Pergen names: Difference between revisions

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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 &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;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;**

For example, 250/243

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names.

Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.

Most rank-2 temperaments require an additional pair of accidentals, [[Ups and Downs Notation|ups and downs]]. Certain rank-2 temperaments require another additional pair, highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.

||= " ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3 &lt;/span&gt;``=`` D ||= P11/3 = ^P4 = vvP5 ||= C F^ Cv F ||=  ||= " ||

Every double-split pergen is either a **true double** or a **false double**. A true double, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and requirea double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multi-gen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts.

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is **explicitly false**. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires R ~ 50¢, perhaps 33/32, and the comma is 4*R - M2 = (33/32)^4 * (9/8)^-1.

If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3. The comma splits both the octave and the fifth.

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) can be constructed from 128/125 and 49/48, which split the octave and the 4th respectively.

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;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;

For example, 250/243&lt;br /&gt;

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names.&lt;br /&gt;

Also, pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments can be notated identically. They require only conventional notation: 7 nominals, plus sharps and flats.&lt;br /&gt;

Most rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. Certain rank-2 temperaments require another additional pair, highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y, g, r, b, j, a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are (P8, P5/2). Using y and g implies dicot, using j and a implies mohajira, but using ^ and v implies neither, and is a more general notation.&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;C F^ Cv F&lt;br /&gt;

Every double-split pergen is either a &lt;strong&gt;true double&lt;/strong&gt; or a &lt;strong&gt;false double&lt;/strong&gt;. A true double, like half-everything (P8/2, P4/2) or third-everything (P8/3, P4/3), can only arise when at least two commas are tempered out, and requirea double pair notation. A false double, like half-octave quarter-tone (P8/2, M2/4), can arise from a single comma, and can be notated with single pair notation. Thus a false double behaves like a single-split, and is easier to construct and easier to notate. In a false double, the multi-gen split automatically splits the octave as well: if M2 = 4 gen, then P8 = M9 - M2 = 2*P5 - 4 gen = (P5 - 2 gen) / 2. In general, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts.&lt;br /&gt;

Finding the comma(s) for a double pergen is trickier. As previously noted, if a pergen's multi-gen is (a,b), the octave is split into at least |b| parts. Therefore if a pergen (P8/m, (a,b)/n) has m = |b|, it is &lt;strong&gt;explicitly false&lt;/strong&gt;. If so, proceed as if the octave were unsplit: (P8/2, M2/4) requires R ~ 50¢, perhaps 33/32, and the comma is 4*R - M2 = (33/32)^4 * (9/8)^-1.&lt;br /&gt;

If the pergen is not explicitly false, put the pergen in its &lt;strong&gt;unreduced&lt;/strong&gt; form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) = (P8/m, P8/m - M/n) = (P8/m, (n*P8 - m*M)/n*m). The new multigen is the product of the outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert the new multi-gen if descending, and reduce if possible. The new multi-gen will have either a larger fraction, or a larger size in cents, or both, hence the name unreduced.&lt;br /&gt;

For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2*P8 - 3*P5)/(3*2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This &lt;u&gt;is&lt;/u&gt; explicitly false, thus the comma can be found from m3/6 alone. R is about 50¢, and the comma is 6*R - m3. The comma splits both the octave and the fifth.&lt;br /&gt;

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) can be constructed from 128/125 and 49/48, which split the octave and the 4th respectively.&lt;br /&gt;