Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.

For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).

For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity (P8/2, (5/4)/1), or if a colon is used (P8/2, 5:4).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.

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Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups.

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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

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.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be ~~1 or -1~~. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).

=__Derivation__=

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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. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.

For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.

For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), or (P8/5, 5:4), the same as Blackwood.

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 may need to be discarded, as in the previous (P8/5,y3) 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 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, 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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n 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 (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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n 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 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:

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||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||

||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||

Thus 2/1 = P, 3/1 = P + ~~2 G1~~, 5/1 = P + G1 + ~~2 G2~~, and 7/1 = ~~2 P~~ + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:

||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||

||~ period ||= 1 ||= 1 ||= 1 ||

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||~ 3/1 ||= 0 ||= 2 ||= -1 || ||

||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||

Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)~~/4 = yy15~~/4 = (25/6)/4.

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19~~/2 = (6/1)~~/2, both of which are much larger, so the best gen1 is P5/2.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)~~/4 = gg15~~/4 = (96/25)/4. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be ~~gg7/4 =~~ (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)~~/4 = Lyy3~~/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

~~Alternatively~~, ~~we could discard~~ the 3rd column and keep the 4th one:

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

||~ ||~ 2/1 ||~ 3/1 ||~ 7/1 ||

||~ period ||= 1 ||= 1 ||= 2 ||

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||~ 3/1 ||= 0 ||= 1 ||= -1 || ||

||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 ~~= r2~~. Adding an octave and subtracting 4 half-5ths makes 64/63 ~~= r1~~. The pergen is (P8, P5/2, r1) ~~= half-5th with red~~. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, ~~gg7~~/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

=__Applications__=

One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), but the tone isn't a generator.

One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (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 triple yellow, and the second one is triple yellow and red. 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.

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||= superpyth ||= (12,-9,1) = m2 ||= c = 9¢ to 10¢ ||= A2 ||= C D# G ||= vM3 ||= C Ev G ||= vm2 ||= 81/80 = m2 ||= 100¢ - 5c = 50-55¢ ||

The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch.

==Notating Blackwood-like pergens==

By "Blackwood-like" is meant a temperament that equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5) or 25/16 or

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.

Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)

12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)

17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \

==Notating rank-3 pergens==

Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.

All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.

Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80).

Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80).

Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =

Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = ``\\``A1

(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is

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To prove: n is always a multiple of b, and n = |b| if and only if n = 1

b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)

The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) ~~= GCD (-3, -6) = 3~~

The GCD is defined here as always positive: GCD (c,d) = GCD (|c|,|d|)

n = zb, and since n > 0, n = |z|·|b|

if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1

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[//unfinished proof//]

Although not rigorously proven, these last two tests have been empirically verified by alt-~~pergenlister~~.

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

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

<h4>Original HTML content:</h4>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div>

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

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

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

</div>

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

<br />

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The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.<br />

<br />

For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation</a> can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).<br />

For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation</a> can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity (P8/2, (5/4)/1), or if a colon is used (P8/2, 5:4).<br />

<br />

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is <strong>highs and lows</strong>, written / and \.<br />

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Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups.<br />

<br />

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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br />

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.<br />

<br />

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be ~~1 or -1~~. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br />

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br />

<br />

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<h1 id="toc2"><a name="Derivation"></a><u>Derivation</u></h1>

<br />

For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br />

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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. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br />

<br />

For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.<br />

For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), or (P8/5, 5:4), the same as Blackwood.<br />

<br />

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 <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) 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.<br />

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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 are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br />

<br />

For example, 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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).<br />

For example, porcupine (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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).<br />

<br />

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br />

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</table>

Thus 2/1 = P, 3/1 = P + ~~2 G1~~, 5/1 = P + G1 + ~~2 G2~~, and 7/1 = ~~2 P~~ + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:<br />

Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:<br />

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</table>

Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)~~/4 = yy15~~/4 = (25/6)/4.<br />

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.<br />

<br />

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19~~/2 = (6/1)~~/2, both of which are much larger, so the best gen1 is P5/2.<br />

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.<br />

<br />

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)~~/4 = gg15~~/4 = (96/25)/4. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be ~~gg7/4 =~~ (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)~~/4 = Lyy3~~/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.<br />

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.<br />

<br />

~~Alternatively~~, ~~we could discard~~ the 3rd column and keep the 4th one:<br />

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:<br />

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</table>

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 ~~= r2~~. Adding an octave and subtracting 4 half-5ths makes 64/63 ~~= r1~~. The pergen is (P8, P5/2, r1) ~~= half-5th with red~~. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, ~~gg7~~/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br />

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br />

<br />

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<h1 id="toc3"><a name="Applications"></a><u>Applications</u></h1>

<br />

One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), but the tone isn't a generator.<br />

One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.<br />

<br />

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 triple yellow, and the second one is triple yellow and red. 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.<br />

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<br />

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

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

<br />

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

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<br />

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<br />

<h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

<h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

<br />

So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br />

<br />

<h2 id="toc7"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<h2 id="toc7"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.<br />

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A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br />

<br />

<h2 id="toc8"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<h2 id="toc8"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<br />

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 exponent ±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 class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br />

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There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br />

<br />

<h2 id="toc9"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<h2 id="toc9"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. 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 a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.<br />

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<br />

<h2 id="toc10"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<h2 id="toc10"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br />

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This is a lot of math, but it only needs to be done once for each pergen!<br />

<br />

<h2 id="toc11"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc11"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br />

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<br />

<h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<h2 id="toc12"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<br />

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> 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.<br />

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<br />

<h2 id="toc13"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<h2 id="toc13"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<br />

As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament &quot;tips over&quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.<br />

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<br />

<h2 id="toc14"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<h2 id="toc14"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<br />

An unsplit pergen doesn't <u>require</u> ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out the mapping comma, such as meantone or archy (2.3.7 and 64/63). For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.<br />

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<br />

<h2 id="toc15"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<h2 id="toc15"><a name="Further Discussion-Notating Blackwood-like pergens"></a>Notating Blackwood-like pergens</h2>

<br />

By &quot;Blackwood-like&quot; is meant a temperament that equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5) or 25/16 or<br />

<br />

Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.<br />

<br />

Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)<br />

12edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)<br />

17edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \<br />

<br />

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens"></a>Notating rank-3 pergens</h2>

<br />

Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.<br />

<br />

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.<br />

<br />

All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.<br />

<br />

Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). <br />

Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). <br />

Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =<br />

Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = \\A1<br />

<br />

(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is<br />

<br />

<h2 id="toc17"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

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

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<br />

<h2 id="~~toc16~~"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc18"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br />

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<br />

<h2 id="~~toc17~~"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

<h2 id="toc19"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

<br />

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><br />

<br />

Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><br />

<br />

Screenshot of the first 38 pergens:<br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<br />

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br />

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<br />

<h2 id="~~toc18~~"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<h2 id="toc20"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<br />

Given:<br />

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To prove: n is always a multiple of b, and n = |b| if and only if n = 1<br />

b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)<br />

The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) ~~= GCD (-3, -6) = 3~~<br />

The GCD is defined here as always positive: GCD (c,d) = GCD (|c|,|d|)<br />

n = zb, and since n &gt; 0, n = |z|·|b|<br />

if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1<br />

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[<em>unfinished proof</em>]<br />

<br />

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

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

<br />

<h2 id="~~toc19~~"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<h2 id="toc21"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<br />

<u><strong>Combining pergens</strong></u><br />

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-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-02 04:11:00 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-02 06:27:50 UTC</tt>.<br>
-: The original revision id was <tt>625770697</tt>.<br>
+: The original revision id was <tt>625774477</tt>.<br>
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@@ Line 45: / Line 45: @@
 The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.
-For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).
+For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity (P8/2, (5/4)/1), or if a colon is used (P8/2, 5:4).
 Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.
@@ Line 51: / Line 51: @@
 Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups.
-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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.
+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.
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).
 =__Derivation__=
@@ Line 61: / Line 61: @@
 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. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.
-For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.
+For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), or (P8/5, 5:4), the same as Blackwood.
 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 may need to be discarded, as in the previous (P8/5,y3) 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 76: / Line 76: @@
 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 are fairly rare. Less than 4% of all pergens have an imperfect multigen.
-For example, 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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n 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 (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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n 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 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&amp;limit=7|x31.com]] gives us this matrix:
@@ Line 83: / Line 83: @@
 ||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||
 ||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||
-Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:
+Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:
 ||~   ||~ 2/1 ||~ 3/1 ||~ 5/1 ||
 ||~ period ||= 1 ||= 1 ||= 1 ||
@@ Line 93: / Line 93: @@
 ||~ 3/1 ||= 0 ||= 2 ||= -1 ||   ||
 ||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||
-Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.
+Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.
-Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.
+Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.
-The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.
+The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.
-Alternatively, we could discard the 3rd column and keep the 4th one:
+Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:
 ||~   ||~ 2/1 ||~ 3/1 ||~ 7/1 ||
 ||~ period ||= 1 ||= 1 ||= 2 ||
@@ Line 109: / Line 109: @@
 ||~ 3/1 ||= 0 ||= 1 ||= -1 ||   ||
 ||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||
-Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.
+Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.
 =__Applications__=
-One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), but the tone isn't a generator.
+One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (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 triple yellow, and the second one is triple yellow and red. 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.
@@ Line 502: / Line 502: @@
 ||= superpyth ||= (12,-9,1) = m2 ||= c = 9¢ to 10¢ ||= A2 ||= C D# G ||= vM3 ||= C Ev G ||= vm2 ||= 81/80 = m2 ||= 100¢ - 5c = 50-55¢ ||
 The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch.
+==Notating Blackwood-like pergens==
+By "Blackwood-like" is meant a temperament that equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5) or 25/16 or
+Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.
+Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)
+edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)
+edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \
+==Notating rank-3 pergens==
+Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.
+All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.
+Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80).
+Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80).
+Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =
+Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = ``\\``A1
+(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is
@@ Line 695: / Line 722: @@
 To prove: n is always a multiple of b, and n = |b| if and only if n = 1
 b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)
-The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3
+The GCD is defined here as always positive: GCD (c,d) = GCD (|c|,|d|)
 n = zb, and since n &gt; 0, n = |z|·|b|
 if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1
@@ Line 739: / Line 766: @@
 [//unfinished proof//]
-Although not rigorously proven, these last two tests have been empirically verified by alt-pergenlister.
+Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.
@@ Line 782: / Line 809: @@
 Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>
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+&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Extremely large multigens"&gt;Extremely large multigens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens"&gt;Notating Blackwood-like pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
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   &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,044: / Line 1,073: @@
 The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.&lt;br /&gt;
 &lt;br /&gt;
-For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).&lt;br /&gt;
+For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-8ve, yellow-3rd. Ratios could be used instead, if enclosed in parentheses for clarity (P8/2, (5/4)/1), or if a colon is used (P8/2, 5:4).&lt;br /&gt;
 &lt;br /&gt;
 Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \.&lt;br /&gt;
@@ Line 1,050: / Line 1,079: @@
 Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups.&lt;br /&gt;
 &lt;br /&gt;
-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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81:80, 64:63, ...). The higher prime's exponent in the comma's monzo must be ±1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:57:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:57 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;
@@ Line 1,060: / Line 1,089: @@
 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. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.&lt;br /&gt;
 &lt;br /&gt;
-For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.&lt;br /&gt;
+For example, consider 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 multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), or (P8/5, 5:4), the same as Blackwood.&lt;br /&gt;
 &lt;br /&gt;
 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 &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank"&gt;x31eq.com/temper/uv.html&lt;/a&gt; that will find such a matrix, it's the reduced mapping. Next make a &lt;strong&gt;square mapping&lt;/strong&gt; by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) 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.&lt;br /&gt;
@@ Line 1,075: / Line 1,104: @@
 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 are fairly rare. Less than 4% of all pergens have an imperfect multigen.&lt;br /&gt;
 &lt;br /&gt;
-For example, 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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &amp;lt;= n &amp;lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).&lt;br /&gt;
+For example, porcupine (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, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &amp;lt;= n &amp;lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).&lt;br /&gt;
 &lt;br /&gt;
 Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;amp;limit=7" rel="nofollow"&gt;x31.com&lt;/a&gt; gives us this matrix:&lt;br /&gt;
@@ Line 1,131: / Line 1,160: @@
 &lt;/table&gt;
-Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:&lt;br /&gt;
+Thus 2/1 = P, 3/1 = P + 2·G1, 5/1 = P + G1 + 2·G2, and 7/1 = 2·P + G1 + G2. Discard the last column to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:&lt;br /&gt;
@@ Line 1,231: / Line 1,260: @@
 &lt;/table&gt;
-Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.&lt;br /&gt;
+Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.&lt;br /&gt;
 &lt;br /&gt;
-Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.&lt;br /&gt;
+Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.&lt;br /&gt;
 &lt;br /&gt;
-The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.&lt;br /&gt;
+The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.&lt;br /&gt;
 &lt;br /&gt;
-Alternatively, we could discard the 3rd column and keep the 4th one:&lt;br /&gt;
+Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:&lt;br /&gt;
@@ Line 1,337: / Line 1,366: @@
 &lt;/table&gt;
-Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
+Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
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-One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), but the tone isn't a generator.&lt;br /&gt;
+One obvious application is to name 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, because either it contains more than 2 primes, or because the 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 of (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.&lt;br /&gt;
 &lt;br /&gt;
 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 triple yellow, and the second one is triple yellow and red. 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.&lt;br /&gt;
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   &lt;br /&gt;
 Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
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 So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
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   &lt;br /&gt;
 If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.&lt;br /&gt;
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 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;
 &lt;br /&gt;
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 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 exponent ±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&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 ÷ (4/3) = 250/243.&lt;br /&gt;
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 There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;
 &lt;br /&gt;
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 In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. 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 a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;
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 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
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 This is a lot of math, but it only needs to be done once for each pergen!&lt;br /&gt;
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 Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2.&lt;br /&gt;
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 Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; 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.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:79:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:79 --&gt;Tipping points and sweet spots&lt;/h2&gt;
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 As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament &amp;quot;tips over&amp;quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Notating unsplit pergens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:81:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:81 --&gt;Notating unsplit pergens&lt;/h2&gt;
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 An unsplit pergen doesn't &lt;u&gt;require&lt;/u&gt; ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out the mapping comma, such as meantone or archy (2.3.7 and 64/63). For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;Pergens and MOS scales&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;Notating Blackwood-like pergens&lt;/h2&gt;
+ &lt;br /&gt;
+By &amp;quot;Blackwood-like&amp;quot; is meant a temperament that equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5) or 25/16 or&lt;br /&gt;
+&lt;br /&gt;
+Such a pergen is in effect multiple copies of an edo. Its notation can be based on the edo's notation, expanded with an additional microtonal accidental pair.&lt;br /&gt;
+&lt;br /&gt;
+Blackwood 5edo+y perchain: D E=F G A B=C D (E = m2), genchain: ... Gb^^ Bb^ D F#v A#vv... (^1 = 81/80, no E)&lt;br /&gt;
+edo+j perchain: D D#=Eb E F F#=Gb... C#=Db D (E = d2), genchain = C F^ Bb^^ (^1 = 33/32, no E)&lt;br /&gt;
+edo+y perchain: C C^ Dv D... (E = vm2), genchain: same as blackwood, but with / and \&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;Notating rank-3 pergens&lt;/h2&gt;
+ &lt;br /&gt;
+Rank-1 edos sometimes require ups and downs, and sometimes don't. Rank-2 pergens sometimes require double-pair notation, and sometimes require single-pair, and occasionally (meantone) don't. Rank-3 pergens sometimes require triple-pair notation, and sometimes only require double- or single-pair notation. They can't ever be notated conventionally.&lt;br /&gt;
+&lt;br /&gt;
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two pairs. There may be very false triples that can arise from a single comma, who knows? All true triples require triple-pair notation, but some false triples and some double-splits will require triple-pair as well, to avoid awkward enharmonics of a 3rd or more.&lt;br /&gt;
+&lt;br /&gt;
+All the previous rank-3 examples had the 2nd generator (gen2 or G2) be a comma, represented by a up or a high. There is no enharmonic for that accidental, just as JI has no enharmonics.&lt;br /&gt;
+&lt;br /&gt;
+Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). &lt;br /&gt;
+Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-8ve with highs (/1 = 81/80). &lt;br /&gt;
+Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th with ups. G1 =&lt;br /&gt;
+Breedsmic (P8, P5/2, ^1) half-5th with ups, G2 = ^1 = 64/63, G1 = /m3 = 49/40, E = &lt;!-- ws:start:WikiTextRawRule:052:``\\`` --&gt;\\&lt;!-- ws:end:WikiTextRawRule:052 --&gt;A1&lt;br /&gt;
+&lt;br /&gt;
+(2.3.5.7 and 686/675) equates two 5/4's to three 7/6's, and divides 5/4 into three 15/14's. Its pergen is (P8, P5, y3/3) or (P8, P5, (5:4)/3) or maybe (P8, P5, vM3/3). The enharmonic is&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Pergens and MOS scales&lt;/h2&gt;
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 Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;Pergens and EDOs&lt;/h2&gt;
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 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Supplemental materials"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Supplemental materials&lt;/h2&gt;
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 This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:5238:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5238 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:5270:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5270 --&gt;&lt;br /&gt;
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 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:5239:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5239 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:5271:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:5271 --&gt;&lt;br /&gt;
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 Screenshot of the first 38 pergens:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3073:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3073 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3080:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3080 --&gt;&lt;br /&gt;
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 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;Various proofs (unfinished)&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;Various proofs (unfinished)&lt;/h2&gt;
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 To prove: n is always a multiple of b, and n = |b| if and only if n = 1&lt;br /&gt;
 b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)&lt;br /&gt;
-The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3&lt;br /&gt;
+The GCD is defined here as always positive: GCD (c,d) = GCD (|c|,|d|)&lt;br /&gt;
 n = zb, and since n &amp;gt; 0, n = |z|·|b|&lt;br /&gt;
 if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1&lt;br /&gt;
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 [&lt;em&gt;unfinished proof&lt;/em&gt;]&lt;br /&gt;
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-Although not rigorously proven, these last two tests have been empirically verified by alt-pergenlister.&lt;br /&gt;
+Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:95:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Miscellaneous Notes&lt;/h2&gt;
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 &lt;u&gt;&lt;strong&gt;Combining pergens&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;