Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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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 pseudo-pergens, because the multigen isn't 3-limit, and might not ~~even~~ actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.

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 pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.

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 JI-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] is a JI-agnostic way to name the Porcupine [7] scale.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments~~. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats~~.

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments.

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 high-five. 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.

For example, 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 high-five. 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.

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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G.

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 because the schisma does not map to a unison, 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. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.

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

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

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

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

The **genchain** (chain of generators) in the table is only a short section of the full genchain.

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||= (P8, P11/3)

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

||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = ~~vvP5~~ ||= C - F^ - Cv - F ||= ||

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

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

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

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=__Further Discussion__=

~~==Cents of the accidentals==~~

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.

==Extremely large multigens==

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==Singles and doubles==

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

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

Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = ~~4 gen~~, then P8 = M9 - M2 = 2⋅P5 - ~~4 gen~~ = (P5 - ~~2 gen~~) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

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

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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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 / (4/3) = 250/243.

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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.

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

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

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).

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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.

There are also alternate enharmonics, see below.

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

==Ratio and cents of the accidentals==

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

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.

==Finding a notation for a pergen==

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Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>``//``ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2.

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.

Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = ``//``d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^``//``d4.

The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3.

From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3.

Equivalent enharmonics: E + E' = v<span style="vertical-align: super;">12</span>/<span style="vertical-align: super;">3</span>~~dd4~~ = ~~3·v~~<span style="vertical-align: super;">4</span>/m2~~, and E - 2·E'~~ = v~~<span style="vertical-align: super;">12</span>\<span style="vertical-align: super;">6</span>AA1 = 3·v~~<span style="vertical-align: super;">4</span>\\A1.

~~Equivalent generators: G = ^\M3~~ =

~~P1 --~~ \M3 ~~-- \\A5~~=~~/m6-- P8~~

~~C -- E\ -- Ab/ -- C~~

~~P1 -- ^\M3 -- ^^\\A5=^^/m6=vv/M6=vv``//``d7 --~~ v~~/m9 -- F~~

~~C -- E^\ -- Ab^^/=Avv/ -- Dbv/ -- F//~~

~~P1 -- v<span style="vertical-align: super;">4</span>M3 -- v<span style="vertical-align: super;">8</span>A5=^~~<span style="vertical-align: super;">4</span~~>m6-- P8~~

~~C -- Ev<span style="vertical-align: super;"~~>4~~</span> -- Ab^<span style="vertical-align: super;">4</span> -- C~~

~~P1 -- v<span style="vertical-align: super;">3</span>/M3 -- v<span style~~=~~"vertical-align: super;">6</span>~~``//``A5=^~~<span style="vertical-align: super;">6</span>``//``m6=^<span style="vertical-align: super;">6</span>~~\~~\d7 -- ^<span style~~=~~"vertical-align: super;">3</span>\m9 -- F~~

~~C -- Ev~~<span style="vertical-align: super;">3</span>~~/ -- G#v<span style~~=~~"vertical-align: super;">6</span>``//``=Ab~~^~~<span style="vertical-align: super;">6</span>~~``//``~~=Bbb^<span style="vertical-align: super;">6</span>\\ -- Db^<span style="vertical-align: super;">3</span>\ -- F//~~

This is a lot of math, but it only needs to be done once for each pergen.

==Alternate enharmonics==

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</span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span>

~~Sometimes~~ the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it.

Approaching pergens in a JI-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-fourth. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-fourth. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.

~~//The comma equals xE and/or yE'.//~~

~~//If M' = [a,b]~~, ~~then G' = [round~~(a/n'), ~~round(b~~/~~n')] makes the smallest zE"~~, ~~but not always~~ the ~~smallest E"//~~

~~//bbT = 49/48 =~~ m2 = ~~36¢~~ = ~~half~~-~~fourth~~. ~~E = vvm2.//~~

//(-22, 11, 2) ~~= -dd2 = 94¢~~ is ~~LLyyT =~~ half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.//

~~//These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime~~ is ~~mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on~~ the ~~staff. By definition,~~ the ~~only commas that map to P1 are notational ones~~, ~~and those that are the sum or difference of notational ones. There is widespread agreement that~~ 5~~'s notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13~~, ~~is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8~~ is an augmented ~~4th~~.//

~~//An alternate enharmonic will arise if~~ the ~~notational~~ comma ~~changes. For example, 11's notational comma can be either 33~~/32, ~~with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4~~. ~~The keyspan of all 11~~-~~limit intervals will reflect this choice of notational comma. For (P8~~, ~~P5/2), G ~ 350¢~~. ~~If G = 11/9, the (vanishing, not notational) comma~~ is ~~P5 - 2·G = 243/242~~. ~~For the first notational comma,~~ 11~~/9 is a m3,~~ and the ~~comma is an A1~~. ~~For the 2nd,~~ 11/~~9= M3~~, and the ~~comma is a d1~~.//

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-fourth commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.

//For ~~single~~-~~comma pergens~~, ~~the enharmonic should equal the comma's mapping. For example~~, ~~(P8~~, ~~P5/2~~) ~~might arise~~ from ~~243~~/~~242~~, ~~which splits~~ the ~~5th into two~~ 11/~~9 halves~~.//

For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-eleventh pergen. The generator is 11/8. If 11/8 is notated as ^P4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.

~~//(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic~~ is ~~P11~~ - ~~4·G = [1,2] = dd3~~. ~~It must be downed, thus E = v<span style="vertical~~-~~align: super;">4</span>dd3~~, and ~~G = ^M3. The enharmonic is unfortunately not a unison or 2nd~~. ~~Note that the generator's stepspan could have been rounded up instead of down~~, ~~making G = [4~~,~~3] = d4~~. ~~This would make E = [~~-1,~~2] = d43~~. ~~Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred~~. ~~However, there~~ are ~~exceptions~~.//

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

==Alternate keyspans and stepspans==

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In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)

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

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

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

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

~~== ==~~

==Combining pergens==

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

Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.

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==~~Misc notes~~==

==Supplemental materials==

~~Pergens were discovered by Kite Giedraitis in 2017~~, ~~and developed with~~ the ~~help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces~~.~~com/pergen+names~~

This PDF shows the full lattice for the first 15 pergens, up through the third-something block.

This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.

(links coming soon)

~~__**Extra paragraphs:**__~~

==Misc notes==

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the ~~notational~~ comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:

k = 12a + 19b + 28c + 34d

s = 7a + 11b + 14c + 20d

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__**Extra paragraphs:**__

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As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.

The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. ~~The~~ enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div>

~~[//Question: what if there are highs and lows?//]~~</pre></div>

<h4>Original HTML content:</h4>

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

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

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

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

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

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

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

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

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

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

<div style="margin-left: 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-~~Cents of the accidentals~~">~~Cents of the accidentals~~</a></div>

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

<div style="margin-left: 2em;"><a href="#Further Discussion-~~Extremely large multigens~~">~~Extremely large multigens~~</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-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-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div>

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

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

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

<div style="margin-left: 2em;"><a href="#~~toc13~~"> </a></div>

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

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

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

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

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

</div>

</div>

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

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

<br />

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Untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br />

<br />

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

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

<br />

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

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Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.<br />

<br />

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

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

<br />

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit, and might not ~~even~~ actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.<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 pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.<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 JI-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] is a JI-agnostic way to name the Porcupine [7] scale.<br />

<br />

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments~~. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats~~.<br />

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. <br />

<br />

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 high-five. 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 />

For example, 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 high-five. 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 />

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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G.<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 because the schisma does not map to a unison, 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. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.<br />

<br />

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

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

<br />

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

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

<br />

The <strong>genchain</strong> (chain of generators) in the table is only a short section of the full genchain.<br />

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</td>

</td>

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<br />

<h1 id="toc4"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<h1 id="toc4"><a name="Further Discussion"></a><u>Further Discussion</u></h1>

<br />

<h2 id="toc5~~"><a name="Further Discussion-Cents of the accidentals"></a>Cents of the accidentals</h2>~~

<h2 id="toc5"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

~~<br />~~

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br />

~~<br />~~

~~<h2 id="toc6~~"><a name="Further Discussion-Extremely large multigens"></a>Extremely large multigens</h2>

<br />

So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.<br />

<br />

<h2 id="~~toc7~~"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

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

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

<br />

Every double-split pergen is either a <strong>true double</strong> or a <strong>false double</strong>. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = ~~4 gen~~, then P8 = M9 - M2 = 2⋅P5 - ~~4 gen~~ = (P5 - ~~2 gen~~) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.<br />

Every double-split pergen is either a <strong>true double</strong> or a <strong>false double</strong>. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.<br />

<br />

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

<br />

<h2 id="~~toc8~~"><a name="Further Discussion-Finding an example temperament"></a>Finding an example temperament</h2>

<h2 id="toc7"><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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 / (4/3) = 250/243.<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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br />

<br />

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

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

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Line 2,082:

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 />

<br />

There are also alternate enharmonics, see below.<br />

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

<br />

<h2 id="toc8"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

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

<br />

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br />

<br />

<h2 id="toc9"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<h2 id="toc9"><a name="Further Discussion-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

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

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Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>//ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br />

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2.~~<br />~~

One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.<br />

~~<br />~~

Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = //d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^//d4.<br />

The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. ~~<br />~~

~~<br />~~

From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3.~~<br />~~

Equivalent enharmonics: E + E' ~~= v<span style="vertical-align: super;">12</span>/<span style="vertical-align: super;">3</span>dd4 = 3·v<span style="vertical-align: super;">4</span>/m2,~~ and E - 2·E' ~~= v<span style="vertical-align: super;">12</span>\<span style="vertical-align: super;">6</span>AA1 = 3·v<span style="vertical-align: super;">4</span>\\A1~~.~~<br />~~

Equivalent generators~~: G = ^\M3 =<br />~~

~~<br />~~

~~P1 -- \M3 -- \\A5=/m6-- P8<br />~~

~~C -- E\ -- Ab/ -- C<br />~~

~~P1 -- ^\M3 -- ^^\\A5=^^/m6=vv/M6=vv//d7 -- v/m9 -- F~~<br />

~~C -- E^\ -- Ab^^/~~=~~Avv/ -- Dbv/ -- F<em><br />~~

~~<br />~~

~~P1 --~~ v<span style="vertical-align: super;">4</span>~~M3 -- v~~<span style="vertical-align: super;">8</span>A5=^<span style="vertical-align: super;">4</span>~~m6-- P8<br~~ /~~>~~

~~C -- Ev<span style~~=~~"vertical-align: super;">4</span> -- Ab^~~<span style="vertical-align: super;">4</span> ~~-- C<br />~~

~~P1 -- v<span style~~=~~"vertical-align: super;">3</span>/~~M3 -- v<span style="vertical-align: super;">6</span>~~//A5~~=~~^<span style="vertical-align: super;">6</span>~~//m6=^~~<span style="vertical-align: super;">6</span>~~\~~\d7 -- ^<span style~~=~~"vertical-align: super;">3</span>\m9 -- F<br />~~

~~C -- Ev~~<span style="vertical-align: super;">3</span>~~/ -- G#v<span style="vertical-align: super;">6</span>//~~=Ab^~~<span style="vertical-align: super;">6</span>~~//~~=Bbb^<span style="vertical-align: super;">6</span>\\ -- Db^<span style="vertical-align: super;">3</span>\ -- F</em>~~<br />

<br />

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

<br />

<h2 id="toc10"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc10"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br />

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Line 2,136:

</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">P1 -- /m2 -- //d3=\\A2 -- \M3 -- P4<br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb//=D#\\ -- E\ -- F</span><br />

~~Sometimes~~ the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it.~~<br />~~

Approaching pergens in a JI-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-fourth. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-fourth. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br />

~~<br />~~

~~<em>The comma equals xE and/or yE'.</em><br />~~

~~<em>If M' = [a~~,~~b], then G' = [round~~(a/n'), ~~round(b~~/~~n')] makes the smallest zE&quot;~~, ~~but not always~~ the ~~smallest E&quot;</em><br />~~

~~<br />~~

~~<em>bbT = 49/48 =~~ m2 = ~~36¢~~ = ~~half~~-~~fourth~~. ~~E = vvm2.</em><br />~~

~~<em>~~(-22, 11, 2) ~~= -dd2 = 94¢~~ is ~~LLyyT =~~ half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.~~</em><br />~~

~~<br />~~

~~<em>These commas are called <strong>notational commas</strong>. They are not necessarily tempered out, they merely determine how a higher prime~~ is ~~mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on~~ the ~~staff. By definition,~~ the ~~only commas that map to P1 are notational ones~~, ~~and those that are the sum or difference of notational ones. There is widespread agreement that~~ 5~~'s notational comma is 81/80. But the choice of notational commas for other primes~~, ~~especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8~~ is an augmented ~~4th~~.~~</em>~~<br />

<br />

~~<em>An alternate enharmonic will arise if~~ the ~~notational~~ comma ~~changes. For example, 11's notational comma can be either 33~~/32, ~~with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4~~. ~~The keyspan of all 11~~-~~limit intervals will reflect this choice of notational comma. For (P8~~, ~~P5/2), G ~ 350¢~~. ~~If G = 11/9, the (vanishing, not notational) comma~~ is ~~P5 - 2·G = 243/242~~. ~~For the first notational comma,~~ 11~~/9 is a m3,~~ and the ~~comma is an A1~~. ~~For the 2nd,~~ 11/~~9= M3~~, and the ~~comma is a d1~~.~~</em>~~<br />

Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-fourth commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br />

<br />

~~<em>~~For ~~single~~-~~comma pergens~~, ~~the enharmonic should equal the comma's mapping. For example~~, ~~(P8~~, ~~P5/2~~) ~~might arise~~ from ~~243~~/~~242~~, ~~which splits~~ the ~~5th into two~~ 11/~~9 halves.~~</em><br />

For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-eleventh pergen. The generator is 11/8. If 11/8 is notated as ^P4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.<br />

<br />

~~<em>(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic~~ is ~~P11~~ - ~~4·G = [1,2] = dd3~~. ~~It must be downed, thus E = v<span style="vertical~~-~~align: super;">4</span>dd3~~, and ~~G = ^M3. The enharmonic is unfortunately not a unison or 2nd~~. ~~Note that the generator's stepspan could have been rounded up instead of down~~, ~~making G = [4~~,~~3] = d4~~. ~~This would make E = [~~-1,~~2] = d43~~. ~~Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred~~. ~~However, there~~ are ~~exceptions~~.~~</em>~~<br />

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

<br />

<h2 id="toc11"><a name="Further Discussion-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<h2 id="toc11"><a name="Further Discussion-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<br />

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br />

<br />

<h2 id="toc12"><a name="Further Discussion-Chord names"></a>Chord names</h2>

<h2 id="toc12"><a name="Further Discussion-Chord names"></a>Chord names</h2>

<br />

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

Line 2,192:

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In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)<br />

<br />

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

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

<br />

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

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

<~~!-- ws:start:WikiTextHeadingRule:59:&lt;h2&gt; --><h2 id="toc13"> <~~/h2>

<br />

<h2 id="~~toc14~~"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

<h2 id="toc13"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

<br />

Tempering out 250/243 creates third-fourth, and 49/48 creates half-fourth, and tempering out both commas creates sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br />

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However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br />

<br />

<h2 id="~~toc15~~"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc14"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

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

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

<br />

Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br />

Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br />

<br />

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br />

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<br />

<h2 id="~~toc16~~"><a name="Further Discussion-~~Misc notes~~"></a>~~Misc notes~~</h2>

<h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a>Supplemental materials</h2>

<br />

~~Pergens were discovered by Kite Giedraitis in 2017~~, ~~and developed with~~ the ~~help of Praveen Venkataramana. Earlier drafts of this article can be found at ~~<br />

This PDF shows the full lattice for the first 15 pergens, up through the third-something block.<br />

<br />

This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />

<br />

(links coming soon)<br />

<br />

<u><~~strong~~>~~Extra paragraphs~~:</~~strong~~></u><br />

<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

<br />

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><br />

<br />

~~<br />~~

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br />

Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the ~~notational~~ comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br />

k = 12a + 19b + 28c + 34d<br />

s = 7a + 11b + 14c + 20d<br />

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<br />

<u><strong>Extra paragraphs:</strong></u><br />

<br />

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As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.<br />

<br />

The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. ~~The~~ enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.~~<br />~~

The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</body></html></pre></div>

~~<br />~~

~~[<em>Question: what if there are highs and lows?</em>]~~</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 05:43:33 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 08:25:21 UTC</tt>.<br>
-: The original revision id was <tt>624773549</tt>.<br>
+: The original revision id was <tt>624778785</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 111: / Line 111: @@
 =__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 pseudo-pergens, because the multigen isn't 3-limit, and might not even actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.
+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 pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.
 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 JI-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 117: / Line 117: @@
 Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic way to name the Porcupine [7] scale.
-The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.
+The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments.
-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 high-five. 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.
+For example, 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 high-five. 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.
-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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G.
+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 because the schisma does not map to a unison, 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. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
@@ Line 127: / Line 127: @@
 Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.
-The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.
+The enharmonic interval, or more briefly the **enharmonic**, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.
 The **genchain** (chain of generators) in the table is only a short section of the full genchain.
@@ Line 210: / Line 210: @@
 ||= (P8, P11/3)
 third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||=   ||
-||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vvP5 ||= C - F^ - Cv - F ||=   ||
+||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||=   ||
 ||= (P8/3, P4/2)
 third-8ve, half-4th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;A2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D# ||= P8/3 = ^^m3
@@ Line 279: / Line 279: @@
 =__Further Discussion__=
-==Cents of the accidentals==
-We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.
 ==Extremely large multigens==
@@ Line 290: / Line 286: @@
 ==Singles and doubles==
-If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** pergen. A single-split pergens can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger.
+If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.
-Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2⋅P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.
+Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.
 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.
@@ Line 298: / Line 294: @@
 ==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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 / (4/3) = 250/243.
+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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 ÷ (4/3) = 250/243.
-An even easier way is to simply assume that the up symbol equals a convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 27/20.
+If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 27/20. The comma must have only one higher prime, with exponent ±1.
 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).
@@ Line 316: / Line 312: @@
 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.
-There are also alternate enharmonics, see below.
+There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.
+==Ratio and cents of the accidentals==
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.
+We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.
 ==Finding a notation for a pergen==
@@ Line 350: / Line 352: @@
 Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;``//``ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F&lt;/span&gt;
-One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2.
+One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.
+Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = ``//``d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^``//``d4.
-The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3.
-From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3.
-Equivalent enharmonics: E + E' = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd4 = 3·v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2, and E - 2·E' = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;\&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;AA1 = 3·v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1.
-Equivalent generators: G = ^\M3 =
-P1 -- \M3 -- \\A5=/m6-- P8
-C -- E\ -- Ab/ -- C
-P1 -- ^\M3 -- ^^\\A5=^^/m6=vv/M6=vv``//``d7 -- v/m9 -- F
-C -- E^\ -- Ab^^/=Avv/ -- Dbv/ -- F//
-P1 -- v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M3 -- v&lt;span style="vertical-align: super;"&gt;8&lt;/span&gt;A5=^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m6-- P8
-C -- Ev&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- Ab^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- C
-P1 -- v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/M3 -- v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;``//``A5=^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;``//``m6=^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;\\d7 -- ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;\m9 -- F
-C -- Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/ -- G#v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;``//``=Ab^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;``//``=Bbb^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;\\ -- Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;\ -- F//
+&lt;span style="display: block; text-align: center;"&gt;P1 — \M3 — \\A5=/m6 — P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E\ — Ab/ — C&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;8=v/m9 — F&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E^\ — Ab^^/=Avv\ — Dbv/ — F&lt;/span&gt;
+This is a lot of math, but it only needs to be done once for each pergen.
 ==Alternate enharmonics==
@@ Line 382: / Line 371: @@
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F&lt;/span&gt;
-Sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it.
+Approaching pergens in a JI-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-fourth. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-fourth. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
-//The comma equals xE and/or yE'.//
-//If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE", but not always the smallest E"//
-//bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.//
-//(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.//
-//These commas are called **notational commas**. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.//
-//An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2·G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1.//
+Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-fourth commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.
-//For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.//
+For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-eleventh pergen. The generator is 11/8. If 11/8 is notated as ^P4, the enharmonic is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2, but if 11/8 is notated as vA4, the enharmonic is ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2.
-//(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4·G = [1,2] = dd3. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.//
+This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, tend to have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. So alternate enharmonics are needed sometimes.
 ==Alternate keyspans and stepspans==
@@ Line 409: / Line 389: @@
 In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)
-Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
+Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
+A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;gT) is also half-octave. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
-A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;gT) is also half-octave. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
-== ==
 ==Combining pergens==
@@ Line 434: / Line 414: @@
 How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.
-Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.
+Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.
 This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.
@@ Line 477: / Line 457: @@
-==Misc notes==
+==Supplemental materials==
-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names
+This PDF shows the full lattice for the first 15 pergens, up through the third-something block.
+This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.
+(links coming soon)
-__**Extra paragraphs:**__
+==Misc notes==
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names
-Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:
+Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:
 k = 12a + 19b + 28c + 34d
 s = 7a + 11b + 14c + 20d
@@ Line 498: / Line 481: @@
+__**Extra paragraphs:**__
@@ Line 538: / Line 524: @@
 As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.
-The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.
+The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div>
-[//Question: what if there are highs and lows?//]</pre></div>
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+&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Extremely large multigens"&gt;Extremely large multigens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:70 --&gt;&lt;!-- ws:start:WikiTextTocRule:71: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:71 --&gt;&lt;!-- ws:start:WikiTextTocRule:72: --&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:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&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:72 --&gt;&lt;!-- ws:start:WikiTextTocRule:73: --&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:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&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:73 --&gt;&lt;!-- ws:start:WikiTextTocRule:74: --&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:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&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:74 --&gt;&lt;!-- ws:start:WikiTextTocRule:75: --&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:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate keyspans and stepspans"&gt;Alternate keyspans and stepspans&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:75 --&gt;&lt;!-- ws:start:WikiTextTocRule:76: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Alternate keyspans and stepspans"&gt;Alternate keyspans and stepspans&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names"&gt;Chord names&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:76 --&gt;&lt;!-- ws:start:WikiTextTocRule:77: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Chord names"&gt;Chord names&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#toc13"&gt; &lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:77 --&gt;&lt;!-- ws:start:WikiTextTocRule:78: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Combining pergens"&gt;Combining pergens&lt;/a&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:81 --&gt;&lt;!-- ws:start:WikiTextTocRule:82: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Combining pergens"&gt;Combining pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:78 --&gt;&lt;!-- ws:start:WikiTextTocRule:79: --&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:82 --&gt;&lt;!-- ws:start:WikiTextTocRule:83: --&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:79 --&gt;&lt;!-- ws:start:WikiTextTocRule:80: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:80 --&gt;&lt;!-- ws:start:WikiTextTocRule:81: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Misc notes"&gt;Misc notes&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:82 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
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 Untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:37:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:37 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 For any comma containing primes 2 and 3, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;
@@ Line 1,094: / Line 1,078: @@
 Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, the multigen2. A double half-fifth is a fifth = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:39:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:39 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&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 pseudo-pergens, because the multigen isn't 3-limit, and might not even actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.&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 pseudo-pergens, because the multigen isn't 3-limit, and might not actually be a generator. Meantone (mean = average, tone = major 2nd) implies (5/4)/2.&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 JI-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 1,102: / Line 1,086: @@
 Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] is a JI-agnostic way to name the Porcupine [7] scale.&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 exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.&lt;br /&gt;
+The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. &lt;br /&gt;
 &lt;br /&gt;
-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 high-five. 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;
+For example, 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 high-five. 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;
-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 because the schisma does not map to a unison, this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G.&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 because the schisma does not map to a unison, 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. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.&lt;br /&gt;
 &lt;br /&gt;
 Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.&lt;br /&gt;
@@ Line 1,112: / Line 1,096: @@
 Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to quarter-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.&lt;br /&gt;
 &lt;br /&gt;
-The enharmonic interval, or more briefly the &lt;strong&gt;enharmonic&lt;/strong&gt;, can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.&lt;br /&gt;
+The enharmonic interval, or more briefly the &lt;strong&gt;enharmonic&lt;/strong&gt;, can be added to or subtracted from any note (or interval), renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, and A4 with d5. In a single-comma temperament, the comma usually maps to the pergen's enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.&lt;br /&gt;
 &lt;br /&gt;
 The &lt;strong&gt;genchain&lt;/strong&gt; (chain of generators) in the table is only a short section of the full genchain.&lt;br /&gt;
@@ Line 1,426: / Line 1,410: @@
          &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:012:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:012 --&gt; D&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;P11/3 = ^4 = vvP5&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;P11/3 = ^4 = vv5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F^ - Cv - F&lt;br /&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:41:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:41 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Further Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;u&gt;Further Discussion&lt;/u&gt;&lt;/h1&gt;
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-&lt;!-- ws:start:WikiTextHeadingRule:43:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Further Discussion-Cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:43 --&gt;Cents of the accidentals&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Further Discussion-Extremely large multigens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Extremely large multigens&lt;/h2&gt;
- &lt;br /&gt;
-We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:45:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Extremely large multigens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:45 --&gt;Extremely large multigens&lt;/h2&gt;
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 So far, the largest multigen has been a 12th. As the multigen fractions get larger, the multigen gets quite wide. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 32/9 = Wm7, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), the multigen can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:47:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:47 --&gt;Singles and doubles&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;Singles and doubles&lt;/h2&gt;
   &lt;br /&gt;
-If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergens can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger.&lt;br /&gt;
+If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.&lt;br /&gt;
 &lt;br /&gt;
-Every double-split pergen is either a &lt;strong&gt;true double&lt;/strong&gt; or a &lt;strong&gt;false double&lt;/strong&gt;. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2⋅P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.&lt;br /&gt;
+Every double-split pergen is either a &lt;strong&gt;true double&lt;/strong&gt; or a &lt;strong&gt;false double&lt;/strong&gt;. A true double, like third-everything (P8/3, P4/3) or half-octave quarter-fourth (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.&lt;br /&gt;
 &lt;br /&gt;
 A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:49:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:49 --&gt;Finding an example temperament&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Finding an example temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 / (4/3) = 250/243.&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. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - P4 = (10/9)^3 ÷ (4/3) = 250/243.&lt;br /&gt;
 &lt;br /&gt;
-An even easier way is to simply assume that the up symbol equals a convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 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 convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 27/20. The comma must have only one higher prime, with exponent ±1.&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;
@@ Line 2,102: / Line 2,082: @@
 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;
 &lt;br /&gt;
-There are also alternate enharmonics, see below.&lt;br /&gt;
+There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
+ &lt;br /&gt;
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.&lt;br /&gt;
+&lt;br /&gt;
+We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢ and ^1 = (A1)/2. But A1 = 100¢ + 7c, so ^1 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:51:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:51 --&gt;Finding a notation for a pergen&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Finding a notation for a pergen&lt;/h2&gt;
   &lt;br /&gt;
 There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;
@@ Line 2,133: / Line 2,119: @@
 Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\d3 = 2·vv\m2, and E - E' = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:025:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:025 --&gt;ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:&lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F&lt;/span&gt;&lt;br /&gt;
-One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2.&lt;br /&gt;
+One might think with (P8/3, P11/4), that one pair would be needed to split the octave, and another two pairs to split the 11th, making three in all. But only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.&lt;br /&gt;
-&lt;br /&gt;
+Enharmonic = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;m3 = /&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1. Period = \M3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 = &lt;!-- ws:start:WikiTextRawRule:026:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:026 --&gt;d4. Generator = ^\M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;4 = ^&lt;!-- ws:start:WikiTextRawRule:027:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:027 --&gt;d4.&lt;br /&gt;
-The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. &lt;br /&gt;
-&lt;br /&gt;
-From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3.&lt;br /&gt;
-Equivalent enharmonics: E + E' = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;/&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd4 = 3·v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;/m2, and E - 2·E' = v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;\&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;AA1 = 3·v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;\\A1.&lt;br /&gt;
-Equivalent generators: G = ^\M3 =&lt;br /&gt;
-&lt;br /&gt;
-P1 -- \M3 -- \\A5=/m6-- P8&lt;br /&gt;
-C -- E\ -- Ab/ -- C&lt;br /&gt;
-P1 -- ^\M3 -- ^^\\A5=^^/m6=vv/M6=vv&lt;!-- ws:start:WikiTextRawRule:026:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:026 --&gt;d7 -- v/m9 -- F&lt;br /&gt;
-C -- E^\ -- Ab^^/=Avv/ -- Dbv/ -- F&lt;em&gt;&lt;br /&gt;
-&lt;br /&gt;
-P1 -- v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M3 -- v&lt;span style="vertical-align: super;"&gt;8&lt;/span&gt;A5=^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m6-- P8&lt;br /&gt;
-C -- Ev&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- Ab^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; -- C&lt;br /&gt;
-P1 -- v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/M3 -- v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:027:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:027 --&gt;A5=^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:028:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:028 --&gt;m6=^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;\\d7 -- ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;\m9 -- F&lt;br /&gt;
-C -- Ev&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;/ -- G#v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:029:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:029 --&gt;=Ab^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;&lt;!-- ws:start:WikiTextRawRule:030:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:030 --&gt;=Bbb^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;\\ -- Db^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;\ -- F&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
+&lt;span style="display: block; text-align: center;"&gt;P1 — \M3 — \\A5=/m6 — P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E\ — Ab/ — C&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;8=v/m9 — F&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C — E^\ — Ab^^/=Avv\ — Dbv/ — F&lt;/span&gt;&lt;br /&gt;
+This is a lot of math, but it only needs to be done once for each pergen.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:53:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:53 --&gt;Alternate enharmonics&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Alternate enharmonics&lt;/h2&gt;
   &lt;br /&gt;
 Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2.&lt;br /&gt;
@@ Line 2,163: / Line 2,136: @@
 &lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;
 &lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:031:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:031 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;P1 -- /m2 -- &lt;!-- ws:start:WikiTextRawRule:028:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:028 --&gt;d3=\\A2 -- \M3 -- P4&lt;br /&gt;
-&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:032:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:032 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
+&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Db/ -- Ebb&lt;!-- ws:start:WikiTextRawRule:029:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:029 --&gt;=D#\\ -- E\ -- F&lt;/span&gt;&lt;br /&gt;
-Sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it.&lt;br /&gt;
+Approaching pergens in a JI-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-fourth. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-fourth. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;The comma equals xE and/or yE'.&lt;/em&gt;&lt;br /&gt;
-&lt;em&gt;If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE&amp;quot;, but not always the smallest E&amp;quot;&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.&lt;/em&gt;&lt;br /&gt;
-&lt;em&gt;(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.&lt;/em&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;em&gt;These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. They are not necessarily tempered out, they merely determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is widespread agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th.&lt;/em&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;An alternate enharmonic will arise if the notational comma changes. For example, 11's notational comma can be either 33/32, with 11/8 notated as a P4, or 729/704, with 11/8 notated as an A4. The keyspan of all 11-limit intervals will reflect this choice of notational comma. For (P8, P5/2), G ~ 350¢. If G = 11/9, the (vanishing, not notational) comma is P5 - 2·G = 243/242. For the first notational comma, 11/9 is a m3, and the comma is an A1. For the 2nd, 11/9= M3, and the comma is a d1.&lt;/em&gt;&lt;br /&gt;
+Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-fourth commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.&lt;/em&gt;&lt;br /&gt;
+For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-eleventh pergen. The generator is 11/8. If 11/8 is notated as ^P4, the enharmonic is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2, but if 11/8 is notated as vA4, the enharmonic is ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2.&lt;br /&gt;
 &lt;br /&gt;
-&lt;em&gt;(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4·G = [1,2] = dd3. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.&lt;/em&gt;&lt;br /&gt;
+This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, tend to have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. So alternate enharmonics are needed sometimes.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:55:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:55 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
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 One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Chord names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;Chord names&lt;/h2&gt;
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 Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&lt;br /&gt;
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 In certain pergens, one spelling isn't always clearly better. For example, in half-fourth, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)&lt;br /&gt;
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-Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).&lt;br /&gt;
+Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-octave, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).&lt;br /&gt;
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-A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;amp;gT) is also half-octave. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. The square mapping (the first two columns) are the same, hence the pergen is the same, but the other columns are different, hence the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.&lt;br /&gt;
+A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&amp;amp;gT) is also half-octave. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Combining pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;Combining pergens&lt;/h2&gt;
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 Tempering out 250/243 creates third-fourth, and 49/48 creates half-fourth, and tempering out both commas creates sixth-fourth. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).&lt;br /&gt;
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 However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;Pergens and EDOs&lt;/h2&gt;
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 Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;
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 How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.&lt;br /&gt;
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-Given an edo, a period, and a generator, what is the pergen? For 12edo, P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.&lt;br /&gt;
+Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.&lt;br /&gt;
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 This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Supplemental materials"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;Supplemental materials&lt;/h2&gt;
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-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3137:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3137 --&gt;&lt;br /&gt;
+This PDF shows the full lattice for the first 15 pergens, up through the third-something block.&lt;br /&gt;
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+This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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+(links coming soon)&lt;br /&gt;
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-&lt;u&gt;&lt;strong&gt;Extra paragraphs:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Misc notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Misc notes&lt;/h2&gt;
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+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3100:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3100 --&gt;&lt;br /&gt;
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+Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:&lt;br /&gt;
-Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the notational comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:&lt;br /&gt;
 k = 12a + 19b + 28c + 34d&lt;br /&gt;
 s = 7a + 11b + 14c + 20d&lt;br /&gt;
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 As a side note, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.&lt;br /&gt;
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-The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;br /&gt;
+The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;/body&gt;&lt;/html&gt;</pre></div>
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-[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;/body&gt;&lt;/html&gt;</pre></div>