Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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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 **~~multi-gen~~**. The 3-limit ~~multi-gen~~ 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, which means "semi-fourth", is of course half-fourth.

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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**.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the ~~multi-gen~~, and as a tie-breaker, to minimize the size in cents of the ~~multi-gen~~. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

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).

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

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

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[xenharmonic/Kite's color notation|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.

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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.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the ~~multi-gen~~ does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF|notation guide]]. It even allows every pergen to be numbered.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF|notation guide]]. It even allows every pergen to be numbered.

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.

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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.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? ~~Let ¢~~(R) be the ~~cents~~ of the ~~ratio R~~, and ~~let ¢[M] be~~ the cents of ~~some monzo M~~. If ~~we allow large commas~~, we can ~~specify that Q~~ = 5.

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 = 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 = 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.

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).

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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

A square mapping [(x, 0), (y, 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

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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

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A pergen (pronounced &quot;peer-gen&quot;) 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 ~~multi-gen~~. The 3-limit ~~multi-gen~~ 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, which means &quot;semi-fourth&quot;, is of course half-fourth.

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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 &gt; 3 (a higher prime), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called unsplit.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the ~~multi-gen~~, and as a tie-breaker, to minimize the size in cents of the ~~multi-gen~~. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.

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).

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

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

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation">Color notation</a> (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.

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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.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the ~~multi-gen~~ does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF">notation guide</a>. It even allows every pergen to be numbered.

Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF">notation guide</a>. It even allows every pergen to be numbered.

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.

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finish proofs

link from: ups and downs page, Kite Giedraitis page, MOS scale names page,

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments">http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments">http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments</a>

<h3 id="toc22"><a name="Further Discussion-Supplemental materials*-Notaion guide PDF"></a>Notaion guide PDF</h3>

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.

<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>

<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>

(screenshot)

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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.

<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>

<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>

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:

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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 &gt; 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.

Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? ~~Let ¢~~(R) be the ~~cents~~ of the ~~ratio R~~, and ~~let ¢[M] be~~ the cents of ~~some monzo M~~. If ~~we allow large commas~~, we can ~~specify that Q~~ = 5.

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 = GCD (a+b, b) = GCD (a, b) = 1

Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 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 = 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.

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).

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Given:

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

A square mapping [(x, 0), (y, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x

To prove: if |z| = 1, n = 1

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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 &gt; 1, therefore |b| &gt; 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

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-02 14:59:50 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-03 05:10:43 UTC</tt>.<br>
-: The original revision id was <tt>627126229</tt>.<br>
+: The original revision id was <tt>627137807</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 13: / Line 13: @@
 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 **multi-gen**. The 3-limit multi-gen 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, which means "semi-fourth", is of course half-fourth.
@@ Line 21: / Line 21: @@
 The largest category contains all single-comma temperaments with a comma of the form 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, where P is a prime &gt; 3 (a **higher prime**), e.g. 81/80 or 135/128. It also includes all commas in which the higher-prime exponents are setwise coprime. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**.
-Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.
+Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.
 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).
@@ Line 43: / Line 43: @@
 (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-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).
+The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).
 Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The pergen always uses at least one higher prime. [[xenharmonic/Kite's color notation|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.
@@ Line 133: / Line 133: @@
 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.
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF|notation guide]]. It even allows every pergen to be numbered.
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a [[pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF|notation guide]]. It even allows every pergen to be numbered.
 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.
@@ Line 911: / Line 911: @@
 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 &gt; 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.
-Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? Let ¢(R) be the cents of the ratio R, and let ¢[M] be the cents of some monzo M. If we allow large commas, we can specify that Q = 5.
+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 = GCD (a+b, b) = GCD (a, b) = 1
+Since the pergen is a double-split, m &gt; 1, therefore |b| &gt; 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 = 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.
+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).
@@ Line 952: / Line 976: @@
 Given:
-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
+A square mapping [(x, 0), (y, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |i| &lt;= x
 To prove: if |z| = 1, n = 1
@@ Line 967: / Line 991: @@
 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 &gt; 1, therefore |b| &gt; 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
@@ Line 1,083: / Line 1,089: @@
 A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) 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.&lt;br /&gt;
 &lt;br /&gt;
-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 &lt;strong&gt;split&lt;/strong&gt; into N parts. The interval which is split into multiple generators is the &lt;strong&gt;multi-gen&lt;/strong&gt;. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.&lt;br /&gt;
+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 &lt;strong&gt;split&lt;/strong&gt; into N parts. The interval which is split into multiple generators is the &lt;strong&gt;multigen&lt;/strong&gt;. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.&lt;br /&gt;
 &lt;br /&gt;
 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 &amp;quot;semi-fourth&amp;quot;, is of course half-fourth.&lt;br /&gt;
@@ Line 1,091: / Line 1,097: @@
 The largest category contains all single-comma temperaments with a comma of the form 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P or 2&lt;span style="vertical-align: super;"&gt;x &lt;/span&gt;3&lt;span style="vertical-align: super;"&gt;y &lt;/span&gt;P&lt;span style="vertical-align: super;"&gt;-1&lt;/span&gt;, where P is a prime &amp;gt; 3 (a &lt;strong&gt;higher prime&lt;/strong&gt;), 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 &lt;strong&gt;unsplit&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
-Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multi-gen, and as a tie-breaker, to minimize the size in cents of the multi-gen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.&lt;br /&gt;
+Every temperament has at least one alternate generator, and more, if the octave is split. To avoid ambiguity, the generator is chosen to minimize the amount of splitting of the multigen, and as a tie-breaker, to minimize the size in cents of the multigen. There is only one exception to this rule: the fifth is preferred over the fourth, to follow historical precedent.&lt;br /&gt;
 &lt;br /&gt;
 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 &lt;u&gt;not&lt;/u&gt; preferred over P4/2. For example, decimal is (P8/2, P4/2), not (P8/2, P5/2).&lt;br /&gt;
@@ Line 1,316: / Line 1,322: @@
 (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.&lt;br /&gt;
 &lt;br /&gt;
-The multi-gen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).&lt;br /&gt;
+The multigen is usually some voicing of the 4th or 5th, but can be any 3-limit interval, as in the second to last example. The color name indicates the amount of splitting: deep (double) splits something into two parts, triple into three parts, etc. For a comma with monzo (a,b,c,d...), the color depth is GCD (c,d...).&lt;br /&gt;
 &lt;br /&gt;
 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. &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; (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.&lt;br /&gt;
@@ Line 1,635: / Line 1,641: @@
 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.&lt;br /&gt;
 &lt;br /&gt;
-Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multi-gen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF"&gt;notation guide&lt;/a&gt;. It even allows every pergen to be numbered.&lt;br /&gt;
+Below is a table that lists all the rank-2 pergens that contain primes 2 and 3, up to third-splits. They are grouped into blocks by the size of the larger splitting fraction, and grouped within each block into sections by the smaller fraction. Most sections have two halves. In the first half, the octave has the larger fraction, in the second, the multigen does. Within each half, the pergens are sorted by multigen size. This is a convenient lexicographical ordering of rank-2 pergens that enables one to easily look up a pergen in a &lt;a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials-Notaion%20guide%20PDF"&gt;notation guide&lt;/a&gt;. It even allows every pergen to be numbered.&lt;br /&gt;
 &lt;br /&gt;
 The enharmonic interval, or more briefly the &lt;strong&gt;enharmonic&lt;/strong&gt;, 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.&lt;br /&gt;
@@ Line 5,717: / Line 5,723: @@
 finish proofs&lt;br /&gt;
 link from: ups and downs page, Kite Giedraitis page, MOS scale names page,&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7127:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7127 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7133:http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments"&gt;http://xenharmonic.wikispaces.com/Proposed+names+for+rank+2+temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7133 --&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7128:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7128 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7134:http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments --&gt;&lt;a href="http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments"&gt;http://xenharmonic.wikispaces.com/Tour+of+Regular+Temperaments&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7134 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:101:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc22"&gt;&lt;a name="Further Discussion-Supplemental materials*-Notaion guide PDF"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:101 --&gt;Notaion guide PDF&lt;/h3&gt;
   &lt;br /&gt;
 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.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextUrlRule:7129:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7129 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextUrlRule:7135:http://www.tallkite.com/misc_files/pergens.pdf --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow"&gt;http://www.tallkite.com/misc_files/pergens.pdf&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:7135 --&gt;&lt;br /&gt;
 (screenshot)&lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,733: / Line 5,739: @@
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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.&lt;br /&gt;
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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:&lt;br /&gt;
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 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 &amp;gt; 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.&lt;br /&gt;
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-Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q? Let ¢(R) be the cents of the ratio R, and let ¢[M] be the cents of some monzo M. If we allow large commas, we can specify that Q = 5.&lt;br /&gt;
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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?&lt;br /&gt;
+(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5&lt;br /&gt;
+Because n is a multiple of b, n/b is an integer&lt;br /&gt;
+M/b = (n/b)·M/n = (n/b)·G&lt;br /&gt;
+(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)&lt;br /&gt;
+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&lt;br /&gt;
+Since the pergen is a double-split, m &amp;gt; 1, therefore |b| &amp;gt; 1, therefore c ≠ 0&lt;br /&gt;
+c·(a+b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;
+(1 - d·b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;
+P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)&lt;br /&gt;
+P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)&lt;br /&gt;
+Therefore P8 is split into m periods&lt;br /&gt;
+Therefore if m = |b|, the pergen is explicitly false&lt;br /&gt;
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+Assuming r = 1, can we prove the existence of C = (u, v, w) for some prime Q?&lt;br /&gt;
+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.&lt;br /&gt;
+a·P8 + C = (n·x, n·y - b, ±n) = (qbx, qby - b, ±qb) = b·(q·x, q·y - 1, ±q)&lt;br /&gt;
+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.&lt;br /&gt;
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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).&lt;br /&gt;
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 Given:&lt;br /&gt;
-A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &amp;gt; 0, z ≠ 0, and |i| &amp;lt;= x&lt;br /&gt;
+A square mapping [(x, 0), (y, z)] creates the pergen (P8/x, (i·z - y, x) / xz), with x &amp;gt; 0, z ≠ 0, and |i| &amp;lt;= x&lt;br /&gt;
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 To prove: if |z| = 1, n = 1&lt;br /&gt;
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 Therefore multigens like M9/3 or M3/4 never occur&lt;br /&gt;
 Therefore a and b must be coprime, otherwise M/n could be simplified by GCD (a,b)&lt;br /&gt;
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-To prove: test for explicitly false&lt;br /&gt;
-If m = |b|, is the pergen explicitly false?&lt;br /&gt;
-Does (a,b)/n split P8 into m periods?&lt;br /&gt;
-(a+b)·P8 = (a+b,0) = (a,b) - (-b,b) = M - b·P5&lt;br /&gt;
-Because n is a multiple of b, n/b is an integer&lt;br /&gt;
-M/b = (n/b)·M/n = (n/b)·G&lt;br /&gt;
-(a+b)·P8 = b·(M/b - P5) = b·((n/b)·G - P5)&lt;br /&gt;
-Let c and d be the bezout pair of a+b and b, with c·(a+b) + d·b = 1&lt;br /&gt;
-Since the pergen is a double-split, m &amp;gt; 1, therefore |b| &amp;gt; 1, therefore c ≠ 0&lt;br /&gt;
-c·(a+b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;
-(1 - d·b)·P8 = c·b·((n/b)·G - P5)&lt;br /&gt;
-P8 = d·b·P8 + c·b·((n/b)·G - P5) = b · (d·P8 + c·(n/b)·G - c·P5)&lt;br /&gt;
-P8/m = P8/|b| = sign (b) · (d·P8 - c·P5 + c·(n/b)·G)&lt;br /&gt;
-Therefore P8 is split into m periods&lt;br /&gt;
-Therefore if m = |b|, the pergen is explicitly false&lt;br /&gt;
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 To prove: true/false test&lt;br /&gt;