pergen names
edited
... The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E…
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The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.
[This 13b part needs
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played, but they can't be
(table is under construction)
pergen
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C^^ = D
C - F^=Gv - C
128/121, j4=P4
^1 = 33/32
"
{P8, P4/2}
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C - D^=Ebv - F
semaphore
^1 = 64/63
14, 15*, 18b*, 19, 20*,
23, 24, 25*, 28*, 29,
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v4A1
C^4 = C#
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Evv F^ G^G
{P8, P11/4}
P11/4 = ^M3 = v3dd5
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v5dd3
[Question: how to find all possible pergens?]
or downed. The implied edo is just the 3-factor of the bare enharmonic interval.
bare enharmonic interval
implied edo
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If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.
G (enharmonic) = N * G (gen) - G(multi-gen)
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1 = -5-7 = m2d8, inverts to A1
P5/2 = M3, enh = 2(4) - 1 = A1
To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.
[Question: how to find all possible pergens?]
Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) 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¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus ups and downs may need to be swapped, depending on the size of the 5th in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.
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sweet spot? Example?]Example?//]
Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.
bare enharmonic interval
pergen names
edited
... large quintuple yellow
Ly5T
... WWM2, etc.
[Question: at what point is "W" a…
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large quintuple yellow
Ly5T
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WWM2, etc.
[Question: at what point is "W" actually needed?]
For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. Color notation is used to indicate higher primes. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third (y3 = 5/4).
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The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
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heptatonic notation.
[This
[This 13b part needs clarification]clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.]
(table is under construction)
pergen
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C^^ = B#
C - F#v=Gb^ - C
srutal
^1 = 81/80
12, 14, 16, 18b, 20*,
or downed.
enharmonic
bare enharmonic interval
implied edo
edo's 5th
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C - C#
7-edo 686¢~686¢
600-686¢
686¢-720¢
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C - Eb3
17-edo 706¢~706¢
706-720¢
600-706¢
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C - Db3
19-edo 695¢~695¢
695-720¢
600-695¢
...
C - Db4
26-edo 692¢~692¢
692-720¢
600-692¢
...
C - Fb4
29-edo 703¢~703¢
703-720¢
600-703¢
...
C - Eb5
31-edo 697¢~697¢
697-720¢
600-697¢
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[Question: how many entries does this table realistically need?]
As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.
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be needed.
Not all enharmonics work with all pergens. The implied edo is always a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d32. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.
[Check this last paragraph!]
[Question: what if there are highs and lows?]
Not all enharmonics work with all pergens. The implied edo must be a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d32. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.
Every rank-2 interval has a genspan, which is the number of generators needed to create the interval. It's also the position of the interval on the relative genchain. For conventional (un-upped) intervals, the genspan is the interval's position on the relative chain of 5ths, which runs ...d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... It equals the 3-factor of the interval's monzo.
If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.
G (enharmonic) = N * G (gen) - G(multi-gen)
P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2
P5/2 = M3, enh = 2(4) - 1 = A1
To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.
[Question: how to find the notation for multi-comma tempers?]
pergen names
edited
... Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double ge…
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Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, but only if the smaller prime is > 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.
Applications
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another additional pair,pair. One possibility is highs and
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and \. v\D is down-low D, and /P5 is high-five. Alternatively, color
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be spelled properly as C Ev
Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.
The enharmonic interval 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, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.
The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
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heptatonic notation. [This
[This 13b part needs
(table is under construction)
pergen
B#3
C - Db^ - Ebb^ -- E#vv - F#v - G Removing"
v5dd3
[Question: how to find all possible pergens?]
Removing the ups
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In the previous table, this
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The other pergenspergens' enharmonic intervals are upped
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the sweet spot?]spot? Example?]
Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is 646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.
enharmonic interval
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600-697¢
upped
etc.
[Question: how many entries does this table realistically need?]
As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.
The enharmonic's number of ups or downs is the LCM of
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two splitting fractions. This numberfractions is called
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height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.
Not all enharmonics work with all pergens. The implied
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of the height.octave fraction. Thus a half-anythinghalf-octave pergen can
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or d32. And aA quarter-octave pergen
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must be ^4d2 or v4d2.a d2.
[Check this last paragraph!]
[Question: what if there are highs and lows?]
To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.
[Question: how to find the notation for multi-comma tempers?]