pergen
edited
... vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
More remote inter…
...
vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps.
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any naturally occuringoccurring split of
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the naturally occuringoccurring split of
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we have n·GCD(a',b').n·GCD (a',b'). If the
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convenience, naturally occuringoccurring splits are
pergen
secondary splits of a 12th or less
fill in the 2nd pergens column above
add a mapping commas section somewhere?
finish proofs
link from: ups and downs page, Kite Giedraitis page, MOS scale names page, is there a rank-2 page?
Notaion guide PDF
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.
pergen
edited
... More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most…
...
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps.
For a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occuring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.
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we have n · GCD (a', b').n·GCD(a',b'). If the
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be split. Thus half-5th splits every 3rd, 5th, 7th , 9th, etc. in half. "Every" means every quality, so 3rd includes d3, m3, M3 and A3, 5th includes P5, A5 and d5, etc.
The following
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"all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.
pergen
secondary splits <=of a 12th or less
all pergens
M3/2, A4/3, d5/2, A5/4, m7/2, M9/2, m10/3, A11/2
If the pergen's notation is known, an even easier method is to simply assume that the up symbol equals a comma that maps to P1, such as 81/80 or 64/63. Thus for (P8/4, P5), since P = vm3, P is 32/27 ÷ 64/63 = 7/6. This method is notation-dependent: (P8/2, P5) with P = vA4 and ^1 = 81/80 gives P = 45/32, but if P = ^4, then P = 27/20.
Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multigen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is explicitly false. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires G ~ 50¢, perhaps 33/32, and the comma is 4⋅G - M2 = (33/32)^4 / (9/8) = (-17, 2, 0, 0, 4).
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(n⋅P8 - m⋅M)/nm).m⋅M)/nm) = (P8/m, M'/n'). The new
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m3/6 alone. GG' is about
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comma is 6⋅G6⋅G' - m3.
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2⋅P8 - 4⋅P4) / (2⋅4) = (2⋅M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.
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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:
For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2
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with E occuringoccurring x times
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with E' occuringoccurring y times
For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics
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M'/n'), with M' = n'G' + zE",a new enharmonic E" and new counts, P8 = mP + xE"x'E", and M' = n'G' + y'E"
The keyspan of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The stepspan of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.
Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a gedra, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:
pergen
edited
... vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
More remote inter…
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vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps. GivenFor a pergen (P8, (a,b)/n), any interval generated by n octaves and the multigen splits into at least n parts. For a pergen (P8/m, P5), any interval generated by the octave and m 5ths splits into at least m parts. Thus any naturally occuring split of m parts occurs in all voicings of that interval. For example, M9 naturally splits into two 5ths, therefore (P8/2, P5) splits all voicings of M9, including M2.
Given a pergen
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(proof below). For an unsplit pergen, we have the naturally occuring split of GCD (a', b'). If n
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(a'·m, b'). If m = n (an nth-everything pergen), we have n · GCD (a', b'). If the enharmonic is an A1, every interval with a degree of n+1 will be split.
The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occuring splits are listed too, under "all pergens".
pergen
secondary splits <= 12th
all pergens
M3/2, d5/2, A4/3, d5/2, A5/4, m7/2,
half-splits
(P8/2, P5)
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(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G
Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.
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± a·b', ±b'·n)b'·n)
If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')
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= the "normal"naturally occurring split.
If m = n (nth-everything), we have n · GCD (a', b')
The multigen and the arbitrary interval can be expressed as gedras:
(a,b) = [k,s] = (-11k+19s, 7k-12s)
(a',b') = [k',s'] = (-11k'+19s', 7k'-12s')
a'·b - a·b' works out to be k·s' - k'·s, and we have GCD ((k·s' - k'·s)·m/b, b'·n/b)
If s is a multiple of n (E is an A1) and s' is a multiple of n, let s = x·n and s' = y·n
GCD ((k·y·n - k'·x·n)·m/b, b'·n/b) = (n/b) · GCD (x·m·(y·k - k'), b')
Thus every such interval is split, e.g. half-5th splits every 3rd, 5th, 7th, 9th and 11th, including aug, dim, major and minor ones.
Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately
can we prove r = 1?
pergen
edited
... 20, 28
See the screenshots in the next section for examples of which pergens are supported by…
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20, 28
See the screenshots in the next section for examples of which pergens are supported by a specific edo.
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|g'|\N= |g|\N'. The bezout pair is also either ancestor of N/N' in the scale tree. Alternate generators
edos
octave split
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The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.
Various proofs (unfinished)
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root of 2 =2, which equals 21/2, and
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(1/2, 0). Likewise,In general, the pergen
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 both (a,b) and n could be reduced by GCD (a,b). Since -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.
Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. 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.
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If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')
If m = 1 and n = 1, we have GCD (a', b') = the "normal" split. ===== false starts ============
Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately
can we prove r = 1?
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Given an arbitrary x and y, GCD (x + k"u, y + k"v, k"w) >= r for some k
Assume GCD (x,y) = 1
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P8 into bm parts
if r > 1, it's a true double
a·P8 = (a,0) = (a,b) - (0,b) = M - b·P12
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a·P8 = qrb·G - b·P12 = b·(qr·G - P12)
Let c and d be the bezout pair of a and b, with c·a + d·b = 1
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= 1, let c = 01 and |d|d = 1
Since the pergen is a double-split, m > 1, therefore |b| > 1, therefore±a, to avoid c ≠= 0
ca·P8 = cb·(qr·G - P12)
(1 - d·b)·P8 = c·b·(qr·G - P12)
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- c·P12)
Therefore P8 is split into at least |b| periods = b · (d,-c) + bcqr·G
Given:
A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x > 0, z ≠ 0, and |i| <= x
pergen
edited
... Naming very large intervals
So far, the largest multigen has been a 12th. As the multigen fra…
...
Naming very large intervals
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*splits
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:
P4/3: C - Dv - Eb^ - F
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vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps. ForGiven a pergen (P8/m, P5), any interval (a,b) in which b mod m = 0 splits into at least m parts. For a pergen (P8, (a,b)/n), anyan interval (a',b') for which (a'·b - a·b') mod (n·|b|) = 0 splits into at leastGCD ((a'·b - a·b')·m/b, b'·n/b) parts (proof below). If n parts. A double-split pergen (P8/m, M/n) contains all the secondary splits of both (P8/m, P5) and (P8, M/n), perhaps plus others.
(a,b) = n·G
(a',b') = i·(n·G) + j·(n·P8) = (i·a + j·n, i·b)
a'·b - a·b' = i·a·b + j·b·n - i·a·b = j·b·n
(a'·b - a·b') mod n·|b| = 0
P8 = m·P
(a',b') = i·(m·P) + j·(m·P12) = (i, j·m)
b' mod m = 0
M2/4 = (-3,2)/4
(2a +3b) mod 8 = 0
b must be even
(-6,4) = 81/64 = M3
mod nb/m? (-1,2) = 9/2 = WM9 =
false double: (P8/m, (a,b)/n)
gcd (m,n) = |b|
(a'·b - a·b') mod1, we have GCD (a'·m, b').
The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occuring splits are listed too, under "all pergens".
pergen
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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 both (a,b) and n could be reduced by GCD (a,b). Since -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.
Let p = m/rb and q = n/rb, where p and q are coprime integers, nonzero but possibly negative. 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?
(a',b') = (a'·b, b'·b) / b = (a'·b - a·b', 0) / b + (a·b', b'·b) / b = (a'·b - a·b')·P8 / b + b'·(a,b) / b = (a'·b - a·b')·(m/b)·P + b'·(n/b)·G
Therefore (a',b') is split into GCD (a'·b - a·b')·(m/b), b'·(n/b)) parts.
If m = 1, then b = ±1, and we have GCD (a' ± a·b', ±b'·n)
If n = 1, then a = -1 and b = 1, and we have GCD (a'·m + b'·m, b') = GCD (a'·m, b')
If m = 1 and n = 1, we have GCD (a', b') = the "normal" split.
===== false starts ============
Assume it's a false double, and there's a comma (u,v,w) that splits both P8 and (a,b) appropriately
can we prove r = 1?
pergen
edited
... Naming very large intervals
So far, the largest multigen has been a 12th. As the multigen fra…
...
Naming very large intervals
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 splitssplits*
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:
P4/3: C - Dv - Eb^ - F
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vm9/4: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15)
vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)
...
notes if unbroken.it had no gaps.
For a pergen (P8/m, M/n), let x beP5), any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R(a,b) in which b mod m = 0 splits into xat least m parts. Let y beFor a pergen (P8, (a,b)/n), any number that divides n, the interval z·M ± y·R(a',b') for which (a'·b - a·b') mod (n·|b|) = 0 splits into yat least n parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).
AA double-split perhgen like (P8/3. P4/2)pergen (P8/m, M/n) contains all
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of both (P8/3,(P8/m, P5) and (P8, P4/2),M/n), perhaps plus others. The
(a,b) = n·G
(a',b') = i·(n·G) + j·(n·P8) = (i·a + j·n, i·b)
a'·b - a·b' = i·a·b + j·b·n - i·a·b = j·b·n
(a'·b - a·b') mod n·|b| = 0
P8 = m·P
(a',b') = i·(m·P) + j·(m·P12) = (i, j·m)
b' mod m = 0
M2/4 = (-3,2)/4
(2a +3b) mod 8 = 0
b must be even
(-6,4) = 81/64 = M3
mod nb/m? (-1,2) = 9/2 = WM9 =
false double: (P8/m, (a,b)/n)
gcd (m,n) = |b|
(a'·b - a·b') mod
The following table
36edo
edited
... The following table gives an overview of all degrees of 36edo.
Degree
Size
in cents
App…
...
The following table gives an overview of all degrees of 36edo.
Degree
Size
in cents
Approximate
ratios of 2.3.7
Additional ratios
of 2.3.7.13.17
ups and downs notation
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Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chord names in other EDOs.
Music
Exponentially More Lost and Forgetful by Stephen Weigel played by flautists Orlando Cela and Wei Zhao
Something by Herman Klein
Hay by Joe Hayseed
pergen
edited
... P4/3: C - Dv - Eb^ - F
A4:/3 C - D - E - F# (the lack of ups and downs indicates that this in…
...
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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- Bb or(also m7/6: C
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Ab^ - BbBb)
M7/3: C - Ev - G^ - B
m10/3: C - F - Bb - Eb (also already split) (m10/9 also occurs)
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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.
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as well. 11-edo and 23-edo.
pergen
supporting edos (12-31 only)
Tenney-Euclidean metrics
edited
... Temperamental complexity
Suppose now A is a matrix whose rows are vals defining a p-limit reg…
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Temperamental complexity
Suppose now A is a matrix whose rows are vals defining a p-limit regular temperament. Then the corresponding weighted matrix is V = AW. The TE tuning projection matrix is then V`V, where V` is the pseudoinverse. If the rows of V (or equivalently, A) are linearly independent, then we have V` = V*(VV*)^(-1), where V* denotes the transpose. In terms of vals, the tuning projection matrix is P = V`V = V*(VV*)^(-1)V = WA*(AW^2A*)^(-1)AW. P is a positive semidefinite matrix, so it defines a positive semidefinite bilinear form. In terms of weighted monzos m1 and m2, m1*Pm2 defines the semidefinite form on weighted monzos, and hence b1*W^(-1)PW^(-1)b2 defines a semidefinite form on unweighted monzos, in terms of the matrix W^(-1)WA*(AW^2A*)^(-1)AWW^(-1) = A*(AW^2A*)^(-1)A = P. From the semidefinite form we obtain an associated semidefinite quadratic form b*Pb and from this the seminorm sqrt(b*Pb).
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monzo b, bA*Ab represents the
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it is sqrt(tPt*)sqrt(t*Pt) where t
...
t = bA*Ab
The OETES
Instead of starting from a matrix of vals, we may start from a matrix of monzos. If B is a matrix with rows of monzos spanning the commas of a regular temperament, then M = BW^(-1) is the corresponding weighted matrix. Q = M`M is a projection matrix dual to P = I-Q, where I is the identity matrix, and P is the same symmetric matrix as in the previous section. If the rows define a basis for the commas of the temperament, and are therefor linearly independent, then P = I - M*(MM*)^(-1)M = I - W^(-1)B*(BW^(-2)B*)^(-1)BW^(-1), and mPm* = bW^(-1)PW^(-1)b*, or b(W^(-2) - W^(-2)B*(BW^(-2)B*)^(-1)BW^(-2))b*, so that the terms inside the parenthesis define a formula for P in terms of the matrix of monzos B.
pergen
edited
... More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most…
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More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if unbroken.
For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).
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the third-splits. TheA split intervalsinterval is only included if it falls in the range from
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genchain of 5ths.5ths, others are too remote. For convenience,
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pergen