Kite's thoughts on pergens: Difference between revisions

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. Both fractions are 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 '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

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 &gt; 3 (a '''higher prime'''), e.g. 81/80 or 135/128. 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 multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

{| class="wikitable" style="text-align:center;"   

| | unsplit

| | 81/80

| | gT

|-

| | "

| | 64/63

| | rT

|-

| | "

| | (-14,8,1)

| | LyT

|-

| | half-8ve

| | (11, -4, -2)

| | sggT

|-

| | "

| | 81/80, 50/49

| | g&rryyT

|-

| | half-5th

| | 25/24

| | yyT

|-

| | "

| | (-1,5,0,0,-2)

| | 1uuT

|-

| | half-4th

| | 49/48

| | zzT

|-

| | half-everything

| | 25/24, 49/48

| | yy&amp;zzT

|-

| | third-4th

| | 250/243

| | y<span style="vertical-align: super;">3</span>T

|-

| | third-11th

| | (12,-1,0,0,-3)

| | s1u<span style="vertical-align: super;">3</span>T

|-

| | quarter-8ve

| | (3,4,-4)

| | g<span style="vertical-align: super;">4</span>T

|-

| | half-8ve quarter-tone

| | (-17,2,0,0,4)

| | L1o<span style="vertical-align: super;">4</span>T

|-

| | fifth-12th

| | (-10,-1,5)

| | Ly<span style="vertical-align: super;">5</span>T

|}

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[Kite's_color_notation|Color notation]] (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yo, (a,b,-1) = gu, (a,b,0,1) = zo, (a,b,0,-1) = ru, (a,b,0,0,1) = lo/ilo, (a,b,0,0,-1) = lu, and (a,b) = wa. Examples: 5/4 = y3 = yo 3rd, 7/5 = zg5 = zogu 5th, etc.

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

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

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 Discussion-Chord names and scale names|Chord names and scale names]] below.

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

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

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.

| | none

| | C - G

|-

! |  

| | P8/2 = vA4 = ^d5

| | C - F#v=Gb^ - C

^1 = 81/80

|-

| | P8/2 = ^A4 = vd5

| | C - F#^=Gbv - C

^1 = 64/63

| | P8/2 = ^4 = vP5

| | C - F^=Gv - C

^1 = 27/26

| | P4/2 = ^M2 = vm3

| | C - D^=Ebv - F

^1 = 64/63

| | P4/2 = vA2 = ^d3

| | C - D#v=Ebb^ - F

^1 = 81/80

| | P5/2 = ^m3 = vM3

| | C - Eb^=Ev - G

^1 = 33/32

C - F^/=Gv\ - C

^1 = 33/32

C - Eb^/=Ev\ - G

^1 = 81/80

C - Dv/=Eb^\ - F

^1 = 81/80

| | P8/3 = vM3 = ^^d4

| | C - Ev - Ab^ - C

^1 = 81/80

| | P4/3 = vM2 = ^^m2

| | C - Dv - Eb^ - F

^1 = 81/80

| | P5/3 = ^M2 = vvm3

| | C - D^ - Fv - G

^1 = 64/63

| | P11/3 = vA4 = ^^dd5

| | C - F#v - Cb^ - F

^1 = 729/704

| | P11/3 = ^4 = vv5

| | C - F^ - Cv - F

^1 = 33/32

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

^1 = 33/32

C - D/=Eb\ - F

^1 = 81/80, /1 = 64/63

C - Eb/=E\ - G

^1 = 49/48, /1 = 343/324

C - D\ - Eb/ - F

^1 = 1029/1024, /1 = 49/48

C - D#vv - Fb^^ - G

^1 = 81/80

C - D/ - F\ - G

^1 = (10,-6,1,-1), /1 = 64/63

C - F^^ - Cvv - F

^1 = 33/32

C - Dv/ - F^\ - G

^1 = 64/63

C - Dv\ - Eb^/ - F

^1 = 81/80

C - Ev/ - Ab^\ - C

^1 = 81/80

| | P8/4 = vm3 = ^<span style="vertical-align: super;">3</span>A2

| | C Ebv Gbvv=F#^^ A^ C

|-

| | 17

| | P4/4 = ^m2 = v<span style="vertical-align: super;">3</span>AA1

| | C Db^ Ebb^^=D#vv Ev F

|-

| | 18

| | P5/4 = vM2 = ^<span style="vertical-align: super;">3</span>m2

| | C Dv Evv=Eb^^ F^ G

|-

| | 19

| | P11/4 = ^M3 = v<span style="vertical-align: super;">3</span>dd5

| | C E^ G#^^ Dbv F

|-

| | 20

| | P12/4 = v4 = ^<span style="vertical-align: super;">3</span>M3

| | C Fv Bbvv=A^^ D^ G

|-

| |  

* 21-edo: ^ = 57¢ (if used, # = 0¢)

* 22-edo: # = 164¢, ^ = 55¢ (^ = 1/3 #)

==Finding a notation for a pergen==

It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

<span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4

For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-2,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3d<span style="vertical-align: super;">3</span>2 = 3·M2 + 1·m2 = P5. (Diminish three A2's once and augment one d<span style="vertical-align: super;">3</span>2 three times.) Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. Because d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, ^ = (-d<span style="vertical-align: super;">3</span>2) / 3 = 67¢ + 8.67·c, about a third-tone.

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.

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 three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.

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 issue occurs in 24-edo.

Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [5] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, e.g. half-octave pergens tend to have scales with an even number of notes.

Lifts and drops, like ups and downs, precede the note head and any sharps or flats. Scores for melody instruments could instead have them above or below the staff. This score uses ups and downs, and has chord names. It's for the third-4th pergen, with E = v³A1. As noted above, it can be played in EDOs 15, 22 or 29 as is, because ^1 maps to 1 edostep. But to be played in EDOs 30, 37 or 44, ^1 must be interpreted as two edosteps.

==Tipping points and sweet spots==

The tipping point depends on the choice of enharmonic. It's not the temperament that tips, it's the notation. Half-8ve could be notated with an E of vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. For single-comma temperaments, E usually equals the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.

Double-pair notation has two enharmonics, and two tipping points to be avoided. (P8/2, P4/2) has three possible notations. The two enharmonics can be either A1, m2 or d2. One can choose to use whichever two enharmonics best avoid tipping.

==Notating unsplit pergens==

! | cents

|-

| | 81/80 = P1

| | c = -3¢ to -5¢

| | ---

|-

| | 135/128 = A1

| | c = -21¢ to -22¢

| | -100¢ - 7c = 47¢-54¢

|-

| | (-15,11,-1) = A1

| | c = -10¢ to -12¢

| | 100¢ + 7c = 26¢-30¢

|-

| | (-15,8,1) = -d2

| | c = 1.7¢ to 2.0¢

| | 12c = 20¢-24¢

|-

| | (-23,16,-1) = -d2

| | c = -0.9¢ to -1.2¢

| | -12c = 10¢-15¢

|-

| | 16/15 = m2

| | c = 56¢ to 58¢

| | -100¢ + 5c = 180-190¢

|-

| | (12,-9,1) = m2

| | c = 9¢ to 10¢

| | 100¢ - 5c = 50-55¢

|}

==Notating rank-3 pergens==

| | ---

|-

| | (P8, P5)

| | rank-2

| | ---

|-

| | (P8/2, P5)

| | rank-2

| | E = ^^d2

|-

| | (P8, P4/2)

| | rank-2

| | E = vvm2

|-

| | (P8/2, P4/2)

| | rank-2

| | ---

|-

| | (P8, P5, ^1)

| | rank-3

| | ---

|-

| | (P8, P5/2, ^1)

| | rank-3

! | enharmonic

|-

| | 225/224

| | (P8, P5, ^1)

| | ^^\d2

|-

| | 50/49

| | (P8/2, P5, ^1)

| | ^^\\d2

|-

| | 1029/1000

| | (P8, P11/3, ^1)

| | ^^^\\\dd3

|-

| | 2401/2400

| | (P8, P5/2, ^1)

| | ^^\<span style="vertical-align: super;">4</span>dd3

|-

| | 686/675

| | (P8, P5, vm3/2)

| | ^^\\\dd3

|}

With this standardization, the enharmonic can be derived directly from the comma. The vanishing comma is written as some 3-limit comma plus some number of mapping commas. For example, 50/49 = (81/80)<span style="vertical-align: super;">-2</span> · (64/63)<span style="vertical-align: super;">2</span> · (-19,12). This can be rewritten as vv1 + //1 - d2 = vv//-d2 = -^^\\d2. The comma is negative (i.e.descending), but the enharmonic never is, therefore E = ^^\\d2. The period is found by adding/subtracting E from the 8ve and dividing by two, thus P = v/A4 = ^\d5. Both E and P become more complex, but the ratio for /1 becomes simpler.

Unlike the previous examples, Demeter's gen2 can't be expressed as a mapping comma. It divides 5/4 into three 15/14 generators, and 7/6 into two generators. Its pergen is (P8, P5, vm3/2). It could also be called (P8, P5, vM3/3), but the pergen with a smaller fraction is preferred. Because the 8ve and 5th are unsplit, single-pair notation is possible, with gen2 = ^m2 and no E. But the 4:5:6:7 chord would be spelled C -- Fbbb^^^ -- G -- Bbb^^, very awkward! Standard double-pair notation is better. Gen2 = v/A1, E = ^^\\\dd3, and C^^\\\ = A##. Genchain2 is C -- C#v/ -- Eb\ -- Ev -- Gb\\ -- Gv\ -- G#vv=Bbb\\\ -- Bbv\\... Unlike other genchains we've seen, the additional accidentals get progressively more complex. Whenever an accidental has its own enharmonic, with no other accidentals in it, it always adds up to something simpler eventually. If it doesn't have its own enharmonic, it's infinitely stackable. A case can be made for a convention that colors are used only for infinitely stackable accidentals, and ups/downs/lifts/drops only for the other kind of accidentals.

If ^1 = 81/80, possible half-split gen2's are vM3/2, vM6/2, and their octave inverses ^m6/2 and ^m3/2. Possible third-split gen2's are vM3/3, vM6/3, vM2/3, and their inverses, plus vM9/3, ^m10/3 and vM10/3. If ^1 = 64/63, possible third-splits are ^M2/3, vm3/3, ^M3/3, vm6/3, ^M6/3, vm7/3, ^M9/3, vm10/3 and ^M10/3. Analogous to rank-2 pergens with imperfect multigens, there will be occasional double-up or double-down multigen2's.

! | ^1 ratio

|-

| | 2.3.7

| | (-14,0,0,5)

| | 49/48

|-

| | 2.3.7

| | (22,-5,0,-5)

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

Conventional notation assumes the 2.3 prime subgroup. Non-8ve and no-5ths pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. A violinist or vocalist will immediately have a rough idea of the pitch. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.

For example, for 7-edo and 17-edo, the edomappings are (7, 4, 2) and (17, 10, 5). Their cross product is (0, -1, 2). This comma equates two generators with a 5th. The pergen is obviously (P8, P5/2).

If the two edos have the same 5th, such as 12edo and 24edo do, the 5th is some multiple of the period, and the pergen is a multi-EDO pergen.

<u>'''Credits'''</u>