Pergen names: 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:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-17 18:27:51 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-18 00:13:59 UTC</tt>.

: The original revision id was <tt>~~623960989~~</tt>.

: The original revision id was <tt>623967575</tt>.

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To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multi-gen as before. Then deduce the period from the enharmonic. If the multi-gen was changed by unreducing, deduce the alternate generator from the enharmonic.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. ~~Deduce~~ the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, ~~and~~ G' = ^5M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Next, deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, thus G' = ^5M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.

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P1 - ^5M2=v5m3 - P4C - D^5=Ebv5 - F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For ~~example,~~ (P8/2, P4/2~~) has from P8/2~~ P = vA4 and E = ^^d2, and ~~from P4/2~~ G = /M2 and E' = \\m2.

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For (P8/2, P4/2, the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.

A false double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).

A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).

==Alternate enharmonics==

For single-comma pergens, the enharmonic should equal the comma's mapping.

For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.

==Alternate keyspans==

One might wonder, 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 7-edo 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.

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Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo)~~. In addition, smaller degrees are always preferred. It should be either a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting~~.

Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).

For {P8/M, multi-gen/N}, an octave = M periods ± some number of enharmonics, and a multi-gen = N generators ± some number of enharmonics.

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The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

[//Question: what if there are highs and lows?//]

[//Question: what if there are highs and lows?//]</pre></div>

~~(to be continued)~~</pre></div>

<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 names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a> | <a href="#Further Discussion">Further Discussion</a>

<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 names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a> | <a href="#Further Discussion">Further Discussion</a>

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

A pergen (pronounced &quot;peer-gen&quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

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To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multi-gen as before. Then deduce the period from the enharmonic. If the multi-gen was changed by unreducing, deduce the alternate generator from the enharmonic.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. ~~Deduce~~ the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, ~~and~~ G' = ^5M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.

For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v10m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Next, deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, thus G' = ^5M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.

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P1 - ^5M2=v5m3 - P4C - D^5=Ebv5 - F

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For ~~example,~~ (P8/2, P4/2~~) has from P8/2~~ P = vA4 and E = ^^d2, and ~~from P4/2~~ G = /M2 and E' = \\m2.

To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For (P8/2, P4/2, the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.

A false double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).

A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).

<h2 id="toc8"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

For single-comma pergens, the enharmonic should equal the comma's mapping.~~<br~~ /~~>~~

For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.

<h2 id="toc9"><a name="Further Discussion-Alternate keyspans"></a>Alternate keyspans</h2>

One might wonder, 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 7-edo 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.

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Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo)~~. In addition, smaller degrees are always preferred. It should be either a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting~~.

Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).

For {P8/M, multi-gen/N}, an octave = M periods ± some number of enharmonics, and a multi-gen = N generators ± some number of enharmonics.

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The LCM of the pergen's two splitting fractions is called the height of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

[Question: what if there are highs and lows?]~~ ~~

[Question: what if there are highs and lows?]</body></html></pre></div>

~~ ~~

~~(to be continued)~~</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-17 18:27:51 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-18 00:13:59 UTC</tt>.<br>
-: The original revision id was <tt>623960989</tt>.<br>
+: The original revision id was <tt>623967575</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 320: / Line 320: @@
 To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multi-gen as before. Then deduce the period from the enharmonic. If the multi-gen was changed by unreducing, deduce the alternate generator from the enharmonic.
-For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;10&lt;/span&gt;m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M2. Deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, and G' = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.
+For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;10&lt;/span&gt;m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M2. Next, deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, thus G' = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.
@@ Line 327: / Line 328: @@
 &lt;span style="display: block; text-align: center;"&gt;P1 - ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2=v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m3 - P4&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C - D^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;=Ebv&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; - F&lt;/span&gt;
-To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For example, (P8/2, P4/2) has from P8/2 P = vA4 and E = ^^d2, and from P4/2 G = /M2 and E' = \\m2.
+To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For (P8/2, P4/2, the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.
-A false double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).
+A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).
 ==Alternate enharmonics==
-For single-comma pergens, the enharmonic should equal the comma's mapping.
+For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.
+==Alternate keyspans==
+One might wonder, 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 7-edo 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.
@@ Line 342: / Line 345: @@
-Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo). In addition, smaller degrees are always preferred. It should be either a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting.
+Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).
 For {P8/M, multi-gen/N}, an octave = M periods ± some number of enharmonics, and a multi-gen = N generators ± some number of enharmonics.
@@ Line 388: / Line 391: @@
 The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.
-[//Question: what if there are highs and lows?//]
+[//Question: what if there are highs and lows?//]</pre></div>
-(to be continued)</pre></div>
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 A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) is a way of identifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.&lt;br /&gt;
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 To find the single-pair notation for a false double pergen, find an explicitly false form of the pergen, and find the generator and enharmonic from the fractional multi-gen as before. Then deduce the period from the enharmonic. If the multi-gen was changed by unreducing, deduce the alternate generator from the enharmonic.&lt;br /&gt;
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-For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;10&lt;/span&gt;m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M2. Deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, and G' = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.&lt;br /&gt;
+For example, (P8/5, P4/2) isn't explicitly false, so we must unreduce it to (P8/5, m2/10). The generator is [1,1]/10 = [0,0] = P1. The bare enharmonic is m2 - 10*P1 = m2. It must be downed, thus E = v&lt;span style="vertical-align: super;"&gt;10&lt;/span&gt;m2. Since m2 = 10*G + E, G is ^1. This is much too small to be half a 4th, so we'll find an alternate generator later. The octave plus or minus some number of enharmonics must equal 5 periods, thus (P8 + x*m2) must be divisible by 5, and ([12,7] + x[1,1]) mod 5 must be 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;M2. Next, deduce the alternate generator G' as we did for the period: P4 plus or minus some number of enharmonics must equal 2*G', and ([5,3] + y[1,1]) mod 2 must be 0. The smallest y is either 1 or -1. Choose -1, to get a smaller G', since the other one is equivalent. 2*G' = P4 - E, thus G' = ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.&lt;br /&gt;
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 &lt;span style="display: block; text-align: center;"&gt;P1 - ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;M2=v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;m3 - P4&lt;/span&gt;&lt;span style="display: block; text-align: center;"&gt;C - D^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;=Ebv&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; - F&lt;/span&gt;&lt;br /&gt;
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-To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For example, (P8/2, P4/2) has from P8/2 P = vA4 and E = ^^d2, and from P4/2 G = /M2 and E' = \\m2.&lt;br /&gt;
+To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. For (P8/2, P4/2, the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.&lt;br /&gt;
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-A false double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).&lt;br /&gt;
+A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is a 3rd or a 4th, as with (P8/2, P4/3).&lt;br /&gt;
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-For single-comma pergens, the enharmonic should equal the comma's mapping.&lt;br /&gt;
+For single-comma pergens, the enharmonic should equal the comma's mapping. For example, (P8, P5/2) might arise from 243/242, which splits the 5th into two 11/9 halves.&lt;br /&gt;
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+One might wonder, 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 7-edo 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.&lt;br /&gt;
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-Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo). In addition, smaller degrees are always preferred. It should be either a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting.&lt;br /&gt;
+Not all enharmonics work with all pergens. There are two logical restrictions on the enharmonic, one based on its degree and the other on its 3-exponent (implied edo).&lt;br /&gt;
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 For {P8/M, multi-gen/N}, an octave = M periods ± some number of enharmonics, and a multi-gen = N generators ± some number of enharmonics.&lt;br /&gt;
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 The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;br /&gt;
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-[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;br /&gt;
+[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;/body&gt;&lt;/html&gt;</pre></div>
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-(to be continued)&lt;/body&gt;&lt;/html&gt;</pre></div>