Kite's thoughts on pergens - Revision history

TallKite: /* PergenLister */ update the screenshots

2026-05-26T01:53:53Z

PergenLister: update the screenshots

← Older revision		Revision as of 01:53, 26 May 2026
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.		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.		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:		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:

TallKite: /* Addenda (Spring 2026) */

2026-05-14T23:41:05Z

Addenda (Spring 2026)

← Older revision		Revision as of 23:41, 14 May 2026
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	== Addenda (Spring 2026) ==		== 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.		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.

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

TallKite: /* Addenda (Spring 2026) */ completed

2026-05-10T08:42:53Z

Addenda (Spring 2026): completed

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TallKite: /* Addenda (Spring 2026) */ alternative algorithm for finding EUs -- look for a nearby small scale step and equate that to an up-arrow

2026-05-04T00:54:57Z

Addenda (Spring 2026): alternative algorithm for finding EUs -- look for a nearby small scale step and equate that to an up-arrow

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AthiTrydhen: /* Addenda (Spring 2026) */

2026-05-03T21:30:01Z

Addenda (Spring 2026)

← Older revision		Revision as of 21:30, 3 May 2026
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	!19		!19
	!(P8, P11/4)		!(P8, P11/4)
	\|		\|dd3/4
	\|		\|25¢ - 4.25c
	\|		\| -17G
	\|		\|17
	\|		\|
	\|		\|
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	!20		!20
	!(P8, P12/4)		!(P8, P12/4)
	\|		\|m2/4
	\|		\|25¢ - 1.25c
	\|		\| -5G
	\|		\|5
	\|		\|
	\|		\|
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	!22		!22
	!(P8/2, M2/4)		!(P8/2, M2/4)
	\|		\|M2/4
	\|		\|50¢ + c/2
	\|		\|G
	\|		\|2
	\|		\|
	\|		\|

AthiTrydhen: /* Addenda (Spring 2026) */

2026-05-03T21:17:24Z

Addenda (Spring 2026)

← Older revision		Revision as of 21:17, 3 May 2026
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	\|		\|
	\|-		\|-
	!		!19
	!(P8, P5)		!(P8, P11/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!20
	!(P8, P5)		!(P8, P12/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!21
	!(P8, P5)		!(P8/4, P4/2)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!22
	!(P8, P5)		!(P8/2, M2/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!23
	!(P8, P5)		!(P8/2, P4/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!24
	!(P8, P5)		!(P8/2, P5/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!25
	!(P8, P5)		!(P8/4, P4/3)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!26
	!(P8, P5)		!(P8/4, P5/3)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!27
	!(P8, P5)		!(P8/4, P11/3)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!28
	!(P8, P5)		!(P8/3, P4/4)
	\|		\|
	\|		\|
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	\|		\|
	\|-		\|-
	!		!29
	!(P8, P5)		!(P8/3, P5/4)
	\|		\|
	\|		\|

TallKite: update temperament names with -ti, add an addenda

2026-05-03T21:13:32Z

update temperament names with -ti, add an addenda

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Xenllium: /* Notating rank-3 pergens */clearfying a tie-breaker of same DOL intervals for multigen2

2026-04-10T10:56:06Z

Notating rank-3 pergens: clearfying a tie-breaker of same DOL intervals for multigen2

← Older revision		Revision as of 10:56, 10 April 2026
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	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-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.

	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.		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:		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		\| \| 10
	\| \| (P8/2, P5, ^m3/2)		\| \| (P8/2, P5, ^m3/2)
	\| \| half-8ve half-upminor 3rd		\| \| half-8ve half-upminor-3rd
	\| \| (P8/2, P5, ^M2/2)		\| \| (P8/2, P5, ^M2/2)
	\| \| half-8ve half-upmajor-2nd		\| \| half-8ve half-upmajor-2nd
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	\| \| 11		\| \| 11
	\| \| (P8/2, P5, vM3/2)		\| \| (P8/2, P5, vM3/2)
	\| \| half-8ve half-downmajor 3rd		\| \| half-8ve half-downmajor-3rd
	\| \| (P8/2, P5, vm3/2)		\| \| (P8/2, P5, vm3/2)
	\| \| ~~etc.~~		\| \| half-8ve half-downminor-3rd
	\|-		\|-
	\| \| 12		\| \| 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		\| \| 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)		\| \| (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)		\| \| (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)		\| \| (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)		\| \| (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)		\| \| (P8, P5/2, vm7/2)
	\| \|		\| \| half-5th half-downminor-7th
	\|-		\|-
	\| \| 22		\| \| 16
	\| \| (P8/2, P4/2, vM3/2)		\| \| (P8/2, P4/2, vM3/2)
	\| \| half-everything half-downmajor-3rd		\| \| half-everything half-downmajor-3rd
	\| \| (P8/2, P4/2, ^M2/2)		\| \| (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.		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.

Xenllium: /* Pergens and MOS scales */

2026-02-12T23:13:57Z

Pergens and MOS scales

← Older revision		Revision as of 23:13, 12 February 2026
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	\|-		\|-
	\| \| 5L 5s		\| \| 5L 5s
	\| \| (P8/2, P5) [10]		\| \| (P8/5, P5) [10]
	\| \| ~~half~~-8ve decatonic		\| \| fifth-8ve decatonic
	\| \| (lopsided unless 5th is quite flat)		\| \| (lopsided unless 5th is quite flat)
	\|-		\|-

TallKite at 22:39, 8 January 2026

2026-01-08T22:39:23Z

← Older revision		Revision as of 22:39, 8 January 2026
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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-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.

	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 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 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 the octave.)

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

	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.		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:		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:
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	\| \| 10		\| \| 10
	\| \| (P8/2, P5, ^m3/2)		\| \| (P8/2, P5, ^m3/2)
	\| \| half-8ve half-upminor 3rd		\| \| half-8ve half-upminor-3rd
	\| \| (P8/2, P5, ^M2/2)		\| \| (P8/2, P5, ^M2/2)
	\| \| half-8ve half-upmajor-2nd		\| \| half-8ve half-upmajor-2nd
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	\| \| 11		\| \| 11
	\| \| (P8/2, P5, vM3/2)		\| \| (P8/2, P5, vM3/2)
	\| \| half-8ve half-downmajor 3rd		\| \| half-8ve half-downmajor-3rd
	\| \| (P8/2, P5, vm3/2)		\| \| (P8/2, P5, vm3/2)
	\| \| ~~etc.~~		\| \| half-8ve half-downminor-3rd
	\|-		\|-
	\| \| 12		\| \| 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		\| \| 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)		\| \| (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)		\| \| (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)		\| \| (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)		\| \| (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)		\| \| (P8, P5/2, vm7/2)
	\| \|		\| \| half-5th half-downminor-7th
	\|-		\|-
	\| \| 22		\| \| 16
	\| \| (P8/2, P4/2, vM3/2)		\| \| (P8/2, P4/2, vM3/2)
	\| \| half-everything half-downmajor-3rd		\| \| half-everything half-downmajor-3rd
	\| \| (P8/2, P4/2, ^M2/2)		\| \| (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.		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.