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Pergen names: Difference between revisions

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**Imported revision 623918267 - Original comment: **
Wikispaces>TallKite
**Imported revision 623937571 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2017-12-16 05:53:14 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-17 04:42:52 UTC</tt>.<br>
: The original revision id was <tt>623918267</tt>.<br>
: The original revision id was <tt>623937571</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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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.  
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.  


For example, (P8/5, P4/2) unreduces 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, we'll address that 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')
For example, (P8/5, P4/2) unreduces 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, we'll address that 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. Our small generator G can generate m2, but not P4. Any number of periods plus or minus a single generator makes an alternate generator. We can deduce the 4th's generator G' as we did for the period: the 4th 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 1 or -1. We choose -1, to get a smaller G', to which E can be added. 2*G' = P4 - E, and G' = ^5M2, which happens to be P + G.
P4 = 2G' + E , perhaps P + G. But that would be a triple-upped interval, and we want to minimize the ups, so instead we use P - G = ^^M2 - ^1 = ^M2.


For example, (P8/5, P4/2) is a false double because GCD(5,2) = 1 = |-1|. The generator of (P8, P4/2) is [5,3]/2 = [2,1] = M2. Both the keyspan and the stepspan could have been rounded up, not down, creating [3,1] = A2, [3,2] = m3, [2,2] = d3. 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, we'll address that 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')
(P8/5, P4/2): P = ^4M2, G = ^1, G' = ^5M2, E = v10m2
 
 
(P8/5, P4/2): P = ^4M2, G = ^5M2, E = v10m2
perchain: P1 - ^4M2=v6m3 - vvP4 - ^^P5 - ^6M6=v4m7 - P8, or C - D^4=Ebv6 - Fvv - G^^ - A^6=Bbv4 - C
perchain: P1 - ^4M2=v6m3 - vvP4 - ^^P5 - ^6M6=v4m7 - P8, or C - D^4=Ebv6 - Fvv - G^^ - A^6=Bbv4 - C
genchain: P1 - ^5M2=v5m3 - P4, or C - D^5=Ebv5 - F
genchain: P1 - ^5M2=v5m3 - P4, or C - D^5=Ebv5 - F
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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. &lt;br /&gt;
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. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
For example, (P8/5, P4/2) unreduces 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, we'll address that 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')&lt;br /&gt;
For example, (P8/5, P4/2) unreduces 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, we'll address that 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. Our small generator G can generate m2, but not P4. Any number of periods plus or minus a single generator makes an alternate generator. We can deduce the 4th's generator G' as we did for the period: the 4th 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 1 or -1. We choose -1, to get a smaller G', to which E can be added. 2*G' = P4 - E, and G' = ^5M2, which happens to be P + G.&lt;br /&gt;
P4 = 2G' + E , perhaps P + G. But that would be a triple-upped interval, and we want to minimize the ups, so instead we use P - G = ^^M2 - ^1 = ^M2.&lt;br /&gt;
&lt;br /&gt;
For example, (P8/5, P4/2) is a false double because GCD(5,2) = 1 = |-1|. The generator of (P8, P4/2) is [5,3]/2 = [2,1] = M2. Both the keyspan and the stepspan could have been rounded up, not down, creating [3,1] = A2, [3,2] = m3, [2,2] = d3. 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, we'll address that 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 = 0. The smallest (least absolute value) x that satisfies this equation is -2, and P8 = 5*P + 2*E, and P = ^4M2. Any number of periods plus or minus a single generator makes an alternate generator. Because G is too small, we look for a larger generator G'. 2(xP + yG')&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
(P8/5, P4/2): P = ^4M2, G = ^5M2, E = v10m2&lt;br /&gt;
(P8/5, P4/2): P = ^4M2, G = ^1, G' = ^5M2, E = v10m2&lt;br /&gt;
perchain: P1 - ^4M2=v6m3 - vvP4 - ^^P5 - ^6M6=v4m7 - P8, or C - D^4=Ebv6 - Fvv - G^^ - A^6=Bbv4 - C&lt;br /&gt;
perchain: P1 - ^4M2=v6m3 - vvP4 - ^^P5 - ^6M6=v4m7 - P8, or C - D^4=Ebv6 - Fvv - G^^ - A^6=Bbv4 - C&lt;br /&gt;
genchain: P1 - ^5M2=v5m3 - P4, or C - D^5=Ebv5 - F&lt;br /&gt;
genchain: P1 - ^5M2=v5m3 - P4, or C - D^5=Ebv5 - F&lt;br /&gt;
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