Kite's thoughts on pergens: Difference between revisions

Line 995:

==Naming very large intervals==

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, '''widening''' by an 8ve is indicated by "W". Thus 10/3 = ~~WM6~~ = wide major 6th, 9/2 = ~~WWM2~~ or ~~WM9~~, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ~~WWP4~~ or ~~WWP5~~.

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, '''widening''' by an 8ve is indicated by "c". Thus 10/3 = cM6 = wide major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5.

==Secondary splits==

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

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:

P4/3: C - Dv - Eb^ - F

Line 1,230:

==Alternate enharmonics==

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ~~WWM6~~/12), a false double. The bare alternate generator is ~~WWM6~~/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The bare alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.

P1 -- ^4m3 -- v4M6 -- C

Line 1,266:

In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara (2.3.5.7 with 50/49 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ~~WWm7~~, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).

Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara (2.3.5.7 with 50/49 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).

A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.

Line 1,904:

! | 3.5 (M6 = 5/3)

! | 3.7 (M3 = 9/7)

! | 5.7 (~~WWM3~~ = 5/1, d5 = 7/5)

! | 5.7 (ccM3 = 5/1, d5 = 7/5)

|-

| | 1

Line 1,912:

| | (P12, M6)

| | (P12, M3)

| | (~~WWM3~~, d5)

| | (ccM3, d5)

|-

! | half-splits

Line 1,936:

| | (P12, M6/2)

| | (P12, M2)*

| | (~~WWM3~~, m3)*

| | (ccM3, m3)*

|-

| | 4

Line 1,944:

| | (P12, P4)*

| | (P12, m10/2)

| | (~~WWM3~~, M7)*

| | (ccM3, M7)*

|-

| | 5

Line 1,968:

| | (P12/3, M6)

| | (P12/3, M3)

| | (~~WWM3~~/3, d5)

| | (ccM3/3, d5)

|-

| | 7

Line 1,976:

| | (P12, M6/3)

| | (P12, M3/3)

| | (~~WWM3~~, d5/3)

| | (ccM3, d5/3)

|-

| | 8

Line 1,984:

| | (P12, m7/3)

| | (P12, P4)*

| | (~~WWM3~~, ~~WA6~~/3)

| | (ccM3, cA6/3)

|-

| | 9

Line 1,990:

| | (P8, M10/3)

| | (P8, M9/3)

| | (P12, ~~WWM3~~/3)

| | (P12, ccM3/3)

| | (P12, ~~WM7~~/3)

| | (P12, cM7/3)

| | (~~WWM3~~, ~~WWm7~~/3)

| | (ccM3, ccm7/3)

|-

| | 10

Line 2,000:

| | (P12/3, M6/2)

| | (P12/3, M2)*

| | (~~WWM3~~/3, m3)*

| | (ccM3/3, m3)*

|-

| | 11

Line 2,008:

| | (P12/3, P4)*

| | (P12/3, m10/2)

| | (~~WWM3~~/3, M7)*

| | (ccM3/3, M7)*

|-

| | 12

Line 2,024:

| | (P12/2, m7/3)

| | (P12/2, P4)*

| | (M9, ~~WA6~~/3)*

| | (M9, cA6/3)*

|-

| | 14

Line 2,030:

| | (P8/2, M10/3)

| | (P8/2, M9/3)

| | (P12/2, ~~WWM3~~/3)

| | (P12/2, ccM3/3)

| | (P12/2, ~~WM7~~/3)

| | (P12/2, cM7/3)

| | (M9, ~~WWm7~~/3)*

| | (M9, ccm7/3)*

|-

| | 15

Line 2,040:

| | (P12/3, M6/3)

| | (P12/3, P4)*

| | (~~WWM3~~/3, d5/3)

| | (ccM3/3, d5/3)

|}

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

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (~~WWM3~~/5, d5) can optionally be replaced too.

Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.

{| class="wikitable" style="text-align:center;"

Line 2,062:

| | (P12/5, M6)

| | (P12/5, M3)

| | (~~WWM3~~/5, ^1)

| | (ccM3/5, ^1)

|}

Line 2,085:

C1 --- G1

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.

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 cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.

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

Line 2,243:

|-

| | 411-422¢

| | ~~WWP4~~/7

| | ccP4/7

| |

Line 2,257:

|-

| | 435-446¢

| | ~~WWP5~~/7

| | ccP5/7

| |

Line 2,273:

| | P4 = P5

| | 480-492¢

| | ~~WWP4~~/6

| | ccP4/6

| | 508-520¢

| | ~~WWP5~~/6

| | ccP5/6

|-

| | 560-585¢

Line 2,285:

|-

| | 576-591¢

| | ~~WWP4~~/5

| | ccP4/5

| | 583-593¢

| | ~~WWWP4~~/7

| | cccP4/7

| |

|}

There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ~~WWP4~~.

There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.

The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.

Line 2,733:

| | (P8, P11/4) [8]

| | quarter-11th octotonic

| | seventh-~~WW4th~~, seventh-~~WW5th~~

| | seventh-cc4th, seventh-cc5th

|-

| | 4L 4s

Line 2,766:

|-

| | 2L 7s

| | (P8, W3P5/8) [9]

| | (P8, c3P5/8) [9]

| | eighth-W35th nonatonic

| | eighth-c35th nonatonic

| | third-11th, fifth-~~WW4th~~

| | third-11th, fifth-cc4th

|-

| | 3L 6s

Line 2,791:

|-

| | 7L 2s

| | (P8, ~~WWP5~~/6)[9]

| | (P8, ccP5/6)[9]

| | sixth-~~WW5th~~ nonatonic

| | sixth-cc5th nonatonic

| | (lopsided unless 5th is sharp)

|-

Line 2,818:

| | (P8, P12/5) [10]

| | fifth-12th decatonic

| | eighth-~~WW4th~~, eighth-~~WW5th~~

| | eighth-cc4th, eighth-cc5th

|-

| | 4L 6s

Line 2,838:

| | (P8, P5/2) [10]

| | half-5th decatonic

| | ninth-~~WW5th~~

| | ninth-cc5th

|-

| | 8L 2s

Line 2,851:

|}

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, ~~WWP5~~/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, ccP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.

Line 2,861:

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ~~WWP5~~/31),... (P8, (i-1,1)/n), where n = 12i+7.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.

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

Line 3,476:

| | P12/6

| | P12/5

| | ~~WWP5~~/7

| | ccP5/7

| | P5

| | P11/3

Line 3,492:

| | P11/4

| | -

| | W3P5/8

| | c3P5/8

| |

Line 3,547:

| | -

| | W3P4/9

| | c3P4/9

| | -

| | P11/3

Line 3,616:

| | P4/2

| | P12/6

| | ~~WWP4~~/8

| | ccP4/8

| | ~~WWP4~~/7

| | ccP4/7

| | P12/4

| | P5

Line 3,634:

| | -

| | W4P5/10

| | c4P5/10

|-

! | "

Line 3,720:

! | 11

|}

Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, W5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c5P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).

'''EDO-pair names'''

Line 3,783:

! | 15-edo

| | (P8/3, P5)

| | (P8, W5P5/12)

| | (P8, c5P5/12)

| | (P8, P4/6)

| |

Line 3,805:

! | 17-edo

| | (P8, P5)

| | (P8, ~~WWP5~~/11)

| | (P8, ccP5/11)

| | (P8, P11/4)

| | (P8, P11/3)

Line 3,838:

! | 20-edo

| | (P8/4, P5)

| | (P8, ~~WWP4~~/16)

| | (P8, ccP4/16)

| | (P8/2, P5/4)

| | (P8/5, ^1)

Line 3,849:

! | 21-edo

| | (P8/3, P5)

| | (P8, W3P4/9)

| | (P8, c3P4/9)

| | (P8/7, ^1)

| | (P8/3, P4/3)

Line 3,860:

! | 22-edo

| | (P8/2, P5)

| | (P8, W3P4/15)

| | (P8, c3P4/15)

| | (P8/2, P4/3)

| | (P8, P4/3)

Line 3,871:

! | 23-edo

| | (P8, P4/5)

| | (P8, ~~WWP4~~/8)

| | (P8, ccP4/8)

| | (P8, P4/2)

| | (P8, P12/12)

Line 3,878:

| | (P8, P12/4)

| | (P8, P12/6)

| | (P8, W5P5/16)

| | (P8, c5P5/16)

|-

! | 24-edo

| | (P8/12, ^1)

| | (P8, W6P4/14)

| | (P8, c6P4/14)

| | (P8/2, P4/2)

| | (P8/3, P4/2)

@@ Line 995: / Line 995: @@
 ==Naming very large intervals==
-So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, '''widening''' by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.
+So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, '''widening''' by an 8ve is indicated by "c". Thus 10/3 = cM6 = wide major 6th, 9/2 = ccM2 or cM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen M is less than 3 octaves, and can be P4, P5, P11, P12, ccP4 or ccP5.
 ==Secondary splits==
-Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:
+Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve ccP8. Stacking 4ths gives these intervals: P4, m7, m10, cm6, cm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives ccP8, ccP5, cM9, cM6, M10, M7, A4... The first four are too large, this leaves us with:
 P4/3: C - Dv - Eb^ - F
@@ Line 1,230: / Line 1,230: @@
 ==Alternate enharmonics==
-Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.
+Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, ccM6/12), a false double. The bare alternate generator is ccM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.
 <span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- C
@@ Line 1,266: / Line 1,266: @@
 In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same issue occurs in 24-edo.
-Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara (2.3.5.7 with 50/49 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
+Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, Pajara (2.3.5.7 with 50/49 and 64/63) is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)], which can also be written [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = ccm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
 A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, Injera (2.3.5.7 with 81/80 and 50/49) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while Pajara favors a fifth wider than that, Injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
@@ Line 1,904: / Line 1,904: @@
 ! | 3.5 (M6 = 5/3)
 ! | 3.7 (M3 = 9/7)
-! | 5.7 (WWM3 = 5/1, d5 = 7/5)
+! | 5.7 (ccM3 = 5/1, d5 = 7/5)
 |-
 | | 1
@@ Line 1,912: / Line 1,912: @@
 | | (P12, M6)
 | | (P12, M3)
-| | (WWM3, d5)
+| | (ccM3, d5)
 |-
 ! | half-splits
@@ Line 1,936: / Line 1,936: @@
 | | (P12, M6/2)
 | | (P12, M2)*
-| | (WWM3, m3)*
+| | (ccM3, m3)*
 |-
 | | 4
@@ Line 1,944: / Line 1,944: @@
 | | (P12, P4)*
 | | (P12, m10/2)
-| | (WWM3, M7)*
+| | (ccM3, M7)*
 |-
 | | 5
@@ Line 1,968: / Line 1,968: @@
 | | (P12/3, M6)
 | | (P12/3, M3)
-| | (WWM3/3, d5)
+| | (ccM3/3, d5)
 |-
 | | 7
@@ Line 1,976: / Line 1,976: @@
 | | (P12, M6/3)
 | | (P12, M3/3)
-| | (WWM3, d5/3)
+| | (ccM3, d5/3)
 |-
 | | 8
@@ Line 1,984: / Line 1,984: @@
 | | (P12, m7/3)
 | | (P12, P4)*
-| | (WWM3, WA6/3)
+| | (ccM3, cA6/3)
 |-
 | | 9
@@ Line 1,990: / Line 1,990: @@
 | | (P8, M10/3)
 | | (P8, M9/3)
-| | (P12, WWM3/3)
+| | (P12, ccM3/3)
-| | (P12, WM7/3)
+| | (P12, cM7/3)
-| | (WWM3, WWm7/3)
+| | (ccM3, ccm7/3)
 |-
 | | 10
@@ Line 2,000: / Line 2,000: @@
 | | (P12/3, M6/2)
 | | (P12/3, M2)*
-| | (WWM3/3, m3)*
+| | (ccM3/3, m3)*
 |-
 | | 11
@@ Line 2,008: / Line 2,008: @@
 | | (P12/3, P4)*
 | | (P12/3, m10/2)
-| | (WWM3/3, M7)*
+| | (ccM3/3, M7)*
 |-
 | | 12
@@ Line 2,024: / Line 2,024: @@
 | | (P12/2, m7/3)
 | | (P12/2, P4)*
-| | (M9, WA6/3)*
+| | (M9, cA6/3)*
 |-
 | | 14
@@ Line 2,030: / Line 2,030: @@
 | | (P8/2, M10/3)
 | | (P8/2, M9/3)
-| | (P12/2, WWM3/3)
+| | (P12/2, ccM3/3)
-| | (P12/2, WM7/3)
+| | (P12/2, cM7/3)
-| | (M9, WWm7/3)*
+| | (M9, ccm7/3)*
 |-
 | | 15
@@ Line 2,040: / Line 2,040: @@
 | | (P12/3, M6/3)
 | | (P12/3, P4)*
-| | (WWM3/3, d5/3)
+| | (ccM3/3, d5/3)
 |}
 For prime subgroup p.q, the unsplit pergen has period p/1. The unsplit pergen's generator is found by dividing q by p until it's less than p/1, and period-inverting if it's more than half of p/1.
-Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (WWM3/5, d5) can optionally be replaced too.
+Multi-EDO pergens also appear in this table. For example, in row #33, (P8/5, ^1) replaces both 2.5 (P8/5, M3) and 2.7 (P8/5, M2). This has the advantage of avoiding missing notes. Since one fifth of 5/1 is only 25¢ from 7/5, the 5.7 (ccM3/5, d5) can optionally be replaced too.
 {| class="wikitable" style="text-align:center;"
@@ Line 2,062: / Line 2,062: @@
 | | (P12/5, M6)
 | | (P12/5, M3)
-| | (WWM3/5, ^1)
+| | (ccM3/5, ^1)
 |}
@@ Line 2,085: / Line 2,085: @@
 C1 --- G1
-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.
+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 cm7 (e.g. D1 to C3), a cM9 (e.g. C1 to D3), and many other intervals.
 C2 --- G2 --- D3 --- A3<br>
@@ Line 2,243: / Line 2,243: @@
 |-
 | | 411-422¢
-| | WWP4/7
+| | ccP4/7
 | |
 | |
@@ Line 2,257: / Line 2,257: @@
 |-
 | | 435-446¢
-| | WWP5/7
+| | ccP5/7
 | |
 | |
@@ Line 2,273: / Line 2,273: @@
 | | P4 = P5
 | | 480-492¢
-| | WWP4/6
+| | ccP4/6
 | | 508-520¢
-| | WWP5/6
+| | ccP5/6
 |-
 | | 560-585¢
@@ Line 2,285: / Line 2,285: @@
 |-
 | | 576-591¢
-| | WWP4/5
+| | ccP4/5
 | | 583-593¢
-| | WWWP4/7
+| | cccP4/7
 | |
 | |
 |}
-There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a WWP4.
+There are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a ccP4.
 The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.
@@ Line 2,733: / Line 2,733: @@
 | | (P8, P11/4) [8]
 | | quarter-11th octotonic
-| | seventh-WW4th, seventh-WW5th
+| | seventh-cc4th, seventh-cc5th
 |-
 | | 4L 4s
@@ Line 2,766: / Line 2,766: @@
 |-
 | | 2L 7s
-| | (P8, W<span style="vertical-align: super;">3</span>P5/8) [9]
+| | (P8, c<span style="vertical-align: super;">3</span>P5/8) [9]
-| | eighth-W<span style="vertical-align: super;">3</span>5th nonatonic
+| | eighth-c<span style="vertical-align: super;">3</span>5th nonatonic
-| | third-11th, fifth-WW4th
+| | third-11th, fifth-cc4th
 |-
 | | 3L 6s
@@ Line 2,791: / Line 2,791: @@
 |-
 | | 7L 2s
-| | (P8, WWP5/6)[9]
+| | (P8, ccP5/6)[9]
-| | sixth-WW5th nonatonic
+| | sixth-cc5th nonatonic
 | | (lopsided unless 5th is sharp)
 |-
@@ Line 2,818: / Line 2,818: @@
 | | (P8, P12/5) [10]
 | | fifth-12th decatonic
-| | eighth-WW4th, eighth-WW5th
+| | eighth-cc4th, eighth-cc5th
 |-
 | | 4L 6s
@@ Line 2,838: / Line 2,838: @@
 | | (P8, P5/2) [10]
 | | half-5th decatonic
-| | ninth-WW5th
+| | ninth-cc5th
 |-
 | | 8L 2s
@@ Line 2,851: / Line 2,851: @@
 |}
-The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
+The pentatonic MOS scales don't include fifth-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, ccP5/7) [5]. The rationale would be that two sensei generators equals 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
 Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2) = (P8, M3/2) = (P8, M2), where 5·G = A6 = 7/4.
@@ Line 2,861: / Line 2,861: @@
 Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.
-How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.
+How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, ccP5/31),... (P8, (i-1,1)/n), where n = 12i+7.
 How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.
@@ Line 3,476: / Line 3,476: @@
 | | P12/6
 | | P12/5
-| | WWP5/7
+| | ccP5/7
 | | P5
 | | P11/3
@@ Line 3,492: / Line 3,492: @@
 | | P11/4
 | | -
-| | W<span style="vertical-align: super;">3</span>P5/8
+| | c<span style="vertical-align: super;">3</span>P5/8
 | |
 | |
@@ Line 3,547: / Line 3,547: @@
 | | -
 | | -
-| | W<span style="vertical-align: super;">3</span>P4/9
+| | c<span style="vertical-align: super;">3</span>P4/9
 | | -
 | | P11/3
@@ Line 3,616: / Line 3,616: @@
 | | P4/2
 | | P12/6
-| | WWP4/8
+| | ccP4/8
-| | WWP4/7
+| | ccP4/7
 | | P12/4
 | | P5
@@ Line 3,634: / Line 3,634: @@
 | | -
 | | -
-| | W<span style="vertical-align: super;">4</span>P5/10
+| | c<span style="vertical-align: super;">4</span>P5/10
 |-
 ! | "
@@ Line 3,720: / Line 3,720: @@
 ! | 11
 |}
-Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, W<span style="vertical-align: super;">5</span>P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).
+Larger edos can have very awkward pergens. For example, 31edo with G = 14/31 is (P8, c<span style="vertical-align: super;">5</span>P4/12). It's much simpler to think of the generator as the downfifth = 17\31, and the pergen as the pseudopergen (P8, v5).
 <u>'''EDO-pair names'''</u>
@@ Line 3,783: / Line 3,783: @@
 ! | 15-edo
 | | (P8/3, P5)
-| | (P8, W<span style="vertical-align: super;">5</span>P5/12)
+| | (P8, c<span style="vertical-align: super;">5</span>P5/12)
 | | (P8, P4/6)
 | |
@@ Line 3,805: / Line 3,805: @@
 ! | 17-edo
 | | (P8, P5)
-| | (P8, WWP5/11)
+| | (P8, ccP5/11)
 | | (P8, P11/4)
 | | (P8, P11/3)
@@ Line 3,838: / Line 3,838: @@
 ! | 20-edo
 | | (P8/4, P5)
-| | (P8, WWP4/16)
+| | (P8, ccP4/16)
 | | (P8/2, P5/4)
 | | (P8/5, ^1)
@@ Line 3,849: / Line 3,849: @@
 ! | 21-edo
 | | (P8/3, P5)
-| | (P8, W<span style="vertical-align: super;">3</span>P4/9)
+| | (P8, c<span style="vertical-align: super;">3</span>P4/9)
 | | (P8/7, ^1)
 | | (P8/3, P4/3)
@@ Line 3,860: / Line 3,860: @@
 ! | 22-edo
 | | (P8/2, P5)
-| | (P8, W<span style="vertical-align: super;">3</span>P4/15)
+| | (P8, c<span style="vertical-align: super;">3</span>P4/15)
 | | (P8/2, P4/3)
 | | (P8, P4/3)
@@ Line 3,871: / Line 3,871: @@
 ! | 23-edo
 | | (P8, P4/5)
-| | (P8, WWP4/8)
+| | (P8, ccP4/8)
 | | (P8, P4/2)
 | | (P8, P12/12)
@@ Line 3,878: / Line 3,878: @@
 | | (P8, P12/4)
 | | (P8, P12/6)
-| | (P8, W<span style="vertical-align: super;">5</span>P5/16)
+| | (P8, c<span style="vertical-align: super;">5</span>P5/16)
 |-
 ! | 24-edo
 | | (P8/12, ^1)
-| | (P8, W<span style="vertical-align: super;">6</span>P4/14)
+| | (P8, c<span style="vertical-align: super;">6</span>P4/14)
 | | (P8/2, P4/2)
 | | (P8/3, P4/2)