Kite's thoughts on pergens: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
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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. | 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. | ||
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, | 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, a pun on "semi-fourth", is of course half-fourth. | ||
Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to a few dozen categories. It focuses on the melodic properties of the temperament, not the harmonic properties. MOS scales in both srutal 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 Notation|ups and downs]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]]. | 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 Notation|ups and downs]]. See the notation guide below, under [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]]. | ||
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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: | 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 | * **A double-split pergen is __explicitly false__ if m = |b|, and not 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 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 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. | 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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||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ??? ||= 81/80 || | ||= 10edo+y ||= (P8/10, /1) ||= rank-2 10-edo ||= E = m2, E' = vvA1 = vvM2 ||= D D^=Ev E=F F^=Gv G... ||= D F#\=G\ B\\... ||= ??? ||= 81/80 || | ||
||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- || | ||= 12edo+j ||= (P8/12, ^1) ||= rank-2 12-edo ||= E = d2 ||= D D#=Eb E F F#=Gb... ||= D G^ C^^ ||= 33/32 ||= --- || | ||
||= | ||= " ||= " ||= " ||= " ||= " ||= D G#v=Abv Dvv... ||= 729/704 = ||= --- || | ||
||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 || | ||= 17edo+y ||= (P8/17, /1) ||= rank-2 17-edo ||= E = dd3, E' = vm2 = vvA1 ||= D D^=Eb D#=Ev E F... ||= D F#\ A#\\=Bv\\... ||= 256/243 ||= 81/80 || | ||
||= ||= ||= ||= ||= ||= ||= ||= || | ||= ||= ||= ||= ||= ||= ||= ||= || | ||
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||= 560-585¢ ||= P11/3 ||= ||= ||= ||= || | ||= 560-585¢ ||= P11/3 ||= ||= ||= ||= || | ||
||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= || | ||= 576-591¢ ||= WWP4/5 ||= 583-593¢ ||= WWWP4/7 ||= ||= || | ||
There 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 of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a WWP4. | |||
Splitting the octave creates alternate generators. For example, if P = 400¢ and G = | The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen. | ||
See also the [[Map of rank-2 temperaments|map of rank-2 temperaments]]. | See also the [[Map of rank-2 temperaments|map of rank-2 temperaments]]. | ||
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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. | 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. | ||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 11-edo and 23-edo could also be considered ambiguous. | ||
||||~ pergen ||~ supporting edos (12-31 only) || | ||||~ pergen ||~ supporting edos (12-31 only) || | ||
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||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= **1\22= 1\24**, 10\22 = 11\24 ||= 13\22 = 14\24 ||= half-8ve quarter-tone ||= || | ||= 22 & 24 ||= 2 ||= 11\22 = 12\24 ||= **1\22= 1\24**, 10\22 = 11\24 ||= 13\22 = 14\24 ||= half-8ve quarter-tone ||= || | ||
A specific pergen can be converted to an edo pair by | A specific pergen can be converted to an edo pair by finding the range of its generator cents in the [[pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator|arbitrary generator]] table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2). | ||
==Supplemental materials*== | ==Supplemental materials*== | ||
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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 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. | 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: | 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: | ||
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G = (a/n, b/n, 0) | G = (a/n, b/n, 0) | ||
C = (u, v, w) | 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. | 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. | ||
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Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C) | Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C) | ||
Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in | Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| > 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular **//[I think, not sure, could it be i or 1/i for some integer i?]//**, and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have: | ||
2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C) | ||
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Let ¢(R) be the cents of the ratio R, and let ¢[M] be the cents of some monzo M. If we limit |v| to < 6, but allow large commas of up to 100¢, we can specify that Q = 5. | |||
¢(C) = u·¢(2) + v·¢(3) + w·¢(Q) < 100¢ | |||
( | Octave-reduce: | ||
¢(C) = u·1200¢ + v·1200¢ + v·702¢ + w·2400¢ + w·386¢ < 100¢ | |||
¢(C) mod 1200 = v·702¢ + w·386¢ < 100¢ | |||
GCD (v, w) = m | |||
GCD (u ± 1, v, w) = m (thus u mod m ≠ 0) | |||
Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? | Assuming r = 1, can we prove the existence of C = (u, v, w) for some 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. | 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) | a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q) | ||
Thus a·P8 splits | Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into 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). | 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). | ||
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Given: | Given: | ||
A square mapping [(x, | 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 | ||
To prove: if |z| = 1, n = 1 | To prove: if |z| = 1, n = 1 | ||
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Therefore multigens like M9/3 or M3/4 never occur | 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) | Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b) | ||
To prove: true/false test | To prove: true/false test | ||
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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 <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multigen</strong>. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | 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 <strong>split</strong> into N parts. The interval which is split into multiple generators is the <strong>multigen</strong>. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.<br /> | ||
<br /> | <br /> | ||
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, | 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, a pun on &quot;semi-fourth&quot;, is of course half-fourth.<br /> | ||
<br /> | <br /> | ||
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 <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a>.<br /> | 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 <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. See the notation guide below, under <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a>.<br /> | ||
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<br /> | <br /> | ||
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 <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize:<br /> | 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 <u>true doubles require commas of at least 7-limit</u>, whereas false doubles require only 5-limit. To summarize:<br /> | ||
<ul><li><strong>A double-split pergen is <u>explicitly false</u> if and | <ul><li><strong>A double-split pergen is <u>explicitly false</u> if m = |b|, and not explicitly false if m &gt; |b|.</strong></li><li><strong>A double-split pergen is a <u>true double</u> if and only if</strong> <strong>neither it nor its unreduced form is explicitly false</strong><strong>.</strong></li><li><strong>A double-split pergen is a <u>true double</u> if</strong> <strong>GCD (m, n) &gt; |b|,</strong> <strong>and a false double if GCD (m, n) = |b|.</strong></li></ul><br /> | ||
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.<br /> | 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.<br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;">&quot;<br /> | ||
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<td style="text-align: center;">D G#v=Abv Dvv...<br /> | <td style="text-align: center;">D G#v=Abv Dvv...<br /> | ||
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<td style="text-align: center;">729/704 =<br /> | <td style="text-align: center;">729/704 =<br /> | ||
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<td style="text-align: center;"><br /> | <td style="text-align: center;">---<br /> | ||
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</table> | </table> | ||
There 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 of possible generators is mostly well covered, providing convenient notation options. In particular, if the tuning's generator is just over 720¢, to avoid descending 2nds, instead of calling the generator a 5th, one can call its inverse a quarter-12th. The generator is notated as an up-5th, and four of them make a WWP4.<br /> | |||
<br /> | <br /> | ||
Splitting the octave creates alternate generators. For example, if P = 400¢ and G = | The table assumes unsplit octaves. Splitting the octave creates alternate generators. For example, if P = P8/3 = 400¢ and G = 300¢, alternate generators are 100¢ and 500¢. Any of these can be used to find a convenient multigen.<br /> | ||
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See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br /> | See also the <a class="wiki_link" href="/Map%20of%20rank-2%20temperaments">map of rank-2 temperaments</a>.<br /> | ||
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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.<br /> | 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.<br /> | ||
<br /> | <br /> | ||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 11-edo and 23-edo could also be considered ambiguous.<br /> | ||
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A specific pergen can be converted to an edo pair by | A specific pergen can be converted to an edo pair by finding the range of its generator cents in the <a class="wiki_link" href="/pergen#Further%20Discussion-Notating%20tunings%20with%20an%20arbitrary%20generator">arbitrary generator</a> table, looking up that cents in the scale tree, and finding a conveniently-sized parent-child pair of edos in that range. For example, half-5th has a generator in the 320-360¢ range, and that part of the scale tree has among others 2\7, 3\10 and 5\17. Any two of those three edos defines (P8, P5/2).<br /> | ||
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finish proofs<br /> | finish proofs<br /> | ||
link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | link from: ups and downs page, Kite Giedraitis page, MOS scale names page,<br /> | ||
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<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7144:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --><a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a><!-- ws:end:WikiTextUrlRule:7144 --><br /> | ||
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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.<br /> | 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.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7145:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:7145 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
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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.<br /> | 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.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7146:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:7146 --><br /> | ||
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Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:<br /> | Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red. Screenshots of the first 38 pergens:<br /> | ||
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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 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.<br /> | 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.<br /> | ||
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If m = |b|, is the pergen explicitly false? Does splitting (a,b) into n generators also split P8 into m periods?<br /> | |||
(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5<br /> | |||
Because n is a multiple of b, n/b is an integer<br /> | |||
M/b = (n/b)·M/n = (n/b)·G<br /> | |||
(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)<br /> | |||
Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1<br /> | |||
Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 1, therefore c ≠ 0<br /> | |||
c·(a+b)·P8 = c·b·((n/b)·G - P5)<br /> | |||
(1 - d·b)·P8 = c·b·((n/b)·G - P5)<br /> | |||
P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)<br /> | |||
P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)<br /> | |||
Therefore P8 is split into m periods<br /> | |||
Therefore if m = |b|, the pergen is explicitly false<br /> | |||
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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:<br /> | 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:<br /> | ||
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G = (a/n, b/n, 0)<br /> | G = (a/n, b/n, 0)<br /> | ||
C = (u, v, w)<br /> | C = (u, v, w)<br /> | ||
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<em>If the pergen is explicitly false, and m = |b|, let w = ±n.</em><br /> | |||
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Here u, v and w are integers. If GCD (u, v, w) &gt; 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.<br /> | Here u, v and w are integers. If GCD (u, v, w) &gt; 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.<br /> | ||
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Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)<br /> | Q = Q/1 = ((av-bu)m/wb, -vn/wb, 1/w) · (P, G, C)<br /> | ||
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Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. To avoid fractions in | Fractions are allowed in the first two rows of A but not the 3rd row. Fractions are allowed in the last column of A-inverse, but not the first two columns. Every pergen except the unsplit one requires |w| &gt; 1, so the last column almost always has a fraction. To avoid fractions in the first two columns, A must be unimodular <strong><em>[I think, not sure, could it be i or 1/i for some integer i?]</em></strong>, and we have wb/mn = ±1, and w = ±mn/b. Substituting for w, we have:<br /> | ||
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2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | 2 = 2/1 = P8 = (m, 0, 0) · (P, G, C)<br /> | ||
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Let ¢(R) be the cents of the ratio R, and let ¢[M] be the cents of some monzo M. If we limit |v| to &lt; 6, but allow large commas of up to 100¢, we can specify that Q = 5.<br /> | |||
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¢(C) = u·¢(2) + v·¢(3) + w·¢(Q) &lt; 100¢<br /> | |||
( | Octave-reduce:<br /> | ||
¢(C) = u·1200¢ + v·1200¢ + v·702¢ + w·2400¢ + w·386¢ &lt; 100¢<br /> | |||
¢(C) mod 1200 = v·702¢ + w·386¢ &lt; 100¢<br /> | |||
( | <br /> | ||
GCD (v, w) = m<br /> | |||
GCD (u ± 1, v, w) = m (thus u mod m ≠ 0)<br /> | |||
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Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?<br /> | Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? <br /> | ||
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.<br /> | 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.<br /> | ||
a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)<br /> | a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)<br /> | ||
Thus a·P8 splits | Thus a·P8 splits into b parts, and since m = |b|, a·P8 splits into m parts. Proceed as before with a bezout pair to find the monzo for P8/m.<br /> | ||
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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).<br /> | 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).<br /> | ||
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Given:<br /> | Given:<br /> | ||
A square mapping [(x, | A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x<br /> | ||
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To prove: if |z| = 1, n = 1<br /> | To prove: if |z| = 1, n = 1<br /> | ||
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Therefore multigens like M9/3 or M3/4 never occur<br /> | Therefore multigens like M9/3 or M3/4 never occur<br /> | ||
Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)<br /> | Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)<br /> | ||
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To prove: true/false test<br /> | To prove: true/false test<br /> | ||