Pergen names: Difference between revisions

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The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.

The genchain shown is a short section of the full genchain.

The **genchain** (chain of generators) shown is a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...

If the octave is split, the genchain ~~shows~~ the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

If the octave is split, the genchain is a **perchain** ("peer-chain") that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.

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||~ pergen ||~ enharmonic

interval(s) ||~ equiva-

lence(s) ||~ split interval(s) ||~ genchain(s) ||~ examples ||~ compatible edos

lence(s) ||~ split interval(s) ||~ genchain(s) and

perchains(s) ||~ examples ||~ compatible edos

(12-31 only) ||

||= (P8, P5) ||= none ||= none ||= none ||= C - G ||= meantone ||= 12, 13b, 14*, 15*, 16,

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==Extremely large multi-gens==

So far, the largest multi-gen has been a 12th. ~~But for~~ fractions ~~of 5 or more~~, the multi-gen ~~may be wider~~. To avoid cumbersome degree names like 16th or ~~18th~~, for degrees above 12, the widening is indicated with one "W" per octave. Thus 32/9 = Wm7, 9/2 = WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, ~~the~~ multi-~~gen can be~~ any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.

So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one "W" per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, valid multi-gens include any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.

==Singles and doubles==

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.

~~Each~~ double-split pergen is either a **true double** or a **false double**. ~~True doubles~~, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and ~~require~~ double pair notation. ~~False doubles~~, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. ~~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 |b| parts.

Every double-split pergen is either a **true double** or a **false double**. A true double, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and requirea 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.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. ~~See below for another~~ test.

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.

==Finding an example of a pergen==

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, simply find a ratio R that contains only one higher prime, of appropriate cents to add up to the larger interval. The comma is the difference between the stacked R's and the larger interval. For example, (P8/4, P5) requires an R of about 300¢. R is the period P. This suggests a comma of 4*(6/5) - P8, or P8 - 4*(7/6), but not P8 - 4*(13/11) or P8 - 4*(32/27), too many and too few primes respectively. (P8, P4/3) requires an R of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*(10/9) - P4, and the comma's ratio is (10/9)^3 * (4/3)^-1 = 250/243. R is the generator G.

Finding the comma for a double pergen is ~~harder~~. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into |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 R ~ 50¢, perhaps 33/32, and the comma is 4*(33/32) - ~~M2. If not, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double~~.

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, simply find a ratio R that contains only one higher prime, of appropriate cents to add up to approximately the larger interval. The comma is the difference between the stacked R's and the larger interval. For example, (P8/4, P5) requires an R of about 300¢. R is the period P, and the comma is the difference between 4*R and P8. If R is 6/5, the comma is 4*R - P8 = (6/5)^4 * (2/1)^-1 = 648/625. If R is 7/6, the comma is P8 - 4*R = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work, too many and too few primes respectively. (P8, P4/3) requires an R of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*R - P4 = (10/9)^3 * (4/3)^-1 = 250/243. R is the generator G.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen 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 R ~ 50¢, perhaps 33/32, and the comma is 4*R - M2 = (33/32)^4 * (9/8)^-1.

The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.

If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced form has (2*P8 - 3*P5)/3*2 = m3/6, and the new pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced form has (2*P8 - 3*P5)/(3*2) = m3/6, and the new pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3. The comma splits both the octave and the fifth.

This creates an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/3, P4/3) isn't explicitly false. Its unreduced form has (3*P8 - 3*P4)/3*3 = (3*P5)/9, which reduces to P5/3. The pergen ~~becomes~~ (P8/3, P5/3), which also isn't explicitly false, thus (P8/3, P4/3) is a true double. It requires two commas, ~~and~~ each fraction ~~implies a separate comma~~. ~~(P8/4~~, ~~P4/3) can be created by tempering out both~~ 648/625 and ~~250~~/~~243. This approach also works for false doubles~~.

This creates an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which reduces to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48.

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) can be constructed from 128/125 and 49/48, which split the octave and the 4th respectively.

==Finding a notation for a pergen==

~~Proceed~~ as in the ~~previous section~~, and ~~convert~~ (P8/m, (a,b)/n) ~~into~~ P8 = m*P and M = n*G

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. There is an easy method for finding such a pergen, if one exists. First, some basic concepts:

The **keyspan** of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The **stepspan** of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P4 and A4 both have stepspan 3. The stepspan can be thought of as the 7-edo keyspan. Again, this concept can be expanded to include pentatonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a **gedra**, analogous to a monzo, but written in brackets not parentheses: 3/2 is a 7-semitone 5th, thus 3/2 = (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval (a,b), there is a unique gedra [k,s], and vice versa:

k = 12a + 19b

s = 7a + 11b

The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

a = -11k + 19b

b = 7a - 12b

Gedras can be manipulated exactly like monzos. Just as (a,b) + (a',b') = (a+a',b+b'), likewise [k,s] + [k',s'] = [k+k',s+s']. If (a,b) is a stack of n intervals, with (a,b) = n(a',b') = (na', nb'), then [k,s] = n[k',s'] = [nk',ns'].

A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will __always__ produce an enharmonic interval with the smallest possible keyspan and stepspan, which is usually the preferred enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the number of periods, thus E = v5m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. Equipped with the period and the enharmonic, the perchain is easily found:

(P8/5, P5) perchain: P1 - ^^M2=v3m3 - v4 - ^5 - ^3M6=vvm7 - P8, or C - D^^=Ebv3 - Fv - G^ - A^3=Bbvv - C

All single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [round(5/5), round(3/5)] = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^5d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:

(P8, P4/5) genchain: P1 - ^^m2=v3A1 - vM2 - ^m3 - ^3d4=vvM3 - P4, or C - Db^^=C#v3 - Dv - Eb^ - Fb^3=Evv - F

(P8, P5/2) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v4dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because it makes it easier to find the alternate generator G' = G + E. Otherwise G' would be G -E, and it's harder to subtract than add. For example (P8, P5/2) has G = ^m3 and E = vvA1

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.

For example, (P8/5, P4/2) unreduces to (P8/5, m2/10). The generator 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. This is much too small to be half a 4th, we'll address that later. 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')

P4 = 2G' + E , perhaps P + G. But that would be a triple-upped interval, and we want to minimize the ups, so instead we use P - G = ^^M2 - ^1 = ^M2.

For example, (P8/5, P4/2) is a false double because GCD(5,2) = 1 = |-1|. The generator of (P8, P4/2) is [5,3]/2 = [2,1] = M2. Both the keyspan and the stepspan could have been rounded up, not down, creating [3,1] = A2, [3,2] = m3, [2,2] = d3. 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. This is much too small to be half a 4th, we'll address that later. 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')

(P8/5, P4/2): P = ^4M2, G = ^5M2, E = v10m2

perchain: P1 - ^4M2=v6m3 - vvP4 - ^^P5 - ^6M6=v4m7 - P8, or C - D^4=Ebv6 - Fvv - G^^ - A^6=Bbv4 - C

genchain: P1 - ^5M2=v5m3 - P4, or C - D^5=Ebv5 - F

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The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.

The genchain shown is a short section of the full genchain.

The genchain (chain of generators) shown is a short section of the full genchain.

C - G implies ...Eb Bb F C G D A E B F# C#...

C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E...

If the octave is split, the genchain ~~shows~~ the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

If the octave is split, the genchain is a perchain (&quot;peer-chain&quot;) that show the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with (P8, P5), because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.

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<th>split interval(s)

</th>

<th>genchain(s)

<th>genchain(s) and

perchains(s)

</th>

<th>examples

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<h2 id="toc4"><a name="Further Discussion-Extremely large multi-gens"></a>Extremely large multi-gens</h2>

So far, the largest multi-gen has been a 12th. ~~But for~~ fractions ~~of 5 or more~~, the multi-gen ~~may be wider~~. To avoid cumbersome degree names like 16th or ~~18th~~, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, ~~the~~ multi-~~gen can be~~ any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.

So far, the largest multi-gen has been a 12th. As the multi-gen fractions get larger, the multi-gen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave and a multi-gen split into n parts, valid multi-gens include any voicing of the fifth that is less than n/2 octaves. For (P8, multi-gen/6), the multi-gen can be P4, P5, P11, P12, WWP4 or WWP5.

<h2 id="toc5"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

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.

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.

~~Each~~ double-split pergen is either a true double or a false double. ~~True doubles~~, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and ~~require~~ double pair notation. ~~False doubles~~, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. ~~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 |b| parts.

Every double-split pergen is either a true double or a false double. A true double, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and requirea 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.

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. ~~See below for another~~ test.

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.

<h2 id="toc6"><a name="Further Discussion-Finding an example of a pergen"></a>Finding an example of a pergen</h2>

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, simply find a ratio R that contains only one higher prime, of appropriate cents to add up to the larger interval. The comma is the difference between the stacked R's and the larger interval. For example, (P8/4, P5) requires an R of about 300¢. R is the period P. ~~This suggests a~~ comma of 4*(6/5) - P8, or P8 - 4*(7/6)~~, but not P8~~ - 4*(13/11~~) or P8 - 4*(~~32/27), too many and too few primes respectively. (P8, P4/3) requires an R of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*~~(10/9)~~ - P4~~, and the comma's ratio is~~ (10/9)^3 * (4/3)^-1 = 250/243. R is the generator G.

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, simply find a ratio R that contains only one higher prime, of appropriate cents to add up to approximately the larger interval. The comma is the difference between the stacked R's and the larger interval. For example, (P8/4, P5) requires an R of about 300¢. R is the period P, and the comma is the difference between 4*R and P8. If R is 6/5, the comma is 4*R - P8 = (6/5)^4 * (2/1)^-1 = 648/625. If R is 7/6, the comma is P8 - 4*R = (2/1) * (7/6)^-4. Neither 13/11 nor 32/27 would work, too many and too few primes respectively. (P8, P4/3) requires an R of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3*R - P4 = (10/9)^3 * (4/3)^-1 = 250/243. R is the generator G.

Finding the comma for a double pergen is ~~harder~~. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into |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 R ~ 50¢, perhaps 33/32, and the comma is 4*(33/32) - ~~M2. If not, put the pergen in its unreduced form, which is always explicitly false if the pergen is a false double~~.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen 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 R ~ 50¢, perhaps 33/32, and the comma is 4*R - M2 = (33/32)^4 * (9/8)^-1.

The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.

If the pergen is not explicitly false, put the pergen in its unreduced form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced form has (2*P8 - 3*P5)/3*2 = m3/6, and the new pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced form has (2*P8 - 3*P5)/(3*2) = m3/6, and the new pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3. The comma splits both the octave and the fifth.

This creates an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/3, P4/3) isn't explicitly false. Its unreduced form has (3*P8 - 3*P4)/3*3 = (3*P5)/9, which reduces to P5/3. The pergen ~~becomes~~ (P8/3, P5/3), which also isn't explicitly false, thus (P8/3, P4/3) is a true double. It requires two commas, ~~and~~ each fraction ~~implies a separate comma~~. ~~(P8/4~~, ~~P4/3) can be created by tempering out both~~ 648/625 and ~~250~~/~~243. This approach also works for false doubles~~.

This creates an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which reduces to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48.

A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) can be constructed from 128/125 and 49/48, which split the octave and the 4th respectively.

<h2 id="toc7"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

~~Proceed~~ as in the ~~previous section~~, and ~~convert~~ (P8/m, (a,b)/n) ~~into~~ P8 = m*P and M = n*G

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. There is an easy method for finding such a pergen, if one exists. First, some basic concepts:

The keyspan of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The stepspan of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P4 and A4 both have stepspan 3. The stepspan can be thought of as the 7-edo keyspan. Again, this concept can be expanded to include pentatonicism, etc., but we'll assume heptatonicism for now.

Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a gedra, analogous to a monzo, but written in brackets not parentheses: 3/2 is a 7-semitone 5th, thus 3/2 = (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval (a,b), there is a unique gedra [k,s], and vice versa:

k = 12a + 19b

s = 7a + 11b

The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:

a = -11k + 19b

b = 7a - 12b

Gedras can be manipulated exactly like monzos. Just as (a,b) + (a',b') = (a+a',b+b'), likewise [k,s] + [k',s'] = [k+k',s+s']. If (a,b) is a stack of n intervals, with (a,b) = n(a',b') = (na', nb'), then [k,s] = n[k',s'] = [nk',ns'].

A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. One fifth of an octave [12,7] is approximately [round(12/5), round(7/5)] = [2,1] = (-3,2) = M2. This approximation will always produce an enharmonic interval with the smallest possible keyspan and stepspan, which is usually the preferred enharmonic. In this example, the bare enharmonic is the difference between P8 and 5*M2, which is [12,7] - 5*[2,1] = [12,7] - [10,5] = [2,2] = 2*[1,1] = 2*m2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the number of periods, thus E = v5m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. Equipped with the period and the enharmonic, the perchain is easily found: