Pergen names: Difference between revisions
Wikispaces>TallKite **Imported revision 624089859 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624089881 - 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>2017-12-20 01: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-20 01:33:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624089881</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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(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions. | (P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions. | ||
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 | 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 original generator from the period and the alternate generator. | ||
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<span style="vertical-align: super;">10</span>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 = ^<span style="vertical-align: super;">4</span>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' = ^<span style="vertical-align: super;">5</span>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<span style="vertical-align: super;">10</span>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 = ^<span style="vertical-align: super;">4</span>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' = ^<span style="vertical-align: super;">5</span>M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2. | ||
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(P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | (P8, P11/4) has a bare generator [17,10]/4 = [4,2] = M3. The bare enharmonic is P11 - 4*G = [1,2] = dd3. It must be downed, thus E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. The enharmonic is unfortunately not a unison or 2nd. Note that the generator's stepspan could have been rounded up instead of down, making G = [4,3] = d4. This would make E = [-1,2] = d43. Rounding down is clearly preferable! In general, rounding down is better, because the smaller of two equivalent generators or periods is preferred. However, there are exceptions.<br /> | ||
<br /> | <br /> | ||
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 | 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 original generator from the period and the alternate generator.<br /> | ||
<br /> | <br /> | ||
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<span style="vertical-align: super;">10</span>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 = ^<span style="vertical-align: super;">4</span>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' = ^<span style="vertical-align: super;">5</span>M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.<br /> | 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<span style="vertical-align: super;">10</span>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 = ^<span style="vertical-align: super;">4</span>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' = ^<span style="vertical-align: super;">5</span>M2, which happens to be P + G. Thus P = ^4M2, G = ^1, G' = ^5M2, and E = v10m2.<br /> | ||