Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-~~14 21~~:17:15 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-15 04:24:54 UTC</tt>.<br>

: The original revision id was <tt>~~626439971~~</tt>.<br>

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The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[xenharmonic/Kite's color notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.

For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated ~~with only~~ an up symbol, and we have (P8, P5, ^1) = unsplit ~~with ups~~.

For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = third-11th ~~with ups~~. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = half-8ve ~~with ups~~ (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

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.

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||~ 3/1 ||= 0 ||= 2 ||= -1 || ||

||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

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||~ 3/1 ||= 0 ||= 1 ||= -1 || ||

||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, ~~64:63~~)~~. Let ^~~1 = 64/63, ~~and the pergen is (P8, P5/2, ^1),~~ half-5th ~~with ups~~. This is far better than (P8, P5/2, (96:25)/4).

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

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

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes backwards compatibility ~~to be essential~~.

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes that backwards compatibility is desirable.

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

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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This ~~represents~~ a ~~natural~~ ordering of rank-2 pergens.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that allows every pergen to be numbered.

The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.

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If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C

||~ pergen ||~ enharmonic

||~ ||~ pergen ||~ enharmonic

interval(s) ||~ equiva-

lence(s) ||~ split

interval(s) ||~ perchain(s) and

genchains(s) ||~ examples ||

||= (P8, P5)

||= 1 ||= (P8, P5)

unsplit ||= none ||= none ||= none ||= C - G ||= meantone,

schismic ||

||~ halves ||~ ||~ ||~ ||~ ||~ ||

||~ ||~ halves ||~ ||~ ||~ ||~ ||~ ||

||= (P8/2, P5)

||= 2 ||= (P8/2, P5)

half-8ve ||= ^^d2 (if 5th

``>`` 700¢ ||= C^^ = B# ||= P8/2 = vA4 = ^d5 ||= C - F#v=Gb^ - C ||= srutal

^1 = 81/80 ||

||= " ||= vvd2 (if 5th

||= ||= " ||= vvd2 (if 5th

< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red

^1 = 64/63 ||

||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121

||= ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121

^1 = 33/32 ||

||= (P8, P4/2)

||= 3 ||= (P8, P4/2)

half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore

^1 = 64/63 ||

||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||

||= ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||

||= (P8, P5/2)

||= 4 ||= (P8, P5/2)

half-5th ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira

^1 = 33/32 ||

||= (P8/2, P4/2)

||= 5 ||= (P8/2, P4/2)

half-

everything ||= \\m2,

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^1 = 33/32

/1 = 64/63 ||

||= " ||= ^^d2,

||= ||= " ||= ^^d2,

\\m2,

vv\\A1 ||= C^^ = B#

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^1 = 81/80

/1 = 64/63 ||

||= " ||= ^^d2,

||= ||= " ||= ^^d2,

\\A1,

^^\\m2 ||= C^^ = B#

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^1 = 81/80

/1 = 33/32 ||

||~ thirds ||~ ||~ ||~ ||~ ||~ ||

||~ ||~ thirds ||~ ||~ ||~ ||~ ||~ ||

||= (P8/3, P5)

||= 6 ||= (P8/3, P5)

third-8ve ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||

||= (P8, P4/3)

||= 7 ||= (P8, P4/3)

third-4th ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||

||= (P8, P5/3)

||= 8 ||= (P8, P5/3)

third-5th ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||

||= (P8, P11/3)

||= 9 ||= (P8, P11/3)

third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,

with 11/8 = A4 ||

||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||

||= ||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||

||= (P8/3, P4/2)

||= 10 ||= (P8/3, P4/2)

third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4

P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3 ||= C - Eb^^ - Avv - C

C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F ||= sixfold jade ||

||= " ||= ^<span style="vertical-align: super;">3</span>d2,

||= ||= " ||= ^<span style="vertical-align: super;">3</span>d2,

\\m2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B#

C``//`` = Db ||= P8/3 = vM3 = ^^d4

P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C

C - D/=Eb\ - F ||= 128/125 & 49/48 ||

||= (P8/3, P5/2)

||= 11 ||= (P8/3, P5/2)

third-8ve, half-5th ||= ^<span style="vertical-align: super;">3</span>d2

\\A1 ||= C^<span style="vertical-align: super;">3</span> ``=`` B#

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P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C

C - Eb/=E\ - G ||= small sixfold blue ||

||= (P8/2, P4/3)

||= 12 ||= (P8/2, P4/3)

half-8ve, third-4th ||= ^^d2

\<span style="vertical-align: super;">3</span>A1 ||= C^^ ``=`` Dbb

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P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C

C - D\ - Eb/ - F ||= large sixfold red ||

||= (P8/2, P5/3)

||= 13 ||= (P8/2, P5/3)

half-8ve,

third-5th ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5

P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C

C - D#vv - Fb^^ - G ||= large sixfold yellow ||

||= " ||= ^^d2,

||= ||= " ||= ^^d2,

\\\m2 ||= C^^ = B#

C``///`` = Db ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C

C - /D - \F - G ||= 50/49 & 1029/1024 ||

||= (P8/2, P11/3)

||= 14 ||= (P8/2, P11/3)

half-8ve,

third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5

P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5 ||= C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C

C - F^^ - Cvv - F ||= large sixfold jade ||

||= (P8/3, P4/3)

||= 15 ||= (P8/3, P4/3)

third-

everything ||= v<span style="vertical-align: super;">3</span>d2,

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^1 = 64/63

/1 = 81/80 ||

||= " ||= ^<span style="vertical-align: super;">3</span>d2,

||= ||= " ||= ^<span style="vertical-align: super;">3</span>d2,

\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B#

C/<span style="vertical-align: super;">3</span> ``=`` Db ||= P8/3 = vM3 = ^^d4

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^1 = 81/80

/1 = 64/63 ||

||= " ||= v<span style="vertical-align: super;">3</span>A1,

||= ||= " ||= v<span style="vertical-align: super;">3</span>A1,

\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` C#

C/3 ``=`` Db ||= P4/3 = vM2 = ^^m2

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^1 = 81/80

/1 = 64/63 ||

||~ quarters ||~ ||~ ||~ ||~ ||~ ||

||~ ||~ quarters ||~ ||~ ||~ ||~ ||~ ||

||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished ||

||= 16 ||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished ||

||= (P8, P4/4) ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= negri ||

||= 17 ||= (P8, P4/4) ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= negri ||

||= (P8, P5/4) ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||

||= 18 ||= (P8, P5/4) ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||

||= (P8, P11/4) ||= v<span style="vertical-align: super;">4</span>dd3 ||= C^<span style="vertical-align: super;">4</span> ``=`` Eb<span style="vertical-align: super;">3</span> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= squares ||

||= 19 ||= (P8, P11/4) ||= v<span style="vertical-align: super;">4</span>dd3 ||= C^<span style="vertical-align: super;">4</span> ``=`` Eb<span style="vertical-align: super;">3</span> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= squares ||

||= (P8, P12/4) ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture ||

||= 20 ||= (P8, P12/4) ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture ||

||= etc. ||= ||= ||= ||= ||= ||

||= ||= etc. ||= ||= ||= ||= ||= ||

==Tipping points==

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A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.

==Notating rank-3 pergens *==

==Notating rank-3 pergens*==

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:

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

||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one ~~may wish to~~ simply use colors.

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one can simply use colors.

All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows:

||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ notation ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic ||

||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit ~~with ups~~ ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||

||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||

||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve ~~with ups~~ ||= double-pair ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ~~``//``~~d2 ||

||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= double-pair ||= v/A4 = 10/7 ||= P5 ||= ^1 = 81/80 ||= ^^\\d2 ||

||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th ~~with ups~~ ||= double-pair ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= \\\dd3 ||

||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= rank-3 third-11th ||= double-pair ||= P8 ||= ^\d5 = 7/5 ||= ^1 = 81/80 ||= ^^^\\\dd3 ||

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

||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\<span style="vertical-align: super;">4</span>dd3 ||

||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, ^1 = 81/80, and /1 = 64/63. Gen2 =v/A1 ~~and~~ E = ^^\\\dd3. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains, the additional accidentals get progressively more complex.

Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, where ^1 = 81/80, and /1 = 64/63. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains we've seen, the additional accidentals get progressively more complex.

~~To avoid E being a 3rd~~, ~~7/4 could instead~~ be ~~mapped~~ to ~~an A6~~, ~~making gen2 = v/m2~~

In double-pair rank-3 notation, the accidentals generally represent mapping commas, with ups and downs applying to the lowest prime > 3 and highs and lows applying to the other prime. It would be possible to have them represent the sum or difference of the mapping commas, but this would make the notation harder to read, IMO. Probably makes it more tippy, too.

G = ^m2, C - Db^ - Ebb^^ - ~~Fbbb^^^~~, chord C~~- Fbbb^^^ -~~ G - Bbb^^

The E is an awkward dd3 because that's what the comma maps to. It's possible to find an E that's a 2nd or even a unison, but chords will be spelled improperly:

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

gen2 = vM3/3 = v[4,2]/3 = [1,1] = /m2, E = [-1,1] = ^``///``dd2, genchain = C -- Db/ -- Ebb``//``=D#v\ -- Ev, 4:5:6:7 chord = C Ev G Bbb``//``

gen2 = \m3/2 = \[3,2]/2 = [1,1] = ^m2 = v\M2, E = [1,0] = vv\A1, genchain = C -- Db^ -- Eb\ -- Fb^\, 4:5:6:7 chord = C Fb^\ G Bb\

~~gen2 = vM3/3~~ = ~~[4,2~~,-~~1,0]/3 = [1,1,0,1] = /m2, E = [~~-~~1,1,1~~,~~3] = ^``~~//~~/``dd2~~, ~~C^``///``~~ = ~~B##, C -- Db/ -- Ebb/~~/=D#

The dd3 enharmonic implies a tipping point of 10\17 = 706¢, which falls within demeter's sweet spot of fifth-comma to eighth-comma. However, neither 81/80 nor 64/63 vanish, and demeter doesn't tip. The commas only vanish when the 5th = 10\17 and the vM3 (5/4) = 6\17 and the \m7 (7/4) = 14\17. These are the 17b-edo values, which tempers out both 81/80 and 64/63. But a 5/4 of 423¢ is far from sweet! So demeter doesn't tip. (what a cheapskate she is!)

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

~~gen2 =~~ \m3/2 = ~~[3,2,0,~~-1]/2 = ~~[1,0,~~-~~1,1] = v~~/A1, E = [1,2,2,-3~~] = ^^\\\dd3~~

The true meaning of the ^^\\\dd3 enharmonic is that 686/675 = ^^1 - ///1 + dd3 = (81/80)<span style="vertical-align: super;">2</span> · (64/63)<span style="vertical-align: super;">-3</span> · (27,-17) = 0¢. We write the comma as a 3-limit comma plus or minus some number of mapping commas. The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes. In a rank-2 temperament, the mapping comma must vanish too, because some number of them plus the 3-limit comma add up to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to 0¢.

Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. So rank-3 temperaments don't tip?

Deep reddish doesn't tip unless w5 = 700¢, y3 = 400¢ and b7 = 1000¢. Deep reddish is double-pair, So 50/49 = -\\d2 = //1 - d2 = R · R · (19,-12), and R = (-9,6,1,-1) and 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7

50/49 = vv//-d2 = -^^\\d2 = //1 - ^^1 - d2 = 64/63 · 64/63 / (81/80 · 81/80 · (19,-12)) = 0¢.

Tell Praveen; "half-8ve with ups" changed to "rank-3 half-8ve"

rank-3 pergens don't tip!

complex enharmonics and pergens are better than accidentals not being mapping commas?

==Notating Blackwood-like pergens*==

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17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \

==Notating arbitrary P and G tunings*==

==Notating tunings with an arbitrary generator==

Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is set. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.

The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. use the octave inverse if G > 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.

||||~ primary choice ||||~ secondary choice ||||~ tertiary choice ||

||~ generator ||~ possible multigen ||~ generator ||~ multigen ||~ generator ||~ multigen ||

||= 23-60¢ ||= M2/4 (requires P8/2) ||= ||= ||= ||= ||

||= 69-79¢ ||= P4/7 ||= ||= ||= ||= ||

||= 80-92¢ ||= P4/6 ||= ||= ||= ||= ||

||= 92-103¢ ||= P5/7 ||= ||= ||= ||= ||

||= 96-111¢ ||= P4/5 ||= ||= ||= ||= ||

||= 108-120¢ ||= P5/6 ||= ||= ||= ||= ||

||= 120-138¢ ||= P4/4 ||= ||= ||= ||= ||

||= 129-144¢ ||= P5/5 ||= ||= ||= ||= ||

||= 160-185¢ ||= P4/3 ||= 162-180¢ ||= P5/4 ||= ||= ||

||= 215-240¢ ||= P5/3 ||= ||= ||= ||= ||

||= 240-277¢ ||= P4/2 ||= 240-251¢ ||= P11/7 ||= 264-274¢ ||= P12/7 ||

||= 280-292¢ ||= P11/6 ||= ||= ||= ||= ||

||= 308-320¢ ||= P12/6 ||= ||= ||= ||= ||

||= 323-360¢ ||= P5/2 ||= 336-351¢ ||= P11/5 ||= ||= ||

||= 369-384¢ ||= P12/5 ||= ||= ||= ||= ||

||= 411-422¢ ||= WWP4/7 ||= ||= ||= ||= ||

||= 420-438¢ ||= P11/4 ||= ||= ||= ||= ||

||= 435-446¢ ||= WWP5/7 ||= ||= ||= ||= ||

||= 462-480¢ ||= P12/4 ||= ||= ||= ||= ||

||= 480-554¢ ||= P4 = P5 ||= 480-492¢ ||= WWP4/6 ||= 508-520¢ ||= WWP5/6 ||

||= 560-585¢ ||= P11/3 ||= ||= ||= ||= ||

||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= ||

The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with heptatonic notation.

~~Given only the period as some fraction of~~ the octave, and ~~the generator's cents~~, ~~it's often possible~~ to ~~work backwards and~~ find ~~an appropriate~~ multigen.

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

==Pergens and MOS scales==

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==Miscellaneous Notes==

__**Staff notation**__

Highs and lows can be added to the score just like ups and downs can.

__**Combining pergens**__

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

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 2em;"><a href="#Applications-Tipping points">Tipping points</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Tipping points and sweet spots">Tipping points and sweet spots</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating unsplit pergens">Notating unsplit pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens *">Notating rank-3 pergens *</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating rank-3 pergens*">Notating rank-3 pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens*">Notating Blackwood-like pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating Blackwood-like pergens*">Notating Blackwood-like pergens*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating arbitrary P and G tunings*">Notating arbitrary P and G tunings*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials*">Supplemental materials*</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Miscellaneous Notes">Miscellaneous Notes</a></div>

</div>

</div>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).<br />

<br />

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation">Color notation</a> (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc. <br />

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation">Color notation</a> (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.<br />

<br />

For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated ~~with only~~ an up symbol, and we have (P8, P5, ^1) = unsplit ~~with ups~~.<br />

For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.<br />

<br />

More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = third-11th ~~with ups~~. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = half-8ve ~~with ups~~ (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. <br />

More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.<br />

<br />

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.<br />

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<br />

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

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

<br />

For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.<br />

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</table>

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.<br />

Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.<br />

<br />

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.<br />

<br />

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.<br />

The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.<br />

<br />

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:<br />

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</table>

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, ~~64:63~~)~~. Let ^~~1 = 64/63, ~~and the pergen is (P8, P5/2, ^1),~~ half-5th ~~with ups~~. This is far better than (P8, P5/2, (96:25)/4).<br />

Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).<br />

<br />

The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br />

<br />

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

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

<br />

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.<br />

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

<br />

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes backwards compatibility ~~to be essential~~.<br />

One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes that backwards compatibility is desirable.<br />

<br />

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

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Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.<br />

<br />

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This ~~represents~~ a ~~natural~~ ordering of rank-2 pergens.<br />

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that allows every pergen to be numbered.<br />

<br />

The enharmonic interval, or more briefly the <strong>enharmonic</strong>, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.<br />

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<tr>

</th>

<th>pergen<br />

</th>

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</tr>

<tr>

</td>

unsplit<br />

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</tr>

<tr>

</th>

<th>halves<br />

</th>

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</tr>

<tr>

</td>

half-8ve<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

half-4th<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

half-5th<br />

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</tr>

<tr>

</td>

half-<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</th>

<th>thirds<br />

</th>

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</tr>

<tr>

</td>

third-8ve<br />

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</tr>

<tr>

</td>

third-4th<br />

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</tr>

<tr>

</td>

third-5th<br />

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</tr>

<tr>

</td>

third-11th<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

third-8ve, half-4th<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

third-8ve, half-5th<br />

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</tr>

<tr>

</td>

half-8ve, third-4th<br />

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</tr>

<tr>

</td>

half-8ve,<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

half-8ve,<br />

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</tr>

<tr>

</td>

third-<br />

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</th>

<th>quarters<br />

</th>

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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<tr>

</td>

</td>

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<tr>

</td>

</td>

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</tr>

<tr>

</td>

</td>

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<br />

<h2 id="toc4"><a name="Applications-Tipping points"></a>Tipping points</h2>

<h2 id="toc4"><a name="Applications-Tipping points"></a>Tipping points</h2>

<br />

Removing the ups and downs from an enharmonic interval makes a &quot;bare&quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &quot;tipping point&quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>ups and downs may need to be swapped, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br />

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<br />

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<h1 id="toc5"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<br />

<h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a>Naming very large intervals</h2>

<h2 id="toc6"><a name="Further Discussion-Naming very large intervals"></a>Naming very large intervals</h2>

<br />

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 &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.<br />

<br />

<h2 id="toc7"><a name="Further Discussion-Secondary splits"></a>Secondary splits</h2>

<h2 id="toc7"><a name="Further Discussion-Secondary splits"></a>Secondary splits</h2>

<br />

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 (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:<br />

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For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).<br />

<br />

<h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.<br />

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A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br />

<br />

<h2 id="toc9"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<h2 id="toc9"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<br />

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.<br />

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There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.<br />

<br />

<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<h2 id="toc10"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.<br />

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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br />

<br />

<h2 id="toc11"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<h2 id="toc11"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br />

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This is a lot of math, but it only needs to be done once for each pergen!<br />

<br />

<h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc12"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

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.<br />

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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br />

<br />

<h2 id="toc13"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<h2 id="toc13"><a name="Further Discussion-Chord names and scale names"></a>Chord names and scale names</h2>

<br />

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br />

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Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.<br />

<br />

<h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<h2 id="toc14"><a name="Further Discussion-Tipping points and sweet spots"></a>Tipping points and sweet spots</h2>

<br />

As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament &quot;tips over&quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.<br />

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An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^<span style="vertical-align: super;">4</span>dd2 or v<span style="vertical-align: super;">4</span>dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^<span style="vertical-align: super;">4</span>dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.<br />

<br />

<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<h2 id="toc15"><a name="Further Discussion-Notating unsplit pergens"></a>Notating unsplit pergens</h2>

<br />

An unsplit pergen doesn't <u>require</u> ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.<br />

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A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.<br />

<br />

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens *"></a>Notating rank-3 pergens *</h2>

<h2 id="toc16"><a name="Further Discussion-Notating rank-3 pergens*"></a>Notating rank-3 pergens*</h2>

<br />

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:<br />

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</table>

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one ~~may wish to~~ simply use colors.<br />

A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &quot;superfalse&quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one can simply use colors.<br />

<br />

All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows:<br />

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</td>

<td style="text-align: center;">unsplit ~~with ups~~<br />

<td style="text-align: center;">rank-3 unsplit<br />

</td>

<td style="text-align: center;">single-pair<br />

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<td style="text-align: center;">double-pair<br />

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</td>

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</td>

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</td>

<td style="text-align: center;">third-11th ~~with ups~~<br />

<td style="text-align: center;">rank-3 third-11th<br />

</td>

<td style="text-align: center;">double-pair<br />

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</table>

Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, ^1 = 81/80, and /1 = 64/63. Gen2 =v/A1 ~~and~~ E = ^^\\\dd3. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains, the additional accidentals get progressively more complex.<br />

Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, where ^1 = 81/80, and /1 = 64/63. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains we've seen, the additional accidentals get progressively more complex.<br />

<br />

~~To avoid E being a 3rd~~, ~~7/4 could instead~~ be ~~mapped~~ to ~~an A6~~, ~~making gen2 = v/m2~~<br />

In double-pair rank-3 notation, the accidentals generally represent mapping commas, with ups and downs applying to the lowest prime &gt; 3 and highs and lows applying to the other prime. It would be possible to have them represent the sum or difference of the mapping commas, but this would make the notation harder to read, IMO. Probably makes it more tippy, too.<br />

<br />

G = ^m2, C - Db^ - Ebb^^ - ~~Fbbb^^^~~, chord C- ~~Fbbb^^^~~ - G - ~~Bbb^^~~<br />

The E is an awkward dd3 because that's what the comma maps to. It's possible to find an E that's a 2nd or even a unison, but chords will be spelled improperly:<br />

~~fifth-comma: c~~ = ~~9.5¢~~, P5 = ~~700¢ + c~~, ~~vM3~~ = ~~400¢ + d~~, d = -~~10¢~~<br />

gen2 = vM3/3 = v[4,2]/3 = [1,1] = /m2, E = [-1,1] = ^///dd2, genchain = C -- Db/ -- Ebb//=D#v\ -- Ev, 4:5:6:7 chord = C Ev G Bbb//<br />

gen2 = \m3/2 = \[3,2]/2 = [1,1] = ^m2 = v\M2, E = [1,0] = vv\A1, genchain = C -- Db^ -- Eb\ -- Fb^\, 4:5:6:7 chord = C Fb^\ G Bb\<br />

<br />

~~gen2 = vM3/3~~ = ~~[4,2~~,-~~1,0]/3 = [1,1,0,1] = /m2, E = [~~-1,~~1,1,3] = ^~~//~~/dd2~~, ~~C^///~~ = ~~B##, C -- Db/ -- Ebb/~~/=~~D#<br />~~

The dd3 enharmonic implies a tipping point of 10\17 = 706¢, which falls within demeter's sweet spot of fifth-comma to eighth-comma. However, neither 81/80 nor 64/63 vanish, and demeter doesn't tip. The commas only vanish when the 5th = 10\17 and the vM3 (5/4) = 6\17 and the \m7 (7/4) = 14\17. These are the 17b-edo values, which tempers out both 81/80 and 64/63. But a 5/4 of 423¢ is far from sweet! So demeter doesn't tip. (what a cheapskate she is!)<br />

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

<br />

~~gen2 =~~ \m3/2 = ~~[3,2,0,~~-1]/2 = ~~[1,0,~~-~~1,1] = v~~/A1, E = [1,2,2,-3~~] = ^^\\\dd3~~<br />

The true meaning of the ^^\\\dd3 enharmonic is that 686/675 = ^^1 - <em>/1 + dd3 = (81/80)<span style="vertical-align: super;">2</span> · (64/63)<span style="vertical-align: super;">-3</span> · (27,-17) = 0¢. We write the comma as a 3-limit comma plus or minus some number of mapping commas. The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes. In a rank-2 temperament, the mapping comma must vanish too, because some number of them plus the 3-limit comma add up to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to 0¢.<br />

<br />

Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. So rank-3 temperaments don't tip?<br />

<br />

Deep reddish doesn't tip unless w5 = 700¢, y3 = 400¢ and b7 = 1000¢. Deep reddish is double-pair, So 50/49 = -\\d2 = </em>1 - d2 = R · R · (19,-12), and R = (-9,6,1,-1) and 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7<br />

50/49 = vv<em>-d2 = -^^\\d2 = </em>1 - ^^1 - d2 = 64/63 · 64/63 / (81/80 · 81/80 · (19,-12)) = 0¢.<br />

<br />

<h2 id="toc17"><a name="Further Discussion-Notating Blackwood-like pergens*"></a>Notating Blackwood-like pergens*</h2>

Tell Praveen; &quot;half-8ve with ups&quot; changed to &quot;rank-3 half-8ve&quot;<br />

rank-3 pergens don't tip!<br />

complex enharmonics and pergens are better than accidentals not being mapping commas?<br />

<br />

<h2 id="toc17"><a name="Further Discussion-Notating Blackwood-like pergens*"></a>Notating Blackwood-like pergens*</h2>

<br />

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.<br />

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17edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \<br />

<br />

<h2 id="toc18"><a name="Further Discussion-Notating arbitrary P and G tunings*"></a>Notating arbitrary P and G tunings*</h2>

<h2 id="toc18"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<br />

Given only the period as some fraction of the octave~~, and the generator's cents~~, it's often possible to work backwards and find an appropriate multigen.<br />

Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is set. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.<br />

<br />

~~<!~~-~~- ws:start:WikiTextHeadingRule:94:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Pergens~~ and ~~MOS scales"></~~a&~~gt;&lt~~;~~!-- ws:end:WikiTextHeadingRule:94 --&~~gt;~~Pergens and MOS scales</h2>~~

The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. use the octave inverse if G &gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.<br />

~~<br />~~

~~Every rank-2 pergen generates certain MOS scales~~. ~~This of course depends on the exact size of the generator~~. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the ~~scale doesn't actually contain any 4ths. Such scales are marked with an asterisk~~.<br />

<br />

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<tr>

<th colspan="2">~~pergen~~<br />

<th colspan="2">primary choice<br />

</th>

<th colspan="2">secondary choice<br />

</th>

<th colspan="2">tertiary choice<br />

</th>

</tr>

<tr>

<th>generator<br />

</th>

<th ~~colspan="2"~~>~~MOS scales of 5-12 notes~~<br />

<th>possible multigen<br />

</th>

<th>generator<br />

</th>

<th>multigen<br />

</th>

<th>generator<br />

</th>

<th>multigen<br />

</th>

</tr>

<tr>

</td>

<td style="text-align: center;">~~unsplit~~<br />

<td style="text-align: center;">M2/4 (requires P8/2)<br />

</td>

</td>

</td>

</td>

<td~~&gt~~;~~<br />~~

~~</td>~~

~~<td~~><br />

</td>

</tr>

<tr>

<th colspan="2">halves<br />

</th>

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</table>

The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with heptatonic notation.<br />

<br />

Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.<br />

<br />

<h2 id="toc19"><a name="Further Discussion-Pergens and MOS scales"></a>Pergens and MOS scales</h2>

<br />

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br />

<br />

<tr>

<th colspan="2">pergen<br />

</th>

<th colspan="2">MOS scales of 5-12 notes<br />

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

<td style="text-align: center;">unsplit<br />

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<th colspan="2">halves<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

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<tr>

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

<br />

<h2 id="toc20"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc20"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br />

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See the next section for examples of which pergens are supported by a specific edo.<br />

<br />

<h2 id="toc21"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<h2 id="toc21"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<br />

needs more screenshots, including 12-edo's pergens<br />

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<br />

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.<br />

<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><br />

<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><br />

<br />

Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />

<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><br />

<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><br />

<br />

Screenshot of the first 38 pergens:<br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<br />

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br />

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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br />

<br />

<h2 id="toc22"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<h2 id="toc22"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<br />

The interval P8/2 has a &quot;ratio&quot; of the square root of 2 = 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). Likewise, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the <strong>pergen matrix</strong> [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.<br />

Line 4,588:

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Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.<br />

<br />

<h2 id="toc23"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<h2 id="toc23"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<br />

<u><strong>Staff notation</strong></u><br />

<br />

Highs and lows can be added to the score just like ups and downs can.<br />

<br />

<u><strong>Combining pergens</strong></u><br />

<br />

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+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-15 04:24:54 UTC</tt>.<br>
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 The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).
 Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[xenharmonic/Kite's color notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.
-For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups.
+For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.
-More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = third-11th with ups. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = half-8ve with ups (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.
+More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.
 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.
@@ Line 94: / Line 94: @@
 ||~ 3/1 ||= 0 ||= 2 ||= -1 ||   ||
 ||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||
-Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.
+Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.
 Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.
-The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.
+The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.
 Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:
@@ Line 110: / Line 110: @@
 ||~ 3/1 ||= 0 ||= 1 ||= -1 ||   ||
 ||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||
-Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4).
+Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).
 The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.
@@ Line 127: / Line 127: @@
 All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, **highs and lows**, written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.
-One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes backwards compatibility to be essential.
+One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes that backwards compatibility is desirable.
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See [[pergen#Further%20Discussion-Notating%20unsplit%20pergens|Notating unsplit pergens]] below.
@@ Line 133: / Line 133: @@
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that allows every pergen to be numbered.
 The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.
@@ Line 141: / Line 141: @@
 If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C
-||~ pergen ||~ enharmonic
+||~   ||~ pergen ||~ enharmonic
 interval(s) ||~ equiva-
 lence(s) ||~ split
 interval(s) ||~ perchain(s) and
 genchains(s) ||~ examples ||
-||= (P8, P5)
+||= 1 ||= (P8, P5)
 unsplit ||= none ||= none ||= none ||= C - G ||= meantone,
 schismic ||
-||~ halves ||~   ||~   ||~   ||~   ||~   ||
+||~   ||~ halves ||~   ||~   ||~   ||~   ||~   ||
-||= (P8/2, P5)
+||= 2 ||= (P8/2, P5)
 half-8ve ||= ^^d2 (if 5th
 ``&gt;`` 700¢ ||= C^^ = B# ||= P8/2 = vA4 = ^d5 ||= C - F#v=Gb^ - C ||= srutal
 ^1 = 81/80 ||
-||= " ||= vvd2 (if 5th
+||=   ||= " ||= vvd2 (if 5th
 &lt; 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red
 ^1 = 64/63 ||
-||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121
+||=   ||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121
 ^1 = 33/32 ||
-||= (P8, P4/2)
+||= 3 ||= (P8, P4/2)
 half-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore
 ^1 = 64/63 ||
-||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||
+||=   ||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||
-||= (P8, P5/2)
+||= 4 ||= (P8, P5/2)
 half-5th ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira
 ^1 = 33/32 ||
-||= (P8/2, P4/2)
+||= 5 ||= (P8/2, P4/2)
 half-
 everything ||= \\m2,
@@ Line 182: / Line 182: @@
 ^1 = 33/32
 /1 = 64/63 ||
-||= " ||= ^^d2,
+||=   ||= " ||= ^^d2,
 \\m2,
 vv\\A1 ||= C^^ = B#
@@ Line 193: / Line 193: @@
 ^1 = 81/80
 /1 = 64/63 ||
-||= " ||= ^^d2,
+||=   ||= " ||= ^^d2,
 \\A1,
 ^^\\m2 ||= C^^ = B#
@@ Line 204: / Line 204: @@
 ^1 = 81/80
 /1 = 33/32 ||
-||~ thirds ||~   ||~   ||~   ||~   ||~   ||
+||~   ||~ thirds ||~   ||~   ||~   ||~   ||~   ||
-||= (P8/3, P5)
+||= 6 ||= (P8/3, P5)
 third-8ve ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||
-||= (P8, P4/3)
+||= 7 ||= (P8, P4/3)
 third-4th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3 ``=`` &lt;/span&gt;C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||
-||= (P8, P5/3)
+||= 8 ||= (P8, P5/3)
 third-5th ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
-||= (P8, P11/3)
+||= 9 ||= (P8, P11/3)
 third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,
 with 11/8 = A4 ||
-||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||
+||=   ||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||
-||= (P8/3, P4/2)
+||= 10 ||= (P8/3, P4/2)
 third-8ve, half-4th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;A2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D# ||= P8/3 = ^^m3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A4
 P4/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= C - Eb^^ - Avv - C
 C - Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - F ||= sixfold jade ||
-||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
+||=   ||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
 \\m2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B#
 C``//`` = Db ||= P8/3 = vM3 = ^^d4
 P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C
 C - D/=Eb\ - F ||= 128/125 &amp; 49/48 ||
-||= (P8/3, P5/2)
+||= 11 ||= (P8/3, P5/2)
 third-8ve, half-5th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2
 \\A1 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B#
@@ Line 230: / Line 230: @@
 P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C
 C - Eb/=E\ - G ||= small sixfold blue ||
-||= (P8/2, P4/3)
+||= 12 ||= (P8/2, P4/3)
 half-8ve, third-4th ||= ^^d2
 \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^^ ``=`` Dbb
@@ Line 236: / Line 236: @@
 P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C
 C - D\ - Eb/ - F ||= large sixfold red ||
-||= (P8/2, P5/3)
+||= 13 ||= (P8/2, P5/3)
 half-8ve,
 third-5th ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P8/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5
 P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; C
 C - D#vv - Fb^^ - G ||= large sixfold yellow ||
-||= " ||= ^^d2,
+||=   ||= " ||= ^^d2,
 \\\m2 ||= C^^ = B#
 C``///`` = Db ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
 C - /D - \F - G ||= 50/49 &amp; 1029/1024 ||
-||= (P8/2, P11/3)
+||= 14 ||= (P8/2, P11/3)
 half-8ve,
 third-11th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D ||= P8/2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;5
 P11/3 = ^^4 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;5 ||= C - F^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; - C
 C - F^^ - Cvv - F ||= large sixfold jade ||
-||= (P8/3, P4/3)
+||= 15 ||= (P8/3, P4/3)
 third-
 everything ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
@@ Line 262: / Line 262: @@
 ^1 = 64/63
 /1 = 81/80 ||
-||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
+||=   ||= " ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2,
 \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` B#
 C/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` Db ||= P8/3 = vM3 = ^^d4
@@ Line 271: / Line 271: @@
 ^1 = 81/80
 /1 = 64/63 ||
-||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1,
+||=   ||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1,
 \&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt; ``=`` C#
 C/3 ``=`` Db ||= P4/3 = vM2 = ^^m2
@@ Line 280: / Line 280: @@
 ^1 = 81/80
 /1 = 64/63 ||
-||~ quarters ||~   ||~   ||~   ||~   ||~   ||
+||~   ||~ quarters ||~   ||~   ||~   ||~   ||~   ||
-||= (P8/4, P5) ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` B# ||= P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished ||
+||= 16 ||= (P8/4, P5) ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` B# ||= P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished ||
-||= (P8, P4/4) ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` B## ||= P4/4 = ^m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= negri ||
+||= 17 ||= (P8, P4/4) ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` B## ||= P4/4 = ^m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= negri ||
-||= (P8, P5/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` C# ||= P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||
+||= 18 ||= (P8, P5/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` C# ||= P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||
-||= (P8, P11/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P11/4 = ^M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5 ||= C E^ G#^^ Dbv F ||= squares ||
+||= 19 ||= (P8, P11/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= P11/4 = ^M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5 ||= C E^ G#^^ Dbv F ||= squares ||
-||= (P8, P12/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` Db ||= P12/4 = v4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture ||
+||= 20 ||= (P8, P12/4) ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ``=`` Db ||= P12/4 = v4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= C Fv Bbvv=A^^ D^ G ||= vulture ||
-||= etc. ||=   ||=   ||=   ||=   ||=   ||
+||=   ||= etc. ||=   ||=   ||=   ||=   ||=   ||
 ==Tipping points==
@@ Line 517: / Line 517: @@
 A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.
-==Notating rank-3 pergens *==
+==Notating rank-3 pergens*==
 Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:
@@ Line 533: / Line 533: @@
 ||= breedsmic ||= rank-3 ||= double-pair ||= rank-4 ||= 1 ||= E = \\dd3 ||
 ||= 7-limit JI ||= rank-4 ||= double-pair ||= rank-4 ||= 0 ||= --- ||
-A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one may wish to simply use colors.
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even "superfalse" triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one can simply use colors.
 All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows:
 ||~ 7-limit temperament ||~ comma ||~ pergen ||~ spoken pergen ||~ notation ||~ period ||~ gen1 ||~ gen2 ||~ enharmonic ||
-||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= unsplit with ups ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||
+||= marvel ||= 225/224 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= single-pair ||= P8 ||= P5 ||= ^1 = 81/80 ||= --- ||
-||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= half-8ve with ups ||= double-pair ||= /d5 = 7/5 ||= P5 ||= ^1 = 81/80 ||= ``//``d2 ||
+||= deep reddish ||= 50/49 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= double-pair ||= v/A4 = 10/7 ||= P5 ||= ^1 = 81/80 ||= ^^\\d2 ||
-||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= third-11th with ups ||= double-pair ||= P8 ||= \d5 = 7/5 ||= ^1 = 81/80 ||= \\\dd3 ||
+||= triple bluish ||= 1029/1000 ||= (P8, P11/3, ^1) ||= rank-3 third-11th ||= double-pair ||= P8 ||= ^\d5 = 7/5 ||= ^1 = 81/80 ||= ^^^\\\dd3 ||
-||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= half-5th with ups ||= double-pair ||= P8 ||= \d4 = 49/40 ||= ^1 = 64/63 ||= \\dd3 ||
+||= breedsmic ||= 2401/2400 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= double-pair ||= P8 ||= v``//``A2 = 60/49 ||= /1 = 64/63 ||= ^^\&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||
 ||= demeter ||= 686/675 ||= (P8, P5, vM3/3) ||= third-downmajor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||
-Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, ^1 = 81/80, and /1 = 64/63. Gen2 =v/A1 and E = ^^\\\dd3. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains, the additional accidentals get progressively more complex.
+Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, where ^1 = 81/80, and /1 = 64/63. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains we've seen, the additional accidentals get progressively more complex.
-To avoid E being a 3rd, 7/4 could instead be mapped to an A6, making gen2 = v/m2
+In double-pair rank-3 notation, the accidentals generally represent mapping commas, with ups and downs applying to the lowest prime &gt; 3 and highs and lows applying to the other prime. It would be possible to have them represent the sum or difference of the mapping commas, but this would make the notation harder to read, IMO. Probably makes it more tippy, too.
-G = ^m2, C - Db^ - Ebb^^ - Fbbb^^^, chord C- Fbbb^^^ - G - Bbb^^
+The E is an awkward dd3 because that's what the comma maps to. It's possible to find an E that's a 2nd or even a unison, but chords will be spelled improperly:
-fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢
+gen2 = vM3/3 = v[4,2]/3 = [1,1] = /m2, E = [-1,1] = ^``///``dd2, genchain = C -- Db/ -- Ebb``//``=D#v\ -- Ev, 4:5:6:7 chord = C Ev G Bbb``//``
+gen2 = \m3/2 = \[3,2]/2 = [1,1] = ^m2 = v\M2, E = [1,0] = vv\A1, genchain = C -- Db^ -- Eb\ -- Fb^\, 4:5:6:7 chord = C Fb^\ G Bb\
-gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^``///``dd2, C^``///`` = B##, C -- Db/ -- Ebb//=D#
+The dd3 enharmonic implies a tipping point of 10\17 = 706¢, which falls within demeter's sweet spot of fifth-comma to eighth-comma. However, neither 81/80 nor 64/63 vanish, and demeter doesn't tip. The commas only vanish when the 5th = 10\17 and the vM3 (5/4) = 6\17 and the \m7 (7/4) = 14\17. These are the 17b-edo values, which tempers out both 81/80 and 64/63. But a 5/4 of 423¢ is far from sweet! So demeter doesn't tip. (what a cheapskate she is!)
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3
+The true meaning of the ^^\\\dd3 enharmonic is that 686/675 = ^^1 - ///1 + dd3 = (81/80)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;-3&lt;/span&gt; · (27,-17) = 0¢. We write the comma as a 3-limit comma plus or minus some number of mapping commas. The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes. In a rank-2 temperament, the mapping comma must vanish too, because some number of them plus the 3-limit comma add up to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to 0¢.
+Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. So rank-3 temperaments don't tip?
+Deep reddish doesn't tip unless w5 = 700¢, y3 = 400¢ and b7 = 1000¢. Deep reddish is double-pair, So 50/49 = -\\d2 = //1 - d2 = R · R · (19,-12), and R = (-9,6,1,-1) and 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7
+/49 = vv//-d2 = -^^\\d2 = //1 - ^^1 - d2 = 64/63 · 64/63 / (81/80 · 81/80 · (19,-12)) = 0¢.
+Tell Praveen; "half-8ve with ups" changed to "rank-3 half-8ve"
+rank-3 pergens don't tip!
+complex enharmonics and pergens are better than accidentals not being mapping commas?
 ==Notating Blackwood-like pergens*==
@@ Line 566: / Line 573: @@
 edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \
-==Notating arbitrary P and G tunings*==
+==Notating tunings with an arbitrary generator==
+Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is set. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.
+The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. use the octave inverse if G &gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.
+||||~ primary choice ||||~ secondary choice ||||~ tertiary choice ||
+||~ generator ||~ possible multigen ||~ generator ||~ multigen ||~ generator ||~ multigen ||
+||= 23-60¢ ||= M2/4 (requires P8/2) ||=   ||=   ||=   ||=   ||
+||= 69-79¢ ||= P4/7 ||=   ||=   ||=   ||=   ||
+||= 80-92¢ ||= P4/6 ||=   ||=   ||=   ||=   ||
+||= 92-103¢ ||= P5/7 ||=   ||=   ||=   ||=   ||
+||= 96-111¢ ||= P4/5 ||=   ||=   ||=   ||=   ||
+||= 108-120¢ ||= P5/6 ||=   ||=   ||=   ||=   ||
+||= 120-138¢ ||= P4/4 ||=   ||=   ||=   ||=   ||
+||= 129-144¢ ||= P5/5 ||=   ||=   ||=   ||=   ||
+||= 160-185¢ ||= P4/3 ||= 162-180¢ ||= P5/4 ||=   ||=   ||
+||= 215-240¢ ||= P5/3 ||=   ||=   ||=   ||=   ||
+||= 240-277¢ ||= P4/2 ||= 240-251¢ ||= P11/7 ||= 264-274¢ ||= P12/7 ||
+||= 280-292¢ ||= P11/6 ||=   ||=   ||=   ||=   ||
+||= 308-320¢ ||= P12/6 ||=   ||=   ||=   ||=   ||
+||= 323-360¢ ||= P5/2 ||= 336-351¢ ||= P11/5 ||=   ||=   ||
+||= 369-384¢ ||= P12/5 ||=   ||=   ||=   ||=   ||
+||= 411-422¢ ||= WWP4/7 ||=   ||=   ||=   ||=   ||
+||= 420-438¢ ||= P11/4 ||=   ||=   ||=   ||=   ||
+||= 435-446¢ ||= WWP5/7 ||=   ||=   ||=   ||=   ||
+||= 462-480¢ ||= P12/4 ||=   ||=   ||=   ||=   ||
+||= 480-554¢ ||= P4 = P5 ||= 480-492¢ ||= WWP4/6 ||= 508-520¢ ||= WWP5/6 ||
+||= 560-585¢ ||= P11/3 ||=   ||=   ||=   ||=   ||
+||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||=   ||=   ||
+The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with heptatonic notation.
-Given only the period as some fraction of the octave, and the generator's cents, it's often possible to work backwards and find an appropriate multigen.
+Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.
 ==Pergens and MOS scales==
@@ Line 844: / Line 881: @@
 ==Miscellaneous Notes==
+__**Staff notation**__
+Highs and lows can be added to the score just like ups and downs can.
 __**Combining pergens**__
@@ Line 879: / Line 920: @@
 Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div>
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+&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:111 --&gt;&lt;!-- ws:start:WikiTextTocRule:112: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Naming very large intervals"&gt;Naming very large intervals&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding an example temperament"&gt;Finding an example temperament&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Finding a notation for a pergen"&gt;Finding a notation for a pergen&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Tipping points and sweet spots"&gt;Tipping points and sweet spots&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating unsplit pergens"&gt;Notating unsplit pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:120 --&gt;&lt;!-- ws:start:WikiTextTocRule:121: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens *"&gt;Notating rank-3 pergens *&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating rank-3 pergens*"&gt;Notating rank-3 pergens*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:121 --&gt;&lt;!-- ws:start:WikiTextTocRule:122: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens*"&gt;Notating Blackwood-like pergens*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating Blackwood-like pergens*"&gt;Notating Blackwood-like pergens*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:122 --&gt;&lt;!-- ws:start:WikiTextTocRule:123: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating arbitrary P and G tunings*"&gt;Notating arbitrary P and G tunings*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating tunings with an arbitrary generator"&gt;Notating tunings with an arbitrary generator&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:123 --&gt;&lt;!-- ws:start:WikiTextTocRule:124: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and MOS scales"&gt;Pergens and MOS scales&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:124 --&gt;&lt;!-- ws:start:WikiTextTocRule:125: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:125 --&gt;&lt;!-- ws:start:WikiTextTocRule:126: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*"&gt;Supplemental materials*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:126 --&gt;&lt;!-- ws:start:WikiTextTocRule:127: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Various proofs (unfinished)"&gt;Various proofs (unfinished)&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:127 --&gt;&lt;!-- ws:start:WikiTextTocRule:128: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Miscellaneous Notes"&gt;Miscellaneous Notes&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:59:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:59 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,145: / Line 1,186: @@
 The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).&lt;br /&gt;
 &lt;br /&gt;
-Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc. &lt;br /&gt;
+Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.&lt;br /&gt;
 &lt;br /&gt;
-For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups.&lt;br /&gt;
+For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.&lt;br /&gt;
 &lt;br /&gt;
-More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = third-11th with ups. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = half-8ve with ups (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd. &lt;br /&gt;
+More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, deep reddish minus white (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.&lt;br /&gt;
 &lt;br /&gt;
 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.&lt;br /&gt;
@@ Line 1,156: / Line 1,197: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:61:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:61 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;
@@ Line 1,333: / Line 1,374: @@
 &lt;/table&gt;
-Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25/6)/4.&lt;br /&gt;
+Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.&lt;br /&gt;
 &lt;br /&gt;
 Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multigen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multigen. The alternate gens are P11/2 and P19/2, both of which are much larger, so the best gen1 is P5/2.&lt;br /&gt;
 &lt;br /&gt;
-The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96/25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128/75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675/512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.&lt;br /&gt;
+The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.&lt;br /&gt;
 &lt;br /&gt;
 Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:&lt;br /&gt;
@@ Line 1,439: / Line 1,480: @@
 &lt;/table&gt;
-Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, 64:63). Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th with ups. This is far better than (P8, P5/2, (96:25)/4).&lt;br /&gt;
+Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).&lt;br /&gt;
 &lt;br /&gt;
 The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.&lt;br /&gt;
@@ Line 1,456: / Line 1,497: @@
 All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. Certain rank-2 temperaments require another additional pair, &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.&lt;br /&gt;
 &lt;br /&gt;
-One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes backwards compatibility to be essential.&lt;br /&gt;
+One can avoid additional accidentals for all rank-1 and rank-2 tunings (but not rank-3 or higher ones) by sacrificing backwards compatibility with conventional notation, defined as octave-equivalent, heptatonic, and fifth-generated. Porcupine can be notated without ups and downs if the notation is 2nd-generated. Half-octave can be notated decatonically. However, one would sacrifice the interval arithmetic and staff notation one has spent years internalizing, and naming chords becomes impossible. The sacrifice is too great to take lightly, and this discussion assumes that backwards compatibility is desirable.&lt;br /&gt;
 &lt;br /&gt;
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. See &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20unsplit%20pergens"&gt;Notating unsplit pergens&lt;/a&gt; below.&lt;br /&gt;
@@ Line 1,462: / Line 1,503: @@
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.&lt;br /&gt;
 &lt;br /&gt;
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.&lt;br /&gt;
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that allows every pergen to be numbered.&lt;br /&gt;
 &lt;br /&gt;
 The enharmonic interval, or more briefly the &lt;strong&gt;enharmonic&lt;/strong&gt;, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. The pergen and the enharmonic together define the notation.&lt;br /&gt;
@@ Line 1,474: / Line 1,515: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
          &lt;th&gt;pergen&lt;br /&gt;
 &lt;/th&gt;
@@ Line 1,492: / Line 1,535: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
 unsplit&lt;br /&gt;
@@ Line 1,508: / Line 1,553: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
          &lt;th&gt;halves&lt;br /&gt;
 &lt;/th&gt;
@@ Line 1,522: / Line 1,569: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
 half-8ve&lt;br /&gt;
@@ Line 1,539: / Line 1,588: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,555: / Line 1,606: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,570: / Line 1,623: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
 half-4th&lt;br /&gt;
@@ Line 1,586: / Line 1,641: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,600: / Line 1,657: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P5/2)&lt;br /&gt;
 half-5th&lt;br /&gt;
@@ Line 1,616: / Line 1,675: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P4/2)&lt;br /&gt;
 half-&lt;br /&gt;
@@ Line 1,645: / Line 1,706: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,669: / Line 1,732: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,693: / Line 1,758: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
          &lt;th&gt;thirds&lt;br /&gt;
 &lt;/th&gt;
@@ Line 1,707: / Line 1,774: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/3, P5)&lt;br /&gt;
 third-8ve&lt;br /&gt;
@@ Line 1,722: / Line 1,791: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P4/3)&lt;br /&gt;
 third-4th&lt;br /&gt;
@@ Line 1,737: / Line 1,808: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P5/3)&lt;br /&gt;
 third-5th&lt;br /&gt;
@@ Line 1,752: / Line 1,825: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P11/3)&lt;br /&gt;
 third-11th&lt;br /&gt;
@@ Line 1,768: / Line 1,843: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,782: / Line 1,859: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/3, P4/2)&lt;br /&gt;
 third-8ve, half-4th&lt;br /&gt;
@@ Line 1,799: / Line 1,878: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,817: / Line 1,898: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/3, P5/2)&lt;br /&gt;
 third-8ve, half-5th&lt;br /&gt;
@@ Line 1,836: / Line 1,919: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P4/3)&lt;br /&gt;
 half-8ve, third-4th&lt;br /&gt;
@@ Line 1,855: / Line 1,940: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P5/3)&lt;br /&gt;
 half-8ve,&lt;br /&gt;
@@ Line 1,873: / Line 1,960: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,891: / Line 1,980: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;14&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/2, P11/3)&lt;br /&gt;
 half-8ve,&lt;br /&gt;
@@ Line 1,909: / Line 2,000: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;15&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/3, P4/3)&lt;br /&gt;
 third-&lt;br /&gt;
@@ Line 1,933: / Line 2,026: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,955: / Line 2,050: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,977: / Line 2,074: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
          &lt;th&gt;quarters&lt;br /&gt;
 &lt;/th&gt;
@@ Line 1,991: / Line 2,090: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;16&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8/4, P5)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,005: / Line 2,106: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;17&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P4/4)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,019: / Line 2,122: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P5/4)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,033: / Line 2,138: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;19&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P11/4)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,047: / Line 2,154: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;20&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(P8, P12/4)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,061: / Line 2,170: @@
      &lt;/tr&gt;
      &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;etc.&lt;br /&gt;
 &lt;/td&gt;
@@ Line 2,077: / Line 2,188: @@
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Applications-Tipping points"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;Tipping points&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Applications-Tipping points"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;Tipping points&lt;/h2&gt;
   &lt;br /&gt;
 Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
@@ Line 2,323: / Line 2,434: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:67:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:67 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Naming very large intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;Naming very large intervals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Naming very large intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt;Naming very large intervals&lt;/h2&gt;
   &lt;br /&gt;
 So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &amp;quot;W&amp;quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Secondary splits"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;Secondary splits&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:71:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Secondary splits"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:71 --&gt;Secondary splits&lt;/h2&gt;
   &lt;br /&gt;
 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 (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:&lt;br /&gt;
@@ Line 2,351: / Line 2,462: @@
 For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;Singles and doubles&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt;Singles and doubles&lt;/h2&gt;
   &lt;br /&gt;
 If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.&lt;br /&gt;
@@ Line 2,361: / Line 2,472: @@
 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;Finding an example temperament&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt;Finding an example temperament&lt;/h2&gt;
   &lt;br /&gt;
 To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.&lt;br /&gt;
@@ Line 2,383: / Line 2,494: @@
 There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
 In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. The sharp symbol is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;
@@ Line 2,411: / Line 2,522: @@
 Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;Finding a notation for a pergen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:79:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:79 --&gt;Finding a notation for a pergen&lt;/h2&gt;
   &lt;br /&gt;
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
@@ Line 2,445: / Line 2,556: @@
 This is a lot of math, but it only needs to be done once for each pergen!&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;Alternate enharmonics&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:81:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:81 --&gt;Alternate enharmonics&lt;/h2&gt;
   &lt;br /&gt;
 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&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.&lt;br /&gt;
@@ Line 2,474: / Line 2,585: @@
 This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:82:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:82 --&gt;Chord names and scale names&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;Chord names and scale names&lt;/h2&gt;
   &lt;br /&gt;
 Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&lt;br /&gt;
@@ Line 2,488: / Line 2,599: @@
 Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:84:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:84 --&gt;Tipping points and sweet spots&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Tipping points and sweet spots"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;Tipping points and sweet spots&lt;/h2&gt;
   &lt;br /&gt;
 As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament &amp;quot;tips over&amp;quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.&lt;br /&gt;
@@ Line 2,500: / Line 2,611: @@
 An example of a temperament that tips easily is negri, 2.3.5 and (-14,3,4). Because negri is 5-limit, the mapping is unambiguous, and the comma must be a dd2, implying 19-edo and a 694.74¢ tipping point. This coincides almost exactly with negri's seventh-comma sweet spot, where 6/5 is just and 3/2 = 694.65¢. Negri's pergen is quarter-4th, with ^ = 25¢ + 4.75c, very nearly 0¢. The up symbol represents either 81/80 or 80/81, and E could be either ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 or v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2. When the choice is so arbitrary, it's best to avoid inverting the ratio. 81/80 implies an E of ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 and a G of ^m2. Negri's generator is 16/15, which is indeed a m2 raised by 81/80. In practice, seventh-comma negri's 5th is only 0.085¢ from 19-edo's 5th, the tuning is hardly audibly different than 19edo, and the piece can be written out in 19edo notation.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:86:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:86 --&gt;Notating unsplit pergens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Notating unsplit pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Notating unsplit pergens&lt;/h2&gt;
   &lt;br /&gt;
 An unsplit pergen doesn't &lt;u&gt;require&lt;/u&gt; ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a comma that maps to P1, such as 81/80 or 64/63. For single-comma temperaments, the pergen is unsplit if and only if the vanishing comma's monzo has a final exponent of ±1.&lt;br /&gt;
@@ Line 2,707: / Line 2,818: @@
 A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such comma except for 64/63 might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:88:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens *"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:88 --&gt;Notating rank-3 pergens *&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Notating rank-3 pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;Notating rank-3 pergens*&lt;/h2&gt;
   &lt;br /&gt;
 Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Examples:&lt;br /&gt;
@@ Line 2,897: / Line 3,008: @@
 &lt;/table&gt;
-A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one may wish to simply use colors.&lt;br /&gt;
+A rank-2 pergen is either unsplit, single-split or double-split, and a double-split is either a true double or a false double. A rank-3 pergen can be any of these. Additionally, some rank-3 pergens are triple-splits, which are either true triples or false triples. False triples are like true doubles in that they only require two commas and two new accidental pairs. (There are even &amp;quot;superfalse&amp;quot; triples that can arise from a single comma, but the higher prime's exponent in the comma must be at least 12, making it difficult to pump, and not very useful musically.) All true triples require triple-pair notation, but some false triples and some double-splits may use triple-pair as well, to avoid awkward enharmonics of a 3rd or more. Rather than devising a third pair of symbols, and a third pair of adjectives to describe them, one can simply use colors.&lt;br /&gt;
 &lt;br /&gt;
 All the previous rank-3 examples had the 2nd generator be a P1 comma, represented by an up. However, this isn't always possible, as the last example shows:&lt;br /&gt;
@@ Line 2,930: / Line 3,041: @@
          &lt;td style="text-align: center;"&gt;(P8, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;unsplit with ups&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 unsplit&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;single-pair&lt;br /&gt;
@@ Line 2,950: / Line 3,061: @@
          &lt;td style="text-align: center;"&gt;(P8/2, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-8ve with ups&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-8ve&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;double-pair&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;/d5 = 7/5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;v/A4 = 10/7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
@@ Line 2,960: / Line 3,071: @@
          &lt;td style="text-align: center;"&gt;^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;d2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^\\d2&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,970: / Line 3,081: @@
          &lt;td style="text-align: center;"&gt;(P8, P11/3, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;third-11th with ups&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 third-11th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;double-pair&lt;br /&gt;
@@ Line 2,976: / Line 3,087: @@
          &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;\d5 = 7/5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^\d5 = 7/5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;\\\dd3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^^\\\dd3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,990: / Line 3,101: @@
          &lt;td style="text-align: center;"&gt;(P8, P5/2, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-5th with ups&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-5th&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;double-pair&lt;br /&gt;
@@ Line 2,996: / Line 3,107: @@
          &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;\d4 = 49/40&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;v&lt;!-- ws:start:WikiTextRawRule:053:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:053 --&gt;A2 = 60/49&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;^1 = 64/63&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;/1 = 64/63&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;\\dd3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;^^\&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 3,025: / Line 3,136: @@
 &lt;/table&gt;
-Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, ^1 = 81/80, and /1 = 64/63. Gen2 =v/A1 and E = ^^\\\dd3. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains, the additional accidentals get progressively more complex.&lt;br /&gt;
+Demeter divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, y3/3) or (P8, P5, vM3/3). It could also be called (P8, P5, b3/2) or (P8, P5, vm3/2). Single-pair notation is possible, with gen2 = ^m2 and no E, but the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^! Double-pair notation is better, with a notation similar to 7-limit JI, where ^1 = 81/80, and /1 = 64/63. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\. Unlike other genchains we've seen, the additional accidentals get progressively more complex.&lt;br /&gt;
 &lt;br /&gt;
-To avoid E being a 3rd, 7/4 could instead be mapped to an A6, making gen2 = v/m2&lt;br /&gt;
+In double-pair rank-3 notation, the accidentals generally represent mapping commas, with ups and downs applying to the lowest prime &amp;gt; 3 and highs and lows applying to the other prime. It would be possible to have them represent the sum or difference of the mapping commas, but this would make the notation harder to read, IMO. Probably makes it more tippy, too.&lt;br /&gt;
 &lt;br /&gt;
-G = ^m2, C - Db^ - Ebb^^ - Fbbb^^^, chord C- Fbbb^^^ - G - Bbb^^&lt;br /&gt;
+The E is an awkward dd3 because that's what the comma maps to. It's possible to find an E that's a 2nd or even a unison, but chords will be spelled improperly:&lt;br /&gt;
-fifth-comma: c = 9.5¢, P5 = 700¢ + c, vM3 = 400¢ + d, d = -10¢&lt;br /&gt;
+gen2 = vM3/3 = v[4,2]/3 = [1,1] = /m2, E = [-1,1] = ^&lt;!-- ws:start:WikiTextRawRule:054:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:054 --&gt;dd2, genchain = C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:055:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:055 --&gt;=D#v\ -- Ev, 4:5:6:7 chord = C Ev G Bbb&lt;!-- ws:start:WikiTextRawRule:056:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:056 --&gt;&lt;br /&gt;
+gen2 = \m3/2 = \[3,2]/2 = [1,1] = ^m2 = v\M2, E = [1,0] = vv\A1, genchain = C -- Db^ -- Eb\ -- Fb^\, 4:5:6:7 chord = C Fb^\ G Bb\&lt;br /&gt;
 &lt;br /&gt;
-gen2 = vM3/3 = [4,2,-1,0]/3 = [1,1,0,1] = /m2, E = [-1,1,1,3] = ^&lt;!-- ws:start:WikiTextRawRule:054:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:054 --&gt;dd2, C^&lt;!-- ws:start:WikiTextRawRule:055:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:055 --&gt; = B##, C -- Db/ -- Ebb//=D#&lt;br /&gt;
+The dd3 enharmonic implies a tipping point of 10\17 = 706¢, which falls within demeter's sweet spot of fifth-comma to eighth-comma. However, neither 81/80 nor 64/63 vanish, and demeter doesn't tip. The commas only vanish when the 5th = 10\17 and the vM3 (5/4) = 6\17 and the \m7 (7/4) = 14\17. These are the 17b-edo values, which tempers out both 81/80 and 64/63. But a 5/4 of 423¢ is far from sweet! So demeter doesn't tip. (what a cheapskate she is!)&lt;br /&gt;
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,1,1,0] = ^m2 = v\M2, E = [1,0,-2,-1] = vv\A1&lt;br /&gt;
 &lt;br /&gt;
-gen2 = \m3/2 = [3,2,0,-1]/2 = [1,0,-1,1] = v/A1, E = [1,2,2,-3] = ^^\\\dd3&lt;br /&gt;
+The true meaning of the ^^\\\dd3 enharmonic is that 686/675 = ^^1 - &lt;em&gt;/1 + dd3 = (81/80)&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; · (64/63)&lt;span style="vertical-align: super;"&gt;-3&lt;/span&gt; · (27,-17) = 0¢. We write the comma as a 3-limit comma plus or minus some number of mapping commas. The 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes. In a rank-2 temperament, the mapping comma must vanish too, because some number of them plus the 3-limit comma add up to 0¢. However, a rank-3 temperament has two mapping commas, and neither is forced to 0¢.&lt;br /&gt;
 &lt;br /&gt;
+Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. So rank-3 temperaments don't tip?&lt;br /&gt;
 &lt;br /&gt;
+Deep reddish doesn't tip unless w5 = 700¢, y3 = 400¢ and b7 = 1000¢. Deep reddish is double-pair, So 50/49 = -\\d2 = &lt;/em&gt;1 - d2 = R · R · (19,-12), and R = (-9,6,1,-1) and 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7&lt;br /&gt;
+/49 = vv&lt;em&gt;-d2 = -^^\\d2 = &lt;/em&gt;1 - ^^1 - d2 = 64/63 · 64/63 / (81/80 · 81/80 · (19,-12)) = 0¢.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:90:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:90 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
+Tell Praveen; &amp;quot;half-8ve with ups&amp;quot; changed to &amp;quot;rank-3 half-8ve&amp;quot;&lt;br /&gt;
+rank-3 pergens don't tip!&lt;br /&gt;
+complex enharmonics and pergens are better than accidentals not being mapping commas?&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Further Discussion-Notating Blackwood-like pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Notating Blackwood-like pergens*&lt;/h2&gt;
   &lt;br /&gt;
 A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.&lt;br /&gt;
@@ Line 3,049: / Line 3,167: @@
 edo+y perchain: C C^ Dv D... (E = dd3, E' = vm2 = vvA1), genchain: same as blackwood, but with / and \&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:92:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating arbitrary P and G tunings*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:92 --&gt;Notating arbitrary P and G tunings*&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
   &lt;br /&gt;
-Given only the period as some fraction of the octave, and the generator's cents, it's often possible to work backwards and find an appropriate multigen.&lt;br /&gt;
+Given only the generator's cents and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is set. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:94:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:94 --&gt;Pergens and MOS scales&lt;/h2&gt;
+The next table lists all the ranges for all multigens up to seventh-splits. You can look up your generator in the first column and find a possible multigen. use the octave inverse if G &amp;gt; 600¢. Some ranges overlap. Those that are contained entirely within another range are listed in the righthand columns.&lt;br /&gt;
- &lt;br /&gt;
-Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 3,061: / Line 3,177: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;pergen&lt;br /&gt;
+         &lt;th colspan="2"&gt;primary choice&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="2"&gt;secondary choice&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="2"&gt;tertiary choice&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;generator&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th colspan="2"&gt;MOS scales of 5-12 notes&lt;br /&gt;
+         &lt;th&gt;possible multigen&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;th&gt;generator&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;th&gt;multigen&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;th&gt;generator&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;&lt;br /&gt;
+         &lt;th&gt;multigen&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;23-60¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;unsplit&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;M2/4 (requires P8/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;5 = 2L 3s&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;7 = 5L 2s&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="2"&gt;12 = 7L 5s (or 5L 7s)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;halves&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;69-79¢&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;P4/7&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/th&gt;
+&lt;/td&gt;
-         &lt;th&gt;&lt;br /&gt;
+    &lt;/tr&gt;
-&lt;/th&gt;
+    &lt;tr&gt;
-     &lt;/tr&gt;
+        &lt;td style="text-align: center;"&gt;80-92¢&lt;br /&gt;
-     &lt;tr&gt;
+&lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;P4/6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-8ve&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;6 = 2L 4s&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;8 = 2L 6s&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;10 = 2L 8s&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="3"&gt;12 = 2L 10s (or 10L 2s)&lt;br /&gt;
+    &lt;/tr&gt;
-&lt;/td&gt;
+    &lt;tr&gt;
-     &lt;/tr&gt;
+        &lt;td style="text-align: center;"&gt;92-103¢&lt;br /&gt;
-     &lt;tr&gt;
+&lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;P5/7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;half-4th&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;5 = 4L 1s&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;9 = 5L 4s&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+    &lt;/tr&gt;
-&lt;/td&gt;
+    &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;96-111¢&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;P4/5&lt;br /&gt;
 &lt;/td&gt;
-     &lt;/tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;108-120¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;120-138¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P4/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;129-144¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;160-185¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P4/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;162-180¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;215-240¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;240-277¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;240-251¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P11/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;264-274¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P12/7&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;280-292¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P11/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;308-320¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P12/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;323-360¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;336-351¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P11/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;369-384¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P12/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;411-422¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWP4/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;420-438¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P11/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;435-446¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWP5/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;462-480¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P12/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;480-554¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P4 = P5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;480-492¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWP4/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;508-520¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWP5/6&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;560-585¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P11/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;576-591¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWP4/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;583-593¢&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;WWWP4/7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+The total range of possible generators is fairly well covered, but there are gaps, especially near 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with heptatonic notation.&lt;br /&gt;
+&lt;br /&gt;
+Splitting the octave creates alternate generators. For example, if P = 400¢ and G = 500¢, alternate generators are 100¢ and 300¢. Any of these can be used to find a convenient multigen.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:95:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Pergens and MOS scales&lt;/h2&gt;
+ &lt;br /&gt;
+Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th colspan="2"&gt;pergen&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="2"&gt;MOS scales of 5-12 notes&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;unsplit&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;5 = 2L 3s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;7 = 5L 2s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td colspan="2"&gt;12 = 7L 5s (or 5L 7s)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;th colspan="2"&gt;halves&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+         &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+     &lt;/tr&gt;
+     &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;half-8ve&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;6 = 2L 4s&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;8 = 2L 6s&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;10 = 2L 8s&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td colspan="3"&gt;12 = 2L 10s (or 10L 2s)&lt;br /&gt;
+&lt;/td&gt;
+     &lt;/tr&gt;
+     &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;half-4th&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;5 = 4L 1s&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;9 = 5L 4s&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+     &lt;/tr&gt;
      &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;(P8, P5/2)&lt;br /&gt;
@@ Line 4,161: / Line 4,620: @@
 Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:96:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:96 --&gt;Pergens and EDOs&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:97:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:97 --&gt;Pergens and EDOs&lt;/h2&gt;
   &lt;br /&gt;
 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;
@@ Line 4,469: / Line 4,928: @@
 See the next section for examples of which pergens are supported by a specific edo.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&gt;Supplemental materials*&lt;/h2&gt;
   &lt;br /&gt;
 needs more screenshots, including 12-edo's pergens&lt;br /&gt;
@@ Line 4,475: / Line 4,934: @@
 &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:6171: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:6171 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextUrlRule:6172: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:6172 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Screenshot of the first 38 pergens:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3234:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3234 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3637:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 460px; width: 704px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3637 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;
@@ Line 4,495: / Line 4,954: @@
 The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;
 &lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Various proofs (unfinished)&lt;/h2&gt;
   &lt;br /&gt;
 The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2 = 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). Likewise, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&lt;br /&gt;
@@ Line 4,588: / Line 5,047: @@
 Although not rigorously proven, these last two tests have been empirically verified by alt-pergenLister.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:102:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:102 --&gt;Miscellaneous Notes&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:103:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:103 --&gt;Miscellaneous Notes&lt;/h2&gt;
   &lt;br /&gt;
+&lt;u&gt;&lt;strong&gt;Staff notation&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+&lt;br /&gt;
+Highs and lows can be added to the score just like ups and downs can.&lt;br /&gt;
+&lt;br /&gt;
 &lt;u&gt;&lt;strong&gt;Combining pergens&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
 &lt;br /&gt;