Pergen names: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-19 23:55:58 UTC</tt>.

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To find a temperament's pergen set, first find the **PGM**, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Choose your generators so that all entries below the diagonal are zero. Graham Breed's website has a temperament finder [[@http://x31eq.com/temper/uv.html|x31eq.com/temper/uv.html]] that will find such a matrix. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous example, to ensure that the diagonal has no zeros. Lower primes > 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the 1st generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen set right from the PGM [(x y), (0, z)]. For a period P and a generator G:

For rank-2, we can compute the pergen set right from the PGM = [(x y), (0, z)]. For a period P and a generator G:

P8 = xP and WP5 = yP + zG

P = P8/x

G = [WP5 - y(P8/x)] / z = [-yP8 + xWP5]/xz = (-y, x) / xz

To G, add n periods~~, which are P8/~~x:

To get alternate generators, add n periods to G. n ranges from -x (subtracting a full octave) to +x (adding a full octave).

G = (-y, x) / xz + nP8/x = (nz - y, x) / xz

G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz

~~n ranges from -x (subtracting a full octave) to +x (adding a full octave).~~

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [[http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7|x31.com]] gives us this matrix:

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The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale would translate to a 5-edo subset. Such edos are marked with asterisks.

(table is under construction)

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temperament ||~ compatible edos

(12-31 only) ||

||= {P8, P5} ||= 600-720¢ ||= none ||= none ||= C - G ||= meantone ||= 12, 16, 19, 23, 26 ||

||= {P8, P5} ||= 600-720¢ ||= none ||= none ||= C - G ||= meantone ||= 12, 13b, 14*, 15*, 16,

||= {P8/2, P5} ||= 700-720¢ ||= P8/2 = vA4 = ^d5 ||= ^^d2 ||= C - F#v=Gb^ - C ||= srutal ||= 12, 20, 22, 24, 30 ||

17, 18b*, 19, 20*, 21*,

||= " ||= 600-700¢ ||= P8/2 = ^A4 = vd5 ||= vvd2 ||= C - F#^=Gbv - C ||= ||= 12, 14, 16, 18b, 26 ||

22, 23, 24*, 25*, 26,

27, 28*, 29, 30*, 31 ||

||~ ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/2, P5} ||= 700-720¢ ||= P8/2 = vA4 = ^d5 ||= ^^d2 ||= C - F#v=Gb^ - C ||= srutal ||= 12, 20*, 22, 24*, 30* ||

||= " ||= 600-700¢ ||= P8/2 = ^A4 = vd5 ||= vvd2 ||= C - F#^=Gbv - C ||= ||= 12, 14, 16, 18b,

24*, 26, 28* ||

||= " ||= 600-720¢ ||= P8/2 = ^P4 = vP5 ||= vvM2 ||= C - F^=Gv - C ||= (is this needed?) ||= ||

||= {P8, P5/2} ||= 4\7 - 720¢ ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C - Eb^=Ev - G ||= mohajira ||= 14, 17, 20, 21, 24

||= {P8, P5/2} ||= 4\7 - 720¢ ||= P5/2 = ^m3 = vM3 ||= vvA1 ||= C - Eb^=Ev - G ||= mohajira ||= 14*, 17, 20*, 21*, 24

27, 28, 30, 31 ||

27, 28*, 30*, 31 ||

||= " ||= 600¢ - 4\7 ||= P5/2 = ^M3 = vm3 ||= vvd1 ||= ||= ||= 14, 18b, 21, 28 ||

||= " ||= 600¢ - 4\7 ||= P5/2 = ^M3 = vm3 ||= vvd1 ||= ||= ||= 14*, 18b, 21*, 28* ||

||= {P8, P4/2} ||= ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= D - E^=Fv - G ||= semaphore ||= 22 ||

||= {P8, P4/2} ||= ||= P4/2 = ^M2 = vm3 ||= vvm2 ||= D - E^=Fv - G ||= semaphore ||= ||

||= " ||= ||= P4/2 = vA2 = ^d3 ||= ^^dd2 ||= ||= ||= ||

||= " ||= ||= P4/2 = ^A2 = vd3 ||= vvdd2 ||= ||= ||= ||

~~||= {P8, P11/2} ||= ||= P11/2 = ^m6 = vM6 ||= vvA1 ||= ||= ||= ||~~

~~||= ||= ||= ||= ||= ||= ||= ||~~

~~||= {P8, P12/2} ||= ||= P12/2 = ^M6 = vm7 ||= vvm2 ||= ||= magic ||= ||~~

||= ||= ||= ||= ||= ||= ||= ||

|| || || || || || || ||

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P4/2 = /M2 = \m3 ||= ^^d2,

\\m2 ||= C - F#v=Gb^ - C,

C - D/=Eb\ - F ||= bb&aaT ||= 22 ||

C - D/=Eb\ - F ||= bb&aaT ||= ||

||= {P8/2, ~~P11/2~~} ||= ||= ||= ||= ||= ||= ||

||~ ||~ ||~ ||~ ||~ ||~ ||~ ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/3, P5} ||= ||= ||= ||= ||= ||= ||

||= {P8/2, ~~P12~~/2} ||= ||= ||= ||= ||= ||= ||

||= {P8, P4/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8, P5/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8, P11/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/3, P4/2} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/3, P5/2} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/2, P4/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/2, P5/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/2, P11/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

||~ ||~ ||~ ||~ ||~ ||~ ||~ ||

||= ||= ||= ||= ||= ||= ||= ||

||= {P8/4, P5} ||= ||= ||= ||= ||= ||= ||

||= etc. ||= ||= ||= ||= ||= ||= ||

||= ||= ||= ||= ||= ||= ||= ||

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To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each period/generator. Choose your generators so that all entries below the diagonal are zero. Graham Breed's website has a temperament finder <a class="wiki_link_ext" href="http://x31eq.com/temper/uv.html" rel="nofollow" target="_blank">x31eq.com/temper/uv.html</a> that will find such a matrix. Next make a square matrix by discarding columns, usually the columns for the highest primes. But lower primes may need to be discarded, as in the previous example, to ensure that the diagonal has no zeros. Lower primes &gt; 3 may also be discarded to minimize splitting, see the Breedsmic example below. Then invert the matrix to get the monzos for each period/generator. Add/subtract periods from the 1st generator to get alternate generators. If the interval becomes descending, invert it. For rank-3, add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

For rank-2, we can compute the pergen set right from the PGM [(x y), (0, z)]. For a period P and a generator G:~~ ~~

For rank-2, we can compute the pergen set right from the PGM = [(x y), (0, z)]. For a period P and a generator G:

P8 = xP and WP5 = yP + zG

P = P8/x

G = [WP5 - y(P8/x)] / z = [-yP8 + xWP5]/xz = (-y, x) / xz

To G, add n periods~~, which are P8/~~x:

To get alternate generators, add n periods to G. n ranges from -x (subtracting a full octave) to +x (adding a full octave).

G = (-y, x) / xz + ~~nP8/x~~ = (nz - y, x) / xz~~<br~~ /~~>~~

G = (-y, x) / xz + nP = (-y, x) / xz + nP8/x = (nz - y, x) / xz

~~n ranges from -~~x (~~subtracting a full octave) to +~~x ~~(adding a full octave~~).

Rank-3 example: Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&amp;limit=7" rel="nofollow">x31.com</a> gives us this matrix:

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The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale would translate to a 5-edo subset. Such edos are marked with asterisks.

(table is under construction)

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<td style="text-align: center;">meantone

</td>

<td style="text-align: center;">12, 16, 19, 23, 26

<td style="text-align: center;">12, 13b, 14*, 15*, 16,

17, 18b*, 19, 20*, 21*,

22, 23, 24*, 25*, 26,

27, 28*, 29, 30*, 31

</td>

</tr>

<tr>

<th>

</th>

<th>

</th>

<th>

</th>

<th>

</th>

<th>

</th>

<th>

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<th>

</th>

</tr>

<tr>

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<td style="text-align: center;">srutal

</td>

<td style="text-align: center;">12, 20, 22, 24, 30

<td style="text-align: center;">12, 20*, 22, 24*, 30*

</td>

</tr>

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</td>

<td style="text-align: center;">12, 14, 16, 18b, 26

<td style="text-align: center;">12, 14, 16, 18b,

24*, 26, 28*

</td>

</tr>

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<td style="text-align: center;">mohajira

</td>

<td style="text-align: center;">14, 17, 20, 21, 24

<td style="text-align: center;">14*, 17, 20*, 21*, 24

27, 28, 30, 31

27, 28*, 30*, 31

</td>

</tr>

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</td>

<td style="text-align: center;">14, 18b, 21, 28

<td style="text-align: center;">14*, 18b, 21*, 28*

</td>

</tr>

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<td style="text-align: center;">semaphore

</td>

<td style="text-align: center;">22

</td>

</tr>

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</td>

~~</td>~~