Pergen names: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-20 00:09:30 UTC</tt>.

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For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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]].

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than perhaps a hundred 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]].

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 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**.

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 64/63. 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~~. 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 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.

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, M2/4) ||= half-octave, quarter-tone ||= (-17,2,0,0,4) ||= large quadruple jade ||= large quadruple jade ||= Lj4T ||

||= (P8, P12/5) ||= fifth-twelfth ||= (-10,-1,5) ||= magic ||= large quintuple yellow ||= Ly5T ||

(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum of octaves and fifths, every single 3-limit interval is also split in half.

(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 color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. 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

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For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third.

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid ~~higher primes~~, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. ~~However,~~ if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid colors, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.

Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.

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To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes > 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. ~~For a~~ period P and a generator G:

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n

~~P8 =~~ 2/1 = xP ~~and P12~~ = 3/1 = yP + zG

2/1 = P8 = xP, thus P = P8/x

~~P = P8/x~~

3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).

G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz

__**The rank-2 pergen from the [(x, 0), (y, z)] matrix ~~= {~~P8/x, (nz-y, x)/xz}**__

**The rank-2 pergen from the [(x, 0), (y, z)] square matrix is (P8/x, (nz-y, x)/xz), with -x <= n <= x**

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:

For example, 250/243

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:

||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||

||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||

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For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &quot;semi-fourth&quot;, is of course half-fourth.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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>.

Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than perhaps a hundred 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>.

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 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called unsplit.

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 64/63. 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~~. 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 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.

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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</table>

(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 of octaves and fifths, every single 3-limit interval is also split in half.

(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 color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. 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

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For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation</a> can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third.

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid ~~higher primes~~, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. ~~However,~~ if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is highs and lows, written / and \.

Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid colors, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is highs and lows, written / and \.

Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.

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To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix, it's the reduced mapping. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes &gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. ~~For a~~ period P and a generator G:

For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n

~~P8 =~~ 2/1 = xP ~~and P12 = 3/1 = yP + zG ~~

2/1 = P8 = xP, thus P = P8/x

P = P8/x

3/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz

To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).

G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz

<~~strong~~>The rank-2 pergen from the [(x, 0), (y, z)] matrix ~~= {~~P8/x, (nz-y, x)/xz}

The rank-2 pergen from the [(x, 0), (y, z)] square matrix is (P8/x, (nz-y, x)/xz), with -x &lt;= n &lt;= x

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:

For example, 250/243

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:

@@ 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>2017-12-19 23:50:06 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-20 00:09:30 UTC</tt>.<br>
-: The original revision id was <tt>624088505</tt>.<br>
+: The original revision id was <tt>624088725</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 15: / Line 15: @@
 For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means "semi-fourth", is of course half-fourth.
-Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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]].
+Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than perhaps a hundred 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]].
-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 64/63. The period is the octave, and the generator is the fifth: (P8, P5). Such temperaments are called **unsplit**.
+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 64/63. 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. 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 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.
 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 40: / Line 40: @@
 ||= (P8/2, M2/4) ||= half-octave, quarter-tone ||= (-17,2,0,0,4) ||= large quadruple jade ||= large quadruple jade ||= Lj&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;T ||
 ||= (P8, P12/5) ||= fifth-twelfth ||= (-10,-1,5) ||= magic ||= large quintuple yellow ||= Ly&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;T ||
-(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 of octaves and fifths, every single 3-limit interval is also split in half.
+(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 color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. 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
@@ Line 46: / Line 46: @@
 For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. [[Kite's color notation|Color notation]] can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third.
-Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid higher primes, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. However, if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.
+Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid colors, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is **highs and lows**, written / and \.
 Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.
@@ Line 64: / Line 64: @@
 To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix, it's the reduced mapping. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes &gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.
-For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. For a period P and a generator G:
+For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n
-P8 = 2/1 = xP and P12 = 3/1 = yP + zG
+/1 = P8 = xP, thus P = P8/x
-P = P8/x
+/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz
-G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz
 To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).
 G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz
-&lt;span style="display: block; text-align: center;"&gt;__**The rank-2 pergen from the [(x, 0), (y, z)] matrix = {P8/x, (nz-y, x)/xz}**__&lt;/span&gt;
+&lt;span style="display: block; text-align: center;"&gt;**&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, 0), (y, z)] square matrix is (P8/x, (nz-y, x)/xz), with -x &lt;= n &lt;= x&lt;/span&gt;**
-Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7|x31.com]] gives us this matrix:
+&lt;/span&gt;
+For example, 250/243
+Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7|x31.com]] gives us this matrix:
 ||~   ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||
 ||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||
@@ Line 402: / Line 404: @@
 For example, the srutal temperament splits the octave in two, and its pergen name is half-octave. The pergen is written (P8/2, P5). Not only the srutal temperament, but also the srutal comma is said to split the octave. The dicot temperament splits the fifth in two, and is called half-fifth, written (P8, P5/2). Porcupine is third-fourth, (P8, P4/3). Semaphore, which means &amp;quot;semi-fourth&amp;quot;, is of course half-fourth.&lt;br /&gt;
 &lt;br /&gt;
-Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than a hundred 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 &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;.&lt;br /&gt;
+Many temperaments share the same pergen. This has the advantage of reducing the thousands of temperament names to less than perhaps a hundred 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 &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-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 64/63. 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;
+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 64/63. 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. 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 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;
 &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 643: / Line 645: @@
 &lt;/table&gt;
-(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 of octaves and fifths, every single 3-limit interval is also split in half.&lt;br /&gt;
+(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 color name indicates the amount of splitting: deep splits something into two parts, triple into three parts, etc. 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&lt;br /&gt;
@@ Line 649: / Line 651: @@
 For non-standard prime groups, the period uses the first prime only, and the multi-gen usually (see the 1st example in the Derivation section) uses the first two primes only. &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Color notation&lt;/a&gt; can be used to indicate higher primes. 5/4 = y3 = yellow 3rd, 7/4 = b7 = blue 7th, and 11/8 = j4 = jade 4th. 2.5.7 with 50/49 tempered out is (P8/2, y3) = half-octave, yellow-third.&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 unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid higher primes, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. However, if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \.&lt;br /&gt;
+Rank-3 pergens have three intervals, period, gen1 and gen2, any or all of which may be split. The unsplit 2.3.5 subgroup's pergen could be (P8, P5, y3). However, to avoid colors, and more closely mimic conventional notation, it's better to reduce gen2 to g1 = 81/80. Since 81/80 is a perfect unison, it can be notated with an up symbol, and we have (P8, P5, ^1) = unsplit with ups. But if either the period or gen1 is split, ups and downs will be needed to notate the fractional intervals, and an additional pair of accidentals must be used for gen2. One possibility is &lt;strong&gt;highs and lows&lt;/strong&gt;, written / and \.&lt;br /&gt;
 &lt;br /&gt;
 Examples: Marvel (2.3.5.7 and 225/224) is (P8, P5, ^1) = unsplit with ups (^1 = 81/80). Deep reddish (2.3.5.7 and 50/49) is (P8/2, P5, /1) = half-octave with highs (/1 = 81/80). Triple bluish (2.3.5.7 and 1029/1000) is (P8, P11/3, ^1) = third-eleventh with ups.&lt;br /&gt;
@@ Line 667: / Line 669: @@
 To find a temperament's pergen, first find the period-generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Not all such mappings will work, the matrix must be in row echelon form. Graham Breed's website has a temperament finder &lt;a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank"&gt;x31eq.com/temper/uv.html&lt;/a&gt; that will find such a matrix, it's the reduced mapping. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous (P8/5,y3) example, to ensure that the diagonal has no zeros. Lower primes &amp;gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.&lt;br /&gt;
 &lt;br /&gt;
-For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. For a period P and a generator G:&lt;br /&gt;
+For rank-2, we can compute the pergen directly from the square matrix = [(x y), (0, z)]. Let the pergen be (P8/m, M/n), where M is the multi-gen, P is the period P8/m and G is the generator M/n&lt;br /&gt;
-P8 = 2/1 = xP and P12 = 3/1 = yP + zG&lt;br /&gt;
+/1 = P8 = xP, thus P = P8/x &lt;br /&gt;
-P = P8/x&lt;br /&gt;
+/1 = P12 = yP + zG, thus G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz&lt;br /&gt;
-G = [P12 - y(P8/x)] / z = [-yP8 + xP12]/xz = (-y, x) / xz&lt;br /&gt;
 &lt;br /&gt;
 To get alternate generators, add n periods to G, with n ranging from -x (subtracting a full octave) to +x (adding a full octave).&lt;br /&gt;
 G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz&lt;br /&gt;
 &lt;br /&gt;
-&lt;span style="display: block; text-align: center;"&gt;&lt;u&gt;&lt;strong&gt;The rank-2 pergen from the [(x, 0), (y, z)] matrix = {P8/x, (nz-y, x)/xz}&lt;/strong&gt;&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="display: block; text-align: center;"&gt;&lt;strong&gt;&lt;span style="font-size: 110%;"&gt;The rank-2 pergen from the [(x, 0), (y, z)] square matrix is (P8/x, (nz-y, x)/xz), with -x &amp;lt;= n &amp;lt;= x&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
-Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;amp;limit=7" rel="nofollow"&gt;x31.com&lt;/a&gt; gives us this matrix:&lt;br /&gt;
+&lt;/span&gt;&lt;br /&gt;
+For example, 250/243&lt;br /&gt;
+&lt;br /&gt;
+Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. &lt;a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;amp;limit=7" rel="nofollow"&gt;x31.com&lt;/a&gt; gives us this matrix:&lt;br /&gt;