Monthly Tunings
edited
... August 2017: 20edo
Underwater Spontaniety (lo-fi ambient) by Stephen Weigel
Pulling Weeds i…
...
August 2017: 20edo
Underwater Spontaniety (lo-fi ambient) by Stephen Weigel Pulling Weeds in the Dark Ages (xen-pop) by Stephen Weigel
God's Favorite Tuning (xen-pop) by Stephen Weigel
You Lied, and I'm not Mentioning your Name (xen-pop) by Stephen Weigel
...
Diablo by Kewti
Ten-Armed Hunter by Herman Miller
March 2018: Monarda
*Monarda is a 12-note subset of "Tannic" temperament, discovered by Scott Dakota, that is rank-three and observes the 17-limit. 17-limit JI is 7-dimensional, and the commas tempered out are: 273/272, 561/560, 441/440, and 225/224. 7 dimensions - 4 commas = the 3 dimensional tuning. To find temperaments, it's recommended to use the Regular Temperament Finder, instead of the wiki. Check out the FB Monthly Tunings page for specifics.
Cam Taylor's Monarda Impressions
Tannic Acid (comparison), Chris Vaisvil
Purity in Just Intonation often has a connection with small numbered ratios low in the harmonic series. So 3/2 (the fifth), 4/3 (the fourth), 5/3 (the sixth), and 5/4 (the third) are among the most consonant and relaxed sounding in music and these form the basis of the diatonic scale. And by "Just ratios" I simply mean "using numbers low in the harmonic series" or simply "using low numbered fractions".
Now how do we make an entire scale keep as many notes as possible at Just ratios from each other? One way is to try and find numbers with as many low common factors as possible.
...
of a GreatestLeast Common Multiple)?
Here are some numbers with a lot of factors:
For the GCMLCM of 12
For 18 the factors include 2, 3, 6 and 9. Again, all these factors are multiples of 2 or 3.
For 24 the factors include 2, 3, 4, 6, 8 and 12. Yet again, all these factors are multiples of 2 or 3 (go figure as well, the fifth in modern music theory is 3/2).
Purity in Just Intonation often has a connection with small numbered ratios low in the harmonic series. So 3/2 (the fifth), 4/3 (the fourth), 5/3 (the sixth), and 5/4 (the third) are among the most consonant and relaxed sounding in music and these form the basis of the diatonic scale. And by "Just ratios" I simply mean "using numbers low in the harmonic series" or simply "using low numbered fractions".
Now how do we make an entire scale keep as many notes as possible at Just ratios from each other? One way is to try and find numbers with as many low common factors as possible.
...
a common factor?factor (AKA as part of a Greatest Common Multiple)? For example
Here are some numbers with a lot of factors:
For the GCM of 12 the
For 18 the factors include 2, 3, 6 and 9. Again, all these factors are multiples of 2 or 3.
For 24 the factors include 2, 3, 4, 6, 8 and 12. Yet again, all these factors are multiples of 2 or 3 (go figure as well, the fifth in modern music theory is 3/2).
Links
edited
... Dive in and ask questions!
Facebook groups:
... you've composed, etc. or whatever... any…
...
Dive in and ask questions!
Facebook groups:
...
you've composed, etc.or whatever... anything to let us know you're a real person!
The Xenharmonic Alliance II (the most active group and a good one for beginners)
https://www.facebook.com/groups/xenharmonic2/
...
https://www.facebook.com/groups/xenharmonicmath/
"This group is a sister of the main Xenharmonic Alliance group. We created this to have a place to talk about mathematical music theory without bombarding the main group with our (often esoteric) conversation. Unlike the main group, this group is smaller and much more focused on specifically the mathematics of music theory..."
...
and Theory. (the largest group, note the period in the name)
https://www.facebook.com/groups/239947772713025/
"Anything to do with microtonal music and theory."
...
https://www.facebook.com/groups/497105067092502/
"A censorship-free forum for sharing and discussing anything relevant to alternative musical tuning systems and music created with them." Microtonal Projects
https://www.facebook.com/pages/Microtonal-Projects/197602703265
"Microtonal Projects is a not-for profit organisation dedicated to promoting microtonal music."
Out Of Tune Radio
https://www.facebook.com/groups/217903948329949/
"With few microtonal groups out there, this group was created in order to share microtonal music with people outside microtonal communities. Its a small step so microtonal composers/performers can get more exposure. So feel free post as much works as you want in any genre. It can be in any tuning or it can be your grandmas out of tune old piano!
RULES: NO TUNING DISCUSSIONS, NO THEORY, NO MATH, LEAVE INSULTS FOR YOUR WALLS AND PMs"
UnTwelve (Chicago area)
https://www.facebook.com/groups/untwelve/
MICROTONALITALIA (Italy)
https://www.facebook.com/groups/356483631067563/
"Uno spazio dove scambiare informazioni al riguardo dei sistemi di accordatura alternativi al consueto sistema in uso in Occidente."
Alternative Musical Notation Systems
https://www.facebook.com/groups/1545157255740770/
...
https://www.facebook.com/groups/535138486627949/
"This group is here for people to discuss all of Erv Wilson's work."
Xenwiki Work Group
https://www.facebook.com/groups/xenwiki/
"This group is for discussion of matters related to the xenharmonic wiki, http://xenharmonic.wikispaces.com/, including terminology, article content and format, wiki organization and cleanup, and open projects."
Tuning-specific
Just Intonation Network
https://www.facebook.com/groups/2204994219/
(for just intonation as opposed to temperaments or EDOs)
Harmonic Series Study Group
https://www.facebook.com/groups/harmonic.series/
"In this Group we study the Natural Harmonic Series of Ascending and Descending Harmonics (Overtones & Undertones) and their Combinations, exactly as they appear IN NUMERICAL FORM only.
Please post only ratios, whole (rational, integer) numbers, exponents and combinations of these. Thank you for understanding.
No cents, no “note” names, no tempered musical entities and sonic spaces (“intervals”), no staff notation, no 7-white/5-black piano keyboard and no other similar accidents of history and culture. And no “octave equivalence”...
Science, mysticism, quacademic fallacy, spiritual crap, accredited research, ancient mysteries, mainstream studies, forgotten/hidden knowledge, pseudo-science & numerology are all accepted for scholar purposes − as long as they do not insult intelligence and common sense in rampant ridiculous ways."
Small Equal Xenharmonic Temperaments
https://www.facebook.com/groups/small.xen.ets/
"This group is for musicians and composers working in equal temperaments other than 12-tone equal temperament, especially those with 24 notes per octave or fewer (though those working in ETs not too much larger are also welcome)."
17 Tone Theory and Pedagogy
https://www.facebook.com/groups/1802185126694280/
"A group focused on the exploration and study and documentation on a 17 Tone Equal Temperament or Well Temperament. Emphasis on studying 17 and as a result creating teaching and learning materials for others. Sharing compositions intended to expand our understanding of the tuning and documenting such research with an emphasis to build on knowledge and grow in this tuning system."
22-Tone Workgroup
https://www.facebook.com/groups/370122423186931/
...
https://www.facebook.com/groups/421996207912134/
"A working group to explore 31-tone equal temperament and related scales and temperaments. Open to all with an interest in 31 tones!" Xenrhythmic Alliance (rhythm theory, not tuning theory)
https://www.facebook.com/groups/293749277451755/
"A group dedicated to the exploration and study of complex rhythmic ideas including but not limited to fractional time signatures, rhythm progressions, polyrhythms, microrhythm, and polytempo."
Composers Capable of Complicated Rhythms (rhythm theory, not tuning theory)
https://www.facebook.com/groups/complexrhy/
"This site is about sharing your own works of advanced rhythmic structures, or someone else's work. The point is to share, as well as to appreciate rhythmic sophistication."
Xenwiki Work Group
https://www.facebook.com/groups/xenwiki/
"This group is for discussion of matters related to the xenharmonic wiki, http://xenharmonic.wikispaces.com/, including terminology, article content and format, wiki organization and cleanup, and open projects."
Harmonic Series Study Group
https://www.facebook.com/groups/harmonic.series/
"In this Group we study the Natural Harmonic Series of Ascending and Descending Harmonics (Overtones & Undertones) and their Combinations, exactly as they appear IN NUMERICAL FORM only.
Please post only ratios, whole (rational, integer) numbers, exponents and combinations of these. Thank you for understanding.
No cents, no “note” names, no tempered musical entities and sonic spaces (“intervals”), no staff notation, no 7-white/5-black piano keyboard and no other similar accidents of history and culture. And no “octave equivalence”...
Science, mysticism, quacademic fallacy, spiritual crap, accredited research, ancient mysteries, mainstream studies, forgotten/hidden knowledge, pseudo-science & numerology are all accepted for scholar purposes − as long as they do not insult intelligence and common sense in rampant ridiculous ways."
XA - Monthly tunings
https://www.facebook.com/groups/979055042181607/
"This group is meant to be a collaborative workgroup where we vote on a tuning and then work it out together. Share what you find! Harmony, chord progressions, melodies, counterpoint, etc. Soft synths, DAWs, hardware, interfaces, etc." 17 Tone Theory and Pedagogy
https://www.facebook.com/groups/1802185126694280/Instrument making
Microtonal Guitarist
https://www.facebook.com/groups/110164742335730/
"A facebook group focused on the exploration and study and documentation on a 17 Tone Equal Temperament or Well Temperament. Emphasis on studying 17 and as a result creating teaching and learning materials for others. Sharing compositions intended to expand our understanding of the tuning and documenting such research with an emphasis to build on knowledgeJI and grow in this tuning system."meta/microtonal guitarists. http://xenguitarist.com/"
Xenharmonic Musical Instrument Makers and Designers
https://www.facebook.com/groups/273086466412208
"Xenharmonic Instrument design and manufacture."
Regional
Microtonal Projects
https://www.facebook.com/pages/Microtonal-Projects/197602703265
"Microtonal Projects is a not-for profit organisation dedicated to promoting microtonal music."
Out Of Tune Radio
https://www.facebook.com/groups/217903948329949/
"With few microtonal groups out there, this group was created in order to share microtonal music with people outside microtonal communities. Its a small step so microtonal composers/performers can get more exposure. So feel free post as much works as you want in any genre. It can be in any tuning or it can be your grandmas out of tune old piano!
RULES: NO TUNING DISCUSSIONS, NO THEORY, NO MATH, LEAVE INSULTS FOR YOUR WALLS AND PMs"
UnTwelve (Chicago area)
https://www.facebook.com/groups/untwelve/
MICROTONALITALIA (Italy)
https://www.facebook.com/groups/356483631067563/
"Uno spazio dove scambiare informazioni al riguardo dei sistemi di accordatura alternativi al consueto sistema in uso in Occidente."
Xenharmonic Alliance UK:
https://www.facebook.com/groups/1798636200458697/
"Concerts and conferences about microtonal music in the UK. Meet other microtonalists."
Rhythm
Composers Capable of Complicated Rhythms (rhythm theory, not tuning theory)
https://www.facebook.com/groups/complexrhy/
"This site is about sharing your own works of advanced rhythmic structures, or someone else's work. The point is to share, as well as to appreciate rhythmic sophistication."
Xenrhythmic Alliance (rhythm theory, not tuning theory)
https://www.facebook.com/groups/293749277451755/
"A group dedicated to the exploration and study of complex rhythmic ideas including but not limited to fractional time signatures, rhythm progressions, polyrhythms, microrhythm, and polytempo.
A handy link to noteworthy facebook threads:
xenharmonic.wikispaces.com/Notable+facebook+threads
one enharmonic, they combinethe first one can be combined with the 2nd to make new enharmonics. Decimal'san equivalent 2nd enharmonic could be written as ^^\\A1, but combining two accidentals in one enharmonic is avoided.enharmonic.
A rank-2
...
useful musically.) All
All true triples
...
use colors.
Quadruple-pair notation is needed for some true triples like (P8/2, P5/2, vM3/2). A true/false test hasn't yet been found for either triple-splits, or double-splits in which multigen2 is split.
Some examples of 7-limit rank-3 temperaments:
7-limit temperament
v/A1 = 15/14
^^\\\dd3
If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.
There's a lot of options in rank-3 double-pair notation for what ratio each accidental pair represents. For example, deep reddish is half-8ve, with a d2 comma, like srutal. Using the same notation as srutal, but with ^1 = 81/80, we have P = \A4 = /d5 and E = //d2. The ratio for /1 is (-9,6,-1,1). 7/4 = 5/4 + 7/5 = vM3 + /d5 = v/m7, and the 4:5:6:7 chord is spelled C Ev G Bbv/. However, double-pair notation for 7-limit rank-3 temperaments can be standardized so that ^1 is always 81/80 and /1 is always 64/63. This ensures the 4:5:6:7 chord is always spelled C Ev G Bb\.
...
rewritten as 50/49 = vv1 +
...
comma, which is tempered to 0¢.vanishes. However, a
...
forced to be 0¢.vanish if the 3-limit comma vanishes. A rank-3
rarely tip.
If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.
Demeter
Demeter is unusual
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= A##. The 4:5:6:7 chord is spelled C Ev G Bb\. Genchain2 is
...
of accidentals.
The next table lists all rank-3 pergens up
There are always many alternate 2nd generators. Any combination of periods, 1st generators and commas can be added to half-splits. Each blockor subtracted from the 2nd generator to make alternates. If gen2 can be expressed as a mapping comma, that is much larger. The notation for 2.3.5 pergenspreferred. For demeter, any combination of vm3, double-8ves and 2.3.7double-5ths (M9's) makes an alternate multigen2. Any 3-limit interval can be added or subtracted twice, because the splitting fraction is 2. Obviously we can't choose the multigen2 with the smallest cents, because any 3-limit comma can be subtracted twice from it. Instead, once the splitting fraction is minimized, choose the multigen2 with the smallest odd limit. In case of two ratios with the same odd limit, as 5/3 and 5/4, the DOL (double odd limit) is minimized. DOL (5/3) = (5,3) and DOL (5/4) = (5,1). Since 1 < 3, 5/4 is preferred.
If ^1 = 81/80, possible half-split gens2'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.
All possible rank-3 pergens can be listed, but the table is oftenmuch longer than for rank-2 pergens. Here are all the same.half-split pergens:
unsplit 2.3.5
2.3.7^1 = 81/80
spoken name
^1 = 64/63
spoken name
1
(P8, P5, ^1)
...
same
6
(P8, P5, vM3/2)
half-downmajor-3rdvm3/2)
half-upminor-3rd
(P8, P5, ^M2/2)
half-upmajor-2nd
7
(P8, P5, vM3/2)
half-downmajor-3rd
(P8, P5, vm3/2)
half-downminor-3rd
8
(P8, P5, ^m6/2)
half-upminor-6th
(P8, P5, ^M6/2)
half-upmajor-6th
9
(P8, P5, vM6/2)
half-downmajor-6th
(P8, P5, vm7/2)
half-downminor-7th 810
(P8/2, P5, ^m3/2)
half-8ve half-upminor 3rd
(P8/2, P5, ^M2/2)
half-8ve half-upmajor-2nd
11
(P8/2, P5, vM3/2)
half-8ve half-downmajor 3rd 9(P8/2, P5, vm3/2)
etc.
12
(P8/2, P5, ^m6/2) 10half-8ve half-upminor 6th
(P8/2, P5, ^M6/2)
13
(P8/2, P5, vM6/2)
half-8ve half-downmajor 6th
(P8/2, P5, vm7/2)
14
(P8, P4/2, ^m3/2)
half-4th half-upminor 3rd
(P8, P4/2, ^M2/2)
15
(P8, P4/2, vM3/2) 11etc.
(P8, P4/2, vm3/2)
16
(P8, P4/2, ^m6/2) 12(P8, P4/2, ^M6/2)
17
(P8, P4/2, vM6/2)
(P8, P4/2, vm7/2)
18
(P8, P5/2, ^m3/2)
(P8, P5/2, ^M2/2)
19
(P8, P5/2, vM3/2) 13(P8, P5/2, vm3/2)
20
(P8, P5/2, ^m6/2) 14(P8, P5/2, ^M6/2)
21
(P8, P5/2, vM6/2)
(P8, P5/2, vm7/2)
22
(P8/2, P4/2, vM3/2)
half-everything half-downmajor-3rd
(P8/2, P4/2, ^M2/2)
There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.
Notating Blackwood-like pergens* Could demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.
another example might work?
A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12.
A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1). The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.
Such a pergen is in effect multiple copies of an edo. Its spoken name is rank-2 N-edo, meaning an edo extended to rank-2. Its notation is based on the edo's notation, expanded with an additional microtonal accidental pair. Examples:
temperament
...
The additional accidental has an equivalent ratio, found by adding the pergen's 3-limit comma onto the ratio. Blackwood's comma is 256/243, and ^1 = 81/80 or equivalently, 16/15.
(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.)
The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like ^m6/2 or ^^A5/4.
When the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...
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.
...
edo-subset notations
P8/6, P8/8
...
non-8ve and non-5thno-5ths pergens*
Just
...
3 are present and independent can
the first two independent primes in the prime subgroup
pergen number
want to deviseuse a personal notation that
Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.
A similar chart could be made of all rank-3 pergen cubes.
pergen
edited
... More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3…
...
More examples: Triple bluish (2.3.5.7 with 1029/1000 tempered out) is (P8, P11/3, ^1) = rank-3 third-11th. Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, ^1) = rank-3 half-8ve (^1 = 81/80). However, deep reddish minus white (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.
...
notated with either microtonal accidentals
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 > m, it will split some 3-limit interval into n parts.
...
of it. Such a prime is dependent on a lower prime. If this
...
is also directly related,dependent, the 4th
...
so forth. In other words, the multigen uses the first two independent primes.
For example,
...
octave. The multigen must use primes 2 and 5. In this case, the pergen is
...
same as Blackwood.Blackwood (see Blackwood-like pergens below).
To find
...
lower primes maythat are dependent need to
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
...
G' = G + i·P = (-y, x) / xz + i·P8/x = (i·z - y, x) / xz
The rank-2 pergen from the [(x, y), (0, z)] square mapping is (P8/x, (i·z-y,x)/xz), with |i| <= x
...
Imperfect multigens like M2 or m3 are fairly
...
is (P8/1, (-3n-2,1)/(-3))(-3i-2,1)/(-3)) = (P8, (3n+2,-1)/3),(3i+2,-1)/3), with -1 <= ni <= 1.
...
value of ni reduces the
Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:
2/1
...
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 OnePergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.
Another obvious application
...
a pseudo-pergen, because either because it contains
...
because the split multigen isn't
...
a pseudo-pergen of (P8, (5/4)/2),
Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a 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 Chord names and scale names below.
The thirdfinal main application,
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. 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.
...
conventional notation, defined aswhich is octave-equivalent, heptatonic,fifth-generated and fifth-generated.heptatonic. Porcupine can
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 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. See Notating unsplit pergens below.
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.
...
C - D/ - F\ - G
C - Dv\ - Eb^/ - F 128/125triple green & 1029/1024large triple blue
^1 = 81/80
/1 = 64/63
...
C - D/ - F\ - G
C - Ev/ - Ab^\ - C 250/243triple yellow & 1029/1024large triple blue
^1 = 81/80
/1 = 64/63
...
vulture
etc.
...
so arguably it isn't all
Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).
Tipping points
...
So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 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), M can be P4, P5, P11, P12, WWP4 or WWP5.
Secondary splits
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example, third-4th (porcupine)(e.g. porcupine) splits intervals
P4/3: C - Dv - Eb^ - F
A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split)
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(P8/2, P4/2)
half-everything (everyevery 3-limit interval)/2interval is split twice as muh as before
third-splits
(P8/3, P5)
...
(P83, P4/3)
third-everything (everyevery 3-limit interval)/3interval is split three times as much as before
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, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.
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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
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(2/1) = 648/625.648/625, the diminished temperament. If P
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(2/1) · (7/6)-4.(7/6)-4, the quadruple red temperament. Neither 13/11
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81/80 or 64/63.64/63 (see mapping commas in the next section). Thus for (P8/4, P5), sinceif P = vm3, and ^1 = 64/63, P is
...
for a doubledouble-split pergen is
If the pergen is not explicitly false, put the pergen in its unreduced form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n⋅P8 - m⋅M)/nm) = (P8/m, M'/n'). 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⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.
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A double-split pergen is a true double if GCD (m, n) > |b|, and a false double if GCD (m, n) = |b|.
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.
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differently. For example: (P8, P5/2) has generator ^m3 and equivalent generator vM3. Another example, half-8ve
...
P5) has generator P5, alternate generators P4 and vA1, period vA4,vA4 and equivalent period ^d5. (P8, P5/2)It has generator ^m3P5 and alternate generators P4 and vA1. vA1 is equivalent generator vM3.to ^m2.
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 ^6dd2.
There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.
Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.
If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. Double-pair notation could be used, for proper spelling. E would be ^^\d2.
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its gen2 isn'tcan't be expressed as a mapping
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(P8, P5, y3/3)b3/2) or (P8, P5, vM3/3).vm3/2). It could
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(P8, P5, b3/2)y3/3) or (P8, P5, vm3/2).vM3/3), but the pergen with a smaller fraction is preferred. Because the
...
of accidentals.
The next table lists all rank-3 pergens up to half-splits. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.
unsplit
2.3.5
2.3.7
1
(P8, P5, ^1)
rank-3 unsplit
same
same
half-splits
2
(P8/2, P5, ^1)
rank-3 half-8ve
same
same
3
(P8, P4/2, ^1)
rank-3 half-4th
same
same
4
(P8, P5/2, ^1)
rank-3 half-5th
same
same
5
(P8/2, P4/2, ^1)
rank-3 half-everything
same
same
6
(P8, P5, vM3/2)
half-downmajor-3rd
(P8, P5, ^M2/2)
half-upmajor-2nd
7
(P8, P5, ^m6/2)
half-upminor-6th
(P8, P5, vm7/2)
half-downminor-7th
8
(P8/2, P5, vM3/2)
half-8ve half-downmajor 3rd
9
(P8/2, P5, ^m6/2)
10
(P8, P4/2, vM3/2)
11
(P8, P4/2, ^m6/2)
12
(P8, P5/2, vM3/2)
13
(P8, P5/2, ^m6/2)
14
(P8/2, P4/2, vM3/2)
half-everything half-downmajor-3rd
Notating Blackwood-like pergens* ACould demeter have ^1 = 15/14 minus A1 = sry1 = r1 - g1? gen2 = ^A1. No, cuz you still need highs/lows for chord spelling. C C###^^^ G G##^^.
another example might work?
A Blackwood-like temperament
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exact fraction of the octave. The 5th is not independent of the
A blackwood-like pergen is a rank-3 pergen plus a 3-limit comma. Adding this comma divides the first term and removes the 2nd term from the rank-3 pergen. For example, Blackwood is 5-limit JI = (P8, P5, ^1) plus 256/243, making (P8/5, ^1).
The generator's ratio contains only 2 and one higher prime. For single-comma temperaments, the generator is simply an octave-reduced prime, with a ratio like 5/4 or 7/4. Multiple-comma temperaments can have split multigens like (8/5)/2 or (25/16)/4.
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D D^=Ev E=F F^=Gv G...
D F#\=G\ B\\... ???(see below)
81/80
12edo+j
...
"
D G#v=Abv Dvv...
729/704 =
---
17edo+y
...
81/80
If the edo's notation uses ups and downs, the up symbol can often be equated to a 3-limit ratio. In 17-edo and 22-edo, ^1 = m2. In 31-edo and 43-edo it's d2. But in edos like 15, 21 and 24, in which the circle of 5ths skips some notes, there is no 3-limit ratio. The ratio depends on the JI interpretation of the edo. For 10-edo, ^1 might equal 16/15 or 12/11 or 13/12.
The additional accidental's ratio can be changedaccidental has an equivalent ratio, found by adding the edo's definingpergen's 3-limit comma onto it. For Blackwood, 5-edothe ratio. Blackwood's comma is defined by 256/243, and /1^1 = 81/80 =or equivalently, 16/15.
(In general, the 1st accidental pair is the mapping comma for the 3rd prime in the subgroup, the 2nd for the 4th, etc.) InWhen the edo implied by the octave split doesn't have a decent 5th, as with P8/3, P8/4, P8/6, P8/8, etc., and the generator isn't ...
In any notation,
...
and genchains. But
But in no-threes"no-5ths" or "minus white" pergens, not every name has a note. For example, deep reddish minus white (2.5.7 with 50/49 tempered out) is (P8/2, M3) = half-8ve, major 3rd. The genchain runs C - E - G# - B# - D##... There is no G or D or A note, in fact 75% of all possible note names have no note. The same is true of relative notation: 75% of all intervals don't exist. There is no perfect 5th or major 2nd.
Note-less names can be avoided in any pergen with a 5th. The full chain of 5ths must appear with neither ups/downs nor highs/lows in either the perchain or the genchain. For this reason, Blackwood's perchain is not the same as fifth-8ve's perchain, even though it sounds the same. Thus the last row of this table is an invalid notation.
tuning
pergen
spoken name
enharmonic
perchain
genchain
notes
large quintuple blue,
small quintuple red
(P8/5, P5)
fifth-8ve
E = v5m2
C D^^ Fv G^ Bbvv C
C G D A E...
a valid notation
Blackwood
(P8/5, ^1)
rank-2 5-edo
E = m2
C D=Eb E=F G A=Bb B=C
C Ev G#vv...
a valid notation
"
(P8/5, M3)
fifth-8ve major 3rd
E = v5m2
C D^^ Fv G^ Bbvv C
C E G#...
invalid, no G, D or A notes
edo-subset notations relative notation: notP8/6, P8/8
Notating non-8ve and non-5th pergens*
Just as all intervals names have intervals. No perfect 5thrank-2 pergens in 6-edo.which 2 and 3 are independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. The pergens are grouped into blocks and sections as before:
the first two independent primes in the prime subgroup
pergen number
2.3
2.5
2.7
3.5
3.7
5.7
1
(P8, P5)
(P8, y3)
(P8, r2)
(P12, y6)
(P12, r3)
(WWy3, bg5)
half-splits
2
(P8/2, P5)
(P8/2, y3)
(P8/2, r2)
3
(P8, P4/2)
(P8, y3/2)
(P8, r2/2)
4
(P8, P5/2)
(P8, g6/2)
(P8, b7/2)
5
(P8/2, P4/2)
(P8/2, y3/2)
(P8/2, r2/2)
third-splits
6
(P8/3, P5)
(P8/3, y3)
7
(P8, P4/3)
(P8, y3/3)
8
(P8, P5/3)
(P8, g6/3)
9
(P8, P11/3)
(P8, y10/3)
10
(P8/3, P4/2)
(P8/3, y3/2)
11
(P8/3, P5/2)
12
(P8/2, P4/3)
13
(P8/2, P5/3)
14
(P8/2, P11/3)
15
(P8/3, P4/3)
Every rank-2 pergen can be identified by the first two independent primes and its pergen number. Blackwood is 2.5 pergen #33. For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1.
Every rank-3 pergen can be identified by its first three independent primes and its pergen number. A similar table can be made for all rank-3 pergens. Each block is much larger. The notation for 2.3.5 pergens and 2.3.7 pergens is often the same.
pergen number
2.3.5
2.3.7
2.5.7
3.5.7
1
(P8, P5, ^1)
rank-3 unsplit
same
(P8, y3, r2)
(P12, y6, r3)
half-splits
2
(P8/2, P5, ^1)
rank-3 half-8ve
same
3
(P8, P4/2, ^1)
same
4
(P8, P5/2, ^1)
same
5
(P8/2, P4/2, ^1)
rank-3 half-everything
same
6
(P8, P5, vM3/2)
half-downmajor-3rd
(P8, P5, ^M2/2)
7
(P8, P5, ^m6/2)
half-upminor-6th
(P8, P5, vm7/2)
8
(P8/2, P5, vM3/2)
half-8ve half-downmajor 3rd
9
(P8/2, P5, ^m6/2)
10
(P8, P4/2, vM3/2)
11
(P8, P4/2, ^m6/2)
12
(P8, P5/2, vM3/2)
13
(P8, P5/2, ^m6/2)
14
(P8/2, P4/2, vM3/2)
rank-3 half-everything
Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus y3 = M3, r2 = M2, bg5 = d5, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a huge number of note-less names. The composer may well want to devise a personal notation that isn't backwards compatible.
Pergen squares are a way to visualize pergens in a way that isn't specific to any primes at all.
A similar chart could be made of all rank-3 pergen cubes.
Notating tunings with an arbitrary generator
Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.
...
http://www.tallkite.com/misc_files/pergens.pdf
(screenshot) Pergen squares pic
One way to visualize pergens...
pergenLister app
Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.
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GCD (v, w) = m
GCD (u ± 1, v, w) = m (thus u mod m ≠ 0)
...
prime Q?
Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)
pergen
edited
... A pergen (pronounced "peer-gen") is a way of identifying a regular temperament solel…
...
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 multigen. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.
...
P4/3). Semaphore, which meansa pun on "semi-fourth", is
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 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. See the notation guide below, under Supplemental materials.
The largest category contains all single-comma temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 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, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2⋅P8 - 3⋅P5) / (3⋅2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This is explicitly false, thus the comma can be found from m3/6 alone. G' is about 50¢, and the comma is 6⋅G' - m3. The comma splits both the octave and the fifth.
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. 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. To summarize:
...
false if m = |b|, and onlynot explicitly false if m => |b|.
A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false.
A double-split pergen is a true double if GCD (m, n) > |b|, and a false double if GCD (m, n) = |b|.
583-593¢
WWWP4/7 TheThere are gaps in the table, especially near 150¢, 200¢, 300¢ and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation. But the total range
...
generators is fairlymostly well covered, providing convenient notation options.
...
over 720¢, to avoid descending 2nds, instead of calling itthe generator a 5th,
...
inverse a quarter-12th, to avoid descending 2nds. Nevertheless, there are gaps in the table, especially near 200¢, 300¢quarter-12th. The generator is notated as an up-5th, and 400¢. Some tunings simply aren't compatible with fifth-generated heptatonic notation.
Splittingfour of them make a WWP4.
The table assumes unsplit octaves. Splitting the octave
...
if P = P8/3 = 400¢
...
G = 500¢,300¢, alternate generators
...
100¢ and 300¢.500¢. Any of
See also the map of rank-2 temperaments.
Pergens and MOS scales
...
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 a multiple of m, and k must be divisible 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? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in 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.
...
and 18-edo. Since bothBoth of these
...
heptatonic notation, and 13edo's half-5th
pergen
supporting edos (12-31 only)
...
13\22 = 14\24
half-8ve quarter-tone
...
pair by looking upfinding the range of its generator
Supplemental materials*
needs more screenshots, including 12-edo's pergens and a page of the pdf
...
The interval P8/2 has a "ratio" of the square root of 2, which equals 21/2, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the pergen matrix [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.
Because we started with a valid pergen, the square mapping must be an integer matrix. Since n/b is an integer, n must be a multiple of |b|. From this it follows that a and b must be coprime, otherwise a, b, and n could all be reduced by GCD (a,b), and the multigen could be simplified. Since GCD (a, b) = 1 and -am/b is an integer, it follows that m must be a multiple of |b| as well. Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.
If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m 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, with 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/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)
Therefore P8 is split into m periods
Therefore if m = |b|, the pergen is explicitly false
Assume the pergen is a false double, and there's a comma C that splits both P8 and (a,b) appropriately. Can we prove r = 1? Let Q = the higher prime that C uses. Express P, G and C as monzos of the prime subgroup 2.3.Q, by expanding the 2x2 pergen matrix to a 3x3 matrix A:
P = (1/m, 0, 0)
G = (a/n, b/n, 0)
C = (u, v, w)
If the pergen is explicitly false, and m = |b|, let w = ±n.
Here u, v and w are integers. If GCD (u, v, w) > 1, simplify C so that it = 1. The inverse of A expresses 2, 3 and Q in terms of P, G and C. If C is tempered out, the C column can be discarded, making the usual 3x2 period-generator mapping. However, if C is not tempered out, the inverse of A is a 3x3 period-generator-comma mapping, which is simply a change of basis. For example, 5-limit JI can be generated by 2/1, 3/2 and 81/80.
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)
Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)
...
two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in thosethe first two columns, A
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)
3 = 3/1 = P12 = (-am/b, n/b, 0) · (P, G, C)
...
[Another try at it:] To split the 8ve into m parts, P8 ± C must be divisible by m, and both v and w must be a multiple of m. To split the multigen into n parts, M ± C must be divisible by n, and w must be a multiple of n. Thus for some nonzero integer k, w = k · LCM (m, n) = k · mn / GCD (m, n) = kmn/br. [end of another try]
For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test. If m = |b|, isLet ¢(R) be the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5
Because n is a multiplecents 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 cthe ratio R, and dlet ¢[M] be the bezout paircents of a+b and b, with c·(a+b) + d·bsome monzo M. If we limit |v| to < 6, but allow large commas of up to 100¢, we can specify that Q = GCD (a+b, b)5.
¢(C) = GCD (a, b)u·¢(2) + v·¢(3) + w·¢(Q) < 100¢
Octave-reduce:
¢(C) = 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·P8u·1200¢ + v·1200¢ + v·702¢ + w·2400¢ + c·b·((n/b)·G - P5)w·386¢ < 100¢
¢(C) mod 1200 = b · (d·P8v·702¢ + c·(n/b)·G - c·P5)
P8/mw·386¢ < 100¢
GCD (v, w) = P8/|b|m
GCD (u ± 1, v, w) = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)
Therefore P8 is split into m periods
Therefore if(thus u mod m = |b|, the pergen is explicitly false≠ 0)
Assuming r
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prime Q?
Assume r = 1 and GCD (m, n) = 1. Let G be the generator, with a 2.3.Q monzo of the form (x, y, ±1) for some unspecified higher prime Q. x and y are chosen so that the cents of (a,b) is about n times the cents of G. If the pergen is explicitly false, with m = |b|, the 2.3.Q comma C can be found from the 2nd half of the pergen: (a,b) + C = n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)
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a·P8 splits intinto b parts,
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a·P8 splits intinto m parts.
Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).
Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.
pergen
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... Definition
A pergen (pronounced "peer-gen") is a way of identifying a regular tempe…
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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.
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is the multi-gen.multigen. The 3-limit multi-genmultigen is referred
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 temperament (2.3.5 and 25/24) splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, or perhaps third-of-a-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 a few dozen 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. See the notation guide below, under Supplemental materials.
The largest category contains all single-comma temperaments with a comma of the form 2x 3y P or 2x 3y P-1, where P is a prime > 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.
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of the multi-gen,multigen, and as
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of the multi-gen.multigen. There is
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
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Ly5T
(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 multi-genmultigen is usually
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. Color notation (or any HEWM notation) can be used to indicate higher primes. Monzos of the form (a,b,1) = yellow, (a,b,-1) = green, (a,b,0,1) = blue, (a,b,0,-1) = red, (a,b,0,0,1) = jade, (a,b,0,0,-1) = amber, and (a,b) = white. Examples: 5/4 = y3 = yellow 3rd, 7/5 = bg5 = blue-green 5th or bluish 5th, etc.
For example, marvel (2.3.5.7 and 225/224) has an unsplit pergen of (P8, P5, y3). Often colors can be replaced with ups and downs. 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.
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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 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. See Notating unsplit pergens below.
Not all combinations of periods and generators are valid. Some are duplicates of other pergens. (P8/2, M2/2) is actually (P8/2, P5). Some combinations are impossible. There is no (P8, M2/2), because no combination of periods and generators equals P5.
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second, the multi-genmultigen does. Within
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. The pergen and the enharmonic together define the notation.
The genchain (chain of generators) in the table is only a short section of the full genchain.
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[Another try at it:] To split the 8ve into m parts, P8 ± C must be divisible by m, and both v and w must be a multiple of m. To split the multigen into n parts, M ± C must be divisible by n, and w must be a multiple of n. Thus for some nonzero integer k, w = k · LCM (m, n) = k · mn / GCD (m, n) = kmn/br. [end of another try]
For v/m to be an integer, v must equal i·m for some integer i. Likewise, av-bu must equal j·n for some integer j. Thus bu = av - jn = iam - jn. Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. Then m = prb and n = qrb. Substituting, we get bu = iaprb - jqrb, and u = r(iap - jq). Furthermore, v = im = iprb and w = ±mn/b = ±pqrrb. Thus u, v and w are all divisible by r. If r > 1, this contradicts the requirement that GCD (u, v, w) = 1, therefore r must be 1, and GCD (m, n) = |b|, and all false doubles pass the false-double test. AssumingIf m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m 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, with c·(a+b) + d·b = GCD (a+b, b) = GCD (a, 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/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)
Therefore P8 is split into m periods
Therefore if m = |b|, the pergen is explicitly false
Assuming r =
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prime Q?
Assume r = 1 and GCD (m, n) = 1. Let ¢(R)G be the centsgenerator, with a 2.3.Q monzo of the ratio R,form (x, y, ±1) for some unspecified higher prime Q. x and let ¢[M] bey are chosen so that the cents of some monzo M.(a,b) is about n times the cents of G. If we allow large commas, wethe pergen is explicitly false, with m = |b|, the 2.3.Q comma C can specify that Qbe found from the 2nd half of the pergen: (a,b) + C = 5.n·G, and C = (n·x - a, n·y - b, ±n). Obviously C splits (a,b) into n parts. Does it split P8 into m parts? Let q = nr/b as before, but with r = 1, it simplifies to q = n/b.
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)
Thus a·P8 splits int b parts, and since m = |b|, a·P8 splits int m parts. Proceed as before with a bezout pair to find the monzo for P8/m.
Next, assume the pergen isn't explicitly false. The unreduced form is (P8/m, (n - am, -bm) / mn). Substituting in m = pb and n = qb, with p and q coprime, we get (P8/m, (q - ap, -pb) / pqb).
Assume the pergen is a true double, and r > 1. Then P8 = m·P = prb·P = r·(pb·P), and P8 splits into (at least) r parts. Furthermore, P12 = (-am/b)P + (n/b)G = -apr·P + qr·G = r·(-ap·P + q·G), and P12 also splits into at least r parts. Thus every 3-limit interval splits into at least r parts.
Given a pergen (P8/m, (a,b)/n), how many parts is an arbitrary interval (a',b') split into?
mapping [(x, y), (0,0), (y, z)] creates
To prove: if |z| = 1, n = 1
If z = 1, let i = y - x, and the pergen = (P8/x, P5)
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Therefore multigens like M9/3 or M3/4 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 m 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, with 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/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)
Therefore P8 is split into m 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?
pergen
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... This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is …
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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. 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. To summarize:
A double-split pergen is explicitly false if and only if m = |b|. A double-split pergen is a true double if and only if GCD (m, n) > |b|.
A double-split pergen is a true double if and only if neither it nor its unreduced form is explicitly false.
A double-split pergen is a true double if GCD (m, n) > |b|, and a false double if GCD (m, n) = |b|.
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.