Pergen names: Difference between revisions

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A **pergen** (pronounced "peer-gen") is a way of identifying a regular temperament solely by its period and generator(s). Pergens group all regular temperaments into broad categories.  

 

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. An interval which is split into multiple generators is called a **multi-gen**. The 3-limit multi-gen is referred to not by its ratio but by its conventional name, e.g. P5, M6, etc.

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 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 \.

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 mapping** 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 example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &lt;= n &lt;= 1. No value of n reduces the fraction, so the best generator is the one with the least cents. The pergen is (P8, P4/3).

Thus the period = (4, 0, 0)/4 = 2/1= P8, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = (25/6) ^ (1/4) = WWyy1/4.

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multi-gen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a __double__ octave to the multi-gen. The alternate gens are WWP5/2 and P11/2, both of which are larger, so the best gen1 is P5/2.

The 2nd multi-gen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be Wgg8/4 (Wgg8 = 96/25). A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 (gg7 = 128/75). Subtracting M9 again, and inverting again, makes gen2 = (-9, 3, 2)/4 = Lyy3/4 (Lyy3 = 675/512). As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.

Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is.

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 \. Dv\ is down-low D, and /5 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.

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 maps to the enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.

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). Pergens group all regular temperaments into broad categories. &lt;br /&gt;

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;

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. An interval which is split into multiple generators is called a &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, etc.&lt;br /&gt;

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;

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;

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 &lt;strong&gt;square mapping&lt;/strong&gt; 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;

For example, 250/243 has a mapping [(1, 2, 3) (0, -3, -5)] and a square mapping of [(1, 2) (0, -3)]. The pergen is (P8/1, (-3n-2,1)/(-3)) = (P8, (3n+2,-1)/3), with -1 &amp;lt;= n &amp;lt;= 1. No value of n reduces the fraction, so the best generator is the one with the least cents. The pergen is (P8, P4/3).&lt;br /&gt;

Thus the period = (4, 0, 0)/4 = 2/1= P8, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = (25/6) ^ (1/4) = WWyy1/4.&lt;br /&gt;

Next, search for alternate generators. Add/subtract the period 2/1 from gen1. Since the multi-gen P5 is split in half, one multi-gen equals two gens, and adding an octave to the gen adds a &lt;u&gt;double&lt;/u&gt; octave to the multi-gen. The alternate gens are WWP5/2 and P11/2, both of which are larger, so the best gen1 is P5/2.&lt;br /&gt;

The 2nd multi-gen is split into quarters, so we must add/subtract quadruple periods and generators. Subtracting a quadruple octave and inverting makes gen2 be Wgg8/4 (Wgg8 = 96/25). A quadruple half-fifth is a double fifth is a M9. Subtracting that makes gen2 be gg7/4 (gg7 = 128/75). Subtracting M9 again, and inverting again, makes gen2 = (-9, 3, 2)/4 = Lyy3/4 (Lyy3 = 675/512). As gen2's cents become smaller, the odd limit becomes greater, and the notation remains awkward.&lt;br /&gt;

Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is (P8, P5/2, r1) = half-fifth with red. Assuming 7's notational comma is 64/63, the pergen is (P8, P5/2, ^1), half-fifth with ups. This is far better than (P8, P5/2, gg7/4). The pergen sometimes uses a larger prime in place of a smaller one in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;

One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo-pergens, because the multigen isn't 3-limit. Meantone (mean = average, tone = major 2nd) implies y3/2. Semisixth (aka sensei) implies y6/2. Sometimes the multi-gen isn't even a multi-gen: y3/2 isn't actually a generator, although y6/2 is.&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 \. Dv\ is down-low D, and /5 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;

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 maps to the enharmonic. This interval is very important. Everything about the notation can be deduced from the pergen and the enharmonic.&lt;br /&gt;