Pergen names: Difference between revisions

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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, etc.

For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.

For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.

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]].

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||= " ||= " ||= 64/63 ||= archy ||= red ||= rT ||

||= " ||= " ||= (-14,8,0,0,1) ||= schismic ||= large yellow ||= LyT ||

||= " ||= " ||= 81/80, 126/125 ||= septimal meantone ||= green and bluish~~-blue~~ ||= g&bg3T ||

||= " ||= " ||= 81/80, 126/125 ||= septimal meantone ||= green and bluish deep green ||= g&bg3T ||

||= (P8/2, P5) ||= half-octave ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||

||= " ||= " ||= 81/80, 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||

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(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.

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

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 ~~multi-gen~~ 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-octave, yellow-third. Ratios could be used instead, if enclosed in parentheses for clarity: ~~(P8/2, y3) =~~ (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-octave, yellow-third. 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 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 \.

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 \.

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-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh 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 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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

~~The preferred pergen for untempered~~ just intonation is 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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...)

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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).

=__Derivation__=

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 > m, it will split some ~~other~~ 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.

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 > m, it will split some 3-limit interval into n parts.

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

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. 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 ~~multi-gen~~ 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), 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 > 3 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.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the ~~multi-gen~~, P is the period P8/m and G is the generator M/n

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = xP, thus P = P8/x

3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

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**The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x <= n <= x**

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 ~~multi-gen~~ is the one with the least cents. The pergen is (P8, P4/3).

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).

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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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.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the ~~multi-gen~~ P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a __double__ octave to the ~~multi-gen~~. 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 multi-gen 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.

The 2nd ~~multi-gen~~ 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-fifth is a double fifth 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. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth 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.

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

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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 ~~multi-gen~~, the ~~multi-gen2~~. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the ~~multi-gen~~ 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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. We can let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth 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-fifth is a fifth = (-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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. We can let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth 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.

=__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 pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies ~~y3/2 =~~ (5/4)/2~~. Semisixth (aka sensei) implies y6/2 = (5/3)/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is~~.

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 pseudo-pergens, because the multigen isn't 3-limit, and might not even actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.

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 JI-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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If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** pergen. A single-split pergens 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 **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** 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.

Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the ~~multi-gen~~ split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's ~~multi-gen~~ is (a,b), the octave is split into at least |b| parts.

Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

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.

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P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- Db^^ -- Dv -- Eb^ -- Evv -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional ~~multi-gen~~ as before. Then deduce the period from the enharmonic. If the ~~multi-gen~~ was changed by unreducing, find the original generator from the period and the alternate generator.

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Next, find the original half-fourth generator. P = P8/5 ~ 240¢, and G = P4/2 ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. The alternate generator is usually simpler than the original generator, and the alternate multigen is usually more complex than the original multigen.

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==Combining pergens==

Tempering out 250/243 creates third-fourth, and 49/48 creates half-fourth, and tempering out both commas creates sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).

General rules for combining pergens:

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Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).

For {P8/M, ~~multi-gen~~/N}, an octave = M periods ± some number of enharmonics, and a ~~multi-gen~~ = N generators ± some number of enharmonics.

For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.

The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:

The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the ~~multi-gen~~ is split into N generators, each spanning y steps, and S = the ~~multi-gen~~'s degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.

The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.

The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-eleventh has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.

For {P8/M, ~~multi-gen~~/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.

For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.

For **{P8/M, P5}**, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)

For {**P8, ~~multi-gen~~/N}**, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the ~~multi-gen~~'s degree - 1

For {**P8, multigen/N}**, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1

For **{P8/M, ~~multi-gen~~/N}**, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1

For **{P8/M, multigen/N}**, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1

or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the ~~multi-gen~~'s enharmonic = |N * round ((S/N) - S ± N| +1

or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1

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For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-octave implies an even-numbered edo.

For {P8, ~~multi-gen~~/N}, since the octave is unsplit, the only possible ~~multi-gens~~ are some voicing of the 5th, and the ~~multi-gen~~'s 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.

For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.

For {P8/M, ~~multi-gen~~/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the ~~multi-gen~~, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.

For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.

For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^6d32, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d34. The perchain would be C - E#^3=Abbv3 - C.

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For **{P8/M, P5}**, the implied edo = Mx

For {**P8, ~~multi-gen~~/N}**, the implied edo = Ny ± 1 (the ~~multi-gen~~ is some voicing of the 5th)

For {**P8, multigen/N}**, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)

For **{P8/M, ~~multi-gen~~/N}**, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the ~~multi-gen~~,

For **{P8/M, multigen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,

or, the 8ve's implied edo = Mx and the ~~multi-gen~~'s implied edo = Ny ± F

or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F

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If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is split into N parts. The interval which is split into multiple generators is the multi-gen. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, etc.

For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.

For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.

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 <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>.

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<td style="text-align: center;">septimal meantone

</td>

<td style="text-align: center;">green and bluish~~-blue~~

<td style="text-align: center;">green and bluish deep green

</td>

<td style="text-align: center;">g&amp;bg3T

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(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.

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

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 ~~multi-gen~~ 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-octave, yellow-third. Ratios could be used instead, if enclosed in parentheses for clarity: ~~(P8/2, y3) =~~ (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. <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-octave, yellow-third. 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 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 \.

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 \.

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-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh 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 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 does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

~~The preferred pergen for untempered~~ just intonation is 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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...)

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 monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).

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

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 ~~other~~ 3-limit interval into n parts. Therefore a comma with only two primes, one of which is 2, always splits the octave (unless the other prime's exponent is ±1, e.g. 32/31). And a comma with only one higher prime will always split something, unless that prime's exponent is ±1.

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.

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

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. 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 ~~multi-gen~~ 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), 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 <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 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 &gt; 3 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.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the ~~multi-gen~~, P is the period P8/m and G is the generator M/n

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.

2/1 = P8 = xP, thus P = P8/x

3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

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The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x &lt;= n &lt;= x

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 ~~multi-gen~~ is the one with the least cents. The pergen is (P8, P4/3).

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).

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:

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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.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the ~~multi-gen~~ P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a double octave to the ~~multi-gen~~. 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 multi-gen 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.

The 2nd ~~multi-gen~~ 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-fifth is a double fifth 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. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth 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.

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

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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 ~~multi-gen~~, the ~~multi-gen2~~. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the ~~multi-gen~~ 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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. We can let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth 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-fifth is a fifth = (-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-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. We can let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth 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.

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

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 pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies ~~y3/2 =~~ (5/4)/2~~. Semisixth (aka sensei) implies y6/2 = (5/3)/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is~~.

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 pseudo-pergens, because the multigen isn't 3-limit, and might not even actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.

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 JI-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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If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a single-split pergen. If it has two fractions, it's a double-split pergen. A single-split pergens 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 single-pair notation because it adds only a single pair of accidentals to conventional notation. Double-pair 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.

Every double-split pergen is either a true double or a false double. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the ~~multi-gen~~ split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's ~~multi-gen~~ is (a,b), the octave is split into at least |b| parts.

Every double-split pergen is either a true double or a false double. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

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.

Line 2,148:

P1 -- ^^m2=v3A1 -- vM2 -- ^m3 -- ^3d4=vvM3 -- P4C -- Db^^ -- Dv -- Eb^ -- Evv -- F

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional ~~multi-gen~~ as before. Then deduce the period from the enharmonic. If the ~~multi-gen~~ was changed by unreducing, find the original generator from the period and the alternate generator.

To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Next, find the original half-fourth generator. P = P8/5 ~ 240¢, and G = P4/2 ~250¢. Because P &lt; G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^4M2 + ^1 = ^5M2. The alternate generator is usually simpler than the original generator, and the alternate multigen is usually more complex than the original multigen.

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<h2 id="toc12"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

Tempering out 250/243 creates third-fourth, and 49/48 creates half-fourth, and tempering out both commas creates sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).

Tempering out 250/243 creates third-fourth, and 49/48 creates half-fourth, and tempering out both commas creates sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).

General rules for combining pergens:

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