Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with ~~either~~ microtonal accidentals (P8, P5, ^1, /1~~,...) or possibly colors (P8, P5, g1, r1~~,...).

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

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It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. It ~~results~~ from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \3A1 and G' = \M2 = ``//``m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. It might result from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \3A1 and G' = \M2 = ``//``m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.

P1 — ^/1=v/m2 — ``//``m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4

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The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1).

All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:

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But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to ~~think in~~ a notation that isn't backwards compatible, but ~~communicate in~~ one that is.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the ~~multigen~~.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). Likewise, 3.5 pergen #4 is (P12, m7/2) = (P12, P4), equivalent to (P12, P8). Such pergens are marked with an asterisk.

||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||

||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||

||= 1 ||= (P8, P5) ||= (P8, M3) ||= (P8, M2) ||= (P12, M6) ||= (P12, M3) ||= (WWM3, d5) ||

||~ half-splits ||~ ||~ ||~ ||~ ||~ ||~ ||

||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (~~WWM3/2~~, d5) ||

||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (M9, d5)* ||

||= 3 ||= (P8, P4/2) ||= (P8, ~~M3/2~~) ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, ~~M3/2~~) ||= (WWM3, ~~d5/2~~) ||

||= 3 ||= (P8, P4/2) ||= (P8, M2)* ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M2)* ||= (WWM3, m3)* ||

||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, ~~m7/2~~) ||= (P12, ~~m7/2~~) ||= (P12, m10/2) ||= (WWM3, ~~WA6/2~~) ||

||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, P5)* ||= (P12, P8)* ||= (P12, m10/2) ||= (WWM3, M7)* ||

||= 5 ||= (P8/2, P4/2) ||= (P8/2, ~~M3/2~~) ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (~~WWM3/2~~, ~~d5/2~~) ||

||= 5 ||= (P8/2, P4/2) ||= (P8/2, M2)* ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (M9, m3)* ||

||~ third-splits ||~ ||~ ||~ ||~ ||~ ||~ ||

||= 6 ||= (P8/3, P5) ||= (P8/3, M3) ||= (P8/3, M2) ||= (P12/3, M6) ||= (P12/3, M3) ||= (WWM3/3, d5) ||

||= 7 ||= (P8, P4/3) ||= (P8, M3/3) ||= (P8, M2/3) ||= (P12, M6/3) ||= (P12, M3/3) ||= (WWM3, d5/3) ||

||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, ~~m10/3~~) ||= (WWM3, WA6/3) ||

||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, P4)* ||= (WWM3, WA6/3) ||

||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) ||

||= 10 ||= (P8/3, P4/2) ||= (P8/3, ~~M3/2~~) ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||

||= 10 ||= (P8/3, P4/2) ||= (P8/3, M2)* ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||

||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, ~~m7/2~~) ||= (P12/3, ~~m7/2~~) ||= ||= ||

||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, P5)* ||= (P12/3, P4)* ||= ||= ||

||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||= ||= ||

||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||= ||= ||

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For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table ~~can~~ be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

==Pergen squares==

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, ~~but~~ assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers:

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2

| |

| . . . . |

C1 -- G1

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C1 --- G1

A pergen square ~~contains~~ all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals.

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a Wm7 (e.g. D1 to C3), a WM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3

F#v1 C#v2 G#v2 D#v3

C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3.

Splitting the 5th adds notes to the horizontal edges of the square:

C2 Ev2 G2

| . . . . . . |

C1 Ev1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 Ev3 G3

||

| . . . . . . |

C2 Ev2 G2

||

| . . . . . . |

C1 Ev1 G1

From G1 to C2 is a 4th, so splitting the 4th adds ~~A^1~~ halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2.

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2

| . A^1 . |

C1 ---- G1

A^1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with ~~either~~ microtonal accidentals (P8, P5, ^1, /1~~,...) or possibly colors (P8, P5, g1, r1~~,...).

In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).

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It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the &quot;half-step glitch&quot;. This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. It ~~results~~ from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \3A1 and G' = \M2 = //m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. It might result from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \3A1 and G' = \M2 = //m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.

P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4

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The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.

All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1).

All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens.

It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:

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But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to ~~think in~~ a notation that isn't backwards compatible, but ~~communicate in~~ one that is.

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a huge number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the ~~multigen~~.

Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). Likewise, 3.5 pergen #4 is (P12, m7/2) = (P12, P4), equivalent to (P12, P8). Such pergens are marked with an asterisk.

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<td style="text-align: center;">(P12/2, M3)

</td>

<td style="text-align: center;">(~~WWM3/2~~, d5)

<td style="text-align: center;">(M9, d5)*

</td>

</tr>

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<td style="text-align: center;">(P8, P4/2)

</td>

<td style="text-align: center;">(P8, ~~M3/2~~)

<td style="text-align: center;">(P8, M2)*

</td>

<td style="text-align: center;">(P8, M2/2)

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<td style="text-align: center;">(P12, M6/2)

</td>

<td style="text-align: center;">(P12, ~~M3/2~~)

<td style="text-align: center;">(P12, M2)*

</td>

<td style="text-align: center;">(WWM3, ~~d5/2~~)

<td style="text-align: center;">(WWM3, m3)*

</td>

</tr>

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<td style="text-align: center;">(P8, m6/2)

</td>

<td style="text-align: center;">(P8, ~~m7/2~~)

<td style="text-align: center;">(P8, P5)*

</td>

<td style="text-align: center;">(P12, ~~m7/2~~)

<td style="text-align: center;">(P12, P8)*

</td>

<td style="text-align: center;">(P12, m10/2)

</td>

<td style="text-align: center;">(WWM3, ~~WA6/2~~)

<td style="text-align: center;">(WWM3, M7)*

</td>

</tr>

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<td style="text-align: center;">(P8/2, P4/2)

</td>

<td style="text-align: center;">(P8/2, ~~M3/2~~)

<td style="text-align: center;">(P8/2, M2)*

</td>

<td style="text-align: center;">(P8/2, M2/2)

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Line 4,271:

<td style="text-align: center;">(P12/2, M3/2)

</td>

<td style="text-align: center;">(~~WWM3/2~~, ~~d5/2~~)

<td style="text-align: center;">(M9, m3)*

</td>

</tr>

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<td style="text-align: center;">(P12, m7/3)

</td>

<td style="text-align: center;">(P12, ~~m10/3~~)

<td style="text-align: center;">(P12, P4)*

</td>

<td style="text-align: center;">(WWM3, WA6/3)

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<td style="text-align: center;">(P8/3, P4/2)

</td>

<td style="text-align: center;">(P8/3, ~~M3/2~~)

<td style="text-align: center;">(P8/3, M2)*

</td>

<td style="text-align: center;">(P8/3, M2/2)

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<td style="text-align: center;">(P8/3. m6/2)

</td>

<td style="text-align: center;">(P8/3, ~~m7/2~~)

<td style="text-align: center;">(P8/3, P5)*

</td>

<td style="text-align: center;">(P12/3, ~~m7/2~~)

<td style="text-align: center;">(P12/3, P4)*

</td>

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For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table ~~can~~ be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).

Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)3 * (64/63)2] = about 60¢.

<h2 id="toc19"><a name="Further Discussion-Pergen squares"></a>Pergen squares</h2>

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, ~~but~~ assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers:

For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).

C2 -- G2

| |

| . . . . |

C1 -- G1

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C1 --- G1

A pergen square ~~contains~~ all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every &quot;downed&quot; note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals.

A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every &quot;downed&quot; note bisects an 8ve, a M2, a Wm7 (e.g. D1 to C3), a WM9 (e.g. C1 to D3), and many other intervals.

C2 --- G2 --- D3 --- A3

F#v1 C#v2 G#v2 D#v3

C1 --- G1 --- D2 --- A2

Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3.

Splitting the 5th adds notes to the horizontal edges of the square:

C2 Ev2 G2

| . . . . . . |

C1 Ev1 G1

Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.

C3 Ev3 G3

||

| . . . . . . |

C2 Ev2 G2

||

| . . . . . . |

C1 Ev1 G1

From G1 to C2 is a 4th, so splitting the 4th adds ~~A^1~~ halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2.

From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.

C2 ---- G2

| . A^1 . |

C1 ---- G1

A^1 also bisects the P12 from C1 to G2.

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.

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

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

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

Screenshots of the first 2 pages:

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Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.

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

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

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-29 02:47:06 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-29 05:39:08 UTC</tt>.<br>
-: The original revision id was <tt>628120879</tt>.<br>
+: The original revision id was <tt>628122853</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or possibly colors (P8, P5, g1, r1,...).
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).
@@ Line 485: / Line 485: @@
 It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.
-Sixth-4th with single-pair notation has an awkward ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;4 enharmonic. It results from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 and G' = \M2 = ``//``m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.
+Sixth-4th with single-pair notation has an awkward ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;4 enharmonic. It might result from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 and G' = \M2 = ``//``m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.
 &lt;span style="display: block; text-align: center;"&gt;P1 — ^/1=v/m2 — ``//``m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4
@@ Line 657: / Line 657: @@
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.
-All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1).
+All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens.
 It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:
@@ Line 673: / Line 673: @@
 But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). Likewise, 3.5 pergen #4 is (P12, m7/2) = (P12, P4), equivalent to (P12, P8). Such pergens are marked with an asterisk.
 ||~ __pergen number__ ||||||||||||~ __prime subgroup__ ||
 ||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) ||
 ||= 1 ||= (P8, P5) ||= (P8, M3) ||= (P8, M2) ||= (P12, M6) ||= (P12, M3) ||= (WWM3, d5) ||
 ||~ half-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
-||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (WWM3/2, d5) ||
+||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (M9, d5)* ||
-||= 3 ||= (P8, P4/2) ||= (P8, M3/2) ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M3/2) ||= (WWM3, d5/2) ||
+||= 3 ||= (P8, P4/2) ||= (P8, M2)* ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M2)* ||= (WWM3, m3)* ||
-||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, m7/2) ||= (P12, m7/2) ||= (P12, m10/2) ||= (WWM3, WA6/2) ||
+||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, P5)* ||= (P12, P8)* ||= (P12, m10/2) ||= (WWM3, M7)* ||
-||= 5 ||= (P8/2, P4/2) ||= (P8/2, M3/2) ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (WWM3/2, d5/2) ||
+||= 5 ||= (P8/2, P4/2) ||= (P8/2, M2)* ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (M9, m3)* ||
 ||~ third-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
 ||= 6 ||= (P8/3, P5) ||= (P8/3, M3) ||= (P8/3, M2) ||= (P12/3, M6) ||= (P12/3, M3) ||= (WWM3/3, d5) ||
 ||= 7 ||= (P8, P4/3) ||= (P8, M3/3) ||= (P8, M2/3) ||= (P12, M6/3) ||= (P12, M3/3) ||= (WWM3, d5/3) ||
-||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, m10/3) ||= (WWM3, WA6/3) ||
+||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, P4)* ||= (WWM3, WA6/3) ||
 ||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) ||
-||= 10 ||= (P8/3, P4/2) ||= (P8/3, M3/2) ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||
+||= 10 ||= (P8/3, P4/2) ||= (P8/3, M2)* ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= etc. ||= etc. ||
-||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, m7/2) ||= (P12/3, m7/2) ||=   ||=   ||
+||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, P5)* ||= (P12/3, P4)* ||=   ||=   ||
 ||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||=   ||=   ||
 ||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||=   ||=   ||
@@ Line 697: / Line 697: @@
 For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
-Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).
+Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.
+Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; * (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;] = about 60¢.
 ==Pergen squares==
-Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.
+Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.
-For (P8, P5), the pergen square has 4 notes, shown here with octave numbers:
+For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).
 C2 -- G2
-| |
+| . . . . |
 C1 -- G1
@@ Line 713: / Line 715: @@
 C1 --- G1
-A pergen square contains all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals.
+A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every "downed" note bisects an 8ve, a M2, a Wm7 (e.g. D1 to C3), a WM9 (e.g. C1 to D3), and many other intervals.
 C2 --- G2 --- D3 --- A3
 F#v1 C#v2 G#v2 D#v3
 C1 --- G1 --- D2 --- A2
-Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3.
+Splitting the 5th adds notes to the horizontal edges of the square:
+C2 Ev2 G2
+| . . . . . . |
+C1 Ev1 G1
+Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.
 C3 Ev3 G3
-||
+| . . . . . . |
 C2 Ev2 G2
-||
+| . . . . . . |
 C1 Ev1 G1
-From G1 to C2 is a 4th, so splitting the 4th adds A^1 halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2.
+From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.
+C2 ---- G2
+| . A^1 . |
+C1 ---- G1
+A^1 also bisects the P12 from C1 to G2.
 Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.
@@ Line 1,431: / Line 1,442: @@
 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.&lt;br /&gt;
 &lt;br /&gt;
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or possibly colors (P8, P5, g1, r1,...).&lt;br /&gt;
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, i.e. the color depth must be 1. Furthermore, the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 2,957: / Line 2,968: @@
 It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the &amp;quot;half-step glitch&amp;quot;. This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.&lt;br /&gt;
 &lt;br /&gt;
-Sixth-4th with single-pair notation has an awkward ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;4 enharmonic. It results from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 and G' = \M2 = &lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.&lt;br /&gt;
+Sixth-4th with single-pair notation has an awkward ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;4 enharmonic. It might result from combining half-4th and third-4th (e.g. tempering out both the semaphore and porcupine commas), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 and G' = \M2 = &lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.&lt;br /&gt;
 &lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 — ^/1=v/m2 — &lt;!-- ws:start:WikiTextRawRule:054:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:054 --&gt;m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4&lt;br /&gt;
@@ Line 4,061: / Line 4,072: @@
 The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and Blackwood's ^1 is 81/80 or equivalently, 16/15.&lt;br /&gt;
 &lt;br /&gt;
-All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1).&lt;br /&gt;
+All Blackwood-like pergens are of the form (P8/m, ^1) or (P8/m, /1), and they can be identified solely by their splitting fraction. Blackwood-like pergens are a small minority of rank-2 pergens.&lt;br /&gt;
 &lt;br /&gt;
 It's possible to have a fifth-8ve pergen with an independent 5th, but there will be small intervals of about 20¢. Here are two such:&lt;br /&gt;
@@ Line 4,139: / Line 4,150: @@
 But in non-8ve and no-5ths pergens, not every name has a note. For example, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... and the perchain runs C - F#v - C. There is no G or D or A note, in fact 75% of all possible note names have no actual note. 75% of all intervals don't exist. There is no perfect 5th or major 2nd. There are missing notes and missing intervals.&lt;br /&gt;
 &lt;br /&gt;
-Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of missing notes and intervals. The composer may want to think in a notation that isn't backwards compatible, but communicate in one that is.&lt;br /&gt;
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.&lt;br /&gt;
 &lt;br /&gt;
-Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the multigen.&lt;br /&gt;
+Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). Likewise, 3.5 pergen #4 is (P12, m7/2) = (P12, P4), equivalent to (P12, P8). Such pergens are marked with an asterisk.&lt;br /&gt;
@@ Line 4,212: / Line 4,223: @@
          &lt;td style="text-align: center;"&gt;(P12/2, M3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(WWM3/2, d5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(M9, d5)*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,220: / Line 4,231: @@
          &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, M3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, M2)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, M2/2)&lt;br /&gt;
@@ Line 4,226: / Line 4,237: @@
          &lt;td style="text-align: center;"&gt;(P12, M6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12, M3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, M2)*&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(WWM3, d5/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, m3)*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,238: / Line 4,249: @@
          &lt;td style="text-align: center;"&gt;(P8, m6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, m7/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5)*&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12, m7/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, P8)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P12, m10/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(WWM3, WA6/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(WWM3, M7)*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,252: / Line 4,263: @@
          &lt;td style="text-align: center;"&gt;(P8/2, P4/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, M3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, M2)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, M2/2)&lt;br /&gt;
@@ Line 4,260: / Line 4,271: @@
          &lt;td style="text-align: center;"&gt;(P12/2, M3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(WWM3/2, d5/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(M9, m3)*&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 4,322: / Line 4,333: @@
          &lt;td style="text-align: center;"&gt;(P12, m7/3)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12, m10/3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12, P4)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(WWM3, WA6/3)&lt;br /&gt;
@@ Line 4,348: / Line 4,359: @@
          &lt;td style="text-align: center;"&gt;(P8/3, P4/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/3, M3/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, M2)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/3, M2/2)&lt;br /&gt;
@@ Line 4,366: / Line 4,377: @@
          &lt;td style="text-align: center;"&gt;(P8/3. m6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/3, m7/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/3, P5)*&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P12/3, m7/2)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P12/3, P4)*&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 4,443: / Line 4,454: @@
 For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.&lt;br /&gt;
 &lt;br /&gt;
-Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).&lt;br /&gt;
+Non-8ve pergens require octave numbers (unless on the staff of course). For example, consider the 1st 3.5 pergen, (P12, M6). The note one M6 above C1 is A1, and the note three P12's above C1 is A5. (There is no A2, A3 or A4.) Referring to a note as simply A is ambiguous.&lt;br /&gt;
+&lt;br /&gt;
+Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table could be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2), with ^M2 = 8/7 and ^1 = √256/245 = √(81/80 * 64/63 * 64/63) = about 38¢. The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3), with ^M3 = 9/7 and ^1 = √6561/6125 = √[(81/80)&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; * (64/63)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;] = about 60¢.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Pergen squares"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Pergen squares&lt;/h2&gt;
   &lt;br /&gt;
-Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.&lt;br /&gt;
+Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all. To understand them, let's assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.&lt;br /&gt;
 &lt;br /&gt;
-For (P8, P5), the pergen square has 4 notes, shown here with octave numbers:&lt;br /&gt;
+For (P8, P5), the pergen square has 4 notes, shown here with octave numbers (ignore the periods).&lt;br /&gt;
 C2 -- G2&lt;br /&gt;
-| |&lt;br /&gt;
+| . . . . |&lt;br /&gt;
 C1 -- G1&lt;br /&gt;
 &lt;br /&gt;
@@ Line 4,459: / Line 4,472: @@
 C1 --- G1&lt;br /&gt;
 &lt;br /&gt;
-A pergen square contains all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every &amp;quot;downed&amp;quot; note bisects an 8ve, a M2, a Wm7, a WM9, and many other intervals.&lt;br /&gt;
+A pergen square shows all alternate gens and multigens. The unreduced form of (P8/2, P5) is (P8/2, M2/2). The M2 becomes visible only after tiling the pergen square. From C2 to D2 is a M2, and C#v2 bisects it. G#v2 bisects the G2-A2 M2. Every &amp;quot;downed&amp;quot; note bisects an 8ve, a M2, a Wm7 (e.g. D1 to C3), a WM9 (e.g. C1 to D3), and many other intervals.&lt;br /&gt;
 C2 --- G2 --- D3 --- A3&lt;br /&gt;
 F#v1 C#v2 G#v2 D#v3&lt;br /&gt;
 C1 --- G1 --- D2 --- A2&lt;br /&gt;
 &lt;br /&gt;
-Splitting the 5th adds notes to the horizontal edges of the square. Each note also bisects a P11, e.g. G1-C3.&lt;br /&gt;
+Splitting the 5th adds notes to the horizontal edges of the square: &lt;br /&gt;
+C2 Ev2 G2&lt;br /&gt;
+| . . . . . . |&lt;br /&gt;
+C1 Ev1 G1&lt;br /&gt;
+&lt;br /&gt;
+Each downed note also bisects a P11 (e.g. G1-C3), as well as many other intervals.&lt;br /&gt;
 C3 Ev3 G3&lt;br /&gt;
-||&lt;br /&gt;
+| . . . . . . |&lt;br /&gt;
 C2 Ev2 G2&lt;br /&gt;
-||&lt;br /&gt;
+| . . . . . . |&lt;br /&gt;
 C1 Ev1 G1&lt;br /&gt;
 &lt;br /&gt;
-From G1 to C2 is a 4th, so splitting the 4th adds A^1 halfway between them, in the center of the square. A^1 also bisects the P12 from C1 to G2.&lt;br /&gt;
+From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square. &lt;br /&gt;
+C2 ---- G2&lt;br /&gt;
+| . A^1 . |&lt;br /&gt;
+C1 ---- G1&lt;br /&gt;
+A^1 also bisects the P12 from C1 to G2.&lt;br /&gt;
 &lt;br /&gt;
 Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;
@@ Line 6,558: / Line 6,580: @@
   &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;
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+&lt;!-- ws:start:WikiTextUrlRule:8179:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8179 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshots of the first 2 pages:&lt;br /&gt;
@@ Line 6,568: / Line 6,590: @@
   &lt;br /&gt;
 Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:8165:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8165 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8180:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8180 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;