Kite's thoughts on pergens: Difference between revisions

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<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>2018-01-~~14 08~~:35:15 UTC</tt>.<br>

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-16 18:40:52 UTC</tt>.<br>

: The original revision id was <tt>~~624838521~~</tt>.<br>

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third-5th ||= v<span style="vertical-align: super;">3</span>m2 ||= C^<span style="vertical-align: super;">3 </span> ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||

||= (P8, P11/3)

third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber ||

third-11th ||= ^<span style="vertical-align: super;">3</span>dd2 ||= C^<span style="vertical-align: super;">3</span> ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,

||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= " ||

with 11/8 = A4 ||

||= " ||= v<span style="vertical-align: super;">3</span>M2 ||= C^<span style="vertical-align: super;">3 </span>``=`` D ||= P11/3 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||

||= (P8/3, P4/2)

third-8ve, half-4th ||= v<span style="vertical-align: super;">6</span>A2 ||= C^<span style="vertical-align: super;">6</span> ``=`` D# ||= P8/3 = ^^m3 = v<span style="vertical-align: super;">4</span>A4

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==Singles and doubles==

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** 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 **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** 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. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs ~~can~~ be ~~freely~~ exchanged with highs/lows.

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** 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 **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** 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. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.

Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve 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 multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.

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==Ratio and cents of the accidentals==

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:

15-edo: # = 240¢, ^ = 80¢

16-edo: # = -75¢

17-edo: # = 141¢, ^ = 71¢

18b-edo: # = -133¢, ^ = 67¢

19-edo: # = 63¢

21-edo: ^ = 57¢

22-edo: # = 164¢, ^ = 55¢

quarter-comma meantone: # = 76¢

fifth-comma meantone: # = 84¢

third-comma archy: # = 177¢

eighth-comma porcupine: # = 157¢, ^ = 52¢

sixth-comma srutal: # = 139¢, ^ = 33¢

third-comma injera: # = 63¢, ^ = 31¢ (third-comma = 1/3 of 81/80)

eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)

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 different.

==Finding a notation for a pergen==

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Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].

Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] ~~= (5,-3)~~ = m3. ~~Here~~ xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.

Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5<span class="nowrap">⋅</span>m2 = [5,3] - 5<span class="nowrap">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span class="nowrap">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span class="nowrap">⋅</span>G - 2<span class="nowrap">⋅</span>E, G must be ^^m2. The genchain is:

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==Misc notes==

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. ~~Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names~~

A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (n·z - y, x) / xz), with x > 0, z ≠ 0, and |n| <= x.

If z = 1, let n = y - x, and the pergen = (P8/x, P5)

If z = -1, let n = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)

A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)].

The GCD is always > 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3.

To prove: inverse of unreducing is also unreducing

Does unreducing a pergen twice result in the original pergen?

(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')

(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)

To prove: n is always a multiple of b, and unless n = 1, n > |b|

b = x/k and n = xz/k, where k = sign (z)·GCD (nz-y, x)

n = zb

To prove: (a,b)/n splits P8 into GCD (b,n) periods

First, assume b > 0

I = M + b·P4 = (a,b) + (2b,-b) = (a+2b,0) = (a+2b)·P8

Let I be the 3-limit interval G - P5 = (M - b·P5) / b

I = M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8

P8 = M/b - P5

Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods

(actually b = x/k and n = xz/k, where k = GCD (nz-y, x), but still GCD (b,n) = x/k = b)

Therefore if m = |b|, the pergen is explicitly false

To prove: true/false test

If GCD (m,n) = b, is the pergen a false double?

To prove: alternate true/false test

if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?

If m > |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?

(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')

Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b

|b'| = m, so the unreduced pergen is explicitly false, and the test works

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.

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<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><h1 id="toc0"> </h1>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#toc0"> </a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Applications">Applications</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Searching for pergens">Searching for pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Searching for pergens">Searching for pergens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div>

<div style="margin-left: ~~2em~~;"><a href="#~~Further Discussion-Finding a notation for a pergen~~">~~Finding a notation for a pergen~~</a></div>

<div style="margin-left: 1em;"><a href="#x240¢, ^"> 240¢, ^ </a></div>

<div style="margin-left: ~~2em~~;"><a href="#~~Further Discussion-Alternate enharmonics">Alternate enharmonics&~~lt;/a></div>

<div style="margin-left: 1em;"><a href="#x141¢, ^"> 141¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div>

<div style="margin-left: 1em;"><a href="#x-133¢, ^"> -133¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 1em;"><a href="#x164¢, ^"> 164¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Combining pergens">Combining pergens</a></div>

<div style="margin-left: 1em;"><a href="#x157¢, ^"> 157¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 1em;"><a href="#x139¢, ^"> 139¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Supplemental materials">Supplemental materials</a></div>

<div style="margin-left: 1em;"><a href="#x63¢, ^"> 63¢, ^ </a></div>

<div style="margin-left: 2em;"><a href="#~~Further Discussion~~-Misc notes">Misc notes</a></div>

<div style="margin-left: 1em;"><a href="#x157¢, ^"> 157¢, ^ </a></div>

</div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Finding a notation for a pergen">Finding a notation for a pergen</a></div>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Alternate enharmonics">Alternate enharmonics</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Alternate keyspans and stepspans">Alternate keyspans and stepspans</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Chord names and scale names">Chord names and scale names</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Combining pergens">Combining pergens</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Pergens and EDOs">Pergens and EDOs</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Supplemental materials">Supplemental materials</a></div>

<div style="margin-left: 2em;"><a href="#x157¢, ^-Misc notes">Misc notes</a></div>

</div>

<h1 id="toc1"><a name="Definition"></a><u><strong>Definition</strong></u></h1>

<br />

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</td>

<td style="text-align: center;">small triple amber<br />

<td style="text-align: center;">small triple amber,<br />

with 11/8 = A4<br />

</td>

</tr>

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</td>

</td>

</tr>

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<h2 id="toc7"><a name="Further Discussion-Singles and doubles"></a>Singles and doubles</h2>

<br />

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> 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 <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> 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. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs ~~can~~ be ~~freely~~ exchanged with highs/lows.<br />

If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a <strong>single-split</strong> pergen. If it has two fractions, it's a <strong>double-split</strong> 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 <strong>single-pair</strong> notation because it adds only a single pair of accidentals to conventional notation. <strong>Double-pair</strong> 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. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.<br />

<br />

Every double-split pergen is either a <strong>true double</strong> or a <strong>false double</strong>. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve 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 multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.<br />

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<h2 id="toc9"><a name="Further Discussion-Ratio and cents of the accidentals"></a>Ratio and cents of the accidentals</h2>

<br />

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.<br />

In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.<br />

<br />

We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br />

<br />

<h2 id="~~toc10~~"><a name="~~Further Discussion~~-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:<br />

<br />

15-edo: # <h1 id="toc10"><a name="x240¢, ^"></a> 240¢, ^ </h1>

80¢<br />

16-edo: # = -75¢<br />

17-edo: # <h1 id="toc11"><a name="x141¢, ^"></a> 141¢, ^ </h1>

71¢<br />

18b-edo: # <h1 id="toc12"><a name="x-133¢, ^"></a> -133¢, ^ </h1>

67¢<br />

19-edo: # = 63¢<br />

21-edo: ^ = 57¢<br />

22-edo: # <h1 id="toc13"><a name="x164¢, ^"></a> 164¢, ^ </h1>

55¢<br />

<br />

quarter-comma meantone: # = 76¢<br />

fifth-comma meantone: # = 84¢<br />

third-comma archy: # = 177¢<br />

eighth-comma porcupine: # <h1 id="toc14"><a name="x157¢, ^"></a> 157¢, ^ </h1>

52¢<br />

sixth-comma srutal: # <h1 id="toc15"><a name="x139¢, ^"></a> 139¢, ^ </h1>

33¢<br />

third-comma injera: # <h1 id="toc16"><a name="x63¢, ^"></a> 63¢, ^ </h1>

31¢ (third-comma = 1/3 of 81/80)<br />

eighth-comma hedgehog: # <h1 id="toc17"><a name="x157¢, ^"></a> 157¢, ^ </h1>

49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)<br />

<br />

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 different.<br />

<br />

<h2 id="toc18"><a name="x157¢, ^-Finding a notation for a pergen"></a>Finding a notation for a pergen</h2>

<br />

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:<br />

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<ul class="quotelist"><li>a = -11k + 19b</li><li>b = 7a - 12b</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].<br />

<br />

Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.<br />

Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. <br />

<br />

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>m2 = d3). The enharmonic's <strong>count</strong> is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:<br />

For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.<br />

<br />

Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5<span class="nowrap">⋅</span>M2 = [12,7] - 5<span class="nowrap">⋅</span>[2,1] = [2,2] = 2<span class="nowrap">⋅</span>[1,1] = 2<span class="nowrap">⋅</span>m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2<span class="nowrap">⋅</span>m2 = d3). The enharmonic's <strong>count</strong> is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v<span style="vertical-align: super;">5</span>m2. Since P8 = 5<span class="nowrap">⋅</span>P + 2<span class="nowrap">⋅</span>E, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:<br />

Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5<span class="nowrap">⋅</span>m2 = [5,3] - 5<span class="nowrap">⋅</span>[1,1] = [5,3] - [5,5] = [0,-2] = -2<span class="nowrap">⋅</span>[0,1] = two descending d2's. The d2 must be upped, and E = ^<span style="vertical-align: super;">5</span>d2. Since P4 = 5<span class="nowrap">⋅</span>G - 2<span class="nowrap">⋅</span>E, G must be ^^m2. The genchain is:<br />

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

<br />

<h2 id="~~toc11~~"><a name="~~Further Discussion~~-Alternate enharmonics"></a>Alternate enharmonics</h2>

<h2 id="toc19"><a name="x157¢, ^-Alternate enharmonics"></a>Alternate enharmonics</h2>

<br />

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<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2.<br />

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Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese is (P8, P11/3), with G = 7/5 = d5. E = 3·d5 - P11 = descending dd3.<br />

<br />

<h2 id="~~toc12~~"><a name="~~Further Discussion~~-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<h2 id="toc20"><a name="x157¢, ^-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

<br />

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br />

<br />

<h2 id="~~toc13~~"><a name="~~Further Discussion~~-Chord names and scale names"></a>Chord names and scale names</h2>

<h2 id="toc21"><a name="x157¢, ^-Chord names and scale names"></a>Chord names and scale names</h2>

<br />

Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> 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.<br />

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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, (5/4)/2), where 5·G = 7/4.<br />

<br />

<h2 id="~~toc14~~"><a name="~~Further Discussion~~-Combining pergens"></a>Combining pergens</h2>

<h2 id="toc22"><a name="x157¢, ^-Combining pergens"></a>Combining pergens</h2>

<br />

Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br />

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However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.<br />

<br />

<h2 id="~~toc15~~"><a name="~~Further Discussion~~-Pergens and EDOs"></a>Pergens and EDOs</h2>

<h2 id="toc23"><a name="x157¢, ^-Pergens and EDOs"></a>Pergens and EDOs</h2>

<br />

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, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br />

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<br />

<h2 id="~~toc16~~"><a name="~~Further Discussion~~-Supplemental materials"></a>Supplemental materials</h2>

<h2 id="toc24"><a name="x157¢, ^-Supplemental materials"></a>Supplemental materials</h2>

<br />

This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br />

<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><br />

<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><br />

<br />

Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><br />

<a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><br />

<br />

Screenshot of the first 38 pergens:<br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><br />

<br />

<h2 id="~~toc17~~"><a name="~~Further Discussion~~-Misc notes"></a>Misc notes</h2>

<h2 id="toc25"><a name="x157¢, ^-Misc notes"></a>Misc notes</h2>

<br />

~~Pergens were discovered by Kite Giedraitis in 2017~~, and ~~developed with~~ the ~~help of Praveen Venkataramana~~. ~~Earlier drafts~~ of ~~this article can be found at~~ <a ~~href~~=~~"http:~~//~~xenharmonic.wikispaces.com~~/pergen~~+names"~~>~~http~~://~~xenharmonic.wikispaces.com~~/pergen~~+names~~</a><br />

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

If z = 1, let n = y - x, and the pergen = (P8/x, P5)<br />

If z = -1, let n = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)<br />

A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)].<br />

The GCD is always &gt; 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3.<br />

<br />

To prove: inverse of unreducing is also unreducing<br />

Does unreducing a pergen twice result in the original pergen?<br />

(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br />

(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)<br />

<br />

To prove: n is always a multiple of b, and unless n = 1, n &gt; |b|<br />

b = x/k and n = xz/k, where k = sign (z)·GCD (nz-y, x)<br />

n = zb<br />

<br />

To prove: (a,b)/n splits P8 into GCD (b,n) periods<br />

First, assume b &gt; 0<br />

<br />

I = M + b·P4 = (a,b) + (2b,-b) = (a+2b,0) = (a+2b)·P8<br />

Let I be the 3-limit interval G - P5 = (M - b·P5) / b<br />

I = M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8<br />

P8 = M/b - P5<br />

<br />

Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods<br />

(actually b = x/k and n = xz/k, where k = GCD (nz-y, x), but still GCD (b,n) = x/k = b)<br />

Therefore if m = |b|, the pergen is explicitly false<br />

<br />

To prove: true/false test<br />

If GCD (m,n) = b, is the pergen a false double?<br />

<br />

To prove: alternate true/false test<br />

if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?<br />

If m &gt; |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?<br />

(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')<br />

Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b<br />

|b'| = m, so the unreduced pergen is explicitly false, and the test works<br />

<br />

Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br />

<br />

@@ 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>2018-01-14 08:35:15 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-16 18:40:52 UTC</tt>.<br>
-: The original revision id was <tt>624838521</tt>.<br>
+: The original revision id was <tt>624957075</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 207: / Line 207: @@
 third-5th ||= 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; ``=`` Db ||= P5/3 = ^M2 = vvm3 ||= C - D^ - Fv - G ||= slendric ||
 ||= (P8, P11/3)
-third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber ||
+third-11th ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ``=`` B## ||= P11/3 = vA4 = ^^dd5 ||= C - F#v - Cb^ - F ||= small triple amber,
-||= " ||= 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 = ^4 = vv5 ||= C - F^ - Cv - F ||= " ||
+with 11/8 = A4 ||
+||= " ||= 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 = ^4 = vv5 ||= C - F^ - Cv - F ||= same, with 11/8 = P4 ||
 ||= (P8/3, P4/2)
 third-8ve, half-4th ||= v&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;A2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; ``=`` D# ||= P8/3 = ^^m3 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A4
@@ Line 349: / Line 350: @@
 ==Singles and doubles==
-If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** 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 **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** 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. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.
+If a pergen has only one fraction, like (P8/2, P5) or (P8, P4/3), the pergen is a **single-split** pergen. If it has two fractions, it's a **double-split** 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 **single-pair** notation because it adds only a single pair of accidentals to conventional notation. **Double-pair** 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. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.
 Every double-split pergen is either a **true double** or a **false double**. A true double, like third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve 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 multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.
@@ Line 379: / Line 380: @@
 ==Ratio and cents of the accidentals==
-In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.
 We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.
+This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:
+-edo: # = 240¢, ^ = 80¢
+-edo: # = -75¢
+-edo: # = 141¢, ^ = 71¢
+b-edo: # = -133¢, ^ = 67¢
+-edo: # = 63¢
+-edo: ^ = 57¢
+-edo: # = 164¢, ^ = 55¢
+quarter-comma meantone: # = 76¢
+fifth-comma meantone: # = 84¢
+third-comma archy: # = 177¢
+eighth-comma porcupine: # = 157¢, ^ = 52¢
+sixth-comma srutal: # = 139¢, ^ = 33¢
+third-comma injera: # = 63¢, ^ = 31¢ (third-comma = 1/3 of 81/80)
+eighth-comma hedgehog: # = 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)
+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 different.
 ==Finding a notation for a pergen==
@@ Line 400: / Line 422: @@
 Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].
-Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.
+Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up.
+For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.
-Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2 = [12,7] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[2,1] = [2,2] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2. Since P8 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P + 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:
+Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2 = [12,7] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[2,1] = [2,2] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = d3). The enharmonic's **count** is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2. Since P8 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P + 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^^M2=v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m3 -- v4 -- ^5 -- ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M6=vvm7 -- P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- D^^=Ebv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- Fv -- G^ -- A^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Bbvv -- C&lt;/span&gt;
 Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = [5,3] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = [5,3] - [5,5] = [0,-2] = -2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[0,1] = two descending d2's. The d2 must be upped, and E = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d2. Since P4 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, G must be ^^m2. The genchain is:
@@ Line 594: / Line 618: @@
 ==Misc notes==
-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at http://xenharmonic.wikispaces.com/pergen+names
+A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (n·z - y, x) / xz), with x &gt; 0, z ≠ 0, and |n| &lt;= x.
+If z = 1, let n = y - x, and the pergen = (P8/x, P5)
+If z = -1, let n = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)
+A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)].
+The GCD is always &gt; 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3.
+To prove: inverse of unreducing is also unreducing
+Does unreducing a pergen twice result in the original pergen?
+(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')
+(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)
+To prove: n is always a multiple of b, and unless n = 1, n &gt; |b|
+b = x/k and n = xz/k, where k = sign (z)·GCD (nz-y, x)
+n = zb
+To prove: (a,b)/n splits P8 into GCD (b,n) periods
+First, assume b &gt; 0
+I = M + b·P4 = (a,b) + (2b,-b) = (a+2b,0) = (a+2b)·P8
+Let I be the 3-limit interval G - P5 = (M - b·P5) / b
+I = M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8
+P8 = M/b - P5
+Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods
+(actually b = x/k and n = xz/k, where k = GCD (nz-y, x), but still GCD (b,n) = x/k = b)
+Therefore if m = |b|, the pergen is explicitly false
+To prove: true/false test
+If GCD (m,n) = b, is the pergen a false double?
+To prove: alternate true/false test
+if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?
+If m &gt; |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?
+(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')
+Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b
+|b'| = m, so the unreduced pergen is explicitly false, and the test works
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.
@@ Line 616: / Line 681: @@
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+&lt;!-- ws:end:WikiTextTocRule:98 --&gt;&lt;!-- ws:start:WikiTextTocRule:99: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Searching for pergens"&gt;Searching for pergens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:99 --&gt;&lt;!-- ws:start:WikiTextTocRule:100: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Extremely large multigens"&gt;Extremely large multigens&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:100 --&gt;&lt;!-- ws:start:WikiTextTocRule:101: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Singles and doubles"&gt;Singles and doubles&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:102 --&gt;&lt;!-- ws:start:WikiTextTocRule:103: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Further Discussion-Ratio and cents of the accidentals"&gt;Ratio and cents of the accidentals&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:103 --&gt;&lt;!-- ws:start:WikiTextTocRule:104: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x240¢, ^"&gt; 240¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:104 --&gt;&lt;!-- ws:start:WikiTextTocRule:105: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x141¢, ^"&gt; 141¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:105 --&gt;&lt;!-- ws:start:WikiTextTocRule:106: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x-133¢, ^"&gt; -133¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:106 --&gt;&lt;!-- ws:start:WikiTextTocRule:107: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x164¢, ^"&gt; 164¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:107 --&gt;&lt;!-- ws:start:WikiTextTocRule:108: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x157¢, ^"&gt; 157¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:108 --&gt;&lt;!-- ws:start:WikiTextTocRule:109: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x139¢, ^"&gt; 139¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:109 --&gt;&lt;!-- ws:start:WikiTextTocRule:110: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x63¢, ^"&gt; 63¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:110 --&gt;&lt;!-- ws:start:WikiTextTocRule:111: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#x157¢, ^"&gt; 157¢, ^ &lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:112 --&gt;&lt;!-- ws:start:WikiTextTocRule:113: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Alternate enharmonics"&gt;Alternate enharmonics&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:113 --&gt;&lt;!-- ws:start:WikiTextTocRule:114: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Alternate keyspans and stepspans"&gt;Alternate keyspans and stepspans&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:114 --&gt;&lt;!-- ws:start:WikiTextTocRule:115: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Chord names and scale names"&gt;Chord names and scale names&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:115 --&gt;&lt;!-- ws:start:WikiTextTocRule:116: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Combining pergens"&gt;Combining pergens&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:116 --&gt;&lt;!-- ws:start:WikiTextTocRule:117: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Pergens and EDOs"&gt;Pergens and EDOs&lt;/a&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:117 --&gt;&lt;!-- ws:start:WikiTextTocRule:118: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#x157¢, ^-Supplemental materials"&gt;Supplemental materials&lt;/a&gt;&lt;/div&gt;
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   &lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,488: / Line 1,561: @@
          &lt;td style="text-align: center;"&gt;C - F#v - Cb^ - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;small triple amber&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;small triple amber,&lt;br /&gt;
+with 11/8 = A4&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 1,502: / Line 1,576: @@
          &lt;td style="text-align: center;"&gt;C - F^ - Cv - F&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;same, with 11/8 = P4&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 2,228: / Line 2,302: @@
 &lt;!-- ws:start:WikiTextHeadingRule:55:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Further Discussion-Singles and doubles"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:55 --&gt;Singles and doubles&lt;/h2&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. In this article, the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs can be freely exchanged with highs/lows.&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. In this article, ups and downs are used for the octave-splitting enharmonic, and highs/lows are used for the multigen-splitting enharmonic. But the choice of which pair of accidentals is used for which enharmonic is arbitrary, and ups/downs could be exchanged with highs/lows.&lt;br /&gt;
 &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 third-everything (P8/3, P4/3) or half-8ve quarter-4th (P8/2, P4/4), can only arise when at least two commas are tempered out, and requires double pair notation. A false double, like half-8ve 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 multigen split automatically splits the octave as well: if M2 = 4·G, then P8 = M9 - M2 = 2⋅P5 - 4·G = (P5 - 2·G) / 2. In general, if a pergen's multigen is (a,b), the octave is split into at least |b| parts.&lt;br /&gt;
@@ Line 2,258: / Line 2,332: @@
 &lt;!-- ws:start:WikiTextHeadingRule:59:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Further Discussion-Ratio and cents of the accidentals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:59 --&gt;Ratio and cents of the accidentals&lt;/h2&gt;
   &lt;br /&gt;
-In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64.&lt;br /&gt;
+In a 5-limit temperament, the up symbol is generally 81/80. However, for diminished (which sets 6/5 = P8/4), ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In a 2.3.7 rank-2 temperament, ^1 is often 64/63, or perhaps 63/64. #1 is always (-11,7) = 2187/2048, by definition.&lt;br /&gt;
 &lt;br /&gt;
 We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:61:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Further Discussion-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:61 --&gt;Finding a notation for a pergen&lt;/h2&gt;
+This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. Ths gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:&lt;br /&gt;
+&lt;br /&gt;
+-edo: #  &lt;!-- ws:start:WikiTextHeadingRule:61:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="x240¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:61 --&gt; 240¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+-edo: # = -75¢&lt;br /&gt;
+-edo: #  &lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="x141¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt; 141¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+b-edo: #  &lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="x-133¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt; -133¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+-edo: # = 63¢&lt;br /&gt;
+-edo: ^ = 57¢&lt;br /&gt;
+-edo: #  &lt;!-- ws:start:WikiTextHeadingRule:67:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="x164¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:67 --&gt; 164¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+&lt;br /&gt;
+quarter-comma meantone: # = 76¢&lt;br /&gt;
+fifth-comma meantone: # = 84¢&lt;br /&gt;
+third-comma archy: # = 177¢&lt;br /&gt;
+eighth-comma porcupine: #  &lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="x157¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt; 157¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+sixth-comma srutal: #  &lt;!-- ws:start:WikiTextHeadingRule:71:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="x139¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:71 --&gt; 139¢, ^ &lt;/h1&gt;
+¢&lt;br /&gt;
+third-comma injera: #  &lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc16"&gt;&lt;a name="x63¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt; 63¢, ^ &lt;/h1&gt;
+¢ (third-comma = 1/3 of 81/80)&lt;br /&gt;
+eighth-comma hedgehog: #  &lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc17"&gt;&lt;a name="x157¢, ^"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt; 157¢, ^ &lt;/h1&gt;
+¢, / = 52¢ (eighth-comma = 1/8 of 250/243)&lt;br /&gt;
+&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 different.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="x157¢, ^-Finding a notation for a pergen"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;Finding a notation for a pergen&lt;/h2&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;
@@ Line 2,272: / Line 2,375: @@
 &lt;ul class="quotelist"&gt;&lt;li&gt;a = -11k + 19b&lt;/li&gt;&lt;li&gt;b = 7a - 12b&lt;/li&gt;&lt;/ul&gt;Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].&lt;br /&gt;
 &lt;br /&gt;
-Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.&lt;br /&gt;
+Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. &lt;br /&gt;
 &lt;br /&gt;
-Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2 = [12,7] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[2,1] = [2,2] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = d3). The enharmonic's &lt;strong&gt;count&lt;/strong&gt; is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2. Since P8 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P + 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, the period must be ^^M2, to make the ups and downs come out even. The period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:&lt;br /&gt;
+For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = m3. The enharmonic can also be found using gedras: xE = M - n&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G = P5 - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m3 = [7,4] - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[3,2] = [7,4] - [6,4] = [1,0] = A1.&lt;br /&gt;
+&lt;br /&gt;
+Next, consider (P8/5, P5). P = [12,7]/5 = [2,1] = M2. xE = P8 - mP = P8 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;M2 = [12,7] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[2,1] = [2,2] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2. Because x = 2, E will occur twice in the perchain, even thought the comma only occurs once (i.e. the comma = 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = d3). The enharmonic's &lt;strong&gt;count&lt;/strong&gt; is 2. The bare enharmonic is m2, which must be downed to vanish. The number of downs equals the splitting fraction, thus E = v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m2. Since P8 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;P + 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, the period must be ^^M2, to make the ups and downs come out even. The number of the period's (or generator's) ups or downs always equals the count. Equipped with the period and the enharmonic, the perchain is easily found:&lt;br /&gt;
 &lt;span style="display: block; text-align: center;"&gt;P1 -- ^^M2=v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m3 -- v4 -- ^5 -- ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M6=vvm7 -- P8&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C -- D^^=Ebv&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; -- Fv -- G^ -- A^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Bbvv -- C&lt;/span&gt;&lt;br /&gt;
 Most single-split pergens are dealt with similarly. For example, (P8, P4/5) has a bare generator [5,3]/5 = [1,1] = m2. The bare enharmonic is P4 - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;m2 = [5,3] - 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[1,1] = [5,3] - [5,5] = [0,-2] = -2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;[0,1] = two descending d2's. The d2 must be upped, and E = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d2. Since P4 = 5&lt;span class="nowrap"&gt;⋅&lt;/span&gt;G - 2&lt;span class="nowrap"&gt;⋅&lt;/span&gt;E, G must be ^^m2. The genchain is:&lt;br /&gt;
@@ Line 2,295: / Line 2,400: @@
 This is a lot of math, but it only needs to be done once for each pergen!&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Further Discussion-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;Alternate enharmonics&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:79:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="x157¢, ^-Alternate enharmonics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:79 --&gt;Alternate enharmonics&lt;/h2&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.&lt;br /&gt;
@@ Line 2,318: / Line 2,423: @@
 Sometimes the temperament implies an enharmonic that isn't even a 2nd. For example, liese is (P8, P11/3), with G = 7/5 = d5. E = 3·d5 - P11 = descending dd3.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Further Discussion-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:81:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="x157¢, ^-Alternate keyspans and stepspans"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:81 --&gt;Alternate keyspans and stepspans&lt;/h2&gt;
   &lt;br /&gt;
 One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:67:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Further Discussion-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:67 --&gt;Chord names and scale names&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="x157¢, ^-Chord names and scale names"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;Chord names and scale names&lt;/h2&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;
@@ Line 2,806: / Line 2,911: @@
 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, (5/4)/2), where 5·G = 7/4.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Further Discussion-Combining pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt;Combining pergens&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc22"&gt;&lt;a name="x157¢, ^-Combining pergens"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;Combining pergens&lt;/h2&gt;
   &lt;br /&gt;
 Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).&lt;br /&gt;
@@ Line 2,814: / Line 2,919: @@
 However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:71:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc15"&gt;&lt;a name="Further Discussion-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:71 --&gt;Pergens and EDOs&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc23"&gt;&lt;a name="x157¢, ^-Pergens and EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Pergens and EDOs&lt;/h2&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, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.&lt;br /&gt;
@@ Line 3,122: / Line 3,227: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Supplemental materials"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt;Supplemental materials&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc24"&gt;&lt;a name="x157¢, ^-Supplemental materials"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;Supplemental materials&lt;/h2&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.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextUrlRule:3968: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:3968 --&gt;&lt;br /&gt;
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 Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. 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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+&lt;!-- ws:start:WikiTextUrlRule:3969:http://www.tallkite.com/misc_files/alt-pergensLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergensLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3969 --&gt;&lt;br /&gt;
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 Screenshot of the first 38 pergens:&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc25"&gt;&lt;a name="x157¢, ^-Misc notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;Misc notes&lt;/h2&gt;
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-Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at &lt;!-- ws:start:WikiTextUrlRule:3881:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3881 --&gt;&lt;br /&gt;
+A square mapping [(x, y), (0, z)] creates the pergen (P8/x, (n·z - y, x) / xz), with x &amp;gt; 0, z ≠ 0, and |n| &amp;lt;= x.&lt;br /&gt;
+If z = 1, let n = y - x, and the pergen = (P8/x, P5)&lt;br /&gt;
+If z = -1, let n = 2x - y, and the pergen = (P8/x, P4) = (P8/x, P5)&lt;br /&gt;
+A pergen (P8/m, (a,b)/n) arises from a square mapping [(m, m-am/b), (0, n/b)].&lt;br /&gt;
+The GCD is always &amp;gt; 0: GCD (-3,6) = GCD (3,-6) = GCD (-3, -6) = 3.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+To prove: inverse of unreducing is also unreducing&lt;br /&gt;
+Does unreducing a pergen twice result in the original pergen?&lt;br /&gt;
+(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')&lt;br /&gt;
+(P8/m, (a',b')/n') unreduced is (P8/m, (n'-a'm, -b'm) / mn') = (P8/m, (mn-(n-am)m, bmm) / mmn) = (P8/m, (a,b)/n)&lt;br /&gt;
+&lt;br /&gt;
+To prove: n is always a multiple of b, and unless n = 1, n &amp;gt; |b|&lt;br /&gt;
+b = x/k and n = xz/k, where k = sign (z)·GCD (nz-y, x)&lt;br /&gt;
+n = zb&lt;br /&gt;
+&lt;br /&gt;
+To prove: (a,b)/n splits P8 into GCD (b,n) periods&lt;br /&gt;
+First, assume b &amp;gt; 0&lt;br /&gt;
+&lt;br /&gt;
+I = M + b·P4 = (a,b) + (2b,-b) = (a+2b,0) = (a+2b)·P8&lt;br /&gt;
+Let I be the 3-limit interval G - P5 = (M - b·P5) / b&lt;br /&gt;
+I = M - b·P5 = (a,b) - (-b,b) = (a+b,0) = (a+b)·P8&lt;br /&gt;
+P8 = M/b - P5&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+Therefore, since b = x and n = xz, (a,b) splits P8 into |b| periods&lt;br /&gt;
+(actually b = x/k and n = xz/k, where k = GCD (nz-y, x), but still GCD (b,n) = x/k = b)&lt;br /&gt;
+Therefore if m = |b|, the pergen is explicitly false&lt;br /&gt;
+&lt;br /&gt;
+To prove: true/false test&lt;br /&gt;
+If GCD (m,n) = b, is the pergen a false double?&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+To prove: alternate true/false test&lt;br /&gt;
+if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?&lt;br /&gt;
+If m &amp;gt; |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?&lt;br /&gt;
+(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, (a',b')/n')&lt;br /&gt;
+Simplify using b = GCD (m,n): a' = (n-am)/b, b' = -m, and n' = mn/b&lt;br /&gt;
+|b'| = m, so the unreduced pergen is explicitly false, and the test works&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.&lt;br /&gt;
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