Kite's thoughts on pergens
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[[toc]]
=__**Definition**__=
A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is **split** into N parts. The interval which is split into multiple generators is the **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, etc.
For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using [[Ups and Downs Notation|ups and downs]].
The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime > 3 (a **higher prime**), e.g. 81/80 or 64/63. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**.
Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.
For example, srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is __not__ preferred over P4/2. For example, decimal is (P8/2, P4/2), not (P8/2, P5/2).
||||~ pergen ||||||||~ example temperaments ||
||~ written ||~ spoken ||~ comma(s) ||~ name ||||~ color name ||
||= (P8, P5) ||= unsplit ||= 81/80 ||= meantone ||= green ||= gT ||
||= " ||= " ||= 64/63 ||= archy ||= red ||= rT ||
||= " ||= " ||= (-14,8,1) ||= schismic ||= large yellow ||= LyT ||
||= (P8/2, P5) ||= half-octave ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||
||= " ||= " ||= 81/80, 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||
||= (P8, P5/2) ||= half-fifth ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||
||= " ||= " ||= (-1,5,0,0,-2) ||= mohajira ||= deep amber ||= aaT ||
||= (P8, P4/2) ||= half-fourth ||= 49/48 ||= semaphore ||= deep blue ||= bbT ||
||= (P8/2, P4/2) ||= half-everything ||= 25/24, 49/48 ||= decimal ||= deep yellow and deep blue ||= yy&bbT ||
||= (P8, P4/3) ||= third-fourth ||= 250/243 ||= porcupine ||= triple yellow ||= y<span style="vertical-align: super;">3</span>T ||
||= (P8, P11/3) ||= third-eleventh ||= (12,-1,0,0,-3) ||= small triple amber ||= small triple amber ||= sa<span style="vertical-align: super;">3</span>T ||
||= (P8/4, P5) ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T ||
||= (P8/2, M2/4) ||= half-octave, quarter-tone ||= (-17,2,0,0,4) ||= large quadruple jade ||= large quadruple jade ||= Lj<span style="vertical-align: super;">4</span>T ||
||= (P8, P12/5) ||= fifth-twelfth ||= (-10,-1,5) ||= magic ||= large quintuple yellow ||= Ly<span style="vertical-align: super;">5</span>T ||
(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.
The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.
For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).
Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.
Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.
A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.
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,...).
=__Derivation__=
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 > m, it will split some 3-limit interval into n parts.
In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.
For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a **square mapping** by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes > 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.
2/1 = P8 = xP, thus P = P8/x
3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz
To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).
G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz
<span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x <= n <= x</span>**
</span>
For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:
||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||
||~ period ||= 1 ||= 1 ||= 1 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 2 ||
Use an [[http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1|online tool]] to invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.
||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||
||~ 2/1 ||= 4 ||= -2 ||= -1 || ||
||~ 3/1 ||= 0 ||= 2 ||= -1 || ||
||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||
Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.
Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.
The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.
Alternatively, we could discard the 3rd column and keep the 4th one:
||~ ||~ 2/1 ||~ 3/1 ||~ 7/1 ||
||~ period ||= 1 ||= 1 ||= 2 ||
||~ gen1 ||= 0 ||= 2 ||= 1 ||
||~ gen2 ||= 0 ||= 0 ||= 1 ||
This inverts to this matrix:
||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||
||~ 2/1 ||= 2 ||= -1 ||= -3 || ||
||~ 3/1 ||= 0 ||= 1 ||= -1 || ||
||~ 7/1 ||= 0 ||= 0 ||= 2 || /2 ||
Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, 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. We can 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 > 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.
=__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.
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.
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.
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.
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to 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 **genchain** (chain of generators) in the table is only a short section of the full genchain.
C - G implies ...Eb Bb F C G D A E B F# C#...
C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...
If the octave is split, the table has a **perchain** ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.
The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.
(table is unfinished)
||~ pergen ||~ enharmonic
interval(s) ||~ equiva-
lence(s) ||~ split
interval(s) ||~ perchain(s) and
genchains(s) ||~ examples ||
||= (P8, P5)
unsplit ||= none ||= none ||= none ||= C - G ||= meantone,
schismic ||
||~ halves ||~ ||~ ||~ ||~ ||~ ||
||= (P8/2, P5)
half-octave ||= ^^d2 (if 5th
``>`` 700¢ ||= C^^ = B# ||= P8/2 = vA4 = ^d5 ||= C - F#v=Gb^ - C ||= srutal
^1 = 81/80 ||
||= " ||= vvd2 (if 5th
< 700¢) ||= C^^ = Db ||= P8/2 = ^A4 = vd5 ||= C - F#^=Gbv - C ||= large deep red
^1 = 64/63 ||
||= " ||= vvM2 ||= C^^ = D ||= P8/2 = ^4 = vP5 ||= C - F^=Gv - C ||= 128/121
^1 = 33/32 ||
||= (P8, P4/2)
half-fourth ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore
^1 = 64/63 ||
||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||
||= (P8, P5/2)
half-fifth ||= vvA1 ||= C^^ = C# ||= P5/2 = ^m3 = vM3 ||= C - Eb^=Ev - G ||= mohajira
^1 = 33/32 ||
||= (P8/2, P4/2)
half-
everything ||= \\m2,
vvA1,
^^\\d2,
vv\\M2 ||= C``//`` = Db
C^^ = C#
C^^``//`` = D ||= P4/2 = /M2 = \m3
P5/2 = ^m3 = vM3
P8/2 = v/A4 = ^\d5
``=`` ^/4 ``=`` v\P5 ||= C - D/=Eb\ - F,
C - Eb^=Ev - G,
C - F#v/=Gb^\ - C,
C - F^/=Gv\ - C ||= 49/48 & 128/121
^1 = 33/32
/1 = 64/63 ||
||= " ||= ^^d2,
\\m2,
vv\\A1 ||= C^^ = B#
C``//`` = Db
C^^``//`` = C# ||= P8/2 = vA4 = ^d5
P4/2 = /M2 = \m3
P5/2 = ^/m3 = v\M3 ||= C - F#v=Gb^ - C,
C - D/=Eb\ - F,
C - Eb^/=Ev\ - G ||= 2048/2025 & 49/48
^1 = 81/80
/1 = 64/63 ||
||= " ||= ^^d2,
\\A1,
^^\\m2 ||= C^^ = B#
C``//`` = C#
C^^\\ = B ||= P8/2 = vA4 = ^d5
P5/2 = /m3 = \M3
P4/2 =v/M2 = ^\m3 ||= C - F#v=Gb^ - C,
C - Eb/=E\ - G,
C - Dv/=Eb^\ - F ||= 2048/2025 & 128/121
^1 = 81/80
/1 = 33/32 ||
||~ thirds ||~ ||~ ||~ ||~ ||~ ||
||= (P8/3, P5)
third-octave ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||
||= (P8, P4/3)
third-fourth ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = ^^m2 = vM2 ||= C - Dv - Eb^ - F ||= porcupine ||
||= (P8, P5/3)
third-fifth ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
||= (P8, P11/3)
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 ||= ||
||= (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
P4/2 = v<span style="vertical-align: super;">3</span>m2 ||= C - Eb^^ - Avv - C
C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F ||= ||
||= (P8/3, P5/2)
third-8ve, half-5th ||= v<span style="vertical-align: super;">6</span>m3 ||= C^<span style="vertical-align: super;">6</span> ``=`` Eb ||= ||= ||= ||
||= (P8/2, P4/3)
half-8ve, third-4th ||= v<span style="vertical-align: super;">6</span>d4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb ||= ||= ||= ||
||= " ||= v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4 ||= C^<span style="vertical-align: super;">6</span> ``=`` Fb3 ||= ||= ||= ||
||= (P8/2, P5/3)
half-8ve,
third-5th ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C
C - D#vv - Fb^^ - G ||= ||
||= " ||= ^^d2,
\\\m2 ||= C^^ = B#
C``///`` = Db ||= P8/2 = vA4 = ^d5
P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
C - /D - \F - G ||= ||
||= (P8/2, P11/3)
half-8ve,
third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= ||= ||= ||
||= (P8/3, P4/3)
third-
everything ||= ||= ||= ||= ||= ||
||~ quarters ||~ ||~ ||~ ||~ ||~ ||
||= (P8/4, P5) ||= ^<span style="vertical-align: super;">4</span>d2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B# ||= P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished,
^1 = 81/80 ||
||= (P8, P4/4) ||= ^<span style="vertical-align: super;">4</span>dd2 ||= C^<span style="vertical-align: super;">4</span> ``=`` B## ||= P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= ||
||= (P8, P5/4) ||= v<span style="vertical-align: super;">4</span>A1 ||= C^<span style="vertical-align: super;">4</span> ``=`` C# ||= P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||
||= (P8, P11/4) ||= v<span style="vertical-align: super;">4</span>dd3 ||= C^<span style="vertical-align: super;">4</span> ``=`` Eb<span style="vertical-align: super;">3</span> ||= P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5 ||= C E^ G#^^ Dbv F ||= ||
||= (P8, P12/4) ||= v<span style="vertical-align: super;">4</span>m2 ||= C^<span style="vertical-align: super;">4</span> ``=`` Db ||= P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3 ||= C Fv Bbvv=A^^ D^ G ||= ||
||= (P8/4, P4/2) ||= ||= ||= ||= ||= ||
||= (P8/2, M2/4) ||= ||= ||= ||= ||= ||
||= (P8/2, P4/4) ||= ||= ||= ||= ||= ||
||= (P8/2, P5/4) ||= ||= ||= ||= ||= ||
||= (P8/4, P4/3) ||= ||= ||= ||= ||= ||
||= (P8/4, P5/3) ||= ||= ||= ||= ||= ||
||= (P8/4, P11/3) ||= ||= ||= ||= ||= ||
||= (P8/3, P4/4) ||= ||= ||= ||= ||= ||
||= (P8/3, P5/4) ||= ||= ||= ||= ||= ||
||= (P8/3, P11/4) ||= ||= ||= ||= ||= ||
||= (P8/3, P12/4) ||= ||= ||= ||= ||= ||
||= (P8/4, P4/4) ||= ||= ||= ||= ||= ||
Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.
The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the "rungspan") is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.
Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-exponent of the bare enharmonic, thus the edo implies the enharmonic.
||||~ bare enharmonic interval . ||~ 3-exponent . ||~ implied edo . ||~ edo's 5th . ||~ upping range . ||~ downing range . ||~ if the 5th is just ||
||= M2 ||= C - D ||= 2 ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||
||= m3 ||= C - Eb ||= -3 ||= 3-edo ||= 800¢ ||= none ||= all ||= downed ||
||= m2 ||= C - Db ||= -5 ||= 5-edo ||= 720¢ ||= none ||= all ||= downed ||
||= A1 ||= C - C# ||= 7 ||= 7-edo ||= ~686¢ ||= 600-686¢ ||= 686¢-720¢ ||= downed ||
||= d2 ||= C - Dbb ||= -12 ||= 12-edo ||= 700¢ ||= 700-720¢ ||= 600-700¢ ||= upped ||
||= dd3 ||= C - Eb<span style="vertical-align: super;">3</span> ||= -17 ||= 17-edo ||= ~706¢ ||= 706-720¢ ||= 600-706¢ ||= downed ||
||= dd2 ||= C - Db<span style="vertical-align: super;">3</span> ||= -19 ||= 19-edo ||= ~695¢ ||= 695-720¢ ||= 600-695¢ ||= upped ||
||= d<span style="vertical-align: super;">3</span>4 ||= C - Fb<span style="vertical-align: super;">3</span> ||= -22 ||= 22-edo ||= ~709¢ ||= 709-720¢ ||= 600-709¢ ||= downed ||
||= d<span style="vertical-align: super;">3</span>2 ||= C - Db<span style="vertical-align: super;">4</span> ||= -26 ||= 26-edo ||= ~692¢ ||= 692-720¢ ||= 600-692¢ ||= upped ||
||= d<span style="vertical-align: super;">4</span>4 ||= C - Fb<span style="vertical-align: super;">4</span> ||= -29 ||= 29-edo ||= ~703¢ ||= 703-720¢ ||= 600-703¢ ||= downed ||
||= d<span style="vertical-align: super;">4</span>3 ||= C - Eb<span style="vertical-align: super;">5</span> ||= -31 ||= 31-edo ||= ~697¢ ||= 697-720¢ ||= 600-697¢ ||= upped ||
||= etc. ||= ||= ||= ||= ||= ||= ||= ||
=__Further Discussion__=
==Extremely large multigens==
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 "W" 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.
==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.
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.
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.
==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*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. 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*G - P4 = (10/9)^3 / (4/3) = 250/243.
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*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.
Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an **alternate** generator. A generator or period plus or minus any number of enharmonics makes an **equivalent** generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-octave (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.
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.
==Finding a notation for a pergen==
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:
* For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2
* For false doubles using single-pair notation, E = E', but x and y are usually different
* The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE", and P8 = mP + xE"
The **keyspan** of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The **stepspan** of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a **gedra**, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:
<span style="display: block; text-align: center;">k = 12a + 19b</span><span style="display: block; text-align: center;">s = 7a + 11b</span>
The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:
<span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span>
Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].
Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-fifth pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n*G = P5 - 2*m3 = [7,4] - 2*[3,2] = [7,4] - [6,4] = [1,0] = A1.
Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - m*P = P8 - 5*M2 = [12,7] - 5*[2,1] = [2,2] = 2*[1,1] = 2*m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:
<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span>
Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:
<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^ -- Dv -- Eb^ -- Evv -- F</span>
To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.
For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10*G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^<span style="vertical-align: super;">4</span>M2. Next, find the original half-fourth generator. P = P8/5 ~ 240¢, and G = P4/2 ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^<span style="vertical-align: super;">4</span>M2 + ^1 = ^<span style="vertical-align: super;">5</span>M2. The alternate generator is usually simpler than the original generator, and the alternate multigen is usually more complex than the original multigen.
<span style="display: block; text-align: center;">P1- - ^<span style="vertical-align: super;">4</span>M2=v<span style="vertical-align: super;">6</span>m3 -- vvP4 -- ^^P5 -- ^<span style="vertical-align: super;">6</span>M6=v<span style="vertical-align: super;">4</span>m7 -- P8</span><span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">4</span>=Ebv<span style="vertical-align: super;">6</span> -- Fvv -- G^^ -- A^<span style="vertical-align: super;">6</span>=Bbv<span style="vertical-align: super;">4</span> -- C</span><span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">5</span>M2=v<span style="vertical-align: super;">5</span>m3 -- P4</span><span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">5</span>=Ebv<span style="vertical-align: super;">5</span> -- F</span>
To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.
A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).
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. But by using double-pair notation, we can get enharmonics that are 2nds. First 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, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6. But ^m6 has an up sign, and there's no such thing as half an up. The answer is to double all ups and downs: P11/2 = ^^m6 or vvM6, and E = v<span style="vertical-align: super;">4</span>A1. The bare generator is ^^m6/2 = ^^[8,5]/2 = ^[4,2] = ^M3, and the bare enharmonic is ^^m6 - 2*(^M3) = [0,1] = d2. For the second enharmonic, we use the second pair of accidentals: E' = //d2, and G = ^\M3 or ^/d4. Here is the genchain:
<span style="display: block; text-align: center;">P1 -- ^\M3=^/d4 -- ^^m6=vvM6 -- v\A8=v/m9 -- P11
</span><span style="display: block; text-align: center;">C -- E^\=Fb^/ -- Ab^^=Avv -- C#v\=Dbv/ -- F
</span>
Using vvM6/2 for 2*G gives a different but equally valid notation: vvM6/2 = vv[9,5]/2 = v[4,2] = vM3, and vvM6 - 2*(vM3) = [1,1] = m2, and E' = \\m2 and G = v/M3 or v\4.
<span style="display: block; text-align: center;">P1 -- v/M3=v\4 -- vvM6=^^m6 -- ^/8=^\m9 -- P11
</span><span style="display: block; text-align: center;">C -- Ev/=Fv\ -- Avv=Ab^^ -- C^/=Db^\ -- F
</span>
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 proceeding as before, we find only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. 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. Before we can divide by 4, we must quadruple all ups and downs: E = ^<span style="vertical-align: super;">12</span>d2 and 4*G' = ^<span style="vertical-align: super;">4</span>m2. The bare alt-generator is ^<span style="vertical-align: super;">4</span>[1,1]/4 = ^[0,0] = ^1, and the bare 2nd enharmonic is ^<span style="vertical-align: super;">4</span>m2 - 4*(^1) = m2. Thus E' = \<span style="vertical-align: super;">4</span>m2 and G' = ^/1.The period can be deduced from 4*G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4*G' = P4 - ^<span style="vertical-align: super;">4</span>m2 = v<span style="vertical-align: super;">4</span>M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = v4M3 + ^/1 = v3/M3.
<span style="display: block; text-align: center;">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
</span><span style="display: block; text-align: center;">C -- Ev<span style="vertical-align: super;">4</span> -- Ab^<span style="vertical-align: super;">4</span> -- C</span><span style="display: block; text-align: center;">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</span><span style="display: block; text-align: center;">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</span>
==Alternate enharmonics==
Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12*[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12*[4,2] = [-15,-5] = -5*[3,1] = -5 * v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.
<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- C
C -- Eb^<span style="vertical-align: super;">4</span> -- Av<span style="vertical-align: super;">4</span> -- C
P1 -- v<span style="vertical-align: super;">3</span>M2 -- v<span style="vertical-align: super;">6</span>M3=^<span style="vertical-align: super;">6</span>m2 -- ^<span style="vertical-align: super;">3</span>m3 -- P4
C -- Dv<span style="vertical-align: super;">3</span> -- Ev<span style="vertical-align: super;">6</span>=Db^<span style="vertical-align: super;">6</span> -- Eb^<span style="vertical-align: super;">3</span> -- F
</span>
Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending are best avoided, and double-pair notation is better for this pergen. We have P = vM3, E = ^3d2, G = /m2, and E' = /4dd2.
<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8
</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C
</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.
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.
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.
(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.
==Alternate keyspans and stepspans==
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.
== ==
==Combining pergens==
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).
General rules for combining pergens:
* (P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')
* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')
* (P8/m, P5) + (P8, M/n) = (P8/m, M/n)
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.
==Pergens and EDOs==
Pergens have much in common with edos. Pergens (rank-2 ones) assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but far fewer than a hundred have been explored. There are an infinite number of pergens, but far fewer than a hundred will suffice most composers.
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported. In practice, if the generator's keyspan is very small, a partially supported pergen. For example, 22edo and 2\22 generator.
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! For example, 12edo supports (P8, P4/5), (P8, P11/17), (P8, WWP4/29), (P8, W<span style="vertical-align: super;">3</span>P4/41), etc.
How many edos support a given pergen? Again, 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 M's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.
Given an edo, a period p\N and a generator g\N, what is the pergen? The octave fraction n is N/p. Let the edo's 5th be f\N. To find the multigen M, we must find a monzo (a,b) such that a*N + b*f is a multiple of g. If n = 1, |b| = 1.
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.
[13b and 18b - needs explanation]
||||~ pergen ||~ supporting edos (12-31 only) ||
||= (P8, P5) ||= unsplit ||= 12, 13b, 14*, 15*, 16, 17, 18b*, 19, 20*, 21*,
22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31 ||
||~ halves ||~ ||~ ||
||= (P8/2, P5) ||= half-octave ||= 12, 14, 16, 18b, 20*, 22, 24*, 26, 28*, 30* ||
||= (P8, P4/2) ||= half-fourth ||= 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30* ||
||= (P8, P5/2) ||= half-fifth ||= 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31 ||
||= (P8/2, P4/2) ||= half-everything ||= 14, 18b, 20*, 24, 28*, 30* ||
||~ thirds ||~ ||~ ||
||= (P8/3, P5) ||= third-octave ||= 12, 15, 18b*, 21, 24*, 27, 30* ||
||= (P8, P4/3) ||= third-fourth ||= 13b, 14*, 15, 21*, 22, 28*, 29, 30* ||
||= (P8, P5/3) ||= third-fifth ||= 15*, 16, 20*, 21, 25*, 26, 30*, 31 ||
||= (P8, P11/3) ||= third-11th ||= 15, 17, 21, 23, 30* ||
||= (P8/3, P4/2) ||= third-8ve, half-4th ||= 15, 18b*, 24, 30* ||
||= (P8/3, P5/2) ||= third-8ve, half-5th ||= 18b, 21, 24, 27, 30 ||
||= (P8/2, P4/3) ||= half-8ve, third-4th ||= 14, 22, 28*, 30 ||
||= (P8/2, P5/3) ||= half-8ve, third-5th ||= 16, 20*, 26, 30* ||
||= (P8/2, P11/3) ||= half-8ve, third-11th ||= 19, 30 ||
||= (P8/3, P4/3) ||= third-everything ||= 15, 21, 30* ||
||~ quarters ||~ ||~ ||
||= (P8/4, P5) ||= quarter-octave ||= 12, 16, 20, 24*, 28 ||
||= (P8, P4/4) ||= quarter-fourth ||= 18b*, 19, 20*, 28, 29, 30* ||
||= (P8, P5/4) ||= quarter-fifth ||= 14*, 20, 21*, 27, 28* ||
||= (P8, P11/4) ||= quarter-eleventh ||= 14, 17, 20, 28*, 31 ||
||= (P8, P12/4) ||= quarter-twelfth ||= 13b, 15*, 18b, 20*, 23, 25*, 28, 30* ||
||= (P8/4, P4/2) ||= quarter-octave, half-fourth ||= 20, 24, 28 ||
||= (P8/2, M2/4) ||= half-octave, quarter-tone ||= 20, 22, 24, 26, 28 ||
||= (P8/2, P4/4) ||= half-octave, quarter-fourth ||= 18b, 20*, 28, 30* ||
||= (P8/2, P5/4) ||= half-octave, quarter-fifth ||= 14, 20, 28* ||
||= (P8/4, P4/3) ||= quarter-octave, third-fourth ||= 28 ||
||= (P8/4, P5/3) ||= quarter-octave, third-fifth ||= 16, 20 ||
||= (P8/4, P11/3) ||= quarter-octave, third-eleventh ||= ||
||= (P8/3, P4/4) ||= third-octave, quarter-fourth ||= 18b*, 30 ||
||= (P8/3, P5/4) ||= third-octave, quarter-fifth ||= 21, 27 ||
||= (P8/3, P11/4) ||= third-octave, quarter-eleventh ||= ||
||= (P8/3, P12/4) ||= third-octave, quarter-twelfth ||= 15, 18b, 30* ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||
==Misc notes==
Pergens were discovered by Kite Giedraitis in 2017. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names
__**Extra paragraphs:**__
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]:
k = 12a + 19b + 28c + 34d
s = 7a + 11b + 14c + 20d
g = -c
r = -d
a = -11k + 19s - 4g + 6r
b = 7k - 12s + 4g - 2r
c = -g
d = -r
Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. There are multiple spellings for many chords. Whenever the enharmonic isn't an A1, even the degree of a chord note can change. It would be possible to spell the chord C Fb^ G, but there's no reason to. But 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#.
Half-octave with a vvM2 enharmonic: 4:5:6 = C Eb^ G. So better to have E = ^^d2.
Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).
For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.
The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:
The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.
The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-eleventh has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.
For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.
For **{P8/M, P5}**, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)
For {**P8, multigen/N}**, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1
For **{P8/M, multigen/N}**, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1
or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1
The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.
For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-octave implies an even-numbered edo.
For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.
For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.
For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d<span style="vertical-align: super;">3</span>4. The perchain would be C - E#^<span style="vertical-align: super;">3</span>=Abbv<span style="vertical-align: super;">3</span> - C.
For **{P8/M, P5}**, the implied edo = Mx
For {**P8, multigen/N}**, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)
For **{P8/M, multigen/N}**, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,
or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F
If the enharmonic is larger than a 2nd, it may be possible to split it into several smaller enharmonics. For example, {P8/5, P5}. If 7/M = 7/5 is rounded to 1, the enharmonic is a 3rd. The enharmonic must be 5x, and also 7y + 4, so the 3-exponent = -10 = dim 3rd. Adding ups and downs, we have enharmonic = v<span style="vertical-align: super;">5</span>d3 and period = ^M2. Fortunately, d3 = m2 + m2, and the 3rd can be reduced to two 2nds. The downs must be doubled, so that the period = ^^M2, and the enharmonic = v<span style="vertical-align: super;">10</span>d3 = 2 *
v<span style="vertical-align: super;">5</span>m2. The enharmonic must be applied twice in the course of an octave: P1 - ^^M2 - ^<span style="vertical-align: super;">4</span>M3=vP4 - ^P5 - ^<span style="vertical-align: super;">3</span>M6=vvm7 - P8
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.
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<html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:31:<h1> --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:31 --> </h1>
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<!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>
<!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>
<!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>
<!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>
<!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>
<!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>
<!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>
<!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div>
<!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div>
<!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div>
<!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 2em;"><a href="#toc11"> </a></div>
<!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div>
<!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div>
<!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div>
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<!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextHeadingRule:33:<h1> --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:33 --><u><strong>Definition</strong></u></h1>
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<br />
A <strong>pergen</strong> (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.<br />
<br />
If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. The fraction is always of the form 1/N, thus the octave and/or the 3-limit interval is <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multi-gen</strong>. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, etc.<br />
<br />
For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma 2048/2025 is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.<br />
<br />
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to fewer than perhaps a hundred categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal and injera sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>.<br />
<br />
The largest category contains all single-comma temperaments with a comma of the form 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P or 2<span style="vertical-align: super;">x </span>3<span style="vertical-align: super;">y </span>P<span style="vertical-align: super;">-1</span>, where P is a prime > 3 (a <strong>higher prime</strong>), e.g. 81/80 or 64/63. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called <strong>unsplit</strong>.<br />
<br />
Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.<br />
<br />
For example, srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is <u>not</u> preferred over P4/2. For example, decimal is (P8/2, P4/2), not (P8/2, P5/2).<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="2">pergen<br />
</th>
<th colspan="4">example temperaments<br />
</th>
</tr>
<tr>
<th>written<br />
</th>
<th>spoken<br />
</th>
<th>comma(s)<br />
</th>
<th>name<br />
</th>
<th colspan="2">color name<br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8, P5)<br />
</td>
<td style="text-align: center;">unsplit<br />
</td>
<td style="text-align: center;">81/80<br />
</td>
<td style="text-align: center;">meantone<br />
</td>
<td style="text-align: center;">green<br />
</td>
<td style="text-align: center;">gT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">64/63<br />
</td>
<td style="text-align: center;">archy<br />
</td>
<td style="text-align: center;">red<br />
</td>
<td style="text-align: center;">rT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">(-14,8,1)<br />
</td>
<td style="text-align: center;">schismic<br />
</td>
<td style="text-align: center;">large yellow<br />
</td>
<td style="text-align: center;">LyT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5)<br />
</td>
<td style="text-align: center;">half-octave<br />
</td>
<td style="text-align: center;">(11, -4, -2)<br />
</td>
<td style="text-align: center;">srutal<br />
</td>
<td style="text-align: center;">small deep green<br />
</td>
<td style="text-align: center;">sggT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">81/80, 50/49<br />
</td>
<td style="text-align: center;">injera<br />
</td>
<td style="text-align: center;">deep reddish and green<br />
</td>
<td style="text-align: center;">rryy&gT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/2)<br />
</td>
<td style="text-align: center;">half-fifth<br />
</td>
<td style="text-align: center;">25/24<br />
</td>
<td style="text-align: center;">dicot<br />
</td>
<td style="text-align: center;">deep yellow<br />
</td>
<td style="text-align: center;">yyT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">(-1,5,0,0,-2)<br />
</td>
<td style="text-align: center;">mohajira<br />
</td>
<td style="text-align: center;">deep amber<br />
</td>
<td style="text-align: center;">aaT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/2)<br />
</td>
<td style="text-align: center;">half-fourth<br />
</td>
<td style="text-align: center;">49/48<br />
</td>
<td style="text-align: center;">semaphore<br />
</td>
<td style="text-align: center;">deep blue<br />
</td>
<td style="text-align: center;">bbT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/2)<br />
</td>
<td style="text-align: center;">half-everything<br />
</td>
<td style="text-align: center;">25/24, 49/48<br />
</td>
<td style="text-align: center;">decimal<br />
</td>
<td style="text-align: center;">deep yellow and deep blue<br />
</td>
<td style="text-align: center;">yy&bbT<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/3)<br />
</td>
<td style="text-align: center;">third-fourth<br />
</td>
<td style="text-align: center;">250/243<br />
</td>
<td style="text-align: center;">porcupine<br />
</td>
<td style="text-align: center;">triple yellow<br />
</td>
<td style="text-align: center;">y<span style="vertical-align: super;">3</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P11/3)<br />
</td>
<td style="text-align: center;">third-eleventh<br />
</td>
<td style="text-align: center;">(12,-1,0,0,-3)<br />
</td>
<td style="text-align: center;">small triple amber<br />
</td>
<td style="text-align: center;">small triple amber<br />
</td>
<td style="text-align: center;">sa<span style="vertical-align: super;">3</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P5)<br />
</td>
<td style="text-align: center;">quarter-octave<br />
</td>
<td style="text-align: center;">(3,4,-4)<br />
</td>
<td style="text-align: center;">diminished<br />
</td>
<td style="text-align: center;">quadruple green<br />
</td>
<td style="text-align: center;">g<span style="vertical-align: super;">4</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, M2/4)<br />
</td>
<td style="text-align: center;">half-octave, quarter-tone<br />
</td>
<td style="text-align: center;">(-17,2,0,0,4)<br />
</td>
<td style="text-align: center;">large quadruple jade<br />
</td>
<td style="text-align: center;">large quadruple jade<br />
</td>
<td style="text-align: center;">Lj<span style="vertical-align: super;">4</span>T<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P12/5)<br />
</td>
<td style="text-align: center;">fifth-twelfth<br />
</td>
<td style="text-align: center;">(-10,-1,5)<br />
</td>
<td style="text-align: center;">magic<br />
</td>
<td style="text-align: center;">large quintuple yellow<br />
</td>
<td style="text-align: center;">Ly<span style="vertical-align: super;">5</span>T<br />
</td>
</tr>
</table>
(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.<br />
<br />
The color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example.<br />
<br />
For non-standard prime groups, the period uses the first prime only, and the multigen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation</a> can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third. Ratios could be used instead, if enclosed in parentheses for clarity: (P8/2, (5/4)/1), or if a colon is used: (P8/2, 5:4).<br />
<br />
Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, colors can be avoided by reducing gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with only an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is <strong>highs and lows</strong>, written / and \.<br />
<br />
Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.<br />
<br />
A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br />
<br />
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 />
<!-- ws:start:WikiTextHeadingRule:35:<h1> --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:35 --><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 > m, it will split some 3-limit interval into n parts.<br />
<br />
In a multi-comma rank-2 or higher temperament, it's possible that one comma will contain only the 1st and 2nd primes, and the 2nd prime is directly related to the 1st prime. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br />
<br />
For example, consider 2.3.5.7 with commas 256/243 and 225/224. The first comma splits the octave into 5 parts, and makes the 5th be exactly 3/5 of the octave. The multigen must use primes 2 and 5. In this case, the pergen is (P8/5, y3), the same as Blackwood.<br />
<br />
To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a <strong>square mapping</strong> by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes > 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose among the alternates to minimize the splitting and the cents.<br />
<br />
For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multigen, P is the period P8/m, and G is the generator M/n.<br />
2/1 = P8 = xP, thus P = P8/x<br />
3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz<br />
<br />
To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).<br />
G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz<br />
<br />
<span style="display: block; text-align: center;"><strong><span style="font-size: 110%;">The rank-2 pergen from the [(x, 0) (y, z)] square mapping is (P8/x, (nz-y,x)/xz), with -x <= n <= x</span></strong><br />
</span><br />
For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 <= n <= 1. No value of n reduces the fraction, so the best multigen is the one with the least cents, (2,-1) = P4. The pergen is (P8, P4/3).<br />
<br />
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7" rel="nofollow">x31.com</a> gives us this matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
<th>7/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2. Discard the last column, to make a square matrix with zeros below the diagonal, and no zeros on the diagonal:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>5/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
</table>
Use an <a class="wiki_link_ext" href="http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1" rel="nofollow">online tool</a> to invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>period<br />
</th>
<th>gen1<br />
</th>
<th>gen2<br />
</th>
<th><br />
</th>
</tr>
<tr>
<th>2/1<br />
</th>
<td style="text-align: center;">4<br />
</td>
<td style="text-align: center;">-2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>3/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>5/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td>/4<br />
</td>
</tr>
</table>
Thus the period = (4,0,0)/4 = 2/1= P8, gen1 = (-2,2,0)/4 = (-1,1,0)/2 = P5/2, and gen2 = (-1,-1,2)/4 = yy15/4 = (25/6)/4.<br />
<br />
Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multigen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19/2 = (6/1)/2, both of which are much larger, so the best gen1 is P5/2.<br />
<br />
The 2nd multigen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = gg15/4 = (96/25)/4. A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 = (128/75)/4. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = Lyy3/4 = (675/512)/4. As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.<br />
<br />
Alternatively, we could discard the 3rd column and keep the 4th one:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>2/1<br />
</th>
<th>3/1<br />
</th>
<th>7/1<br />
</th>
</tr>
<tr>
<th>period<br />
</th>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">2<br />
</td>
</tr>
<tr>
<th>gen1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
<tr>
<th>gen2<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">1<br />
</td>
</tr>
</table>
This inverts to this matrix:<br />
<table class="wiki_table">
<tr>
<th><br />
</th>
<th>period<br />
</th>
<th>gen1<br />
</th>
<th>gen2<br />
</th>
<th><br />
</th>
</tr>
<tr>
<th>2/1<br />
</th>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td style="text-align: center;">-3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>3/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">1<br />
</td>
<td style="text-align: center;">-1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<th>7/1<br />
</th>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">0<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td>/2<br />
</td>
</tr>
</table>
Again, period = P8 and gen1 = P5/2. Gen2 = (-3,-1,2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multigen, 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. We can 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 > 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 />
<!-- ws:start:WikiTextHeadingRule:37:<h1> --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:37 --><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 />
<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 />
<br />
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 />
<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, 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.<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 />
<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 />
<br />
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 />
<br />
The <strong>genchain</strong> (chain of generators) in the table is only a short section of the full genchain.<br />
C - G implies ...Eb Bb F C G D A E B F# C#...<br />
C - Eb^=Ev - G implies ...F -- Ab^=Av -- C -- Eb^=Ev -- G -- Bb^=Bv -- D -- F^=F#v -- A -- C^=C#v -- E...<br />
If the octave is split, the table has a <strong>perchain</strong> ("peer-chain", chain of periods) that shows the octave: In C -- F#v=Gb^ -- C, the last C is an octave above the first one.<br />
<br />
The table lists several possible notations for each pergen. To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination.<br />
<br />
(table is unfinished)<br />
<br />
<table class="wiki_table">
<tr>
<th>pergen<br />
</th>
<th>enharmonic<br />
interval(s)<br />
</th>
<th>equiva-<br />
lence(s)<br />
</th>
<th>split<br />
interval(s)<br />
</th>
<th>perchain(s) and<br />
genchains(s)<br />
</th>
<th>examples<br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8, P5)<br />
unsplit<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">C - G<br />
</td>
<td style="text-align: center;">meantone,<br />
schismic<br />
</td>
</tr>
<tr>
<th>halves<br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5)<br />
half-octave<br />
</td>
<td style="text-align: center;">^^d2 (if 5th<br />
<!-- ws:start:WikiTextRawRule:00:``&gt;`` -->><!-- ws:end:WikiTextRawRule:00 --> 700¢<br />
</td>
<td style="text-align: center;">C^^ = B#<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C<br />
</td>
<td style="text-align: center;">srutal<br />
^1 = 81/80<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">vvd2 (if 5th<br />
< 700¢)<br />
</td>
<td style="text-align: center;">C^^ = Db<br />
</td>
<td style="text-align: center;">P8/2 = ^A4 = vd5<br />
</td>
<td style="text-align: center;">C - F#^=Gbv - C<br />
</td>
<td style="text-align: center;">large deep red<br />
^1 = 64/63<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">vvM2<br />
</td>
<td style="text-align: center;">C^^ = D<br />
</td>
<td style="text-align: center;">P8/2 = ^4 = vP5<br />
</td>
<td style="text-align: center;">C - F^=Gv - C<br />
</td>
<td style="text-align: center;">128/121<br />
^1 = 33/32<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/2)<br />
half-fourth<br />
</td>
<td style="text-align: center;">vvm2<br />
</td>
<td style="text-align: center;">C^^ = Db<br />
</td>
<td style="text-align: center;">P4/2 = ^M2 = vm3<br />
</td>
<td style="text-align: center;">C - D^=Ebv - F<br />
</td>
<td style="text-align: center;">semaphore<br />
^1 = 64/63<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^^dd2<br />
</td>
<td style="text-align: center;">C^^ = B##<br />
</td>
<td style="text-align: center;">P4/2 = vA2 = ^d3<br />
</td>
<td style="text-align: center;">C - D#v=Ebb^ - F<br />
</td>
<td style="text-align: center;">(-22,-11,2)<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/2)<br />
half-fifth<br />
</td>
<td style="text-align: center;">vvA1<br />
</td>
<td style="text-align: center;">C^^ = C#<br />
</td>
<td style="text-align: center;">P5/2 = ^m3 = vM3<br />
</td>
<td style="text-align: center;">C - Eb^=Ev - G<br />
</td>
<td style="text-align: center;">mohajira<br />
^1 = 33/32<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/2)<br />
half-<br />
everything<br />
</td>
<td style="text-align: center;">\\m2,<br />
vvA1,<br />
^^\\d2,<br />
vv\\M2<br />
</td>
<td style="text-align: center;">C<!-- ws:start:WikiTextRawRule:01:``//`` -->//<!-- ws:end:WikiTextRawRule:01 --> = Db<br />
C^^ = C#<br />
C^^<!-- ws:start:WikiTextRawRule:02:``//`` -->//<!-- ws:end:WikiTextRawRule:02 --> = D<br />
</td>
<td style="text-align: center;">P4/2 = /M2 = \m3<br />
P5/2 = ^m3 = vM3<br />
P8/2 = v/A4 = ^\d5<br />
<!-- ws:start:WikiTextRawRule:03:``=`` -->=<!-- ws:end:WikiTextRawRule:03 --> ^/4 <!-- ws:start:WikiTextRawRule:04:``=`` -->=<!-- ws:end:WikiTextRawRule:04 --> v\P5<br />
</td>
<td style="text-align: center;">C - D/=Eb\ - F,<br />
C - Eb^=Ev - G,<br />
C - F#v/=Gb^\ - C,<br />
C - F^/=Gv\ - C<br />
</td>
<td style="text-align: center;">49/48 & 128/121<br />
^1 = 33/32<br />
/1 = 64/63<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^^d2,<br />
\\m2,<br />
vv\\A1<br />
</td>
<td style="text-align: center;">C^^ = B#<br />
C<!-- ws:start:WikiTextRawRule:05:``//`` -->//<!-- ws:end:WikiTextRawRule:05 --> = Db<br />
C^^<!-- ws:start:WikiTextRawRule:06:``//`` -->//<!-- ws:end:WikiTextRawRule:06 --> = C#<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
P4/2 = /M2 = \m3<br />
P5/2 = ^/m3 = v\M3<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C,<br />
C - D/=Eb\ - F,<br />
C - Eb^/=Ev\ - G<br />
</td>
<td style="text-align: center;">2048/2025 & 49/48<br />
^1 = 81/80<br />
/1 = 64/63<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^^d2,<br />
\\A1,<br />
^^\\m2<br />
</td>
<td style="text-align: center;">C^^ = B#<br />
C<!-- ws:start:WikiTextRawRule:07:``//`` -->//<!-- ws:end:WikiTextRawRule:07 --> = C#<br />
C^^\\ = B<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
P5/2 = /m3 = \M3<br />
P4/2 =v/M2 = ^\m3<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C,<br />
C - Eb/=E\ - G,<br />
C - Dv/=Eb^\ - F<br />
</td>
<td style="text-align: center;">2048/2025 & 128/121<br />
^1 = 81/80<br />
/1 = 33/32<br />
</td>
</tr>
<tr>
<th>thirds<br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5)<br />
third-octave<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3 </span> <!-- ws:start:WikiTextRawRule:08:``=`` -->=<!-- ws:end:WikiTextRawRule:08 --> B#<br />
</td>
<td style="text-align: center;">P8/3 = vM3 = ^^d4<br />
</td>
<td style="text-align: center;">C - Ev - Ab^ - C<br />
</td>
<td style="text-align: center;">augmented<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/3)<br />
third-fourth<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>A1<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3 <!-- ws:start:WikiTextRawRule:09:``=`` -->=<!-- ws:end:WikiTextRawRule:09 --> </span>C#<br />
</td>
<td style="text-align: center;">P4/3 = ^^m2 = vM2<br />
</td>
<td style="text-align: center;">C - Dv - Eb^ - F<br />
</td>
<td style="text-align: center;">porcupine<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/3)<br />
third-fifth<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>m2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3 </span> <!-- ws:start:WikiTextRawRule:010:``=`` -->=<!-- ws:end:WikiTextRawRule:010 --> Db<br />
</td>
<td style="text-align: center;">P5/3 = ^M2 = vvm3<br />
</td>
<td style="text-align: center;">C - D^ - Fv - G<br />
</td>
<td style="text-align: center;">slendric<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P11/3)<br />
third-11th<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>dd2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:011:``=`` -->=<!-- ws:end:WikiTextRawRule:011 --> B##<br />
</td>
<td style="text-align: center;">P11/3 = vA4 = ^^dd5<br />
</td>
<td style="text-align: center;">C - F#v - Cb^ - F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>M2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3 </span><!-- ws:start:WikiTextRawRule:012:``=`` -->=<!-- ws:end:WikiTextRawRule:012 --> D<br />
</td>
<td style="text-align: center;">P11/3 = ^4 = vvP5<br />
</td>
<td style="text-align: center;">C - F^ - Cv - F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/2)<br />
third-8ve, half-4th<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>A2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:013:``=`` -->=<!-- ws:end:WikiTextRawRule:013 --> D#<br />
</td>
<td style="text-align: center;">P8/3 = ^^m3<br />
P4/2 = v<span style="vertical-align: super;">3</span>m2<br />
</td>
<td style="text-align: center;">C - Eb^^ - Avv - C<br />
C - Dbv<span style="vertical-align: super;">3</span>=E^<span style="vertical-align: super;">3</span> - F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5/2)<br />
third-8ve, half-5th<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>m3<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:014:``=`` -->=<!-- ws:end:WikiTextRawRule:014 --> Eb<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/3)<br />
half-8ve, third-4th<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>d4<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:015:``=`` -->=<!-- ws:end:WikiTextRawRule:015 --> Fb<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>4<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> Fb3<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5/3)<br />
half-8ve,<br />
third-5th<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:017:``=`` -->=<!-- ws:end:WikiTextRawRule:017 --> B#<span style="vertical-align: super;">3</span><br />
</td>
<td style="text-align: center;">P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5<br />
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3<br />
</td>
<td style="text-align: center;">C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C<br />
C - D#vv - Fb^^ - G<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^^d2,<br />
\\\m2<br />
</td>
<td style="text-align: center;">C^^ = B#<br />
C<!-- ws:start:WikiTextRawRule:018:``///`` -->///<!-- ws:end:WikiTextRawRule:018 --> = Db<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
P5/3 = /M2 = \\m3<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C<br />
C - /D - \F - G<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P11/3)<br />
half-8ve,<br />
third-11th<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">6</span>M2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">6</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> D<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/3)<br />
third-<br />
everything<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<th>quarters<br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P5)<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">4</span>d2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:020:``=`` -->=<!-- ws:end:WikiTextRawRule:020 --> B#<br />
</td>
<td style="text-align: center;">P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2<br />
</td>
<td style="text-align: center;">C Ebv Gbvv=F#^^ A^ C<br />
</td>
<td style="text-align: center;">diminished,<br />
^1 = 81/80<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/4)<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">4</span>dd2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:021:``=`` -->=<!-- ws:end:WikiTextRawRule:021 --> B##<br />
</td>
<td style="text-align: center;">P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1<br />
</td>
<td style="text-align: center;">C Db^ Ebb^^=D#vv Ev F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/4)<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>A1<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:022:``=`` -->=<!-- ws:end:WikiTextRawRule:022 --> C#<br />
</td>
<td style="text-align: center;">P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2<br />
</td>
<td style="text-align: center;">C Dv Evv=Eb^^ F^ G<br />
</td>
<td style="text-align: center;">tetracot<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P11/4)<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>dd3<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:023:``=`` -->=<!-- ws:end:WikiTextRawRule:023 --> Eb<span style="vertical-align: super;">3</span><br />
</td>
<td style="text-align: center;">P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5<br />
</td>
<td style="text-align: center;">C E^ G#^^ Dbv F<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P12/4)<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">4</span>m2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">4</span> <!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> Db<br />
</td>
<td style="text-align: center;">P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3<br />
</td>
<td style="text-align: center;">C Fv Bbvv=A^^ D^ G<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/2)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, M2/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/3)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P5/3)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P11/3)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P11/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P12/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/4)<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the "tipping point": if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u><strong>ups and downs may need to be swapped, depending on the size of the 5th</strong></u> in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.<br />
<br />
The sweet spot is narrower if the comma's cents are smaller, or if the number of lattice rungs it spans (the "rungspan") is larger. If the sweet spot contains the tipping point, and the 5th equals the implied edo's 5th, then the bare enharmonic vanishes without any help from ups or downs needed.<br />
<br />
Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-exponent of the bare enharmonic, thus the edo implies the enharmonic.<br />
<table class="wiki_table">
<tr>
<th colspan="2">bare enharmonic interval .<br />
</th>
<th>3-exponent .<br />
</th>
<th>implied edo .<br />
</th>
<th>edo's 5th .<br />
</th>
<th>upping range .<br />
</th>
<th>downing range .<br />
</th>
<th>if the 5th is just<br />
</th>
</tr>
<tr>
<td style="text-align: center;">M2<br />
</td>
<td style="text-align: center;">C - D<br />
</td>
<td style="text-align: center;">2<br />
</td>
<td style="text-align: center;">2-edo<br />
</td>
<td style="text-align: center;">600¢<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">all<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">m3<br />
</td>
<td style="text-align: center;">C - Eb<br />
</td>
<td style="text-align: center;">-3<br />
</td>
<td style="text-align: center;">3-edo<br />
</td>
<td style="text-align: center;">800¢<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">all<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">m2<br />
</td>
<td style="text-align: center;">C - Db<br />
</td>
<td style="text-align: center;">-5<br />
</td>
<td style="text-align: center;">5-edo<br />
</td>
<td style="text-align: center;">720¢<br />
</td>
<td style="text-align: center;">none<br />
</td>
<td style="text-align: center;">all<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">A1<br />
</td>
<td style="text-align: center;">C - C#<br />
</td>
<td style="text-align: center;">7<br />
</td>
<td style="text-align: center;">7-edo<br />
</td>
<td style="text-align: center;">~686¢<br />
</td>
<td style="text-align: center;">600-686¢<br />
</td>
<td style="text-align: center;">686¢-720¢<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">d2<br />
</td>
<td style="text-align: center;">C - Dbb<br />
</td>
<td style="text-align: center;">-12<br />
</td>
<td style="text-align: center;">12-edo<br />
</td>
<td style="text-align: center;">700¢<br />
</td>
<td style="text-align: center;">700-720¢<br />
</td>
<td style="text-align: center;">600-700¢<br />
</td>
<td style="text-align: center;">upped<br />
</td>
</tr>
<tr>
<td style="text-align: center;">dd3<br />
</td>
<td style="text-align: center;">C - Eb<span style="vertical-align: super;">3</span><br />
</td>
<td style="text-align: center;">-17<br />
</td>
<td style="text-align: center;">17-edo<br />
</td>
<td style="text-align: center;">~706¢<br />
</td>
<td style="text-align: center;">706-720¢<br />
</td>
<td style="text-align: center;">600-706¢<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">dd2<br />
</td>
<td style="text-align: center;">C - Db<span style="vertical-align: super;">3</span><br />
</td>
<td style="text-align: center;">-19<br />
</td>
<td style="text-align: center;">19-edo<br />
</td>
<td style="text-align: center;">~695¢<br />
</td>
<td style="text-align: center;">695-720¢<br />
</td>
<td style="text-align: center;">600-695¢<br />
</td>
<td style="text-align: center;">upped<br />
</td>
</tr>
<tr>
<td style="text-align: center;">d<span style="vertical-align: super;">3</span>4<br />
</td>
<td style="text-align: center;">C - Fb<span style="vertical-align: super;">3</span><br />
</td>
<td style="text-align: center;">-22<br />
</td>
<td style="text-align: center;">22-edo<br />
</td>
<td style="text-align: center;">~709¢<br />
</td>
<td style="text-align: center;">709-720¢<br />
</td>
<td style="text-align: center;">600-709¢<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">d<span style="vertical-align: super;">3</span>2<br />
</td>
<td style="text-align: center;">C - Db<span style="vertical-align: super;">4</span><br />
</td>
<td style="text-align: center;">-26<br />
</td>
<td style="text-align: center;">26-edo<br />
</td>
<td style="text-align: center;">~692¢<br />
</td>
<td style="text-align: center;">692-720¢<br />
</td>
<td style="text-align: center;">600-692¢<br />
</td>
<td style="text-align: center;">upped<br />
</td>
</tr>
<tr>
<td style="text-align: center;">d<span style="vertical-align: super;">4</span>4<br />
</td>
<td style="text-align: center;">C - Fb<span style="vertical-align: super;">4</span><br />
</td>
<td style="text-align: center;">-29<br />
</td>
<td style="text-align: center;">29-edo<br />
</td>
<td style="text-align: center;">~703¢<br />
</td>
<td style="text-align: center;">703-720¢<br />
</td>
<td style="text-align: center;">600-703¢<br />
</td>
<td style="text-align: center;">downed<br />
</td>
</tr>
<tr>
<td style="text-align: center;">d<span style="vertical-align: super;">4</span>3<br />
</td>
<td style="text-align: center;">C - Eb<span style="vertical-align: super;">5</span><br />
</td>
<td style="text-align: center;">-31<br />
</td>
<td style="text-align: center;">31-edo<br />
</td>
<td style="text-align: center;">~697¢<br />
</td>
<td style="text-align: center;">697-720¢<br />
</td>
<td style="text-align: center;">600-697¢<br />
</td>
<td style="text-align: center;">upped<br />
</td>
</tr>
<tr>
<td style="text-align: center;">etc.<br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:39:<h1> --><h1 id="toc4"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:39 --><u>Further Discussion</u></h1>
<br />
<!-- ws:start:WikiTextHeadingRule:41:<h2> --><h2 id="toc5"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:41 -->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 "W" 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 />
<!-- ws:start:WikiTextHeadingRule:43:<h2> --><h2 id="toc6"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:43 -->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 />
<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 />
<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 />
<!-- ws:start:WikiTextHeadingRule:45:<h2> --><h2 id="toc7"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:45 -->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*P and P8. If P is 6/5, the comma is 4*P - P8 = (6/5)^4 / (2/1) = 648/625. If P is 7/6, the comma is P8 - 4*P = (2/1) * (7/6)^-4. 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*G - P4 = (10/9)^3 / (4/3) = 250/243.<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*G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br />
<br />
If the pergen is not explicitly false, put the pergen in its <strong>unreduced</strong> form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen M' is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.<br />
<br />
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6*G - m3. The comma splits both the octave and the fifth.<br />
<br />
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2*P8 - 4*P4)/(2*4) = (2*M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.<br />
<br />
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br />
<br />
Unreducing replaces the generator with an alternate generator. Any number of periods plus or minus a single generator makes an <strong>alternate</strong> generator. A generator or period plus or minus any number of enharmonics makes an <strong>equivalent</strong> generator or period. An equivalent generator is always the same size in cents, since the enharmonic is always 0¢. An equivalent generator is the same interval, merely notated differently. For example, half-octave (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.<br />
<br />
Of the two equivalent generators, the preferred generator is always the smaller one (smaller degree, or if the degrees are the same, more diminished). This is because the equivalent generator can be more easily found by adding the enharmonic, rather than subtracting it. For example, ^m3 + vvA1 = vM3 is an easier calculation than vM3 - vvA1 = ^m3. This is particularly true with complex enharmonics like ^<span style="vertical-align: super;">6</span>dd2.<br />
<br />
There are also alternate enharmonics, see below.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:47:<h2> --><h2 id="toc8"><a name="Further Discussion-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:47 -->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 />
<ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2</li><li>For false doubles using single-pair notation, E = E', but x and y are usually different</li><li>The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE", and P8 = mP + xE"</li></ul><br />
The <strong>keyspan</strong> of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The <strong>stepspan</strong> of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.<br />
<br />
Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a <strong>gedra</strong>, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:<br />
<span style="display: block; text-align: center;">k = 12a + 19b</span><span style="display: block; text-align: center;">s = 7a + 11b</span><br />
The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:<br />
<span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span><br />
Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].<br />
<br />
Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-fifth pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n*G = P5 - 2*m3 = [7,4] - 2*[3,2] = [7,4] - [6,4] = [1,0] = A1.<br />
<br />
Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - m*P = P8 - 5*M2 = [12,7] - 5*[2,1] = [2,2] = 2*[1,1] = 2*m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2*m2 = d3). The enharmonic's <strong>count</strong> is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5*P + 2*E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:<br />
<span style="display: block; text-align: center;">P1 -- ^^M2=v<span style="vertical-align: super;">3</span>m3 -- v4 -- ^5 -- ^<span style="vertical-align: super;">3</span>M6=vvm7 -- P8</span><span style="display: block; text-align: center;">C -- D^^=Ebv<span style="vertical-align: super;">3</span> -- Fv -- G^ -- A^<span style="vertical-align: super;">3</span>=Bbvv -- C</span><br />
Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5*m2 = [5,3] - 5*[1,1] = [5,3] - [5,5] = [0,-2] = -2*[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5*G - 2*E, G must be ^^m2. The genchain is:<br />
<br />
<span style="display: block; text-align: center;">P1 -- ^^m2=v<span style="vertical-align: super;">3</span>A1 -- vM2 -- ^m3 -- ^<span style="vertical-align: super;">3</span>d4=vvM3 -- P4</span><span style="display: block; text-align: center;">C -- Db^^ -- Dv -- Eb^ -- Evv -- F</span><br />
To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multigen as before. Then deduce the period from the enharmonic. If the multigen was changed by unreducing, find the original generator from the period and the alternate generator.<br />
<br />
For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The bare alternate generator G' is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10*G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^<span style="vertical-align: super;">4</span>M2. Next, find the original half-fourth generator. P = P8/5 ~ 240¢, and G = P4/2 ~250¢. Because P < G, G' is not P - G but G - P, and G is not P - G' but P + G', which equals ^<span style="vertical-align: super;">4</span>M2 + ^1 = ^<span style="vertical-align: super;">5</span>M2. The alternate generator is usually simpler than the original generator, and the alternate multigen is usually more complex than the original multigen.<br />
<span style="display: block; text-align: center;">P1- - ^<span style="vertical-align: super;">4</span>M2=v<span style="vertical-align: super;">6</span>m3 -- vvP4 -- ^^P5 -- ^<span style="vertical-align: super;">6</span>M6=v<span style="vertical-align: super;">4</span>m7 -- P8</span><span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">4</span>=Ebv<span style="vertical-align: super;">6</span> -- Fvv -- G^^ -- A^<span style="vertical-align: super;">6</span>=Bbv<span style="vertical-align: super;">4</span> -- C</span><span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">5</span>M2=v<span style="vertical-align: super;">5</span>m3 -- P4</span><span style="display: block; text-align: center;">C -- D^<span style="vertical-align: super;">5</span>=Ebv<span style="vertical-align: super;">5</span> -- F</span><br />
To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.<br />
<br />
A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br />
<br />
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. But by using double-pair notation, we can get enharmonics that are 2nds. First 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, so E = vvA1 and 2*G = ^m6 or vM6. Next we find G = (2*G)/2, using either ^m6 or vM6. But ^m6 has an up sign, and there's no such thing as half an up. The answer is to double all ups and downs: P11/2 = ^^m6 or vvM6, and E = v<span style="vertical-align: super;">4</span>A1. The bare generator is ^^m6/2 = ^^[8,5]/2 = ^[4,2] = ^M3, and the bare enharmonic is ^^m6 - 2*(^M3) = [0,1] = d2. For the second enharmonic, we use the second pair of accidentals: E' = //d2, and G = ^\M3 or ^/d4. Here is the genchain:<br />
<span style="display: block; text-align: center;">P1 -- ^\M3=^/d4 -- ^^m6=vvM6 -- v\A8=v/m9 -- P11<br />
</span><span style="display: block; text-align: center;">C -- E^\=Fb^/ -- Ab^^=Avv -- C#v\=Dbv/ -- F<br />
</span><br />
Using vvM6/2 for 2*G gives a different but equally valid notation: vvM6/2 = vv[9,5]/2 = v[4,2] = vM3, and vvM6 - 2*(vM3) = [1,1] = m2, and E' = \\m2 and G = v/M3 or v\4.<br />
<span style="display: block; text-align: center;">P1 -- v/M3=v\4 -- vvM6=^^m6 -- ^/8=^\m9 -- P11<br />
</span><span style="display: block; text-align: center;">C -- Ev/=Fv\ -- Avv=Ab^^ -- C^/=Db^\ -- F<br />
</span><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 proceeding as before, we find only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. 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. Before we can divide by 4, we must quadruple all ups and downs: E = ^<span style="vertical-align: super;">12</span>d2 and 4*G' = ^<span style="vertical-align: super;">4</span>m2. The bare alt-generator is ^<span style="vertical-align: super;">4</span>[1,1]/4 = ^[0,0] = ^1, and the bare 2nd enharmonic is ^<span style="vertical-align: super;">4</span>m2 - 4*(^1) = m2. Thus E' = \<span style="vertical-align: super;">4</span>m2 and G' = ^/1.The period can be deduced from 4*G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4*G' = P4 - ^<span style="vertical-align: super;">4</span>m2 = v<span style="vertical-align: super;">4</span>M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = v4M3 + ^/1 = v3/M3.<br />
<span style="display: block; text-align: center;">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 />
</span><span style="display: block; text-align: center;">C -- Ev<span style="vertical-align: super;">4</span> -- Ab^<span style="vertical-align: super;">4</span> -- C</span><span style="display: block; text-align: center;">P1 -- v<span style="vertical-align: super;">3</span>/M3 -- v<span style="vertical-align: super;">6</span><!-- ws:start:WikiTextRawRule:025:``//`` -->//<!-- ws:end:WikiTextRawRule:025 -->A5=^<span style="vertical-align: super;">6</span><!-- ws:start:WikiTextRawRule:026:``//`` -->//<!-- ws:end:WikiTextRawRule:026 -->m6=^<span style="vertical-align: super;">6</span>\\d7 -- ^<span style="vertical-align: super;">3</span>\m9 -- F</span><span style="display: block; text-align: center;">C -- Ev<span style="vertical-align: super;">3</span>/ -- G#v<span style="vertical-align: super;">6</span><!-- ws:start:WikiTextRawRule:027:``//`` -->//<!-- ws:end:WikiTextRawRule:027 -->=Ab^<span style="vertical-align: super;">6</span><!-- ws:start:WikiTextRawRule:028:``//`` -->//<!-- ws:end:WikiTextRawRule:028 -->=Bbb^<span style="vertical-align: super;">6</span>\\ -- Db^<span style="vertical-align: super;">3</span>\ -- F</span><br />
<!-- ws:start:WikiTextHeadingRule:49:<h2> --><h2 id="toc9"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:49 -->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 />
<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- C<br />
C -- Eb^<span style="vertical-align: super;">4</span> -- Av<span style="vertical-align: super;">4</span> -- C<br />
P1 -- v<span style="vertical-align: super;">3</span>M2 -- v<span style="vertical-align: super;">6</span>M3=^<span style="vertical-align: super;">6</span>m2 -- ^<span style="vertical-align: super;">3</span>m3 -- P4<br />
C -- Dv<span style="vertical-align: super;">3</span> -- Ev<span style="vertical-align: super;">6</span>=Db^<span style="vertical-align: super;">6</span> -- Eb^<span style="vertical-align: super;">3</span> -- F<br />
</span><br />
Because G is a M2 and E is an A2, the equivalent generator G - E is a descending A1. Ascending intervals that look descending are best avoided, and double-pair notation is better for this pergen. We have P = vM3, E = ^3d2, G = /m2, and E' = /4dd2.<br />
<span style="display: block; text-align: center;">P1 -- vM3 -- ^m6 -- P8<br />
</span><span style="display: block; text-align: center;">C -- Ev -- Ab^ -- C<br />
</span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:029:``//`` -->//<!-- ws:end:WikiTextRawRule:029 -->d3=\\A2 -- \M3 -- P4<br />
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:030:``//`` -->//<!-- ws:end:WikiTextRawRule:030 -->=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 />
<br />
<br />
The comma equals xE and/or yE'.<br />
If M' = [a,b], then G' = [round(a/n'), round(b/n')] makes the smallest zE", but not always the smallest E"<br />
<br />
bbT = 49/48 = m2 = 36¢ = half-fourth. E = vvm2.<br />
(-22, 11, 2) = -dd2 = 94¢ is LLyyT = half-fourth. E = ^^dd2, and the genchain is C -- D#v=Eb^ -- F.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
(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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:51:<h2> --><h2 id="toc10"><a name="Further Discussion-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:51 -->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 />
<!-- ws:start:WikiTextHeadingRule:53:<h2> --><h2 id="toc11"><!-- ws:end:WikiTextHeadingRule:53 --> </h2>
<!-- ws:start:WikiTextHeadingRule:55:<h2> --><h2 id="toc12"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:55 -->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 />
<br />
General rules for combining pergens:<br />
<ul><li>(P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n')</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li></ul><br />
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:57:<h2> --><h2 id="toc13"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:57 -->Pergens and EDOs</h2>
<br />
Pergens have much in common with edos. Pergens (rank-2 ones) assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but far fewer than a hundred have been explored. There are an infinite number of pergens, but far fewer than a hundred will suffice most composers.<br />
<br />
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported. In practice, if the generator's keyspan is very small, a partially supported pergen. For example, 22edo and 2\22 generator.<br />
<br />
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! For example, 12edo supports (P8, P4/5), (P8, P11/17), (P8, WWP4/29), (P8, W<span style="vertical-align: super;">3</span>P4/41), etc.<br />
<br />
How many edos support a given pergen? Again, 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 M's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br />
<br />
Given an edo, a period p\N and a generator g\N, what is the pergen? The octave fraction n is N/p. Let the edo's 5th be f\N. To find the multigen M, we must find a monzo (a,b) such that a*N + b*f is a multiple of g. If n = 1, |b| = 1.<br />
<br />
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.<br />
<br />
[13b and 18b - needs explanation]<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="2">pergen<br />
</th>
<th>supporting edos (12-31 only)<br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8, P5)<br />
</td>
<td style="text-align: center;">unsplit<br />
</td>
<td style="text-align: center;">12, 13b, 14*, 15*, 16, 17, 18b*, 19, 20*, 21*,<br />
22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31<br />
</td>
</tr>
<tr>
<th>halves<br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5)<br />
</td>
<td style="text-align: center;">half-octave<br />
</td>
<td style="text-align: center;">12, 14, 16, 18b, 20*, 22, 24*, 26, 28*, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/2)<br />
</td>
<td style="text-align: center;">half-fourth<br />
</td>
<td style="text-align: center;">13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/2)<br />
</td>
<td style="text-align: center;">half-fifth<br />
</td>
<td style="text-align: center;">14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/2)<br />
</td>
<td style="text-align: center;">half-everything<br />
</td>
<td style="text-align: center;">14, 18b, 20*, 24, 28*, 30*<br />
</td>
</tr>
<tr>
<th>thirds<br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5)<br />
</td>
<td style="text-align: center;">third-octave<br />
</td>
<td style="text-align: center;">12, 15, 18b*, 21, 24*, 27, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/3)<br />
</td>
<td style="text-align: center;">third-fourth<br />
</td>
<td style="text-align: center;">13b, 14*, 15, 21*, 22, 28*, 29, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/3)<br />
</td>
<td style="text-align: center;">third-fifth<br />
</td>
<td style="text-align: center;">15*, 16, 20*, 21, 25*, 26, 30*, 31<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P11/3)<br />
</td>
<td style="text-align: center;">third-11th<br />
</td>
<td style="text-align: center;">15, 17, 21, 23, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/2)<br />
</td>
<td style="text-align: center;">third-8ve, half-4th<br />
</td>
<td style="text-align: center;">15, 18b*, 24, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5/2)<br />
</td>
<td style="text-align: center;">third-8ve, half-5th<br />
</td>
<td style="text-align: center;">18b, 21, 24, 27, 30<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/3)<br />
</td>
<td style="text-align: center;">half-8ve, third-4th<br />
</td>
<td style="text-align: center;">14, 22, 28*, 30<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5/3)<br />
</td>
<td style="text-align: center;">half-8ve, third-5th<br />
</td>
<td style="text-align: center;">16, 20*, 26, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P11/3)<br />
</td>
<td style="text-align: center;">half-8ve, third-11th<br />
</td>
<td style="text-align: center;">19, 30<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/3)<br />
</td>
<td style="text-align: center;">third-everything<br />
</td>
<td style="text-align: center;">15, 21, 30*<br />
</td>
</tr>
<tr>
<th>quarters<br />
</th>
<th><br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P5)<br />
</td>
<td style="text-align: center;">quarter-octave<br />
</td>
<td style="text-align: center;">12, 16, 20, 24*, 28<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P4/4)<br />
</td>
<td style="text-align: center;">quarter-fourth<br />
</td>
<td style="text-align: center;">18b*, 19, 20*, 28, 29, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P5/4)<br />
</td>
<td style="text-align: center;">quarter-fifth<br />
</td>
<td style="text-align: center;">14*, 20, 21*, 27, 28*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P11/4)<br />
</td>
<td style="text-align: center;">quarter-eleventh<br />
</td>
<td style="text-align: center;">14, 17, 20, 28*, 31<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8, P12/4)<br />
</td>
<td style="text-align: center;">quarter-twelfth<br />
</td>
<td style="text-align: center;">13b, 15*, 18b, 20*, 23, 25*, 28, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/2)<br />
</td>
<td style="text-align: center;">quarter-octave, half-fourth<br />
</td>
<td style="text-align: center;">20, 24, 28<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, M2/4)<br />
</td>
<td style="text-align: center;">half-octave, quarter-tone<br />
</td>
<td style="text-align: center;">20, 22, 24, 26, 28<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P4/4)<br />
</td>
<td style="text-align: center;">half-octave, quarter-fourth<br />
</td>
<td style="text-align: center;">18b, 20*, 28, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/2, P5/4)<br />
</td>
<td style="text-align: center;">half-octave, quarter-fifth<br />
</td>
<td style="text-align: center;">14, 20, 28*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/3)<br />
</td>
<td style="text-align: center;">quarter-octave, third-fourth<br />
</td>
<td style="text-align: center;">28<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P5/3)<br />
</td>
<td style="text-align: center;">quarter-octave, third-fifth<br />
</td>
<td style="text-align: center;">16, 20<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P11/3)<br />
</td>
<td style="text-align: center;">quarter-octave, third-eleventh<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/4)<br />
</td>
<td style="text-align: center;">third-octave, quarter-fourth<br />
</td>
<td style="text-align: center;">18b*, 30<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P5/4)<br />
</td>
<td style="text-align: center;">third-octave, quarter-fifth<br />
</td>
<td style="text-align: center;">21, 27<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P11/4)<br />
</td>
<td style="text-align: center;">third-octave, quarter-eleventh<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P12/4)<br />
</td>
<td style="text-align: center;">third-octave, quarter-twelfth<br />
</td>
<td style="text-align: center;">15, 18b, 30*<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/4, P4/4)<br />
</td>
<td style="text-align: center;">quarter-everything<br />
</td>
<td style="text-align: center;">20, 28<br />
</td>
</tr>
</table>
<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:59:<h2> --><h2 id="toc14"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:59 -->Misc notes</h2>
<br />
Pergens were discovered by Kite Giedraitis in 2017. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule:3057:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3057 --><br />
<br />
<br />
<br />
<u><strong>Extra paragraphs:</strong></u><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 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 />
g = -c<br />
r = -d<br />
<br />
a = -11k + 19s - 4g + 6r<br />
b = 7k - 12s + 4g - 2r<br />
c = -g<br />
d = -r<br />
<br />
Chord names: All rank-2 chords can be named using ups and downs, as if they were edos. For example, in half-octave, a 4:5:6 chord is C Ev G = C.v. There are multiple spellings for many chords. Whenever the enharmonic isn't an A1, even the degree of a chord note can change. It would be possible to spell the chord C Fb^ G, but there's no reason to. But 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 />
Half-octave with a vvM2 enharmonic: 4:5:6 = C Eb^ G. So better to have E = ^^d2.<br />
<br />
<br />
Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).<br />
<br />
For {P8/M, multigen/N}, an octave = M periods ± some number of enharmonics, and a multigen = N generators ± some number of enharmonics.<br />
<br />
The first is based on the enharmonic's degree, which can be deduced from the pergen as follows:<br />
<br />
The octave spans 7 steps. If the octave is split into M periods, each spanning x steps, x is roughly 7/M. The enharmonic, or some multiple of it, must span |Mx - 7| steps. Likewise, if the multigen is split into N generators, each spanning y steps, and S = the multigen's degree - 1, then y is roughly S/N, and the enharmonic spans |Ny - S| steps.<br />
<br />
The enharmonic's degree depends on exactly how 7/M or S/N is rounded off, and alternate degrees are possible. For example, third-eleventh has S/N = 10/3 = 3 or 4, implying a generator that's a 4th or a 5th, and an enharmonic that's a 2nd or a 3rd. The lower degree is generally preferred. However, for single-comma temperaments, the enharmonic interval should be the same degree as the comma. So sometimes larger degrees are preferred.<br />
<br />
For {P8/M, multigen/N}, there are two conditions on the enharmonic's degree, which may be mutually exclusive. If so, two unconventional accidental pairs (e.g. ups/downs and highs/lows) must be used, and each accidental pair has its own enharmonic interval.<br />
<br />
For <strong>{P8/M, P5}</strong>, the enharmonic's degree = |M * round (7/M) - 7 + a*M| + 1 (a = 0, or 1 or -1 for the alternate enharmonics)<br />
For {<strong>P8, multigen/N}</strong>, the enharmonic's degree = |N * round ((S/N) - S ± N| + 1, where S = the multigen's degree - 1<br />
For <strong>{P8/M, multigen/N}</strong>, the enharmonic's degree = |M * round (7/M) - 7 ± M| + 1 = |N * round ((S/N) - S ± S| + 1<br />
or, the 8ve's enharmonic = |M * round (7/M) - 7 ± M| + 1 and the multigen's enharmonic = |N * round ((S/N) - S ± N| +1<br />
<br />
<br />
The 2nd restriction is based on the implied edo. The possible edos, and thus the possible enharmonics, can be deduced from the pergen.<br />
<br />
For {P8/M, P5}, the octave is split into M periods. If the period has a 3-exponent of x, then the enharmonic interval's 3-exponent is Mx, and the implied edo is |Mx|. Thus half-octave implies an even-numbered edo.<br />
<br />
For {P8, multigen/N}, since the octave is unsplit, the only possible multigens are some voicing of the 5th, and the multigen's 3-exponent is ±1. If the generator has a 3-exponent of y, the enharmonic's 3-exponent is Ny ± 1, and the implied edo is |Ny ± 1|. Thus half-fourth and half-fifth both imply an odd-numbered edo.<br />
<br />
For {P8/M, multigen/N}, there are two conditions on the enharmonic. If T is the 3-exponent of the multigen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd. Therefore there must be two accidental pairs, each with its own enharmonic interval. In the main table, this pergen is notated with both ups/downs and highs/lows. Since the 8ve and 4th are split, the 5th is too. Each interval has its own genchain. One of these is notated with ups/downs, another with highs/lows, and the third with both. The 3 possible ways of allocating the two accidental pairs are all listed. Furthermore, ups/downs can be exchanged for highs/lows.<br />
<br />
For {P8/2, P5/3}, the edo = 2x = 3y ± 1. The edo must be even, thus y must be odd. Possible edos are 2, 4, 8, 10, 14, 16, 20, 22, 26, 28... The main table has ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2, which implies 26-edo. Most of the other edos aren't practical. 10 and 20 imply the m3. 22-edo suggests a d<span style="vertical-align: super;">3</span>4. The perchain would be C - E#^<span style="vertical-align: super;">3</span>=Abbv<span style="vertical-align: super;">3</span> - C.<br />
<br />
<br />
For <strong>{P8/M, P5}</strong>, the implied edo = Mx<br />
For {<strong>P8, multigen/N}</strong>, the implied edo = Ny ± 1 (the multigen is some voicing of the 5th)<br />
For <strong>{P8/M, multigen/N}</strong>, the implied edo = Mx = Ny ± T, where T is the 3-exponent of the multigen,<br />
or, the 8ve's implied edo = Mx and the multigen's implied edo = Ny ± F<br />
<br />
<br />
<br />
If the enharmonic is larger than a 2nd, it may be possible to split it into several smaller enharmonics. For example, {P8/5, P5}. If 7/M = 7/5 is rounded to 1, the enharmonic is a 3rd. The enharmonic must be 5x, and also 7y + 4, so the 3-exponent = -10 = dim 3rd. Adding ups and downs, we have enharmonic = v<span style="vertical-align: super;">5</span>d3 and period = ^M2. Fortunately, d3 = m2 + m2, and the 3rd can be reduced to two 2nds. The downs must be doubled, so that the period = ^^M2, and the enharmonic = v<span style="vertical-align: super;">10</span>d3 = 2 *<br />
v<span style="vertical-align: super;">5</span>m2. The enharmonic must be applied twice in the course of an octave: P1 - ^^M2 - ^<span style="vertical-align: super;">4</span>M3=vP4 - ^P5 - ^<span style="vertical-align: super;">3</span>M6=vvm7 - P8<br />
<br />
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 />
<br />
[<em>Question: what if there are highs and lows?</em>]</body></html>