Kite's thoughts on pergens: Difference between revisions

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The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.

||~ __5-limit temperament__ ||~ __comma__ ||~ __sweet spot__ ||||~ __no ups or downs__ ||||||~ __with ups and downs__ ||||~ __up symbol__ ||

||~ (pergen is unsplit) ||~  ||~ (5th = 700¢ + c) ||~ 5/4 is ||~ 4:5:6 chord ||~ 5/4 is ||~ 4:5:6 chord ||~ E ||~ ratio ||~ cents ||

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

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

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

Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.

==Notating Blackwood-like pergens*==  

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.

||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* ||

||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||

See the next section for examples of which pergens are supported by a specific edo.

==Supplemental materials*==  

needs pergen squares picture

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.

http://www.tallkite.com/misc_files/pergens.pdf

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.

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

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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;

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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), only 2 primes, but the tone isn't a generator.&lt;br /&gt;

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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 bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &amp;quot;W&amp;quot;. Thus 10/3 = WM6 = wide major 6th, 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), M can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;

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Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:&lt;br /&gt;

For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).&lt;br /&gt;

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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, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.&lt;br /&gt;

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;

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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.&lt;br /&gt;

There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&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. The sharp symbol is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;

Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&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;

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. ^1 = 25¢ + 0.75·c, about an eighth-tone.&lt;br /&gt;

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.&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;

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.&lt;br /&gt;

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The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. 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. 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;

An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.&lt;br /&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 a comma that maps to P1, such as 81/80 or 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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The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate.&lt;br /&gt;

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

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Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:&lt;br /&gt;

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With this standardization, the enharmonic can be derived from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)&lt;span style="vertical-align: super;"&gt;-2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (19,-12)&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;. This can be rewritten as 50/49 = vv1 + &lt;!-- ws:start:WikiTextRawRule:055:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:055 --&gt;1 - d2 = vv&lt;!-- ws:start:WikiTextRawRule:056:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:056 --&gt;-d2 = -^^\\d2. The comma is negative, but the enharmonic never is, therefore E = ^^\\d2. P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.&lt;br /&gt;

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Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.&lt;br /&gt;

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

17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \&lt;br /&gt;

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Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is set. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;

Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.&lt;br /&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;

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.&lt;br /&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;

How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.&lt;br /&gt;

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This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.&lt;br /&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;

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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;

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Screenshot of the first 38 pergens:&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;

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;

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The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2 = 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). Likewise, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&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;u&gt;&lt;strong&gt;Staff notation&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;