Kite's thoughts on pergens: Difference between revisions

A '''pergen''' (pronounced "peer-gen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|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. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.  

{{See also| Rank-2 temperaments by mapping of 3 }}

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.

For example, the Srutal temperament (2.3.5 and 2048/2025) splits the octave in two, and its spoken pergen name is half-octave. The pergen is written (P8/2, P5). Not only the temperament, but also the comma is said to split the octave. The Dicot aka Yoyo temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine aka Triyo is third-fourth, or perhaps third-of-a-fourth, (P8, P4/3). Semaphore aka Zozo, a pun on "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 a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both Srutal aka Sagugu and Injera aka Gu & Biruyo sound the same, although they temper out different commas. In addition, the pergen tells us how to notate the temperament using '''ups and downs''' (^ and v). See the notation guide below, under [[pergen#Further Discussion-Supplemental materials|Supplemental materials]]. Ups and downs are also used in [[Ups and downs notation|EDO notation]] to represent one edostep. Although the symbol is the same, the meaning is different.

The largest category contains all single-comma rank-2 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'''.

 

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).

 

The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: bi- splits something into two parts, tri- into three parts, etc. For a comma with monzo (a,b,c,d...), the '''color depth''' is GCD (c,d...).

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.

For example, Marvel aka Ruyoyo (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). But colors can be replaced with ups and downs, to be higher-prime-agnostic. Here, gen2 can be reduced to g1 = 81/80. Since 81/80 maps to a perfect unison, it can be notated by an up symbol, and we have (P8, P5, ^1) = rank-3 unsplit.

More examples: Trizogu (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Biruyo (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, Biruyo Nowa (2.5.7 and 50/49) is (P8/2, M3) = half-8ve, major 3rd.

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

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

=Derivation=

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &gt; m, it will split some 3-limit interval into n parts.

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

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

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

 

 

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| &lt;= x</span>'''

 

 

 

 

 

 

 

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

|-

|-

! | 2/1

| | -2

! | 3/1

|-

! | 5/1

| | /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 = (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 <u>double</u> octave to the multigen. The alternate gens are P11/2 and P19/2, 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 to it. Subtracting a quadruple octave and inverting makes gen2 be (5,1,-2)/4 = (96:25)/4. The multigen is a diminished double octave. A quadruple half-5th is a double 5th is a M9. Subtracting that makes gen2 be (128:75)/4, quarter-dim-7th. Subtracting M9 again, and inverting again, makes gen2 = (-9,3,2)/4 = (675:512)/4, quarter-aug-3rd. As gen2's cents become smaller, the odd limit becomes greater, the intervals remain obscure, and the notation remains awkward.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

 

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

|-

! | 2/1

! | 3/1

! | 7/1

! | gen2

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

 

 

=Applications=

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

Another obvious application is to categorize regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), Semihemi is (P8/2, P4/2), Triforce is (P8/3, P4/2), both Tetracot and Semihemififths are (P8, P5/4), Fourfives is (P8/4, P5/5), Pental is (P8/5, P5), and Fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, either because it contains more than 2 primes, or because the split multigen isn't actually a generator. For example, Sensei, or Semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen (P8, (5/4)/2), only 2 primes, but the tone isn't a generator.

Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example Porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first Porcupine is Triyo, and the second one is Triyo & Ru. 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.

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

All unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. All other rank-2 temperaments require an additional pair of accidentals, [[Ups and downs notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, '''lifts and drops''', written / and \. v\D is down-drop D, and /5 is a lift-fifth. Alternatively, color accidentals (y, g, r, z, 1o, 1u, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both Mohajira aka Lulu and Dicot aka Yoyo are (P8, P5/2). Using y and g implies Dicot, using 1o and 1u 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 aka Layo is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled not as stacked thirds but as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C vE G. See [[pergen#Further Discussion-Notating unsplit pergens|Notating unsplit pergens]] below.

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 multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further Discussion-Supplemental materials-Notaion guide PDF|notation guide]]. It even allows every pergen to be numbered.

 

The '''[[enharmonic unison]]''', or more briefly the '''EU''', 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 EU. The pergen and the EU together define the notation. (''Edited to add: not quite accurate, see the Addenda.'')

 

 

 

 

&lt; 700¢)

| | ^^C = Db

| | P8/2 = ^A4 = vd5

| | C - ^F#=vGb - C

|-

| | ^^C = D

| | P8/2 = ^4 = v5

| | C - ^F=vG - C

| | Thotho, if 13/8 = M6

|-

| | (P8, P4/2)

 

 

| | (P8, P5/2)

 

 

| | (P8/2, P4/2)

 

everything

| | \\m2,

 

 

vv\\M2

| | //C = Db

 

^^C = C#

 

^^//C = D

| | P4/2 = /M2 = \m3

 

 

= ^/4 = v\5

| | C - /D=\Eb - F,

 

C - ^Eb=vE - G,

 

C - v/F#=^\Gb - C,

 

C - ^/F=v\G - C

| | Zozo &amp; Lulu

 

 

/1 = 64/63

| |  

| | "

| | ^^d2,

 

 

vv\\A1

| | ^^ C= B#

 

//C = Db

 

^^//C = C#

| | P8/2 = vA4 = ^d5

 

 

P5/2 = ^/m3 = v\M3

| | C - vF#=^Gb - C,

 

C - /D=\Eb - F,

 

C - ^/Eb=v\E - G

| | Sagugu &amp; Zozo

 

 

/1 = 64/63

| |  

| | "

| | ^^d2,

 

 

^^\\m2

| | ^^C = B#

 

//C = C#

 

^^\\C = B

| | P8/2 = vA4 = ^d5

 

 

P4/2 =v/M2 = ^\m3

| | C - vF#=^Gb - C,

 

C - /Eb=\E - G,

 

C - v/D=^\Eb - F

| | Sagugu & Lulu

 

 

/1 = 33/32

|-

! |  

! | third-splits

! |  

! |  

! |  

! |  

! |  

|-

| | 6

| | (P8/3, P5)

 

third-8ve

| | ^<span style="vertical-align: super;">3</span>d2

| | ^<span style="vertical-align: super;">3</span>C = B#

| | P8/3 = vM3 = ^^d4

| | C - vE - ^Ab - C

| | Augmented aka Trigu

 

| | 7

| | (P8, P4/3)

 

third-4th

| | v<span style="vertical-align: super;">3</span>A1

| | ^<span style="vertical-align: super;">3</span>C = C#

| | P4/3 = vM2 = ^^m2

| | C - vD - ^Eb - F

| | Porcupine aka Triyo

 

^1 = 81/80

| | (P8, P5/3)

 

third-5th

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

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

| | P5/3 = ^M2 = vvm3

| | C - ^D - vF - G

| | Slendric aka Latrizo

 

^1 = 64/63

|-

| | 9

| | (P8, P11/3)

 

third-11th

| | ^<span style="vertical-align: super;">3</span>dd2

| | ^<span style="vertical-align: super;">3</span>C = B##

| | P11/3 = vA4 = ^^dd5

| | C - vF# - ^Cb - F

| | Satrilu, if 11/8 = A4

 

^1 = 729/704

| |  

| | "

| | v<span style="vertical-align: super;">3</span>M2

| | ^<span style="vertical-align: super;">3</span>C = D

| | P11/3 = ^4 = vv5

| | C - ^F - vC - F

| | Satrilu, if 11/8 = P4

 

| | (P8/3, P4/2)

 

third-8ve, half-4th

| | v<span style="vertical-align: super;">6</span>A2

| | ^<span style="vertical-align: super;">6</span>C = 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 - vvA - C

 

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

| | Tribilo, if 11/8 = P4

 

^1 = 33/32

|-

| | "

| | ^<span style="vertical-align: super;">3</span>d2,

 

| | ^<span style="vertical-align: super;">3</span>C = B#

 

//C = Db

| | P8/3 = vM3 = ^^d4

 

P4/2 = /M2 = \m3

| | C - vE - ^Ab - C

 

C - /D=\Eb - F

| | Triforce aka Trigu & Zozo

 

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

|-

| | 11

| | (P8/3, P5/2)

 

third-8ve, half-5th

| | ^<span style="vertical-align: super;">3</span>d2

 

| | ^<span style="vertical-align: super;">3</span>C = B#

 

| | P8/3 = vM3 = ^^d4

 

P5/2 = /m3 = \M3

| | C - vE - ^Ab - C

 

C - /Eb=\E - G

 

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

|-

| | (P8/2, P4/3)

 

| | ^^d2

 

\<span style="vertical-align: super;">3</span>A1

| | ^^C = Dbb

 

/<span style="vertical-align: super;">3</span>C = C#

| | P8/2 = vA4 = ^d5

 

P4/3 = \M2 = //m2

| | C - vF#=^Gb - C

 

| | Latribiru

 

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

|-

| | 13

| | (P8/2, P5/3)

 

half-8ve, third-5th

| | ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">3</span>2

| | ^<span style="vertical-align: super;">6</span>C = 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 - v<span style="vertical-align: super;">3</span>F<span style="vertical-align: super;">x</span>=^<span style="vertical-align: super;">3</span>Gbb C

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Removing the ups and downs from an EU makes an '''uninflected''' EU, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s EU is ^^d2, the uninflected EU 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 EU vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore <u>'''up may need to be swapped with down, depending on the size of the 5th'''</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' EUs are upped or downed as if the 5th were just.