4,915: / Line 4,894: @@
 | -5G
 |10
-|
-|
 |-
 !14
@@ Line 4,924: / Line 4,901: @@
 |1G
 |2
-|30
-|
 |-
 !15
@@ Line 4,932: / Line 4,907: @@
 | -7G
 |21
-|
-|
 |-
 !16
@@ Line 4,941: / Line 4,914: @@
 |±3G
 |12
-|
-|
 |-
 !17
@@ Line 4,949: / Line 4,920: @@
 |10G
 |10
-|
-|
 |-
 !18
@@ Line 4,958: / Line 4,927: @@
 |7G
 |7
-|
+|-
-|
-|-
 !19
 !(P8, P11/4)
@@ Line 4,967: / Line 4,934: @@
 | -17G
 |17
-|
-|
 |-
 !20
@@ Line 4,976: / Line 4,941: @@
 | -5G
 |5
-|
-|
 |-
 !21
 !(P8/4, P4/2)
-|
+|M2/4
-|
+|50¢ + c/2
-|
+|G
-|
+|4
-|
-|
 |-
 !22
@@ Line 4,994: / Line 4,955: @@
 |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
-|
-|
-|-
-!26
-!(P8/4, P5/3)
-|
-|
-|
-|
-|
-|
-|-
-!27
-!(P8/4, P11/3)
-|
-|
-|
-|
-|
-|
-|-
-!28
-!(P8/3, P4/4)
-|
-|
-|
-|
-|
-|
-|-
-!29
-!(P8/3, P5/4)
-|
-|
-|
-|
-|
-|
 |}
-Note the similarity of the initial comma to the EU divided by the height.
+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.
-The initial comma of (P8, P11/3) is a rather large 67¢, but if there are more than 2 notes per 8ve, the L/s ratio becomes enormous!
-Note the unusability of certain pergens such as (P8/2, P11/3).
 [[Category:Regular temperament theory]]
 [[Category:Notation]]
 [[Category:Pages with proofs]]