Kite's thoughts on pergens: Difference between revisions

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Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using [[Ups and Downs Notation|ups and downs]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]].

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

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).

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.

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.

Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See [[pergen#Further%20Discussion-Notating%20unsplit%20pergens|Notating unsplit pergens]] below.

==Secondary splits==  

P4: C - Dv - Eb^ - F

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.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

==Alternate enharmonics==  

&lt;span style="display: block; text-align: center;"&gt;P1 -- ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 -- v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M6 -- C

C -- Eb^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- Av&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- C

C -- Dv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- Ev&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;=Db^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; -- Eb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- F

&lt;/span&gt;

&lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F&lt;/span&gt;

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see [[pergen#Applications-Tipping%20points|tipping points]] above. Add n**·**count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).

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

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

==Notating unsplit pergens==  

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 ||

||= mavila ||= 135/128 = A1 ||= c = -21¢ to -22¢ ||= m3 ||= C Eb G ||= ^M3 ||= C E^ G ||= ^A1 ||= 80/81 = d1 ||= -100¢ - 7c = 47¢-54¢ ||

||= large green ||= (-15,11,-1) = A1 ||= c = -10¢ to -12¢ ||= A3 ||= C E# G ||= ^M3 ||= C E^ G ||= vA1 ||= 80/81 = A1 ||= 100¢ + 7c = 26¢-30¢ ||

The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

==Notating rank-3 pergens==  

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:

||= 12-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = d2 ||

||= 19-edo ||= rank-1 ||= conventional ||= rank-2 ||= 1 ||= E = dd2 ||

||= 15-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = m2, E' = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||

||= 24-edo ||= rank-1 ||= single-pair ||= rank-3 ||= 2 ||= E = d2, E' = vvA1 = vvm2 ||

||= semaphore ||= rank-2 ||= single-pair ||= rank-3 ||= 1 ||= E = vvm2 ||

||= decimal ||= rank-2 ||= double-pair ||= rank-4 ||= 2 ||= E = vvd2, E' = \\m2 ||

||= marvel ||= rank-3 ||= single-pair ||= rank-3 ||= 0 ||= --- ||

||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||

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 new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.

||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= ``\\``dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, \m3/2) ||= half-lowminor-3rd ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1

==Various proofs (unfinished)==  

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r.

* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')

__**Expanding gedras to 5-limit**__

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

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Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. See the notation guide below, under &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials"&gt;Supplemental materials&lt;/a&gt;.&lt;br /&gt;

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Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.&lt;br /&gt;

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.&lt;br /&gt;

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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;

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;

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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;

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. Certain rank-2 temperaments require another additional pair, &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.&lt;br /&gt;

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Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20unsplit%20pergens"&gt;Notating unsplit pergens&lt;/a&gt; below.&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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P4: C - Dv - Eb^ - F&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;

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Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).&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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&lt;span style="display: block; text-align: center;"&gt;P1 -- ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 -- v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M6 -- C&lt;br /&gt;

C -- Eb^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- Av&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- C&lt;br /&gt;

C -- Dv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- Ev&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;=Db^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; -- Eb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- F&lt;br /&gt;

&lt;/span&gt;&lt;br /&gt;

&lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:050:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:050 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:051:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:051 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;

To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan becomes negative, or if it's zero and the keyspan becomes negative, invert the gedra. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see &lt;a class="wiki_link" href="/pergen#Applications-Tipping%20points"&gt;tipping points&lt;/a&gt; above. Add n&lt;strong&gt;·&lt;/strong&gt;count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into n (or m) parts. Each part is the new generator (or period).&lt;br /&gt;

&lt;br /&gt;

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

&lt;br /&gt;

&lt;br /&gt;

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

&lt;br /&gt;

  &lt;br /&gt;

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;

&lt;br /&gt;

  &lt;br /&gt;

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;

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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  &lt;br /&gt;

&lt;br /&gt;

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;

         &lt;td style="text-align: center;"&gt;C E G&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

&lt;/td&gt;

&lt;/td&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.&lt;br /&gt;

&lt;br /&gt;

&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;

         &lt;th&gt;notation's rank&lt;br /&gt;

&lt;/th&gt;

         &lt;th&gt;enharmonics&lt;br /&gt;

&lt;/th&gt;

         &lt;td style="text-align: center;"&gt;0&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

         &lt;td style="text-align: center;"&gt;0&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

         &lt;td style="text-align: center;"&gt;0&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

&lt;/table&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 new accidental pairs. (There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more.&lt;br /&gt;

&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;^1 = 81/80&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

         &lt;td style="text-align: center;"&gt;^1 = 81/80&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

         &lt;td style="text-align: center;"&gt;^1 = 64/63&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

     &lt;/tr&gt;

&lt;/table&gt;

&lt;br /&gt;

fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢&lt;br /&gt;

&lt;br /&gt;

gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

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;

&lt;br /&gt;

  &lt;br /&gt;

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;

&lt;br /&gt;

  &lt;br /&gt;

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;

&lt;br /&gt;

  &lt;br /&gt;

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;br /&gt;

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;

&lt;br /&gt;

  &lt;br /&gt;

&lt;br /&gt;

Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise both (a,b) and n could be reduced by GCD (a,b). Since -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|)= |b| · r.&lt;br /&gt;

Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;u&gt;&lt;strong&gt;Combining pergens&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;

General rules for combining pergens:&lt;br /&gt;

&lt;ul&gt;&lt;li&gt;(P8/m, M/n) + (P8, P5) = (P8/m, M/n)&lt;/li&gt;&lt;li&gt;(P8/m, P5) + (P8, M/n) = (P8/m, M/n)&lt;/li&gt;&lt;li&gt;(P8/m, P5) + (P8/m', P5) = (P8/m&amp;quot;, P5), where m&amp;quot; = LCM (m,m')&lt;/li&gt;&lt;li&gt;(P8, M/n) + (P8, M/n') = (P8, M/n&amp;quot;), where n&amp;quot; = LCM (n,n')&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;

&lt;br /&gt;

&lt;u&gt;&lt;strong&gt;Expanding gedras to 5-limit&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;