Kite's thoughts on pergens: Difference between revisions

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

<h4>Original Wikitext content:</h4>

This is a lot of math, but it only needs to be done once for each pergen!

==Alternate enharmonics==  

C1 --- G1 --- D2 --- A2

C3 Ev3 G3

||

===Notation guide PDF===  

Screenshots of the first 2 pages:

=== ===  

===alt-pergenLister===  

==Various proofs (unfinished)==  

The interval P8/2 has a "ratio" of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the **pergen matrix** [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.

__**Credits**__

<h4>Original HTML content:</h4>

  &lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

For any comma, let m = the GCD of all the monzo's exponents other than the 2-exponent, and let n = the GCD of all its higher-prime exponents, where GCD (0,x) = x. The comma will split the octave into m parts, and if n &amp;gt; m, it will split some 3-limit interval into n parts.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Pergens allow for a systematic exploration of all posible rank-2 tunings, potentially identifying new musical resources.&lt;br /&gt;

Some pergens are not very musically useful. (P8/2, P11/3) has a period of about 600¢ and a generator of about 566¢, or equivalently 34¢. The generator is much smaller than the period, and MOS scales will have a very lopsided L/s ratio. (P8/3, P5/2) is almost as lopsided (P = 400¢, G = 50¢).&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Removing the ups and downs from an enharmonic interval makes a &amp;quot;bare&amp;quot; enharmonic, a conventional 3-limit interval which vanishes in certain edos. For example, (P8/2, P5)'s enharmonic interval is ^^d2, the bare enharmonic is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. 12-edo's fifth is the &amp;quot;tipping point&amp;quot;: if the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Therefore &lt;u&gt;&lt;strong&gt;up may need to be swapped with down, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the above table, this is shown explicitly for (P8/2, P5), and implied for all the other pergens. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

  &lt;br /&gt;

So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, &lt;strong&gt;widening&lt;/strong&gt; by an 8ve is indicated by &amp;quot;W&amp;quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For a pergen with an unsplit octave, the multigen is some voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M is less than 3 octaves, and can be P4, P5, P11, P12, WWP4 or WWP5.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (e.g. porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a &lt;strong&gt;single-split&lt;/strong&gt; pergen. If it has two fractions, it's a &lt;strong&gt;double-split&lt;/strong&gt; pergen. A single-split pergen can result from tempering out only a single comma, although it can be created by multiple commas. A single-split pergen can be notated with only ups and downs, called &lt;strong&gt;single-pair&lt;/strong&gt; notation because it adds only a single pair of accidentals to conventional notation. &lt;strong&gt;Double-pair&lt;/strong&gt; notation uses both ups/downs and highs/lows. In general, single-pair notation is preferred, because it's simpler. However, double-pair notation may be preferred, especially if the enharmonic for single-pair notation is a 3rd or larger, or if single-pair notation requires using quadruple, quintuple, etc. ups and downs.&lt;br /&gt;

A pergen (P8/m, (a,b)/n) is a false double if and only if GCD (m,n) = |b|. The next section discusses an alternate test.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

To find an example of a temperament with a specific pergen, we must find the comma(s) the temperament tempers out. To construct a comma that creates a single-split pergen, find a ratio for P or G that contains only one higher prime, with color depth of 1 (i.e. exponent of ±1), of appropriate cents to add up to approximately the octave or the multigen. The comma is the difference between the stacked ratios and the larger interval. For example, (P8/4, P5) requires a P of about 300¢. The comma is the difference between 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P and P8. If P is 6/5, the comma is 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P - P8 = (6/5)&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ÷ (2/1) = 648/625, the diminished temperament. If P is 7/6, the comma is P8 - 4&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P = (2/1) · (7/6)&lt;span style="vertical-align: super;"&gt;-4&lt;/span&gt;, the quadruple red temperament. Neither 13/11 nor 32/27 would work for P, too many and too few higher primes respectively.&lt;br /&gt;

There are also alternate enharmonics, see below. For double-pair notation, there are also equivalent enharmonics.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These &lt;strong&gt;mapping commas&lt;/strong&gt; are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio &amp;quot;lands&amp;quot; on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.&lt;br /&gt;

Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:&lt;br /&gt;

This is a lot of math, but it only needs to be done once for each pergen!&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12·[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12·[4,2] = [-15,-5] = -5·[3,1] = -5·v&lt;span style="vertical-align: super;"&gt;12&lt;/span&gt;A2, which is an improvement but still awkward. The period is ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m3 and the generator is v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.&lt;br /&gt;

&lt;span style="display: block; text-align: center;"&gt;P1 -- vM3 -- ^m6 -- P8&lt;br /&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- Ev -- Ab^ -- C&lt;br /&gt;

Because the enharmonic found by rounding off the gedra is only an estimate that may need to be revised, there isn't any point to using gedras with something other than 12-edo keyspans, like 5edo or 19edo. Heptatonic stepspans are best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic.&lt;br /&gt;

&lt;br /&gt;

This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has three possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt; page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.&lt;br /&gt;

Chord progressions can be written out by applying ups and downs to the chord roots as needed, e.g. I.v -- vIII.v -- vVI.^m -- I.v. A porcupine (third-4th) comma pump can be written out like so: C.v -- Av.^m -- Dv.v -- [Bvv=Bb^]^m -- Eb^.v -- G.^m -- G.v -- C.v. Brackets are used to show that Bvv and Bb^ are enharmonically equivalent. The equivalence is shown roughly half-way through the pump. Bvv is written first to show that this root is a vM6 above the previous root, Dv. Bb^ is second to show the P4 relationship to the next root, Eb^. Such an equivalence of course couldn't be used on the staff, where the chord would be written as either Bvv.^m or Bb^.^m, or possibly Bb^.vvM = Bb^ Dv F^.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. As noted above, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But if a single temperament &amp;quot;tips over&amp;quot;, either the up symbol sometimes means down in pitch, or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. Fortunately, the temperament's &amp;quot;sweet spot&amp;quot;, where the damage to those JI ratios likely to occur in chords is minimized, rarely contains the tipping point.&lt;br /&gt;

Another &amp;quot;tippy&amp;quot; temperament is found by adding the mapping comma 81/80 to the negri comma and getting the large triple yellow comma (-18,7,3). This temperament is (P8, P11/3) with G = 45/32. The sweet spot is tenth-comma, even closer to 19edo's 5th.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

An unsplit pergen doesn't &lt;u&gt;require&lt;/u&gt; ups and downs, but they are generally needed for proper chord spellings. The only exception is when tempering out a mapping comma, such as 81/80 or 64/63. For single-comma rank-2 temperaments, the pergen is unsplit if and only if the vanishing comma's color depth is 1 (i.e. the monzo has a final exponent of ±1).&lt;br /&gt;

A similar table could be made for 7-limit commas of the form (a,b,0,±1). Every such temperament except for archy (2.3.7 and 64/63) might use ups and downs to spell 7/4 as a m7. A 7-limit temperament with two commas may need double-pair notation, even though its pergen is unsplit, to avoid spelling the 4:5:6:7 chord something like C D# G A#. However, if both commas map to the same 3-limit comma, only single-pair is needed. For example, 7-limit schismic tempers out both (-15,8,1) = -d2 and (25,-14,0,-1) = d2. The up symbol stands for the pythagorean comma, and the 4:5:6:7 chord is spelled C Ev G Bbv.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one. Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:&lt;br /&gt;

         &lt;th&gt;notation's rank&lt;br /&gt;

&lt;/th&gt;

&lt;/th&gt;

         &lt;th&gt;enharmonics&lt;br /&gt;

         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;

&lt;/td&gt;

&lt;/td&gt;

         &lt;td style="text-align: center;"&gt;/1 = 64/63&lt;br /&gt;

If using single-pair notation, marvel is notated like 5-limit JI, but the 4:5:6:7 chord is spelled C - Ev - G - A#vv. If double-pair notation is used, /1 = 64/63, and the chord is spelled C - Ev - G - Bb\.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

This 3-limit comma defines the tipping point. At the tipping point, the 3-limit comma vanishes too. In a rank-2 temperament, the mapping comma must also vanish, because some number of them plus the 3-limit comma must add up to the original comma, which vanishes. However, a rank-3 temperament has two mapping commas, and neither is forced to vanish if the 3-limit comma vanishes. A rank-3 double-pair notation's tipping point is where both mapping commas are tempered out. For deep reddish, this happens when the tuning is exactly 12edo. This tuning is much farther from just than need be, well outside the sweet spot. Therefore deep reddish doesn't tip. Single-pair rank-3 notation has no enharmonic, and thus no tipping point. Double-pair rank-3 notation has 1 enharmonic, but two mapping commas. Rank-3 notations rarely tip.&lt;br /&gt;

There are at least 100 third-splits and 287 quarter-splits. More columns could be added for ^1 = 33/32, ^1 = 729/704, ^1 = 27/26, etc.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

A Blackwood-like temperament is rank-2 and equates some number of 5ths to some number of 8ves, thus equating the 5th to some exact fraction of the octave. The 5th is not independent of the octave, thus it doesn't appear in the pergen. Such pergens make a lot of sense musically when the octave's splitting fraction corresponds to an edo with a 5th fairly close to just, like P8/5, P8/7, P8/10 and especially P8/12. Pergens which imply an edo which doesn't have a decent 5th, e.g. P8/3, P8/4, P8/6, etc., are covered in the next section.&lt;br /&gt;

Unlike Blackwood, the ups and downs are in the perchain, not the genchain. It would be possible to notate Blackwood similarly. The pergen would be not (P8/5, ^1), but (P8/5, M3). The perchain would be C D^^ Fv G^ Bbvv C and the genchain would be C E G#... But this is not recommended, because it would cause &amp;quot;missing notes&amp;quot; (see next section).&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

In Blackwood-like pergens, the 5th is present but not independent. In non-5th pergens, the 5th is not present, and the prime subgroup doesn't contain 3.&lt;br /&gt;

Every rank-3 pergen can also be identified by its prime subgroup and its pergen number. A similar table can be made for all rank-3 pergens. The 2.3.5 and 2.3.7 subgroups are listed in the section on rank-3 pergens. The 2.5.7 subgroup's unsplit pergen is (P8, M3, ^M2). The 3.5.7 subgroup's unsplit pergen is (P12, M6, ^M3).&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Pergen squares are a way to visualize pergens squares in a way that isn't specific to any primes at all, but assume the standard 2.3 prime subgroup for now. The genchain runs left to right along the top and bottom sides of the square. One horizontal side of the square equals one 5th. The perchain runs up the sides of the square. One vertical side of the square equals one octave. The pergen square is the building block of the rank-2 lattice. The complete lattice is formed by tiling many squares in all directions.&lt;br /&gt;

C1 --- G1 --- D2 --- A2&lt;br /&gt;

&lt;br /&gt;

C3 Ev3 G3&lt;br /&gt;

||&lt;br /&gt;

&lt;br /&gt;

Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;

A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Given only the generator's cents, and the period as some fraction of the octave, it's often possible to work backwards and find an appropriate multigen. Heptatonic notation requires that the 5th be between 600¢ and 720¢, to avoid descending 2nds. However 600¢ makes an extremely lopsided scale, so a more reasonable lower bound of 7\13 = 647¢ is used here. This limit is chosen because 13-edo notation uses the alternate 5th 7\13, and as a bonus it includes 16/11 = 649¢. The 4th is limited to 480-553¢, which includes 11/8. This sets a range for each possible generator, e.g. half-4th's generator ranges from 240¢ to 277¢.&lt;br /&gt;

See also the &lt;a class="wiki_link" href="/Map%20of%20rank-2%20temperaments"&gt;map of rank-2 temperaments&lt;/a&gt;.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.&lt;br /&gt;

Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, y3/2), where 5·G = 7/4.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos and pergens, but only a few dozen of either have been explored.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&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;br /&gt;

Screenshots of the first 2 pages:&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

&lt;br /&gt;

The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;

&lt;br /&gt;

Screenshots of the first 60 pergens:&lt;br /&gt;

Pergens supported by 12edo:&lt;br /&gt;

Pergens supported by 15edo. A red asterisk means partial support.&lt;br /&gt;

Pergens supported by 19edo:&lt;br /&gt;

&lt;br /&gt;

Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;

The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of i from -x to x is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and i makes a valid pergen. If the pergen is of the form (P8/m, P4), it's converted to (P8/m, P5). This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they need to be sorted. Experimenting with allowing y and i to range further does not produce any additional pergens.&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;br /&gt;

The interval P8/2 has a &amp;quot;ratio&amp;quot; of the square root of 2, which equals 2&lt;span style="vertical-align: super;"&gt;1/2&lt;/span&gt;, and its monzo can be written with fractions as (1/2, 0). In general, the pergen (P8/m, (a,b)/n) implies P = (1/m, 0) and G = (a/n, b/n). These equations make the &lt;strong&gt;pergen matrix&lt;/strong&gt; [(1/m 0) (a/n b/n)], which is P and G in terms of P8 and P12. Its inverse is [(m 0) (-am/b n/b)], which is P8 and P12 in terms of P and G, i.e. the square mapping.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

Thus GCD (m,n) = |b| · GCD (m/|b|, n/|b|) = |b| · r. If r = 1, then GCD (m, n) = |b|, and vice versa, which is the proposed test for a false double.&lt;br /&gt;

&lt;br /&gt;

&lt;br /&gt;

  &lt;br /&gt;

&lt;u&gt;&lt;strong&gt;Glossary&lt;/strong&gt;&lt;/u&gt; (to find the definition, use control-F to search for the bolded occurence of the word on this page)&lt;br /&gt;

&lt;u&gt;&lt;strong&gt;Credits&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;

&lt;br /&gt;