Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

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

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

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For any comma, 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, where GCD (0,x) = x. The comma will split the octave into m parts, and if n > m, it will split some 3-limit interval into n parts.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also ~~directly related~~, the 4th prime is used, and so forth.

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is **dependent** on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two **independent** primes.

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The ~~multigen must use primes 2 and 5. In this case, the~~ pergen is (P8/5, ^1), the same as Blackwood.

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Further%20Discussion-Notating%20Blackwood-like%20pergens|Blackwood-like pergens]] below).

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes ~~may~~ need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.

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

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<span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span>**

</span>

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.

For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 <= i <= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:

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=__Applications__=

~~One~~ obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called 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.

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.

Another 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, either because it contains more than 2 primes, or because the split 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 (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

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Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further%20Discussion-Chord%20names%20and%20scale%20names|Chord names and scale names]] below.

The ~~third~~ main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.

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 all notation used here is backwards compatible.

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, which is octave-equivalent, fifth-generated and heptatonic. 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 all notation used here is backwards compatible.

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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P4/3 = v\M2 ||= C - Ev - Ab^ - C

C - D/ - F\ - G

C - Dv\ - Eb^/ - F ||= ~~128/125~~ & ~~1029/1024~~

C - Dv\ - Eb^/ - F ||= triple green & large triple blue

^1 = 81/80

/1 = 64/63 ||

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P8/3 = v/M3 ||= C - Dv - Eb^ - F

C - D/ - F\ - G

C - Ev/ - Ab^\ - C ||= ~~250/243~~ & ~~1029/1024~~

C - Ev/ - Ab^\ - C ||= triple yellow & large triple blue

^1 = 81/80

/1 = 64/63 ||

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||= 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. ||= ||= ||= ||= ||= ||

The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex.

The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't all that complex.

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).

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==Secondary splits==

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

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve 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:

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

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||= (P8, P4/2) ||= half-4th ||= m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2 ||

||= (P8, P5/2) ||= half-5th ||= A1/2, m3/2, M7/2, m9/2, P11/2 ||

||= (P8/2, P4/2) ||= half-everything ||= (every 3-limit interval~~)/2~~ ||

||= (P8/2, P4/2) ||= half-everything ||= every 3-limit interval is split twice as muh as before ||

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

||= (P8/3, P5) ||= third-8ve ||= m3/3, M6/3, d5/6, A11/3, d12/3 ||

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||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve splits, third-5th splits, m6/6, M9/6, A12/6 ||

||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24 ||

||= (P83, P4/3) ||= third-everything ||= (every 3-limit interval~~)/3~~ ||

||= (P83, P4/3) ||= third-everything ||= every 3-limit interval is split three times as much as before ||

==Singles and doubles==

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==Finding an example temperament==

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.

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, the diminished temperament. 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>, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63. Thus for (P8/4, P5), ~~since~~ P = vm3, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).

If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.

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

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-8ve (P8/2, P5) has ~~generator P5, alternate generators P4 and vA1,~~ period vA4, and equivalent period ^d5. ~~(P8, P5/2)~~ has generator ~~^m3~~ and equivalent ~~generator vM3~~.

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^<span style="vertical-align: super;">6</span>dd2.

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||= 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) ||= 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 ||= 686/675 ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||

There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.

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If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.

Demeter is unusual in that its gen2 ~~isn~~'t a mapping comma. It 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). Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.

Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.

The next table lists all rank-3 pergens up to half-splits. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.

||~ unsplit ||~ 2.3.5 ||~ ||~ 2.3.7 ||~ ||

||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= same ||

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

||= 2 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= same ||= same ||

||= 3 ||= (P8, P4/2, ^1) ||= rank-3 half-4th ||= same ||= same ||

||= 4 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= same ||= same ||

||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||= same ||

||= 6 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||

||= 7 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, vm7/2) ||= half-downminor-7th ||

||= 8 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||= ||= ||

||= 9 ||= (P8/2, P5, ^m6/2) ||= ||= ||= ||

||= 10 ||= (P8, P4/2, vM3/2) ||= ||= ||= ||

||= 11 ||= (P8, P4/2, ^m6/2) ||= ||= ||= ||

||= 12 ||= (P8, P5/2, vM3/2) ||= ||= ||= ||

||= 13 ||= (P8, P5/2, ^m6/2) ||= ||= ||= ||

||= 14 ||= (P8/2, P4/2, vM3/2) ||= half-everything half-downmajor-3rd ||= ||= ||

==Notating Blackwood-like pergens*==

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. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.

another example might work?

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 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).

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||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 = 16/15 ||= --- ||

||= Whitewood ||= (P8/7, ^1) ||= rank-2 7-edo ||= E = A1 ||= D E F G A B C D ||= D F^ A^^... ||= 80/81 = 135/128 ||= --- ||

||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ~~???~~ ||= 81/80 ||

||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= (see below) ||= 81/80 ||

||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||

||= " ||= " ||= " ||= " ||= " ||= D G#v=Abv Dvv... ||= 729/704 = ||= --- ||

||= " ||= " ||= " ||= " ||= " ||= D G#v=Abv Dvv... ||= 729/704 ||= --- ||

||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 ||

||= ||= ||= ||= ||= ||= ||= ||= ||

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If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.

The additional accidental's ratio ~~can be changed~~ by adding the ~~edo~~'s ~~defining~~ comma onto it. ~~For~~ Blackwood~~, 5-edo~~ is ~~defined by~~ 256/243, and /1 = 81/80 = 16/15.

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

(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)

~~In any notation~~, ~~every note has a name~~, ~~and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics~~, ~~each note has several names. Generally~~, ~~every name has a note~~. ~~Every possible name~~, and ~~anything that can be written on on~~ the ~~staff, corresponds to one and only one note in the lattice formed by perchains and genchains~~. ~~But in no-threes pergens,~~

//When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...//

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.

But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.

Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.

||~ tuning ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||

||= large quintuple blue,

small quintuple red ||= (P8/5, P5) ||= fifth-8ve ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||

||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||

||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v<span style="vertical-align: super;">5</span>m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G, D or A notes ||

== ==

edo-subset notations

P8/6, P8/8

==Notating non-8ve and non-5th pergens*==

Just as all rank-2 pergens in which 2 and 3 are independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:

||~ ||||||||||||~ __the first two independent primes in the prime subgroup__ ||

||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||

||= 1 ||= (P8, P5) ||= (P8, y3) ||= (P8, r2) ||= (P12, y6) ||= (P12, r3) ||= (WWy3, bg5) ||

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

||= 2 ||= (P8/2, P5) ||= (P8/2, y3) ||= (P8/2, r2) ||= ||= ||= ||

||= 3 ||= (P8, P4/2) ||= (P8, y3/2) ||= (P8, r2/2) ||= ||= ||= ||

||= 4 ||= (P8, P5/2) ||= (P8, g6/2) ||= (P8, b7/2) ||= ||= ||= ||

||= 5 ||= (P8/2, P4/2) ||= (P8/2, y3/2) ||= (P8/2, r2/2) ||= ||= ||= ||

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

||= 6 ||= (P8/3, P5) ||= (P8/3, y3) ||= ||= ||= ||= ||

||= 7 ||= (P8, P4/3) ||= (P8, y3/3) ||= ||= ||= ||= ||

||= 8 ||= (P8, P5/3) ||= (P8, g6/3) ||= ||= ||= ||= ||

||= 9 ||= (P8, P11/3) ||= (P8, y10/3) ||= ||= ||= ||= ||

||= 10 ||= (P8/3, P4/2) ||= (P8/3, y3/2) ||= ||= ||= ||= ||

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

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

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

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

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

~~relative~~ notation~~: not all intervals~~ names ~~have intervals~~. ~~No perfect 5th~~ in 6-~~edo~~.

Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.

Every rank-3 pergen can be identified by its first __three__ independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.

||~ pergen number ||~ 2.3.5 ||~ ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5.7 ||

||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= (P8, y3, r2) ||= (P12, y6, r3) ||

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

||= 2 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= same ||= ||= ||

||= 3 ||= (P8, P4/2, ^1) ||= ||= same ||= ||= ||

||= 4 ||= (P8, P5/2, ^1) ||= ||= same ||= ||= ||

||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||= ||= ||

||= 6 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, ^M2/2) ||= ||= ||

||= 7 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, vm7/2) ||= ||= ||

||= 8 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||= ||= ||= ||

||= 9 ||= (P8/2, P5, ^m6/2) ||= ||= ||= ||= ||

||= 10 ||= (P8, P4/2, vM3/2) ||= ||= ||= ||= ||

||= 11 ||= (P8, P4/2, ^m6/2) ||= ||= ||= ||= ||

||= 12 ||= (P8, P5/2, vM3/2) ||= ||= ||= ||= ||

||= 13 ||= (P8, P5/2, ^m6/2) ||= ||= ||= ||= ||

||= 14 ||= (P8/2, P4/2, vM3/2) ||= rank-3 half-everything ||= ||= ||= ||

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to devise a personal notation that isn't backwards compatible.

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.

A similar chart could be made of all rank-3 pergen cubes.

==Notating tunings with an arbitrary generator==

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http://www.tallkite.com/misc_files/pergens.pdf

(screenshot)

~~===Pergen squares pic===~~

~~One way to visualize pergens...~~

===pergenLister app===

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Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)

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<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 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 tunings with an arbitrary generator~~">~~Notating tunings with an arbitrary generator~~</a></div>

<div style="margin-left: 2em;"><a href="#toc18"> </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-Notating non-8ve and non-5th pergens*">Notating non-8ve and non-5th pergens*</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-Notating tunings with an arbitrary generator">Notating tunings with an arbitrary generator</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-Pergens and MOS scales">Pergens and MOS scales</a></div>

<div style="margin-left: ~~3em~~;"><a href="#Further Discussion-~~Supplemental materials*-Notaion guide PDF~~">~~Notaion guide PDF~~</a></div>

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

<div style="margin-left: ~~3em~~;"><a href="#Further Discussion-Supplemental materials*~~-Pergen squares pic~~">~~Pergen squares pic~~</a></div>

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

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-~~pergenLister app~~">~~pergenLister app~~&~~lt;/a></div>~~

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-Notaion guide PDF">Notaion guide PDF</a></div>

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

<div style="margin-left: 3em;"><a href="#Further Discussion-Supplemental materials*-pergenLister app">pergenLister app</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-Various proofs (unfinished)">Various proofs (unfinished)</a></div>

</div>

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

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

</div>

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

<br />

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

<br />

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

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

<br />

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For any comma, 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, where GCD (0,x) = x. 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 />

<br />

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also ~~directly related~~, the 4th prime is used, and so forth.<br />

In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is <strong>dependent</strong> on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two <strong>independent</strong> primes.<br />

<br />

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The ~~multigen must use primes 2 and 5. In this case, the~~ pergen is (P8/5, ^1), the same as Blackwood.<br />

For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20Blackwood-like%20pergens">Blackwood-like pergens</a> below).<br />

<br />

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes ~~may~~ need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.<br />

To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.<br />

<br />

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

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Line 1,447:

<span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| &lt;= x</span></strong><br />

</span><br />

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br />

A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br />

<br />

For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).<br />

For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 &lt;= i &lt;= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).<br />

<br />

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br />

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<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 />

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.<br />

<br />

Another 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, either because it contains more than 2 primes, or because the split 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 (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.<br />

<br />

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.<br />

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Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See <a class="wiki_link" href="/pergen#Further%20Discussion-Chord%20names%20and%20scale%20names">Chord names and scale names</a> below.<br />

<br />

The ~~third~~ main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.<br />

The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.<br />

<br />

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 all notation used here is backwards compatible.<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, which is octave-equivalent, fifth-generated and heptatonic. 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 all notation used here is backwards compatible.<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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Line 2,286:

C - Dv\ - Eb^/ - F<br />

</td>

<td style="text-align: center;">triple green &amp; large triple blue<br />

^1 = 81/80<br />

/1 = 64/63<br />

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Line 2,310:

C - Ev/ - Ab^\ - C<br />

</td>

<td style="text-align: center;">triple yellow &amp; large triple blue<br />

^1 = 81/80<br />

/1 = 64/63<br />

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Line 2,429:

</table>

The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex.<br />

The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't all that complex.<br />

<br />

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).<br />

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<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 />

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

<br />

P4/3: C - Dv - Eb^ - F<br />

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

</td>

<td style="text-align: center;">(every 3-limit interval~~)/2~~<br />

<td style="text-align: center;">every 3-limit interval is split twice as muh as before<br />

</td>

</tr>

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

</td>

<td style="text-align: center;">(every 3-limit interval~~)/3~~<br />

<td style="text-align: center;">every 3-limit interval is split three times as much as before<br />

</td>

</tr>

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<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 />

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, the diminished temperament. 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>, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.<br />

<br />

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63. Thus for (P8/4, P5), ~~since~~ P = vm3, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.<br />

If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.<br />

<br />

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is <strong>explicitly false</strong>. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br />

Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is <strong>explicitly false</strong>. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br />

<br />

If the pergen is not explicitly false, put the pergen in its <strong>unreduced</strong> form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.<br />

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

<br />

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an <strong>alternate</strong> generator. A generator or period plus or minus any number of enharmonics makes an <strong>equivalent</strong> generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-8ve (P8/2, P5) has ~~generator P5, alternate generators P4 and vA1,~~ period vA4, and equivalent period ^d5. ~~(P8, P5/2)~~ has generator ~~^m3~~ and equivalent ~~generator vM3~~.<br />

Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an <strong>alternate</strong> generator. A generator or period plus or minus any number of enharmonics makes an <strong>equivalent</strong> generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.<br />

<br />

Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^<span style="vertical-align: super;">6</span>dd2.<br />

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</td>

</td>

<td style="text-align: center;">~~third~~-~~downmajor~~-3rd<br />

<td style="text-align: center;">half-downminor-3rd<br />

</td>

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

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If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.<br />

<br />

Demeter is unusual in that its gen2 ~~isn~~'t a mapping comma. It 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). Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals~~.<br />~~

Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.<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. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.<br />

~~<br />~~

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).<br />

<br />

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 />

The next table lists all rank-3 pergens up to half-splits. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.<br />

<br />

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:<br />

<tr>

<th>~~temperament~~<br />

<th>unsplit<br />

</th>

<th>~~pergen~~<br />

</th>

<th>~~spoken~~<br />

</th>

<th>~~enharmonics~~<br />

</th>

<th>~~perchain<br />~~

~~</th>~~

~~<th>genchain<br />~~

~~</th>~~

~~<th>^1<br />~~

~~</th>~~

~~<th>/1~~<br />

</th>

</tr>

<tr>

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

</td>

</td>

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

</td>

</td>

</td>

<~~td style="text~~-~~align: center~~;">~~D F#v=Gv Bvv...~~<br />

</tr>

</td>

<tr>

<th>half-splits<br />

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

~~</tr>~~

~~<tr>~~

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

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

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

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-downmajor-3rd<br />

</td>

</td>

<td style="text-align: center;">~~D D#=Eb E F F#=Gb...<br />~~

<td style="text-align: center;">half-upmajor-2nd<br />

~~</td>~~

~~<td style="text-align: center;">D G^ C^^<br />~~

~~</td>~~

~~<td style="text-align: center;">33/32<br />~~

~~</td>~~

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

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-upminor-6th<br />

</td>

</td>

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

<td style="text-align: center;">half-downminor-7th<br />

~~</td>~~

~~<td style="text-align: center;">D G#v=Abv Dvv...<br />~~

~~</td>~~

~~<td style="text-align: center;">729/704 =<br />~~

~~</td>~~

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

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-8ve half-downmajor 3rd<br />

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

~~<td style="text-align: center;">81/80<br />~~

~~</td>~~

~~</tr>~~

~~<tr>~~

</td>

Line 3,647:

Line 3,723:

</td>

</tr>

<tr>

</td>

</td>

Line 3,659:

Line 3,737:

</tr>

<tr>

</td>

</td>

</td>

Line 3,665:

Line 3,747:

</td>

</tr>

<tr>

</td>

</td>

Line 3,677:

Line 3,761:

</tr>

<tr>

</td>

</td>

Line 3,685:

Line 3,771:

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-everything half-downmajor-3rd<br />

</td>

Line 3,694:

Line 3,784:

</td>

</tr>

<tr>

</table>

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

</td>

<br />

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

<br />

<~~/td~~>

<br />

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

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

</td>

<br />

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

Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.<br />

</td>

another example might work?<br />

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

<br />

</td>

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 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.<br />

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

<br />

</~~td>~~

A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).<br />

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

<br />

</td>

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 />

<br />

</td~~>~~

Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:<br />

~~</tr>~~

~~<tr~~>

</td>

<tr>

<th>temperament<br />

</td>

</th>

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

<th>pergen<br />

<~~/td~~>

</th>

<th>spoken<br />

</th>

<th>enharmonics<br />

</th>

<th>perchain<br />

</th>

<th>genchain<br />

</th>

</th>

</th>

</tr>

<tr>

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

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

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

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

<td style="text-align: center;">(see below)<br />

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

</table>

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.<br />

<br />

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

<br />

(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)<br />

<br />

<em>When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...</em><br />

<br />

In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.<br />

<br />

But in &quot;no-5ths&quot; or &quot;minus white&quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.<br />

<br />

Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.<br />

<tr>

<th>tuning<br />

</th>

<th>pergen<br />

</th>

<th>spoken name<br />

</th>

<th>enharmonic<br />

</th>

<th>perchain<br />

</th>

<th>genchain<br />

</th>

<th>notes<br />

</th>

</tr>

<tr>

<td style="text-align: center;">large quintuple blue,<br />

small quintuple red<br />

</td>

</td>

<td style="text-align: center;">fifth-8ve<br />

</td>

</td>

</td>

</td>

<td style="text-align: center;">a valid notation<br />

</td>

</tr>

<tr>

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

</td>

</td>

</td>

</td>

</td>

</td>

<td style="text-align: center;">a valid notation<br />

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">fifth-8ve major 3rd<br />

</td>

</td>

</td>

</td>

<td style="text-align: center;">invalid, no G, D or A notes<br />

</td>

</tr>

</table>

<h2 id="toc18"> </h2>

edo-subset notations<br />

P8/6, P8/8<br />

<br />

<h2 id="toc19"><a name="Further Discussion-Notating non-8ve and non-5th pergens*"></a>Notating non-8ve and non-5th pergens*</h2>

<br />

Just as all rank-2 pergens in which 2 and 3 are independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:<br />

<tr>

</th>

<th colspan="6"><u>the first two independent primes in the prime subgroup</u><br />

</th>

</tr>

<tr>

<th>pergen number<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<th>half-splits<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<th>third-splits<br />

</th>

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

</table>

<br />

Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.<br />

<br />

Every rank-3 pergen can be identified by its first <u>three</u> independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.<br />

<tr>

<th>pergen number<br />

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

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

</td>

</td>

</td>

</td>

</tr>

<tr>

<th>half-splits<br />

</th>

</th>

</th>

</th>

</th>

</th>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

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

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-downmajor-3rd<br />

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-upminor-6th<br />

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

<td style="text-align: center;">half-8ve half-downmajor 3rd<br />

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

</td>

</td>

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

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

~~</tr>~~

~~<tr>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

~~</td>~~

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

</td>

Line 3,750:

Line 4,663:

</table>

If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.<br />

~~<br />~~

~~The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.<br />~~

~~<br />~~

~~(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)<br />~~

<br />

~~In any~~ notation, ~~every note has a name~~, ~~and no two notes have the same name~~. ~~A note's representation on the musical~~ staff ~~follows from its name~~. ~~If the notation has any enharmonics, each~~ note ~~has several~~ names. ~~Generally, every name has~~ a ~~note. Every possible name, and anything~~ that ~~can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains~~. ~~But in no-threes pergens,~~<br />

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of note-less names. The composer may well want to devise a personal notation that isn't backwards compatible.<br />

<br />

~~edo-subset notations~~<br />

Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.<br />

<br />

~~relative notation: not~~ all ~~intervals names have intervals. No perfect 5th in 6~~-~~edo~~.<br />

A similar chart could be made of all rank-3 pergen cubes.<br />

<br />

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

<h2 id="toc20"><a name="Further Discussion-Notating tunings with an arbitrary generator"></a>Notating tunings with an arbitrary generator</h2>

<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 used here. 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 />

Line 4,107:

Line 5,015:

See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br />

<br />

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

<h2 id="toc21"><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 />

Line 5,212:

Line 6,120:

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="toc22"><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 and pergens, but only a few dozen of either have been explored.<br />

Line 5,723:

Line 6,631:

A specific pergen can be converted to an edo pair by finding the range of its generator cents in the <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator">arbitrary generator</a> table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).<br />

<br />

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

<h2 id="toc23"><a name="Further Discussion-Supplemental materials*"></a>Supplemental materials*</h2>

<br />

needs more screenshots, including 12-edo's pergens and a page of the pdf<br />

Line 5,733:

Line 6,641:

finish proofs<br />

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br />

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><br />

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><br />

<br />

<h3 id="~~toc22~~"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<h3 id="toc24"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

<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 />

(screenshot)<br />

<br />

<h3 id="~~toc23"><a name="Further Discussion-Supplemental materials*-Pergen squares pic"></a>Pergen squares pic</h3>~~

<h3 id="toc25"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

~~<br />~~

~~One way to visualize pergens...<br />~~

~~<br />~~

~~<h3 id="toc24~~"><a name="Further Discussion-Supplemental materials*-pergenLister app"></a>pergenLister app</h3>

<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 />

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Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots 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 />

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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 need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.<br />

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<h2 id="~~toc25~~"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

<h2 id="toc26"><a name="Further Discussion-Various proofs (unfinished)"></a>Various proofs (unfinished)</h2>

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The interval P8/2 has a &quot;ratio&quot; of the square root of 2, which equals 2<span style="vertical-align: super;">1/2</span>, and its monzo can be written with fractions as (1/2, 0). In general, 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 />

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Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? <br />

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?<br />

Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.<br />

a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)<br />

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

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<h2 id="~~toc26~~"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

<h2 id="toc27"><a name="Further Discussion-Miscellaneous Notes"></a>Miscellaneous Notes</h2>

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

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+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-04 22:43:36 UTC</tt>.<br>
-: The original revision id was <tt>627174459</tt>.<br>
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 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, and the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, and the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).
@@ Line 60: / Line 60: @@
 For any comma, 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, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.
-In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.
+In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is **dependent** on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two **independent** primes.
-For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, ^1), the same as Blackwood.
+For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see [[pergen#Further%20Discussion-Notating%20Blackwood-like%20pergens|Blackwood-like pergens]] below).
-To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.
+To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.
 For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.
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 &lt;span style="display: block; text-align: center;"&gt;**&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| &lt;= x&lt;/span&gt;**
 &lt;/span&gt;
-A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.
+A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.
-For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
+For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 &lt;= i &lt;= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
 Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7|x31.com]] gives us this matrix:
@@ Line 117: / Line 117: @@
 =__Applications__=
-One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called 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.
+Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.
+Another 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, either because it contains more than 2 primes, or because the split 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 (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.
 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
@@ Line 123: / Line 125: @@
 Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further%20Discussion-Chord%20names%20and%20scale%20names|Chord names and scale names]] below.
-The third main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.
+The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.
 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 all notation used here is backwards compatible.
+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, which is octave-equivalent, fifth-generated and heptatonic. 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 all notation used here is backwards compatible.
 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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 P4/3 = v\M2 ||= C - Ev - Ab^ - C
 C - D/ - F\ - G
-C - Dv\ - Eb^/ - F ||= 128/125 &amp; 1029/1024
+C - Dv\ - Eb^/ - F ||= triple green &amp; large triple blue
 ^1 = 81/80
 /1 = 64/63 ||
@@ Line 286: / Line 288: @@
 P8/3 = v/M3 ||= C - Dv - Eb^ - F
 C - D/ - F\ - G
-C - Ev/ - Ab^\ - C ||= 250/243 &amp; 1029/1024
+C - Ev/ - Ab^\ - C ||= triple yellow &amp; large triple blue
 ^1 = 81/80
 /1 = 64/63 ||
@@ Line 296: / Line 298: @@
 ||= 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. ||=   ||=   ||=   ||=   ||=   ||
-The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex.
+The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't all that complex.
 Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).
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 ==Secondary splits==
-Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:
+Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve 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:
 P4/3: C - Dv - Eb^ - F
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 ||= (P8, P4/2) ||= half-4th ||= m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2 ||
 ||= (P8, P5/2) ||= half-5th ||= A1/2, m3/2, M7/2, m9/2, P11/2 ||
-||= (P8/2, P4/2) ||= half-everything ||= (every 3-limit interval)/2 ||
+||= (P8/2, P4/2) ||= half-everything ||= every 3-limit interval is split twice as muh as before ||
 ||||~ third-splits ||~   ||
 ||= (P8/3, P5) ||= third-8ve ||= m3/3, M6/3, d5/6, A11/3, d12/3 ||
@@ Line 370: / Line 372: @@
 ||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve splits, third-5th splits, m6/6, M9/6, A12/6 ||
 ||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve splits, third-11th splits, M2/6, M3/12, A4/18, A5/24 ||
-||= (P83, P4/3) ||= third-everything ||= (every 3-limit interval)/3 ||
+||= (P83, P4/3) ||= third-everything ||= every 3-limit interval is split three times as much as before ||
 ==Singles and doubles==
@@ Line 384: / Line 386: @@
 ==Finding an example temperament==
-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.
+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, the diminished temperament. 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;, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.
-If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63. Thus for (P8/4, P5), since P = vm3, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
+If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
-Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
+Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
 If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.
@@ Line 401: / Line 403: @@
 A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.
-Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-8ve (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.
+Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.
 Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;dd2.
@@ Line 586: / Line 588: @@
 ||= 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) ||= 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 ||= 686/675 ||= (P8, P5, vm3/2) ||= half-downminor-3rd ||= double-pair ||= P8 ||= P5 ||= v/A1 = 15/14 ||= ^^\\\dd3 ||
 There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = ``//``d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.
@@ Line 597: / Line 599: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.
-Demeter is unusual in that its gen2 isn't a mapping comma. It 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). Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.
+Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.
+The next table lists all rank-3 pergens up to half-splits. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.
+||~ unsplit ||~ 2.3.5 ||~   ||~ 2.3.7 ||~   ||
+||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= same ||
+||~ half-splits ||~   ||~   ||~   ||~   ||
+||= 2 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= same ||= same ||
+||= 3 ||= (P8, P4/2, ^1) ||= rank-3 half-4th ||= same ||= same ||
+||= 4 ||= (P8, P5/2, ^1) ||= rank-3 half-5th ||= same ||= same ||
+||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||= same ||
+||= 6 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, ^M2/2) ||= half-upmajor-2nd ||
+||= 7 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, vm7/2) ||= half-downminor-7th ||
+||= 8 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||=   ||=   ||
+||= 9 ||= (P8/2, P5, ^m6/2) ||=   ||=   ||=   ||
+||= 10 ||= (P8, P4/2, vM3/2) ||=   ||=   ||=   ||
+||= 11 ||= (P8, P4/2, ^m6/2) ||=   ||=   ||=   ||
+||= 12 ||= (P8, P5/2, vM3/2) ||=   ||=   ||=   ||
+||= 13 ||= (P8, P5/2, ^m6/2) ||=   ||=   ||=   ||
+||= 14 ||= (P8/2, P4/2, vM3/2) ||= half-everything half-downmajor-3rd ||=   ||=   ||
 ==Notating Blackwood-like pergens*==
-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. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
+Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.
+another example might work?
+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 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
 A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).
@@ Line 611: / Line 637: @@
 ||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= D E=F G A B=C D ||= D F#v=Gv Bvv... ||= 81/80 = 16/15 ||= --- ||
 ||= Whitewood ||= (P8/7, ^1) ||= rank-2 7-edo ||= E = A1 ||= D E F G A B C D ||= D F^ A^^... ||= 80/81 = 135/128 ||= --- ||
-||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ??? ||= 81/80 ||
+||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= (see below) ||= 81/80 ||
 ||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- ||
-||= " ||= " ||= " ||= " ||= " ||= D G#v=Abv Dvv... ||= 729/704 = ||= --- ||
+||= " ||= " ||= " ||= " ||= " ||= D G#v=Abv Dvv... ||= 729/704 ||= --- ||
 ||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 ||
 ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 623: / Line 649: @@
 If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.
-The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.
+The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.
 (In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)
-In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens,
+//When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...//
+In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.
+But in "no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.
+Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.
+||~ tuning ||~ pergen ||~ spoken name ||~ enharmonic ||~ perchain ||~ genchain ||~ notes ||
+||= large quintuple blue,
+small quintuple red ||= (P8/5, P5) ||= fifth-8ve ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C G D A E... ||= a valid notation ||
+||= Blackwood ||= (P8/5, ^1) ||= rank-2 5-edo ||= E = m2 ||= C D=Eb E=F G A=Bb B=C ||= C Ev G#vv... ||= a valid notation ||
+||= " ||= (P8/5, M3) ||= fifth-8ve major 3rd ||= E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2 ||= C D^^ Fv G^ Bbvv C ||= C E G#... ||= invalid, no G, D or A notes ||
+== ==
 edo-subset notations
+P8/6, P8/8
+==Notating non-8ve and non-5th pergens*==
+Just as all rank-2 pergens in which 2 and 3 are independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:
+||~   ||||||||||||~ __the first two independent primes in the prime subgroup__ ||
+||~ pergen number ||~ 2.3 ||~ 2.5 ||~ 2.7 ||~ 3.5 ||~ 3.7 ||~ 5.7 ||
+||= 1 ||= (P8, P5) ||= (P8, y3) ||= (P8, r2) ||= (P12, y6) ||= (P12, r3) ||= (WWy3, bg5) ||
+||~ half-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+||= 2 ||= (P8/2, P5) ||= (P8/2, y3) ||= (P8/2, r2) ||=   ||=   ||=   ||
+||= 3 ||= (P8, P4/2) ||= (P8, y3/2) ||= (P8, r2/2) ||=   ||=   ||=   ||
+||= 4 ||= (P8, P5/2) ||= (P8, g6/2) ||= (P8, b7/2) ||=   ||=   ||=   ||
+||= 5 ||= (P8/2, P4/2) ||= (P8/2, y3/2) ||= (P8/2, r2/2) ||=   ||=   ||=   ||
+||~ third-splits ||~   ||~   ||~   ||~   ||~   ||~   ||
+||= 6 ||= (P8/3, P5) ||= (P8/3, y3) ||=   ||=   ||=   ||=   ||
+||= 7 ||= (P8, P4/3) ||= (P8, y3/3) ||=   ||=   ||=   ||=   ||
+||= 8 ||= (P8, P5/3) ||= (P8, g6/3) ||=   ||=   ||=   ||=   ||
+||= 9 ||= (P8, P11/3) ||= (P8, y10/3) ||=   ||=   ||=   ||=   ||
+||= 10 ||= (P8/3, P4/2) ||= (P8/3, y3/2) ||=   ||=   ||=   ||=   ||
+||= 11 ||= (P8/3, P5/2) ||=   ||=   ||=   ||=   ||=   ||
+||= 12 ||= (P8/2, P4/3) ||=   ||=   ||=   ||=   ||=   ||
+||= 13 ||= (P8/2, P5/3) ||=   ||=   ||=   ||=   ||=   ||
+||= 14 ||= (P8/2, P11/3) ||=   ||=   ||=   ||=   ||=   ||
+||= 15 ||= (P8/3, P4/3) ||=   ||=   ||=   ||=   ||=   ||
-relative notation: not all intervals names have intervals. No perfect 5th in 6-edo.
+Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
+Every rank-3 pergen can be identified by its first __three__ independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.
+||~ pergen number ||~ 2.3.5 ||~   ||~ 2.3.7 ||~ 2.5.7 ||~ 3.5.7 ||
+||= 1 ||= (P8, P5, ^1) ||= rank-3 unsplit ||= same ||= (P8, y3, r2) ||= (P12, y6, r3) ||
+||~ half-splits ||~   ||~   ||~   ||~   ||~   ||
+||= 2 ||= (P8/2, P5, ^1) ||= rank-3 half-8ve ||= same ||=   ||=   ||
+||= 3 ||= (P8, P4/2, ^1) ||=   ||= same ||=   ||=   ||
+||= 4 ||= (P8, P5/2, ^1) ||=   ||= same ||=   ||=   ||
+||= 5 ||= (P8/2, P4/2, ^1) ||= rank-3 half-everything ||= same ||=   ||=   ||
+||= 6 ||= (P8, P5, vM3/2) ||= half-downmajor-3rd ||= (P8, P5, ^M2/2) ||=   ||=   ||
+||= 7 ||= (P8, P5, ^m6/2) ||= half-upminor-6th ||= (P8, P5, vm7/2) ||=   ||=   ||
+||= 8 ||= (P8/2, P5, vM3/2) ||= half-8ve half-downmajor 3rd ||=   ||=   ||=   ||
+||= 9 ||= (P8/2, P5, ^m6/2) ||=   ||=   ||=   ||=   ||
+||= 10 ||= (P8, P4/2, vM3/2) ||=   ||=   ||=   ||=   ||
+||= 11 ||= (P8, P4/2, ^m6/2) ||=   ||=   ||=   ||=   ||
+||= 12 ||= (P8, P5/2, vM3/2) ||=   ||=   ||=   ||=   ||
+||= 13 ||= (P8, P5/2, ^m6/2) ||=   ||=   ||=   ||=   ||
+||= 14 ||= (P8/2, P4/2, vM3/2) ||= rank-3 half-everything ||=   ||=   ||=   ||
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of note-less names. The composer may well want to devise a personal notation that isn't backwards compatible.
+Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.
+A similar chart could be made of all rank-3 pergen cubes.
 ==Notating tunings with an arbitrary generator==
@@ Line 851: / Line 937: @@
 http://www.tallkite.com/misc_files/pergens.pdf
 (screenshot)
-===Pergen squares pic===
-One way to visualize pergens...
 ===pergenLister app===
@@ Line 939: / Line 1,021: @@
 Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?
 Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.
 a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)
@@ Line 1,066: / Line 1,148: @@
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+  &lt;!-- ws:start:WikiTextTocRule:113:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#toc0"&gt; &lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Applications-Tipping points"&gt;Tipping points&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Further Discussion"&gt;Further Discussion&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&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;
+&lt;!-- ws:end:WikiTextTocRule:119 --&gt;&lt;!-- ws:start:WikiTextTocRule:120: --&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;
-&lt;!-- ws:end:WikiTextTocRule:118 --&gt;&lt;!-- ws:start:WikiTextTocRule:119: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Secondary splits"&gt;Secondary splits&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-Secondary splits"&gt;Secondary splits&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-Singles and doubles"&gt;Singles and doubles&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-Singles and doubles"&gt;Singles and doubles&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-Finding an example temperament"&gt;Finding an example temperament&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-Finding an example temperament"&gt;Finding an example temperament&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-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&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-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&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-Finding a notation for a pergen"&gt;Finding a notation for a pergen&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-Finding a notation for a pergen"&gt;Finding a notation for a pergen&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-Alternate enharmonics"&gt;Alternate enharmonics&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-Alternate enharmonics"&gt;Alternate enharmonics&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-Chord names and scale names"&gt;Chord names and scale names&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-Chord names and scale names"&gt;Chord names and scale names&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-Tipping points and sweet spots"&gt;Tipping points and sweet spots&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-Tipping points and sweet spots"&gt;Tipping points and sweet spots&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-Notating unsplit pergens"&gt;Notating unsplit pergens&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-Notating unsplit pergens"&gt;Notating unsplit pergens&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-Notating rank-3 pergens"&gt;Notating rank-3 pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&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:128 --&gt;&lt;!-- ws:start:WikiTextTocRule:129: --&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:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&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:129 --&gt;&lt;!-- ws:start:WikiTextTocRule:130: --&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:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc18"&gt; &lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:130 --&gt;&lt;!-- ws:start:WikiTextTocRule:131: --&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:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Notating non-8ve and non-5th pergens*"&gt;Notating non-8ve and non-5th pergens*&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:131 --&gt;&lt;!-- ws:start:WikiTextTocRule:132: --&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:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&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:132 --&gt;&lt;!-- ws:start:WikiTextTocRule:133: --&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:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&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:133 --&gt;&lt;!-- ws:start:WikiTextTocRule:134: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&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:134 --&gt;&lt;!-- ws:start:WikiTextTocRule:135: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Pergen squares pic"&gt;Pergen squares pic&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&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:135 --&gt;&lt;!-- ws:start:WikiTextTocRule:136: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;Notaion guide PDF&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:136 --&gt;&lt;!-- ws:start:WikiTextTocRule:137: --&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:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Further Discussion-Supplemental materials*-pergenLister app"&gt;pergenLister app&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:137 --&gt;&lt;!-- ws:start:WikiTextTocRule:138: --&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:139 --&gt;&lt;!-- ws:start:WikiTextTocRule:140: --&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:138 --&gt;&lt;!-- ws:start:WikiTextTocRule:139: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:140 --&gt;&lt;!-- ws:start:WikiTextTocRule:141: --&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:139 --&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;!-- ws:end:WikiTextTocRule:141 --&gt;&lt;!-- ws:start:WikiTextTocRule:142: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:142 --&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 1,342: / Line 1,425: @@
 A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos or edonois does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.&lt;br /&gt;
 &lt;br /&gt;
-In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, and the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).&lt;br /&gt;
+In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime. The higher prime's exponent in the comma's monzo must be ±1, and the comma should map to P1, e.g. 81/80 or 64/63. The commas are notated with either microtonal accidentals (P8, P5, ^1, /1,...) or colors (P8, P5, g1, r1,...).&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,349: / Line 1,432: @@
 For any comma, 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, where GCD (0,x) = x. 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;
 &lt;br /&gt;
-In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.&lt;br /&gt;
+In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime, i.e. it is the period or a multiple of it. Such a prime is &lt;strong&gt;dependent&lt;/strong&gt; on a lower prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also dependent, the 4th prime is used, and so forth. In other words, the multigen uses the first two &lt;strong&gt;independent&lt;/strong&gt; primes.&lt;br /&gt;
 &lt;br /&gt;
-For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, ^1), the same as Blackwood.&lt;br /&gt;
+For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The pergen is (P8/5, ^1), the same as Blackwood (see &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20Blackwood-like%20pergens"&gt;Blackwood-like pergens&lt;/a&gt; below).&lt;br /&gt;
 &lt;br /&gt;
-To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank"&gt;x31eq.com/temper/uv.html&lt;/a&gt; that will find such a matrix, it's the reduced mapping. Next make a &lt;strong&gt;square mapping&lt;/strong&gt; by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.&lt;br /&gt;
+To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank"&gt;x31eq.com/temper/uv.html&lt;/a&gt; that will find such a matrix, it's the reduced mapping. Next make a &lt;strong&gt;square mapping&lt;/strong&gt; by discarding columns, usually the columns for the highest primes. But lower primes that are dependent need to be discarded, as in the previous (P8/5, ^1) example, to ensure that the diagonal has no zeros. Lower primes may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.&lt;br /&gt;
 &lt;br /&gt;
 For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.&lt;br /&gt;
@@ Line 1,364: / Line 1,447: @@
 &lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| &amp;lt;= x&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
 &lt;/span&gt;&lt;br /&gt;
-A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.&lt;br /&gt;
+A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens like M2 or m3 are fairly rare. Less than 4% of all pergens have an imperfect multigen.&lt;br /&gt;
 &lt;br /&gt;
-For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &amp;lt;= n &amp;lt;= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).&lt;br /&gt;
+For example, porcupine (2.3.5 and 250/243) has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3i-2,1)/(-3)) = (P8, (3i+2,-1)/3), with -1 &amp;lt;= i &amp;lt;= 1. No value of i reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).&lt;br /&gt;
 &lt;br /&gt;
 Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;amp;limit=7" rel="nofollow"&gt;x31.com&lt;/a&gt; gives us this matrix:&lt;br /&gt;
@@ Line 1,635: / Line 1,718: @@
 &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;
+Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.&lt;br /&gt;
+&lt;br /&gt;
+Another 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, either because it contains more than 2 primes, or because the split 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 (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.&lt;br /&gt;
 &lt;br /&gt;
 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.&lt;br /&gt;
@@ Line 1,641: / Line 1,726: @@
 Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Chord%20names%20and%20scale%20names"&gt;Chord names and scale names&lt;/a&gt; below.&lt;br /&gt;
 &lt;br /&gt;
-The third main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.&lt;br /&gt;
+The final main application, which the rest of this article will focus on, is that pergens allow a systematic approach to notating regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion mostly focuses on rank-2 temperaments that include primes 2 and 3.&lt;br /&gt;
 &lt;br /&gt;
 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 all notation used here is backwards compatible.&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, which is octave-equivalent, fifth-generated and heptatonic. 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 all notation used here is backwards compatible.&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 2,201: / Line 2,286: @@
 C - Dv\ - Eb^/ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;128/125 &amp;amp; 1029/1024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;triple green &amp;amp; large triple blue&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
 /1 = 64/63&lt;br /&gt;
@@ Line 2,225: / Line 2,310: @@
 C - Ev/ - Ab^\ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;250/243 &amp;amp; 1029/1024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;triple yellow &amp;amp; large triple blue&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
 /1 = 64/63&lt;br /&gt;
@@ Line 2,344: / Line 2,429: @@
 &lt;/table&gt;
-The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably isn't all that complex.&lt;br /&gt;
+The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't all that complex.&lt;br /&gt;
 &lt;br /&gt;
 Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).&lt;br /&gt;
@@ Line 2,602: / Line 2,687: @@
 &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;
+Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:&lt;br /&gt;
 &lt;br /&gt;
 P4/3: C - Dv - Eb^ - F&lt;br /&gt;
@@ Line 2,675: / Line 2,760: @@
          &lt;td style="text-align: center;"&gt;half-everything&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(every 3-limit interval)/2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;every 3-limit interval is split twice as muh as before&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,761: / Line 2,846: @@
          &lt;td style="text-align: center;"&gt;third-everything&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(every 3-limit interval)/3&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;every 3-limit interval is split three times as much as before&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,779: / Line 2,864: @@
 &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;
+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, the diminished temperament. 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;, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.&lt;br /&gt;
 &lt;br /&gt;
-If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63. Thus for (P8/4, P5), since P = vm3, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.&lt;br /&gt;
+If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63 (see mapping commas in the next section). Thus for (P8/4, P5), if P = vm3, and ^1 = 64/63, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.&lt;br /&gt;
 &lt;br /&gt;
-Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).&lt;br /&gt;
+Finding the comma(s) for a double-split pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).&lt;br /&gt;
 &lt;br /&gt;
 If the pergen is not explicitly false, put the pergen in its &lt;strong&gt;unreduced&lt;/strong&gt; form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P8 - m&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M)/nm) = (P8/m, M'/n'). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &amp;lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.&lt;br /&gt;
@@ Line 2,793: / Line 2,878: @@
 A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.&lt;br /&gt;
 &lt;br /&gt;
-Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an &lt;strong&gt;alternate&lt;/strong&gt; generator. A generator or period plus or minus any number of enharmonics makes an &lt;strong&gt;equivalent&lt;/strong&gt; generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-8ve (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.&lt;br /&gt;
+Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an &lt;strong&gt;alternate&lt;/strong&gt; generator. A generator or period plus or minus any number of enharmonics makes an &lt;strong&gt;equivalent&lt;/strong&gt; generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve (P8/2, P5) has period vA4 and equivalent period ^d5. It has generator P5 and alternate generators P4 and vA1. vA1 is equivalent to ^m2.&lt;br /&gt;
 &lt;br /&gt;
 Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;dd2.&lt;br /&gt;
@@ Line 3,473: / Line 3,558: @@
          &lt;td style="text-align: center;"&gt;686/675&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8, P5, vM3/3)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, vm3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;third-downmajor-3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-downminor-3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;double-pair&lt;br /&gt;
@@ Line 3,500: / Line 3,585: @@
 If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.&lt;br /&gt;
 &lt;br /&gt;
-Demeter is unusual in that its gen2 isn't a mapping comma. It 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). Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.&lt;br /&gt;
+Demeter is unusual in that its gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, b3/2) or (P8, P5, vm3/2). It could also be called (P8, P5, y3/3) or (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, 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^^. Standard double-pair notation is better. 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. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/highs/lows only for the other kind of accidentals.&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. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
-&lt;br /&gt;
-A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).&lt;br /&gt;
 &lt;br /&gt;
-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;
+The next table lists all rank-3 pergens up to half-splits. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.&lt;br /&gt;
 &lt;br /&gt;
-Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:&lt;br /&gt;
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th&gt;temperament&lt;br /&gt;
+         &lt;th&gt;unsplit&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;pergen&lt;br /&gt;
+         &lt;th&gt;2.3.5&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;spoken&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;enharmonics&lt;br /&gt;
+         &lt;th&gt;2.3.7&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;perchain&lt;br /&gt;
+         &lt;th&gt;&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;genchain&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;^1&lt;br /&gt;
-&lt;/th&gt;
-        &lt;th&gt;/1&lt;br /&gt;
 &lt;/th&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 5-edo&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;E = m2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D E=F G A B=C D&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D F#v=Gv Bvv...&lt;br /&gt;
+    &lt;/tr&gt;
-&lt;/td&gt;
+    &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;81/80 = 16/15&lt;br /&gt;
+         &lt;th&gt;half-splits&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;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5, ^1)&lt;br /&gt;
 &lt;/td&gt;
-    &lt;/tr&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-8ve&lt;br /&gt;
-    &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;Whitewood&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/7, ^1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 7-edo&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = A1&lt;br /&gt;
+    &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;D E F G A B C D&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/2, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D F^ A^^...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-4th&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;80/81 = 135/128&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;---&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;10edo+y&lt;br /&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, ^1)&lt;br /&gt;
+&lt;/td&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;(P8/10, /1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 10-edo&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = m2, E' = vvA1 = vvM2&lt;br /&gt;
+    &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;D D^=Ev E=F F^=Gv G...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P4/2, ^1)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D F#\=G\ B\\...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-everything&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;???&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;12edo+j&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/12, ^1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 12-edo&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-downmajor-3rd&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = d2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, ^M2/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D D#=Eb E F F#=Gb...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-upmajor-2nd&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D G^ C^^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;33/32&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;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, ^m6/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-upminor-6th&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5, vm7/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-downminor-7th&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;D G#v=Abv Dvv...&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;729/704 =&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;17edo+y&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;(P8/17, /1)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;rank-2 17-edo&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-8ve half-downmajor 3rd&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;E = dd3, E' = vm2 = vvA1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D D^=Eb D#=Ev E F...&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;D F#\ A#\\=Bv\\...&lt;br /&gt;
+    &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;256/243&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P5, ^m6/2)&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,647: / Line 3,723: @@
          &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;tr&gt;
+         &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P4/2, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,659: / Line 3,737: @@
      &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, P4/2, ^m6/2)&lt;br /&gt;
+&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 3,665: / Line 3,747: @@
          &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;tr&gt;
+         &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8, P5/2, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,677: / Line 3,761: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2, ^m6/2)&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,685: / Line 3,771: @@
          &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;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, P4/2, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;half-everything half-downmajor-3rd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,694: / Line 3,784: @@
 &lt;/td&gt;
      &lt;/tr&gt;
-    &lt;tr&gt;
+&lt;/table&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
-&lt;/td&gt;
+&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+&lt;br /&gt;
-        &lt;td style="text-align: center;"&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;/td&gt;
+ &lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.&lt;br /&gt;
-&lt;/td&gt;
+another example might work?&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&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 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).&lt;br /&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+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;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;br /&gt;
-&lt;/td&gt;
+Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:&lt;br /&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
-&lt;/td&gt;
+    &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+        &lt;th&gt;temperament&lt;br /&gt;
-&lt;/td&gt;
+&lt;/th&gt;
-        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+        &lt;th&gt;pergen&lt;br /&gt;
-&lt;/td&gt;
+&lt;/th&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+        &lt;th&gt;spoken&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;enharmonics&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;perchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;genchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;^1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;/1&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 5-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D E=F G A B=C D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F#v=Gv Bvv...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;81/80 = 16/15&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;Whitewood&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/7, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 7-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = A1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D E F G A B C D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F^ A^^...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;80/81 = 135/128&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;10edo+y&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/10, /1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 10-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = m2, E' = vvA1 = vvM2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D D^=Ev E=F F^=Gv G...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F#\=G\ B\\...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(see below)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;12edo+j&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/12, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 12-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = d2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D D#=Eb E F F#=Gb...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D G^ C^^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;33/32&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;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D G#v=Abv Dvv...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;729/704&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;17edo+y&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/17, /1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 17-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = dd3, E' = vm2 = vvA1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D D^=Eb D#=Ev E F...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D F#\ A#\\=Bv\\...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;256/243&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+    &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;&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;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;&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;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;&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;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;&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;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;&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;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;&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;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;/table&gt;
+If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.&lt;br /&gt;
+&lt;br /&gt;
+The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.&lt;br /&gt;
+&lt;br /&gt;
+(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)&lt;br /&gt;
+&lt;br /&gt;
+&lt;em&gt;When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...&lt;/em&gt;&lt;br /&gt;
+&lt;br /&gt;
+In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains.&lt;br /&gt;
+&lt;br /&gt;
+But in &amp;quot;no-5ths&amp;quot; or &amp;quot;minus white&amp;quot; pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.&lt;br /&gt;
+&lt;br /&gt;
+Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;tuning&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;pergen&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;spoken name&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;enharmonic&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;perchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;genchain&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;notes&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;large quintuple blue,&lt;br /&gt;
+small quintuple red&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifth-8ve&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C D^^ Fv G^ Bbvv C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C G D A E...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;a valid notation&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;Blackwood&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-2 5-edo&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C D=Eb E=F G A=Bb B=C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C Ev G#vv...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;a valid notation&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/5, M3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;fifth-8ve major 3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C D^^ Fv G^ Bbvv C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C E G#...&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;invalid, no G, D or A notes&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt; &lt;/h2&gt;
+ edo-subset notations&lt;br /&gt;
+P8/6, P8/8&lt;br /&gt;
+&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-Notating non-8ve and non-5th pergens*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Notating non-8ve and non-5th pergens*&lt;/h2&gt;
+ &lt;br /&gt;
+Just as all rank-2 pergens in which 2 and 3 are independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th colspan="6"&gt;&lt;u&gt;the first two independent primes in the prime subgroup&lt;/u&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;pergen number&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.3&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.5&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.5&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5.7&lt;br /&gt;
+&lt;/th&gt;
+    &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;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, y3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, r2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12, y6)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12, r3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(WWy3, bg5)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;half-splits&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;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, y3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, r2)&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;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, y3/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, r2/2)&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;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, g6/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, b7/2)&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;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P4/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, y3/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, r2/2)&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;th&gt;third-splits&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;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, P5)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, y3)&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;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, y3/3)&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;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, g6/3)&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;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P11/3)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, y10/3)&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;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, P4/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, y3/2)&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;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, P5/2)&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;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;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P4/3)&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;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;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, 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;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;14&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, 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;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;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, P4/3)&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;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.&lt;br /&gt;
+&lt;br /&gt;
+Every rank-3 pergen can be identified by its first &lt;u&gt;three&lt;/u&gt; independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;pergen number&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.3.5&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.3.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2.5.7&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3.5.7&lt;br /&gt;
+&lt;/th&gt;
+    &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, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-3 unsplit&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;same&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, y3, r2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P12, y6, r3)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;half-splits&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;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5, ^1)&lt;br /&gt;
+&lt;/td&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;same&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;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2, ^1)&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;same&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;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2, ^1)&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;same&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;5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P4/2, ^1)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;rank-3 half-everything&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;same&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;6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, vM3/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-downmajor-3rd&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, ^M2/2)&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;7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, ^m6/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-upminor-6th&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5, vm7/2)&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;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5, vM3/2)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-8ve half-downmajor 3rd&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;9&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5, ^m6/2)&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;10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2, vM3/2)&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;11&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2, ^m6/2)&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;12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2, vM3/2)&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;13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/2, ^m6/2)&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;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;(P8/2, P4/2, vM3/2)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;rank-3 half-everything&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;&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;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 3,750: / Line 4,663: @@
 &lt;/table&gt;
-If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.&lt;br /&gt;
-&lt;br /&gt;
-The additional accidental's ratio can be changed by adding the edo's defining comma onto it. For Blackwood, 5-edo is defined by 256/243, and /1 = 81/80 = 16/15.&lt;br /&gt;
-&lt;br /&gt;
-(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)&lt;br /&gt;
 &lt;br /&gt;
-In any notation, every note has a name, and no two notes have the same name. A note's representation on the musical staff follows from its name. If the notation has any enharmonics, each note has several names. Generally, every name has a note. Every possible name, and anything that can be written on on the staff, corresponds to one and only one note in the lattice formed by perchains and genchains. But in no-threes pergens,&lt;br /&gt;
+Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a &lt;u&gt;huge&lt;/u&gt; number of note-less names. The composer may well want to devise a personal notation that isn't backwards compatible.&lt;br /&gt;
 &lt;br /&gt;
-edo-subset notations&lt;br /&gt;
+Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.&lt;br /&gt;
 &lt;br /&gt;
-relative notation: not all intervals names have intervals. No perfect 5th in 6-edo.&lt;br /&gt;
+A similar chart could be made of all rank-3 pergen cubes.&lt;br /&gt;
 &lt;br /&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;!-- ws:start:WikiTextHeadingRule:97:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="Further Discussion-Notating tunings with an arbitrary generator"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:97 --&gt;Notating tunings with an arbitrary generator&lt;/h2&gt;
   &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 used here. 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;
@@ Line 4,107: / Line 5,015: @@
 See also the &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;map of rank-2 temperaments&lt;/a&gt;.&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;!-- ws:start:WikiTextHeadingRule:99:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="Further Discussion-Pergens and MOS scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:99 --&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;
@@ Line 5,212: / Line 6,120: @@
 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: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;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&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 and pergens, but only a few dozen of either have been explored.&lt;br /&gt;
@@ Line 5,723: / Line 6,631: @@
 A specific pergen can be converted to an edo pair by finding the range of its generator cents in the &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator"&gt;arbitrary generator&lt;/a&gt; table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).&lt;br /&gt;
 &lt;br /&gt;
-&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;!-- ws:start:WikiTextHeadingRule:103:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="Further Discussion-Supplemental materials*"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:103 --&gt;Supplemental materials*&lt;/h2&gt;
   &lt;br /&gt;
 needs more screenshots, including 12-edo's pergens and a page of the pdf&lt;br /&gt;
@@ Line 5,733: / Line 6,641: @@
 finish proofs&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7143:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7143 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8291:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8291 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7144:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7144 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8292:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8292 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Notaion guide PDF&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:105:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:105 --&gt;Notaion guide PDF&lt;/h3&gt;
   &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;
-&lt;!-- ws:start:WikiTextUrlRule:7145: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:7145 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8293: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:8293 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:103:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc23"&gt;&lt;a name="Further Discussion-Supplemental materials*-Pergen squares pic"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:103 --&gt;Pergen squares pic&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:107:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc25"&gt;&lt;a name="Further Discussion-Supplemental materials*-pergenLister app"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:107 --&gt;pergenLister app&lt;/h3&gt;
- &lt;br /&gt;
-One way to visualize pergens...&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:105:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc24"&gt;&lt;a name="Further Discussion-Supplemental materials*-pergenLister app"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:105 --&gt;pergenLister app&lt;/h3&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;
-&lt;!-- ws:start:WikiTextUrlRule:7146: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:7146 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:8294: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:8294 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:4260:&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:4260 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:5045:&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:5045 --&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 5,766: / Line 6,670: @@
 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 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;
-&lt;!-- ws:start:WikiTextHeadingRule:107:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:107 --&gt;Various proofs (unfinished)&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:109:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Further Discussion-Various proofs (unfinished)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:109 --&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, which equals 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). In general, 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 5,822: / Line 6,726: @@
 &lt;br /&gt;
 &lt;br /&gt;
-Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? &lt;br /&gt;
+Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?&lt;br /&gt;
 Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.&lt;br /&gt;
 a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)&lt;br /&gt;
@@ Line 5,908: / Line 6,812: @@
 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:109:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc26"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:109 --&gt;Miscellaneous Notes&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:111:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc27"&gt;&lt;a name="Further Discussion-Miscellaneous Notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:111 --&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;