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, m7, 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]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]].
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-8ve ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||
||= " ||= " ||= 81/80, 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||
||= (P8, P5/2) ||= half-5th ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||
||= " ||= " ||= (-1,5,0,0,-2) ||= mohajira ||= deep amber ||= aaT ||
||= (P8, P4/2) ||= half-4th ||= 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-4th ||= 250/243 ||= porcupine ||= triple yellow ||= y<span style="vertical-align: super;">3</span>T ||
||= (P8, P11/3) ||= third-11th ||= (12,-1,0,0,-3) ||= small triple amber ||= small triple amber ||= sa<span style="vertical-align: super;">3</span>T ||
||= (P8/4, P5) ||= quarter-8ve ||= (3,4,-4) ||= diminished ||= quadruple green ||= g<span style="vertical-align: super;">4</span>T ||
||= (P8/2, M2/4) ||= half-8ve 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-12th ||= (-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-8ve, yellow-3rd. 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-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th 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.
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's 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, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.
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 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 = x·P, thus P = P8/x
3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz
M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave).
G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz
<span style="display: block; text-align: center;">**<span style="font-size: 110%;">The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span>**
</span>
A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.
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. Here "/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 multigen equals two gens, and adding an octave to the gen adds a __double__ octave to the multigen. The alternate gens are P11/2 and P19/2 = (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-5th is a double 5th 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-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th 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 a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), but the tone isn't a generator.
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See [[pergen#Further%20Discussion-Chord%20names%20and%20scale%20names|Chord names and scale names]] below.
The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.
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 a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.
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: C -- F#v=Gb^ -- C
||~ 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-8ve ||= ^^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-4th ||= vvm2 ||= C^^ = Db ||= P4/2 = ^M2 = vm3 ||= C - D^=Ebv - F ||= semaphore
^1 = 64/63 ||
||= " ||= ^^dd2 ||= C^^ = B## ||= P4/2 = vA2 = ^d3 ||= C - D#v=Ebb^ - F ||= (-22,-11,2) ||
||= (P8, P5/2)
half-5th ||= 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-8ve ||= ^<span style="vertical-align: super;">3</span>d2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B# ||= P8/3 = vM3 = ^^d4 ||= C - Ev - Ab^ - C ||= augmented ||
||= (P8, P4/3)
third-4th ||= v<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 ``=`` </span>C# ||= P4/3 = vM2 = ^^m2 ||= C - Dv - Eb^ - F ||= porcupine ||
||= (P8, P5/3)
third-5th ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
||= (P8, P11/3)
third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,
with 11/8 = A4 ||
||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||
||= (P8/3, P4/2)
third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4
P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3 ||= C - Eb^^ - Avv - C
C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F ||= sixfold jade ||
||= " ||= ^<span style="vertical-align: super;">3</span>d2,
\\m2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B#
C``//`` = Db ||= P8/3 = vM3 = ^^d4
P4/2 = /M2 = \m3 ||= C - Ev - Ab^ - C
C - D/=Eb\ - F ||= 128/125 & 49/48 ||
||= (P8/3, P5/2)
third-8ve, half-5th ||= ^<span style="vertical-align: super;">3</span>d2
\\A1 ||= C^<span style="vertical-align: super;">3</span> ``=`` B#
C``//`` = C# ||= P8/3 = vM3 = ^^d4
P5/2 = /m3 = \M3 ||= C - Ev - Ab/ - C
C - Eb/=E\ - G ||= small sixfold blue ||
||= (P8/2, P4/3)
half-8ve, third-4th ||= ^^d2
\<span style="vertical-align: super;">3</span>A1 ||= C^^ ``=`` Dbb
C/<span style="vertical-align: super;">3</span> ``=`` C# ||= P8/2 = vA4 = ^d5
P4/3 = \M2 = ``//``m2 ||= C - F#v=Gb^ - C
C - D\ - Eb/ - F ||= large sixfold red ||
||= (P8/2, P5/3)
half-8ve,
third-5th ||= ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2 ||= C^<span style="vertical-align: super;">6</span> ``=`` B#<span style="vertical-align: super;">3</span> ||= P8/2 = v<span style="vertical-align: super;">3</span>AA4 = ^<span style="vertical-align: super;">3</span>dd5
P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3 ||= C - F<span style="vertical-align: super;">x</span>v<span style="vertical-align: super;">3</span>=Gbb^<span style="vertical-align: super;">3</span> C
C - D#vv - Fb^^ - G ||= large sixfold yellow ||
||= " ||= ^^d2,
\\\m2 ||= C^^ = B#
C``///`` = Db ||= P8/2 = vA4 = ^d5
P5/3 = /M2 = \\m3 ||= C - F#v=Gb^ - C
C - /D - \F - G ||= 50/49 & 1029/1024 ||
||= (P8/2, P11/3)
half-8ve,
third-11th ||= v<span style="vertical-align: super;">6</span>M2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D ||= P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5
P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5 ||= C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C
C - F^^ - Cvv - F ||= large sixfold jade ||
||= (P8/3, P4/3)
third-
everything ||= v<span style="vertical-align: super;">3</span>d2,
\<span style="vertical-align: super;">3</span>A1 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Dbb
C/3 ``=`` C# ||= P8/3 = ^M3 = vvd4
P4/3 = \M2 = ``//``m2
P5/3 = v/M2 ||= C - E^ - Abv - C
C - D\ - Eb/ - F
C - Dv/ - F^\ - G ||= 250/243 & 729/686
^1 = 64/63
/1 = 81/80 ||
||= " ||= ^<span style="vertical-align: super;">3</span>d2,
\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` B#
C/<span style="vertical-align: super;">3</span> ``=`` Db ||= P8/3 = vM3 = ^^d4
P5/3 = /M2 = \\m3
P4/3 = v\M2 ||= C - Ev - Ab^ - C
C - D/ - F\ - G
C - Dv\ - Eb^/ - F ||= 128/125 & 1029/1024
^1 = 81/80
/1 = 64/63 ||
||= " ||= v<span style="vertical-align: super;">3</span>A1,
\<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` C#
C/3 ``=`` Db ||= P4/3 = vM2 = ^^m2
P5/3 = /M2 = \\m3
P8/3 = v/M3 ||= C - Dv - Eb^ - F
C - D/ - F\ - G
C - Ev/ - Ab^\ - C ||= 250/243 & 1029/1024
^1 = 81/80
/1 = 64/63 ||
||~ 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 tipping-point 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¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.
||||~ bare enharmonic
interval ||~ 3-exponent ||~ tipping
point 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__=
==Searching for pergens==
To list all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a viable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:
If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)
If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))
Next we loop through all combinations of x and z in such a way that larger values of x and z come last:
i = 1; loop (maxFraction,
> j = 1; loop (i - 1,
>> makeMapping (i, j); makeMapping (i, -j);
>> makeMapping (j, i); makeMapping (j, -i);
>> j += 1;
> );
> makeMapping (i, i); makeMapping (i, -i);
> i += 1;
);
The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted.
In the [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]] section, a program is linked to that performs these calculations and lists all pergens. It also lists suggested enharmonics. Experimenting with allowing y and i to range further does not produce any additional pergens.
==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 pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.
Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (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-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.
A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.
==Finding an example temperament==
To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.
If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 27/20. The comma must have only one higher prime, with exponent ±1.
Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5) / (3<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4) / (2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>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-8ve (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.
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. For double-pair notation, there are also equivalent enharmonics.
==Ratio and cents of the accidentals==
In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.
We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.
This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:
15-edo: # = 240¢, ^ = 80¢
16-edo: # = -75¢
17-edo: # = 141¢, ^ = 71¢
18b-edo: # = -133¢, ^ = 67¢
19-edo: # = 63¢
21-edo: ^ = 57¢
22-edo: # = 164¢, ^ = 55¢
quarter-comma meantone: # = 76¢
fifth-comma meantone: # = 84¢
third-comma archy: # = 177¢
eighth-comma porcupine: # = 157¢, ^ = 52¢
sixth-comma srutal: # = 139¢, ^ = 33¢
third-comma injera: # = 63¢, ^ = 31¢ (third-comma = 1/3 of 81/80)
eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)
Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.
==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:
> k = 12a + 19b
> s = 7a + 11b
The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:
> a = -11k + 19b
> b = 7a - 12b
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-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.
Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The number of 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<span class="nowrap">⋅</span>m2 = [5,3] - 5<span class="nowrap">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span class="nowrap">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span class="nowrap">⋅</span>G - 2<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P1 = m2. It must be downed, thus E = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10<span class="nowrap">⋅</span>G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, and P = ^<span style="vertical-align: super;">4</span>M2. Next, find the original half-4th 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. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>``//``ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:
<span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span><span style="display: block; text-align: center;">C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- 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 only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.
Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = ``//``d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^``//``d4.
<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span>
This is a lot of math, but it only needs to be done once for each pergen!
==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>
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.
Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.
For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.
Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese is (P8, P11/3), with G = 7/5 = d5. E = 3·d5 - P11 = descending dd3.
==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.
==Chord names and scale names==
Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the [[Ups and Downs Notation|ups and downs]] page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.
In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same ambiguity occurs in 24-edo. (Even in 12-edo, there are chords with ambiguous spellings. B D F Ab = Bdim7, and B D F# G# = Bmin6. But without the 5th, the chord could be spelled either B D Ab or B D G#.)
Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).
A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&gT) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.
MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as any others that could generate the scale. Preference is given to the pergen that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. For example, 3L2s (anti-pentatonic) has a generator in the 400-480¢ range, suggesting both P11/4 and P12/4. But the former, with a just 11th, makes L = 351¢ and s = 73.5¢, and L/s = 4.76, quite large. The latter with a just 12th makes L = 249¢, s = 226.5¢, and L/s = 1.10, much better.
||||||~ Tetratonic MOS scales ||~ secondary examples ||
||= 1L 3s ||= (P8, P4/2) [4] ||= half-4th tetratonic ||< third-4th, third-5th ||
||= 2L 2s ||= (P8/2, P5) [4] ||= half-8ve tetratonic ||< ||
||= 3L 1s ||= (P8, P5/2) [4] ||= half-5th tetratonic ||< ||
||||||~ Pentatonic MOS scales ||~ ||
||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic ||< third-4th, quarter-4th, quarter-5th ||
||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic ||< third-11th ||
||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic ||< quarter-11th ||
||= 4L 1s ||= (P8, P4/2) [5] ||= half-4th pentatonic ||< ||
||||||~ Hexatonic MOS scales ||~ ||
||= 1L 5s ||= (P8, P4/3) [6] ||= third-4th hexatonic ||< quarter-4th, quarter-5th, fifth-4th, fifth-5th ||
||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic ||< ||
||= 3L 3s ||= (P8/3, P5) [6] ||= third-8ve hexatonic ||< ||
||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic ||< ||
||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic ||< ||
||||||~ Heptatonic MOS scales ||~ ||
||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic ||< quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th ||
||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic ||< fifth-WW4th, sixth-WW5th ||
||= 3L 4s ||= (P8, P5/2) [7] ||= half-5th heptatonic ||< fifth-12th ||
||= 4L 3s ||= (P8, P11/5) [7] ||= fifth-11th heptatonic ||< sixth-12th ||
||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic ||< sixth-WW4th ||
||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< ||
||||||~ Octotonic MOS scales ||~ ||
||= 1L 7s ||= (P8, P4/4) [8] ||= quarter-4th octotonic ||< fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th ||
||= 2L 6s ||= (P8/2, P5) [8] ||= half-8ve octotonic ||< ||
||= 3L 5s ||= (P8, P11/4) [8] ||= quarter-11th octotonic ||< seventh-WW4th, seventh-WW5th ||
||= 4L 4s ||= (P8/4, P5) [8] ||= quarter-8ve octotonic ||< ||
||= 5L 3s ||= (P8, P12/4) [8] ||= quarter-12th octotonic ||< (very lopsided, unless 5th is quite flat) ||
||= 6L 2s ||= (P8/2, P4/3) [8] ||= half-8ve third-4th octotonic ||< ||
||= 7L 1s ||= (P8, P4/3) [8] ||= third-4th octotonic ||< ||
||||||~ Nonatonic MOS scales ||~ ||
||= 1L 8s ||= (P8, P4/4) [9] ||= quarter-4th nonatonic ||< fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th ||
||= 2L 7s ||= (P8, W<span style="vertical-align: super;">3</span>P5/8) [9] ||= eighth-W<span style="vertical-align: super;">3</span>5th nonatonic ||< third-11th, fifth-WW4th ||
||= 3L 6s ||= (P8/3, P5) [9] ||= third-8ve nonatonic ||< third-8ve half-5th ||
||= 4L 5s ||= (P8, P12/7) [9] ||= seventh-12th nonatonic ||< sixth-11th ||
||= 5L 4s ||= (P8, P4/2) [9] ||= half-4th nonatonic ||< (lopsided unless 4th is sharp), seventh-11th ||
||= 6L 3s ||= (P8/3, P4/2) [9] ||= third-8ve half-4th nonatonic ||< ||
||= 7L 2s ||= (P8, WWP5/6)[9] ||= sixth-WW5th nonatonic ||< (lopsided unless 5th is sharp) ||
||= 8L 1s ||= (P8, P5/5) [9] ||= fifth-5th nonatonic ||< ||
||||||~ Decatonic MOS scales ||~ ||
||= 1L 9s ||= (P8, P5/6) [10] ||= sixth-5th decatonic ||< fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th ||
||= 2L 8s ||= (P8/2, P5) [10] ||= half-8ve decatonic ||< half-8ve quartertone, half-8ve third-11th ||
||= 3L 7s ||= (P8, P12/5) [10] ||= fifth-12th decatonic ||< eighth-WW4th, eighth-WW5th ||
||= 4L 6s ||= (P8/2, P4/2) [10] ||= half-everything decatonic ||< ||
||= 5L 5s ||= (P8/2, P5) [10] ||= half-8ve decatonic ||< (lopsided unless 5th is quite flat) ||
||= 6L 4s ||= (P8/2, P5/3) [10] ||= half-8ve third-5th decatonic ||< ||
||= 7L 3s ||= (P8, P5/2) [10] ||= half-5th decatonic ||< ninth-WW5th ||
||= 8L 2s ||= (P8/2, P4/4) [10] ||= half-8ve quarter-4th decatonic ||< half-8ve quarter-12th ||
||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< ||
The tetratonic MOS scales don't include quarter-split pergens, because a tetratonic genchain has only 3 steps, and can only divide a multigen into thirds. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.
==Combining pergens==
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).
General rules for combining pergens:
* (P8/m, M/n) + (P8, P5) = (P8/m, M/n)
* (P8/m, P5) + (P8, M/n) = (P8/m, M/n)
* (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')
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.
==Pergens and EDOs==
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.
How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.
Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.
||||~ pergen ||~ supporting edos (12-31 only) ||
||= (P8, P5) ||= unsplit ||= 12, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,
21*, 22, 23, 24*, 25*, 26, 27, 28*, 29, 30*, 31 ||
||~ halves ||~ ||~ ||
||= (P8/2, P5) ||= half-8ve ||= 12, 14, 16, 18, 18b, 20*, 22, 24*, 26, 28*, 30* ||
||= (P8, P4/2) ||= half-4th ||= 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30* ||
||= (P8, P5/2) ||= half-5th ||= 13, 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-8ve ||= 12, 15, 18, 18b*, 21, 24*, 27, 30* ||
||= (P8, P4/3) ||= third-4th ||= 13b, 14*, 15, 21*, 22, 28*, 29, 30* ||
||= (P8, P5/3) ||= third-5th ||= 15*, 16, 20*, 21, 25*, 26, 30*, 31 ||
||= (P8, P11/3) ||= third-11th ||= 13, 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-8ve ||= 12, 16, 20, 24*, 28 ||
||= (P8, P4/4) ||= quarter-4th ||= 18b*, 19, 20*, 28, 29, 30* ||
||= (P8, P5/4) ||= quarter-5th ||= 13, 14*, 20, 21*, 27, 28* ||
||= (P8, P11/4) ||= quarter-11th ||= 14, 17, 20, 28*, 31 ||
||= (P8, P12/4) ||= quarter-12th ||= 13b, 15*, 18b, 20*, 23, 25*, 28, 30* ||
||= (P8/4, P4/2) ||= quarter-8ve, half-4th ||= 20, 24, 28 ||
||= (P8/2, M2/4) ||= half-8ve, quarter-tone ||= 18, 20, 22, 24, 26, 28 ||
||= (P8/2, P4/4) ||= half-8ve, quarter-4th ||= 18b, 20*, 28, 30* ||
||= (P8/2, P5/4) ||= half-8ve, quarter-5th ||= 14, 20, 28* ||
||= (P8/4, P4/3) ||= quarter-8ve, third-4th ||= 28 ||
||= (P8/4, P5/3) ||= quarter-8ve, third-5th ||= 16, 20 ||
||= (P8/4, P11/3) ||= quarter-8ve, third-11th ||= ||
||= (P8/3, P4/4) ||= third-8ve, quarter-4th ||= 18b*, 30 ||
||= (P8/3, P5/4) ||= third-8ve, quarter-5th ||= 21, 27 ||
||= (P8/3, P11/4) ||= third-8ve, quarter-11th ||= ||
||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||
==Supplemental materials==
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.
http://www.tallkite.com/misc_files/pergens.pdf
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.
http://www.tallkite.com/misc_files/alt-pergensLister.zip
Screenshot of the first 38 pergens:
[[image:alt-pergenLister.png width="704" height="460"]]
==Misc notes==
Given:
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x > 0, z ≠ 0, and |i| <= x
To prove: if |z| = 1, n = 1
If z = 1, let i = y - x, and the pergen = (P8/x, P5)
If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)
Therefore if |z| = 1, n = 1
To prove: n is always a multiple of b, and n = |b| if and only if n = 1
b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)
The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3
n = zb, and since n > 0, n = |z|·|b|
if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1
if n = 1, |z|·|b| = 1, therefore |b| = 1, and n = |b|
Therefore multigens like M9/3 or M3/5 never occur
Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)
To prove: test for explicitly false
If m = |b|, is the pergen explicitly false?
Does (a,b)/n split P8 into |b| periods?
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5
Because n is a multiple of b, n/b is an integer
M/b = (n/b)·M/n = (n/b)·G
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)
Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0
c·(a+b)·P8 = c·b·((n/b)·G - P5)
(1 - d·b)·P8 = c·b·((n/b)·G - P5)
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)
P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)
Therefore P8 is split into |b| periods
Therefore if m = |b|, the pergen is explicitly false
To prove: true/false test
If GCD (m,n) = |b|, is the pergen a false double?
If m = |b|, the pergen is explicitly false
Therefore assume m > |b| and unreduce
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn)
Simplify by dividing by b to get (P8/m, (n/b - a(m/b), -m) / m(n/b)) = (P8/m, (a',b')/n')
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false
Therefore the original pergen is a false double
To prove: alternate true/false test
if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?
If m > |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b
Can b' be reduced by simplifying further?
No, because GCD (a', b', n') = GCD (n/b - am/b, -m, mn/b)
GCD (b', n') = m
GCD (n/b, m) = 1
GCD (
|b'| = m, so the unreduced pergen is explicitly false, and the test works
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.
__**Extra stuff, not sure if it should be included:**__
Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. 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
The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.Original HTML content:
<html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:41:<h1> --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:41 --> </h1>
<!-- ws:start:WikiTextTocRule:93:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div>
<!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>
<!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>
<!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>
<!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>
<!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Further Discussion-Searching for pergens">Searching for pergens</a></div>
<!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>
<!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>
<!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>
<!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>
<!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 1em;"><a href="#x240¢, ^"> 240¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 1em;"><a href="#x141¢, ^"> 141¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#x-133¢, ^"> -133¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 1em;"><a href="#x164¢, ^"> 164¢, ^ </a></div>
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<!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#x139¢, ^"> 139¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 1em;"><a href="#x63¢, ^"> 63¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 1em;"><a href="#x157¢, ^"> 157¢, ^ </a></div>
<!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Finding a notation for a pergen">Finding a notation for a pergen</a></div>
<!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Alternate enharmonics">Alternate enharmonics</a></div>
<!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div>
<!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Chord names and scale names">Chord names and scale names</a></div>
<!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Combining pergens">Combining pergens</a></div>
<!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Pergens and EDOs">Pergens and EDOs</a></div>
<!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Supplemental materials">Supplemental materials</a></div>
<!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#x157¢, ^-Misc notes">Misc notes</a></div>
<!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --></div>
<!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextHeadingRule:43:<h1> --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:43 --><u><strong>Definition</strong></u></h1>
<br />
<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, m7, 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>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</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-8ve<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-5th<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-4th<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-4th<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-11th<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-8ve<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-8ve 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-12th<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-8ve, yellow-3rd. 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-8ve with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-11th 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 />
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:45:<h1> --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:45 --><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, i.e. it is the period or a multiple of it. If this happens, the multigen must use the 1st and 3rd primes. If the 3rd prime is also directly related, the 4th prime is used, and so forth.<br />
<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 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 = x·P, thus P = P8/x<br />
3/1 = P12 = y·P + z·G, thus G = [P12 - y·(P8/x)] / z = [-y·P8 + x·P12] / xz = (-y, x) / xz<br />
<br />
M's 3-limit monzo is (-y, x), or (y, -x) if z is negative. To get alternate generators, add i periods to G, with i ranging from -x (subtracting a full octave) to +x (adding a full octave). <br />
G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - 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, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x</span></strong><br />
</span><br />
A corollary of this formula is that if the octave is unsplit, the multigen is some voicing of the fifth, i.e. some perfect interval. Imperfect multigens are fairly rare. Less than 4% of all pergens have an imperfect multigen.<br />
<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. Here "/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 multigen equals two gens, and adding an octave to the gen adds a <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19/2 = (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-5th is a double 5th 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-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-5ths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-5th with red. Let ^1 = 64/63, and the pergen is (P8, P5/2, ^1), half-5th 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:47:<h1> --><h1 id="toc3"><a name="Applications"></a><!-- ws:end:WikiTextHeadingRule:47 --><u>Applications</u></h1>
<br />
One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), but the tone isn't a generator.<br />
<br />
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a higher-primes-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.<br />
<br />
Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-4th heptatonic is a higher-primes-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. See <a class="wiki_link" href="/pergen#Further%20Discussion-Chord%20names%20and%20scale%20names">Chord names and scale names</a> below.<br />
<br />
The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.<br />
<br />
For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. Certain rank-2 temperaments require another additional pair, <strong>highs and lows</strong>, written / and \. Dv\ is down-low D, and /5 is a high-fifth. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.<br />
<br />
Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But because the schisma does not map to a unison, this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G. The 135/128 comma maps to a unison, although not a perfect one, and as a result ups and downs aren't as desirable. 4:5:6 is spelled C Eb G, which while unusual is a stack of thirds.<br />
<br />
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.<br />
<br />
Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This represents a natural ordering of rank-2 pergens.<br />
<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: C -- F#v=Gb^ -- C<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-8ve<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-4th<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-5th<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-8ve<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-4th<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 = vM2 = ^^m2<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-5th<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;">small triple amber,<br />
with 11/8 = A4<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 = vv5<br />
</td>
<td style="text-align: center;">C - F^ - Cv - F<br />
</td>
<td style="text-align: center;">same, with 11/8 = P4<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 = v<span style="vertical-align: super;">4</span>A4<br />
P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3<br />
</td>
<td style="text-align: center;">C - Eb^^ - Avv - C<br />
C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F<br />
</td>
<td style="text-align: center;">sixfold jade<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2,<br />
\\m2<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:014:``=`` -->=<!-- ws:end:WikiTextRawRule:014 --> B#<br />
C<!-- ws:start:WikiTextRawRule:015:``//`` -->//<!-- ws:end:WikiTextRawRule:015 --> = Db<br />
</td>
<td style="text-align: center;">P8/3 = vM3 = ^^d4<br />
P4/2 = /M2 = \m3<br />
</td>
<td style="text-align: center;">C - Ev - Ab^ - C<br />
C - D/=Eb\ - F<br />
</td>
<td style="text-align: center;">128/125 & 49/48<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;">^<span style="vertical-align: super;">3</span>d2<br />
\\A1<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> B#<br />
C<!-- ws:start:WikiTextRawRule:017:``//`` -->//<!-- ws:end:WikiTextRawRule:017 --> = C#<br />
</td>
<td style="text-align: center;">P8/3 = vM3 = ^^d4<br />
P5/2 = /m3 = \M3<br />
</td>
<td style="text-align: center;">C - Ev - Ab/ - C<br />
C - Eb/=E\ - G<br />
</td>
<td style="text-align: center;">small sixfold blue<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;">^^d2<br />
\<span style="vertical-align: super;">3</span>A1<br />
</td>
<td style="text-align: center;">C^^ <!-- ws:start:WikiTextRawRule:018:``=`` -->=<!-- ws:end:WikiTextRawRule:018 --> Dbb<br />
C/<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> C#<br />
</td>
<td style="text-align: center;">P8/2 = vA4 = ^d5<br />
P4/3 = \M2 = <!-- ws:start:WikiTextRawRule:020:``//`` -->//<!-- ws:end:WikiTextRawRule:020 -->m2<br />
</td>
<td style="text-align: center;">C - F#v=Gb^ - C<br />
C - D\ - Eb/ - F<br />
</td>
<td style="text-align: center;">large sixfold red<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:021:``=`` -->=<!-- ws:end:WikiTextRawRule:021 --> 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;">large sixfold yellow<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:022:``///`` -->///<!-- ws:end:WikiTextRawRule:022 --> = 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;">50/49 & 1029/1024<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:023:``=`` -->=<!-- ws:end:WikiTextRawRule:023 --> D<br />
</td>
<td style="text-align: center;">P8/2 = ^<span style="vertical-align: super;">3</span>4 = v<span style="vertical-align: super;">3</span>5<br />
P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5<br />
</td>
<td style="text-align: center;">C - F^<span style="vertical-align: super;">3</span>=Gv<span style="vertical-align: super;">3</span> - C<br />
C - F^^ - Cvv - F<br />
</td>
<td style="text-align: center;">large sixfold jade<br />
</td>
</tr>
<tr>
<td style="text-align: center;">(P8/3, P4/3)<br />
third-<br />
everything<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>d2,<br />
\<span style="vertical-align: super;">3</span>A1<br />
</td>
<td style="text-align: center;">C^<span style="vertical-align: super;">3 </span> <!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 --> Dbb<br />
C/3 <!-- ws:start:WikiTextRawRule:025:``=`` -->=<!-- ws:end:WikiTextRawRule:025 --> C#<br />
</td>
<td style="text-align: center;">P8/3 = ^M3 = vvd4<br />
P4/3 = \M2 = <!-- ws:start:WikiTextRawRule:026:``//`` -->//<!-- ws:end:WikiTextRawRule:026 -->m2<br />
P5/3 = v/M2<br />
</td>
<td style="text-align: center;">C - E^ - Abv - C<br />
C - D\ - Eb/ - F<br />
C - Dv/ - F^\ - G<br />
</td>
<td style="text-align: center;">250/243 & 729/686<br />
^1 = 64/63<br />
/1 = 81/80<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">^<span style="vertical-align: super;">3</span>d2,<br />
\<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:027:``=`` -->=<!-- ws:end:WikiTextRawRule:027 --> B#<br />
C/<span style="vertical-align: super;">3</span> <!-- ws:start:WikiTextRawRule:028:``=`` -->=<!-- ws:end:WikiTextRawRule:028 --> Db<br />
</td>
<td style="text-align: center;">P8/3 = vM3 = ^^d4<br />
P5/3 = /M2 = \\m3<br />
P4/3 = v\M2<br />
</td>
<td style="text-align: center;">C - Ev - Ab^ - C<br />
C - D/ - F\ - G<br />
C - Dv\ - Eb^/ - F<br />
</td>
<td style="text-align: center;">128/125 & 1029/1024<br />
^1 = 81/80<br />
/1 = 64/63<br />
</td>
</tr>
<tr>
<td style="text-align: center;">"<br />
</td>
<td style="text-align: center;">v<span style="vertical-align: super;">3</span>A1,<br />
\<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:029:``=`` -->=<!-- ws:end:WikiTextRawRule:029 --> C#<br />
C/3 <!-- ws:start:WikiTextRawRule:030:``=`` -->=<!-- ws:end:WikiTextRawRule:030 --> Db<br />
</td>
<td style="text-align: center;">P4/3 = vM2 = ^^m2<br />
P5/3 = /M2 = \\m3<br />
P8/3 = v/M3<br />
</td>
<td style="text-align: center;">C - Dv - Eb^ - F<br />
C - D/ - F\ - G<br />
C - Ev/ - Ab^\ - C<br />
</td>
<td style="text-align: center;">250/243 & 1029/1024<br />
^1 = 81/80<br />
/1 = 64/63<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:031:``=`` -->=<!-- ws:end:WikiTextRawRule:031 --> 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:032:``=`` -->=<!-- ws:end:WikiTextRawRule:032 --> 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:033:``=`` -->=<!-- ws:end:WikiTextRawRule:033 --> 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:034:``=`` -->=<!-- ws:end:WikiTextRawRule:034 --> 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:035:``=`` -->=<!-- ws:end:WikiTextRawRule:035 --> 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 tipping-point 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¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.<br />
<table class="wiki_table">
<tr>
<th colspan="2">bare enharmonic<br />
interval<br />
</th>
<th>3-exponent<br />
</th>
<th>tipping<br />
point 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:49:<h1> --><h1 id="toc4"><a name="Further Discussion"></a><!-- ws:end:WikiTextHeadingRule:49 --><u>Further Discussion</u></h1>
<br />
<!-- ws:start:WikiTextHeadingRule:51:<h2> --><h2 id="toc5"><a name="Further Discussion-Searching for pergens"></a><!-- ws:end:WikiTextHeadingRule:51 -->Searching for pergens</h2>
<br />
To list all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a viable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br />
<br />
If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)<br />
If z < 0, then y is at least ceiling (x·3/2) and at most floor (x·(5/3 - z/2))<br />
<br />
Next we loop through all combinations of x and z in such a way that larger values of x and z come last:<br />
i = 1; loop (maxFraction,<br />
<ul class="quotelist"><li>j = 1; loop (i - 1,<ul class="quotelist"><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);<br />
<br />
The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted.<br />
<br />
In the <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a> section, a program is linked to that performs these calculations and lists all pergens. It also lists suggested enharmonics. Experimenting with allowing y and i to range further does not produce any additional pergens.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:53:<h2> --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:53 -->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:55:<h2> --><h2 id="toc7"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:55 -->Singles and doubles</h2>
<br />
If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.<br />
<br />
Every double-split pergen is either a <strong>true double</strong> or a <strong>false double</strong>. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (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-8ve quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.<br />
<br />
A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:57:<h2> --><h2 id="toc8"><a name="Further Discussion-Finding an example temperament"></a><!-- ws:end:WikiTextHeadingRule:57 -->Finding an example temperament</h2>
<br />
To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with exponent ±1, of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4<span class="nowrap">⋅</span>P and P8. If P is 6/5, the comma is 4<span class="nowrap">⋅</span>P - P8 = (6/5)<span style="vertical-align: super;">4</span> ÷ (2/1) = 648/625. If P is 7/6, the comma is P8 - 4<span class="nowrap">⋅</span>P = (2/1) · (7/6)<span style="vertical-align: super;">-4</span>. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively. (P8, P4/3) requires a G of about (498¢)/3 = 166¢, perhaps 10/9. The comma is 3<span class="nowrap">⋅</span>G - P4 = (10/9)^3 ÷ (4/3) = 250/243.<br />
<br />
If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a convenient comma such as 81/80 or 64/63. Thus for (P8/4, P5), since G = vm3, G is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with G = vA4 and ^1 = 81/80 gives G = 45/32, but if G = ^4, then G = 27/20. The comma must have only one higher prime, with exponent ±1.<br />
<br />
Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is <strong>explicitly false</strong>. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4<span class="nowrap">⋅</span>G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).<br />
<br />
If the pergen is not explicitly false, put the pergen in its <strong>unreduced</strong> form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.<br />
<br />
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5) / (3<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4) / (2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>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-8ve (P8/2, P5) has generator P5, alternate generators P4 and vA1, period vA4, and equivalent period ^d5. (P8, P5/2) has generator ^m3 and equivalent generator vM3.<br />
<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. For double-pair notation, there are also equivalent enharmonics.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:59:<h2> --><h2 id="toc9"><a name="Further Discussion-Ratio and cents of the accidentals"></a><!-- ws:end:WikiTextHeadingRule:59 -->Ratio and cents of the accidentals</h2>
<br />
In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.<br />
<br />
We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br />
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This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:<br />
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15-edo: # <!-- ws:start:WikiTextHeadingRule:61:<h1> --><h1 id="toc10"><a name="x240¢, ^"></a><!-- ws:end:WikiTextHeadingRule:61 --> 240¢, ^ </h1>
80¢<br />
16-edo: # = -75¢<br />
17-edo: # <!-- ws:start:WikiTextHeadingRule:63:<h1> --><h1 id="toc11"><a name="x141¢, ^"></a><!-- ws:end:WikiTextHeadingRule:63 --> 141¢, ^ </h1>
71¢<br />
18b-edo: # <!-- ws:start:WikiTextHeadingRule:65:<h1> --><h1 id="toc12"><a name="x-133¢, ^"></a><!-- ws:end:WikiTextHeadingRule:65 --> -133¢, ^ </h1>
67¢<br />
19-edo: # = 63¢<br />
21-edo: ^ = 57¢<br />
22-edo: # <!-- ws:start:WikiTextHeadingRule:67:<h1> --><h1 id="toc13"><a name="x164¢, ^"></a><!-- ws:end:WikiTextHeadingRule:67 --> 164¢, ^ </h1>
55¢<br />
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quarter-comma meantone: # = 76¢<br />
fifth-comma meantone: # = 84¢<br />
third-comma archy: # = 177¢<br />
eighth-comma porcupine: # <!-- ws:start:WikiTextHeadingRule:69:<h1> --><h1 id="toc14"><a name="x157¢, ^"></a><!-- ws:end:WikiTextHeadingRule:69 --> 157¢, ^ </h1>
52¢<br />
sixth-comma srutal: # <!-- ws:start:WikiTextHeadingRule:71:<h1> --><h1 id="toc15"><a name="x139¢, ^"></a><!-- ws:end:WikiTextHeadingRule:71 --> 139¢, ^ </h1>
33¢<br />
third-comma injera: # <!-- ws:start:WikiTextHeadingRule:73:<h1> --><h1 id="toc16"><a name="x63¢, ^"></a><!-- ws:end:WikiTextHeadingRule:73 --> 63¢, ^ </h1>
31¢ (third-comma = 1/3 of 81/80)<br />
eighth-comma hedgehog: # <!-- ws:start:WikiTextHeadingRule:75:<h1> --><h1 id="toc17"><a name="x157¢, ^"></a><!-- ws:end:WikiTextHeadingRule:75 --> 157¢, ^ </h1>
49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)<br />
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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br />
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<!-- ws:start:WikiTextHeadingRule:77:<h2> --><h2 id="toc18"><a name="x157¢, ^-Finding a notation for a pergen"></a><!-- ws:end:WikiTextHeadingRule:77 -->Finding a notation for a pergen</h2>
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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 />
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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 />
<ul class="quotelist"><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:<br />
<ul class="quotelist"><li>a = -11k + 19b</li><li>b = 7a - 12b</li></ul>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 />
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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. <br />
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For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.<br />
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Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The number of 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<span class="nowrap">⋅</span>m2 = [5,3] - 5<span class="nowrap">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span class="nowrap">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span class="nowrap">⋅</span>G - 2<span class="nowrap">⋅</span>E, G must be ^^m2. The genchain is:<br />
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<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 />
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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<span class="nowrap">⋅</span>P1 = m2. It must be downed, thus E = v<span style="vertical-align: super;">10</span>m2. Since m2 = 10<span class="nowrap">⋅</span>G' + E, G' is ^1. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x<span class="nowrap">⋅</span>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<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, and P = ^<span style="vertical-align: super;">4</span>M2. Next, find the original half-4th 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 />
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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 />
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Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span><!-- ws:start:WikiTextRawRule:036:``//`` -->//<!-- ws:end:WikiTextRawRule:036 -->ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br />
<span style="display: block; text-align: center;">P1 -- ^M3=v\4 -- /m6=\M6 -- ^/8=vm9 -- P11</span><span style="display: block; text-align: center;">C -- E^=Fv\ -- Ab/=A\ -- C^/=Dbv -- F</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 only two pairs are needed. First unreduce to get (P8/3, m3/12). m3/12 is the alternate generator G'. We have [3,2]/12 = [0,0] = P1, and G' = ^1 and E = v<span style="vertical-align: super;">12</span>m3. Next find 4·G' = m3/3 = [3,2]/3 = [1,1] = m2. Next, the bare enharmonic: m3 - 3·m2 = [0,-1] = descending d2. Thus E' = /<span style="vertical-align: super;">3</span>d2, and 4·G' = /m2. The period can be deduced from 4·G': P8/3 = (m10 - m3)/3 = (m10)/3 - 4·G' = P4 - /m2 = \M3. From the unreducing, we know that G - P = P11/4 - P8/3 = m3/12, so G = P11/4 = P8/3 + m3/12 = \M3 + ^1 = ^\M3. Equivalent enharmonics are found from E + E' and E - 2·E'. Equivalent periods and generators are found from the many enharmonics, which also allow much freedom in chord spelling.<br />
Enharmonic = v<span style="vertical-align: super;">12</span>m3 = /<span style="vertical-align: super;">3</span>d2 = v<span style="vertical-align: super;">4</span>/m2 = v<span style="vertical-align: super;">4</span>\\A1. Period = \M3 = v<span style="vertical-align: super;">4</span>4 = <!-- ws:start:WikiTextRawRule:037:``//`` -->//<!-- ws:end:WikiTextRawRule:037 -->d4. Generator = ^\M3 = v<span style="vertical-align: super;">3</span>4 = ^<!-- ws:start:WikiTextRawRule:038:``//`` -->//<!-- ws:end:WikiTextRawRule:038 -->d4.<br />
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<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br />
This is a lot of math, but it only needs to be done once for each pergen!<br />
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<!-- ws:start:WikiTextHeadingRule:79:<h2> --><h2 id="toc19"><a name="x157¢, ^-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:79 -->Alternate enharmonics</h2>
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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 />
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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:039:``//`` -->//<!-- ws:end:WikiTextRawRule:039 -->d3=\\A2 -- \M3 -- P4<br />
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:040:``//`` -->//<!-- ws:end:WikiTextRawRule:040 -->=D#\\ -- E\ -- F</span><br />
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2. Thus the vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br />
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Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br />
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For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.<br />
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br />
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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese is (P8, P11/3), with G = 7/5 = d5. E = 3·d5 - P11 = descending dd3.<br />
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<!-- ws:start:WikiTextHeadingRule:81:<h2> --><h2 id="toc20"><a name="x157¢, ^-Alternate keyspans and stepspans"></a><!-- ws:end:WikiTextHeadingRule:81 -->Alternate keyspans and stepspans</h2>
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One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br />
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<!-- ws:start:WikiTextHeadingRule:83:<h2> --><h2 id="toc21"><a name="x157¢, ^-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:83 -->Chord names and scale names</h2>
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br />
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In certain pergens, one spelling isn't always clearly better. For example, in half-4th, C E G A^ and C E G Bbv are the same chord, and either spelling might be used. This exact same 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 />
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Given a specific temperament, the full period/generator mapping gives the notation of higher primes, and thus of any ratio. Thus JI chords can be named. For example, pajara's pergen is (P8/2, P5), half-8ve, with P = vA4 or ^d5, G = P5, and E = ^^d2. The full mapping is [(2 2 7 8) (0 1 -2 -2)] = [(2 0) (2 1) (7 -2) (8 -2)]. This tells us 7/1 = 8·P - 2·G = 4·P8 - 2·P5 = WWm7, and 7/4 = m7. Likewise 5/1 = 7·P - 2·G = 7/1 minus a half-octave. From this it follows that 5/4 = m7 - ^d5 = vM3. A 4:5:6:7 chord is written C Ev G Bb = C7(v3).<br />
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A different temperament may result in the same pergen with the same enharmonic, but may still produce a different name for the same chord. For example, injera (2.3.5.7 with 81/80 and 50/49, or rryy&gT) is also half-8ve. However, the tipping point for the d2 enharmonic is at 700¢, and while pajara favors a fifth wider than that, injera favors a fifth narrower than that. Hence ups and downs are exchanged, and E = vvd2, and P = ^A4 = vd5. The mapping is [(2 2 0 1) (0 1 4 4)] = [(2 0) (2 1) (0 4) (1 4)]. Because the square mapping (the first two columns) are the same, the pergen is the same. Because the other columns are different, the higher primes are mapped differently. 5/4 = M3 and 7/4 = M3 + vd5 = vm7, and 4:5:6:7 = C E G Bbv = C,v7.<br />
<br />
MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as any others that could generate the scale. Preference is given to the pergen that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. For example, 3L2s (anti-pentatonic) has a generator in the 400-480¢ range, suggesting both P11/4 and P12/4. But the former, with a just 11th, makes L = 351¢ and s = 73.5¢, and L/s = 4.76, quite large. The latter with a just 12th makes L = 249¢, s = 226.5¢, and L/s = 1.10, much better.<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="3">Tetratonic MOS scales<br />
</th>
<th>secondary examples<br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 3s<br />
</td>
<td style="text-align: center;">(P8, P4/2) [4]<br />
</td>
<td style="text-align: center;">half-4th tetratonic<br />
</td>
<td style="text-align: left;">third-4th, third-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 2s<br />
</td>
<td style="text-align: center;">(P8/2, P5) [4]<br />
</td>
<td style="text-align: center;">half-8ve tetratonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 1s<br />
</td>
<td style="text-align: center;">(P8, P5/2) [4]<br />
</td>
<td style="text-align: center;">half-5th tetratonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Pentatonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 4s<br />
</td>
<td style="text-align: center;">(P8, P5/3) [5]<br />
</td>
<td style="text-align: center;">third-5th pentatonic<br />
</td>
<td style="text-align: left;">third-4th, quarter-4th, quarter-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 3s<br />
</td>
<td style="text-align: center;">(P8, P5) [5]<br />
</td>
<td style="text-align: center;">unsplit pentatonic<br />
</td>
<td style="text-align: left;">third-11th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 2s<br />
</td>
<td style="text-align: center;">(P8, P12/5) [5]<br />
</td>
<td style="text-align: center;">quarter-12th pentatonic<br />
</td>
<td style="text-align: left;">quarter-11th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 1s<br />
</td>
<td style="text-align: center;">(P8, P4/2) [5]<br />
</td>
<td style="text-align: center;">half-4th pentatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Hexatonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 5s<br />
</td>
<td style="text-align: center;">(P8, P4/3) [6]<br />
</td>
<td style="text-align: center;">third-4th hexatonic<br />
</td>
<td style="text-align: left;">quarter-4th, quarter-5th, fifth-4th, fifth-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 4s<br />
</td>
<td style="text-align: center;">(P8/2, P5) [6]<br />
</td>
<td style="text-align: center;">half-8ve hexatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 3s<br />
</td>
<td style="text-align: center;">(P8/3, P5) [6]<br />
</td>
<td style="text-align: center;">third-8ve hexatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 2s<br />
</td>
<td style="text-align: center;">(P8/2, P4/2) [6]<br />
</td>
<td style="text-align: center;">half-everything hexatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">5L 1s<br />
</td>
<td style="text-align: center;">(P8, P5/3) [6]<br />
</td>
<td style="text-align: center;">third-5th hexatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Heptatonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 6s<br />
</td>
<td style="text-align: center;">(P8, P4/3) [7]<br />
</td>
<td style="text-align: center;">third-4th heptatonic<br />
</td>
<td style="text-align: left;">quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 5s<br />
</td>
<td style="text-align: center;">(P8, P11/3) [7]<br />
</td>
<td style="text-align: center;">third-11th heptatonic<br />
</td>
<td style="text-align: left;">fifth-WW4th, sixth-WW5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 4s<br />
</td>
<td style="text-align: center;">(P8, P5/2) [7]<br />
</td>
<td style="text-align: center;">half-5th heptatonic<br />
</td>
<td style="text-align: left;">fifth-12th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 3s<br />
</td>
<td style="text-align: center;">(P8, P11/5) [7]<br />
</td>
<td style="text-align: center;">fifth-11th heptatonic<br />
</td>
<td style="text-align: left;">sixth-12th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5L 2s<br />
</td>
<td style="text-align: center;">(P8, P5) [7]<br />
</td>
<td style="text-align: center;">unsplit heptatonic<br />
</td>
<td style="text-align: left;">sixth-WW4th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6L 1s<br />
</td>
<td style="text-align: center;">(P8, P5/4) [7]<br />
</td>
<td style="text-align: center;">quarter-5th heptatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Octotonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 7s<br />
</td>
<td style="text-align: center;">(P8, P4/4) [8]<br />
</td>
<td style="text-align: center;">quarter-4th octotonic<br />
</td>
<td style="text-align: left;">fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 6s<br />
</td>
<td style="text-align: center;">(P8/2, P5) [8]<br />
</td>
<td style="text-align: center;">half-8ve octotonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 5s<br />
</td>
<td style="text-align: center;">(P8, P11/4) [8]<br />
</td>
<td style="text-align: center;">quarter-11th octotonic<br />
</td>
<td style="text-align: left;">seventh-WW4th, seventh-WW5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 4s<br />
</td>
<td style="text-align: center;">(P8/4, P5) [8]<br />
</td>
<td style="text-align: center;">quarter-8ve octotonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">5L 3s<br />
</td>
<td style="text-align: center;">(P8, P12/4) [8]<br />
</td>
<td style="text-align: center;">quarter-12th octotonic<br />
</td>
<td style="text-align: left;">(very lopsided, unless 5th is quite flat)<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6L 2s<br />
</td>
<td style="text-align: center;">(P8/2, P4/3) [8]<br />
</td>
<td style="text-align: center;">half-8ve third-4th octotonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">7L 1s<br />
</td>
<td style="text-align: center;">(P8, P4/3) [8]<br />
</td>
<td style="text-align: center;">third-4th octotonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Nonatonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 8s<br />
</td>
<td style="text-align: center;">(P8, P4/4) [9]<br />
</td>
<td style="text-align: center;">quarter-4th nonatonic<br />
</td>
<td style="text-align: left;">fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 7s<br />
</td>
<td style="text-align: center;">(P8, W<span style="vertical-align: super;">3</span>P5/8) [9]<br />
</td>
<td style="text-align: center;">eighth-W<span style="vertical-align: super;">3</span>5th nonatonic<br />
</td>
<td style="text-align: left;">third-11th, fifth-WW4th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 6s<br />
</td>
<td style="text-align: center;">(P8/3, P5) [9]<br />
</td>
<td style="text-align: center;">third-8ve nonatonic<br />
</td>
<td style="text-align: left;">third-8ve half-5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 5s<br />
</td>
<td style="text-align: center;">(P8, P12/7) [9]<br />
</td>
<td style="text-align: center;">seventh-12th nonatonic<br />
</td>
<td style="text-align: left;">sixth-11th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">5L 4s<br />
</td>
<td style="text-align: center;">(P8, P4/2) [9]<br />
</td>
<td style="text-align: center;">half-4th nonatonic<br />
</td>
<td style="text-align: left;">(lopsided unless 4th is sharp), seventh-11th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6L 3s<br />
</td>
<td style="text-align: center;">(P8/3, P4/2) [9]<br />
</td>
<td style="text-align: center;">third-8ve half-4th nonatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">7L 2s<br />
</td>
<td style="text-align: center;">(P8, WWP5/6)[9]<br />
</td>
<td style="text-align: center;">sixth-WW5th nonatonic<br />
</td>
<td style="text-align: left;">(lopsided unless 5th is sharp)<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8L 1s<br />
</td>
<td style="text-align: center;">(P8, P5/5) [9]<br />
</td>
<td style="text-align: center;">fifth-5th nonatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<th colspan="3">Decatonic MOS scales<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td style="text-align: center;">1L 9s<br />
</td>
<td style="text-align: center;">(P8, P5/6) [10]<br />
</td>
<td style="text-align: center;">sixth-5th decatonic<br />
</td>
<td style="text-align: left;">fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">2L 8s<br />
</td>
<td style="text-align: center;">(P8/2, P5) [10]<br />
</td>
<td style="text-align: center;">half-8ve decatonic<br />
</td>
<td style="text-align: left;">half-8ve quartertone, half-8ve third-11th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">3L 7s<br />
</td>
<td style="text-align: center;">(P8, P12/5) [10]<br />
</td>
<td style="text-align: center;">fifth-12th decatonic<br />
</td>
<td style="text-align: left;">eighth-WW4th, eighth-WW5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">4L 6s<br />
</td>
<td style="text-align: center;">(P8/2, P4/2) [10]<br />
</td>
<td style="text-align: center;">half-everything decatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">5L 5s<br />
</td>
<td style="text-align: center;">(P8/2, P5) [10]<br />
</td>
<td style="text-align: center;">half-8ve decatonic<br />
</td>
<td style="text-align: left;">(lopsided unless 5th is quite flat)<br />
</td>
</tr>
<tr>
<td style="text-align: center;">6L 4s<br />
</td>
<td style="text-align: center;">(P8/2, P5/3) [10]<br />
</td>
<td style="text-align: center;">half-8ve third-5th decatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;">7L 3s<br />
</td>
<td style="text-align: center;">(P8, P5/2) [10]<br />
</td>
<td style="text-align: center;">half-5th decatonic<br />
</td>
<td style="text-align: left;">ninth-WW5th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">8L 2s<br />
</td>
<td style="text-align: center;">(P8/2, P4/4) [10]<br />
</td>
<td style="text-align: center;">half-8ve quarter-4th decatonic<br />
</td>
<td style="text-align: left;">half-8ve quarter-12th<br />
</td>
</tr>
<tr>
<td style="text-align: center;">9L 1s<br />
</td>
<td style="text-align: center;">(P8, P4/2) [10]<br />
</td>
<td style="text-align: center;">quarter-4th decatonic<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
</table>
<br />
The tetratonic MOS scales don't include quarter-split pergens, because a tetratonic genchain has only 3 steps, and can only divide a multigen into thirds. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<br />
<br />
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:85:<h2> --><h2 id="toc22"><a name="x157¢, ^-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:85 -->Combining pergens</h2>
<br />
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br />
<br />
General rules for combining pergens:<br />
<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><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></ul><br />
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:87:<h2> --><h2 id="toc23"><a name="x157¢, ^-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:87 -->Pergens and EDOs</h2>
<br />
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br />
<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.<br />
<br />
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinite possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.<br />
<br />
How many edos support a given pergen? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br />
<br />
Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br />
<br />
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br />
<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, 13, 13b, 14*, 15*, 16, 17, 18, 18b*, 19, 20*,<br />
21*, 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-8ve<br />
</td>
<td style="text-align: center;">12, 14, 16, 18, 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-4th<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-5th<br />
</td>
<td style="text-align: center;">13, 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-8ve<br />
</td>
<td style="text-align: center;">12, 15, 18, 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-4th<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-5th<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;">13, 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-8ve<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-4th<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-5th<br />
</td>
<td style="text-align: center;">13, 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-11th<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-12th<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-8ve, half-4th<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-8ve, quarter-tone<br />
</td>
<td style="text-align: center;">18, 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-8ve, quarter-4th<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-8ve, quarter-5th<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-8ve, third-4th<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-8ve, third-5th<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-8ve, third-11th<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-8ve, quarter-4th<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-8ve, quarter-5th<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-8ve, quarter-11th<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-8ve, quarter-12th<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 />
<!-- ws:start:WikiTextHeadingRule:89:<h2> --><h2 id="toc24"><a name="x157¢, ^-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:89 -->Supplemental materials</h2>
<br />
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br />
<!-- ws:start:WikiTextUrlRule:3979:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:3979 --><br />
<br />
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />
<!-- ws:start:WikiTextUrlRule:3980:http://www.tallkite.com/misc_files/alt-pergensLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><!-- ws:end:WikiTextUrlRule:3980 --><br />
<br />
Screenshot of the first 38 pergens:<br />
<!-- ws:start:WikiTextLocalImageRule:2251:<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="" title="" style="height: 460px; width: 704px;" /> --><img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><!-- ws:end:WikiTextLocalImageRule:2251 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:91:<h2> --><h2 id="toc25"><a name="x157¢, ^-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:91 -->Misc notes</h2>
<br />
Given:<br />
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x > 0, z ≠ 0, and |i| <= x<br />
<br />
To prove: if |z| = 1, n = 1<br />
If z = 1, let i = y - x, and the pergen = (P8/x, P5)<br />
If z = -1, let i = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)<br />
Therefore if |z| = 1, n = 1<br />
<br />
To prove: n is always a multiple of b, and n = |b| if and only if n = 1<br />
b = x/k and n = xz/k, where k = sign (z) · GCD (iz-y, x)<br />
The GCD is defined here as always positive: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3<br />
n = zb, and since n > 0, n = |z|·|b|<br />
if n = |b|, then |z| = n/|b| = 1, and from the earlier proof, n = 1<br />
if n = 1, |z|·|b| = 1, therefore |b| = 1, and n = |b|<br />
Therefore multigens like M9/3 or M3/5 never occur<br />
Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)<br />
<br />
To prove: test for explicitly false<br />
If m = |b|, is the pergen explicitly false?<br />
Does (a,b)/n split P8 into |b| periods?<br />
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br />
Because n is a multiple of b, n/b is an integer<br />
M/b = (n/b)·M/n = (n/b)·G<br />
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br />
Let c and d be the bezout pair of a+b and b such that c·(a+b) + d·b = 1<br />
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore c ≠ 0<br />
c·(a+b)·P8 = c·b·((n/b)·G - P5)<br />
(1 - d·b)·P8 = c·b·((n/b)·G - P5)<br />
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br />
P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br />
Therefore P8 is split into |b| periods<br />
Therefore if m = |b|, the pergen is explicitly false<br />
<br />
To prove: true/false test<br />
If GCD (m,n) = |b|, is the pergen a false double?<br />
If m = |b|, the pergen is explicitly false<br />
Therefore assume m > |b| and unreduce<br />
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn)<br />
Simplify by dividing by b to get (P8/m, (n/b - a(m/b), -m) / m(n/b)) = (P8/m, (a',b')/n')<br />
b' = -m, therefore m = |b'|, and the unreduced pergen is explicitly false<br />
Therefore the original pergen is a false double<br />
<br />
To prove: alternate true/false test<br />
if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?<br />
If m > |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?<br />
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br />
Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br />
Can b' be reduced by simplifying further?<br />
No, because GCD (a', b', n') = GCD (n/b - am/b, -m, mn/b)<br />
GCD (b', n') = m<br />
GCD (n/b, m) = 1<br />
GCD (<br />
|b'| = m, so the unreduced pergen is explicitly false, and the test works<br />
<br />
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br />
<br />
<br />
<br />
<br />
<br />
<u><strong>Extra stuff, not sure if it should be included:</strong></u><br />
<br />
Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. 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 />
The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</body></html>