Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624832143 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624832325 - 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>2018-01-13 22: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-13 22:30:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624832325</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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Next we loop through all combinations of x and z in such a way that larger values of x and z come last: | Next we loop through all combinations of x and z in such a way that larger values of x and z come last: | ||
i = 1; loop (maxFraction, | i = 1; loop (maxFraction, | ||
j = 1; loop (i - 1, | > j = 1; loop (i - 1, | ||
makeMapping (i, j); makeMapping (i, -j); | >> makeMapping (i, j); makeMapping (i, -j); | ||
makeMapping (j, i); makeMapping (j, -i); | >> makeMapping (j, i); makeMapping (j, -i); | ||
j += 1; | >> j += 1; | ||
); | > ); | ||
makeMapping (i, i); makeMapping (i, -i); | > makeMapping (i, i); makeMapping (i, -i); | ||
i += 1; | > i += 1; | ||
); | ); | ||
The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of n from -x to x-1 is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and n makes a valid pergen. This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. | The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of n from -x to x-1 is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and n makes a valid pergen. This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted. | ||
In the Supplemental | In the [[pergen#Further%20Discussion-Supplemental%20materials|Supplemental materials]] section, a program is linked to that performs these calculations and lists all pergens. It also lists suggested enharmonics. | ||
It's possible that allowing y and/or n to range further might produce muscally useful pergens. | It's possible that allowing y and/or n to range further might produce muscally useful pergens. Further study is needed. | ||
==Extremely large multigens== | ==Extremely large multigens== | ||
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If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced. | If the pergen is not explicitly false, put the pergen in its **unreduced** form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P < G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced. | ||
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5)/(3<span class="nowrap">⋅</span>2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6<span class="nowrap">⋅</span>G - m3. The comma splits both the octave and the fifth. | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5) / (3<span class="nowrap">⋅</span>2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This __is__ explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6<span class="nowrap">⋅</span>G - m3. The comma splits both the octave and the fifth. | ||
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4)/(2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4) / (2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit. | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively. | ||
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<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span> | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span> | ||
This is a lot of math, but it only needs to be done once for each pergen | This is a lot of math, but it only needs to be done once for each pergen! | ||
==Alternate enharmonics== | ==Alternate enharmonics== | ||
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Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | ||
For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as ^ | For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2. | ||
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | ||
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. | 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. | ||
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General rules for combining pergens: | General rules for combining pergens: | ||
* (P8/m, M/n) + (P8, P5) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8, M/n) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m') | * (P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m') | ||
* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | * (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | ||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed. | ||
==Pergens and EDOs== | ==Pergens and EDOs== | ||
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Next we loop through all combinations of x and z in such a way that larger values of x and z come last:<br /> | Next we loop through all combinations of x and z in such a way that larger values of x and z come last:<br /> | ||
i = 1; loop (maxFraction,<br /> | i = 1; loop (maxFraction,<br /> | ||
j = 1; loop (i - 1,< | <ul class="quotelist"><li>j = 1; loop (i - 1,<ul class="quotelist"><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);<br /> | ||
makeMapping (i, j); makeMapping (i, -j);< | |||
makeMapping (j, i); makeMapping (j, -i);< | |||
j += 1;< | |||
);< | |||
makeMapping (i, i); makeMapping (i, -i);< | |||
i += 1;< | |||
);<br /> | |||
<br /> | <br /> | ||
The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of n from -x to x-1 is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and n makes a valid pergen. This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted.<br /> | The makeMapping function uses the two parameters as x and z, and loops through all valid values of y. Every value of n from -x to x-1 is tested, and the one that minimizes the multigen's splitting fraction and cents is chosen. This combination of x, y, z and n makes a valid pergen. This pergen is added to the list, unless it's a duplicate. The pergens are almost but not quite in the proper order, they will need to be sorted.<br /> | ||
<br /> | <br /> | ||
In the Supplemental | In the <a class="wiki_link" href="/pergen#Further%20Discussion-Supplemental%20materials">Supplemental materials</a> section, a program is linked to that performs these calculations and lists all pergens. It also lists suggested enharmonics.<br /> | ||
<br /> | <br /> | ||
It's possible that allowing y and/or n to range further might produce muscally useful pergens. | It's possible that allowing y and/or n to range further might produce muscally useful pergens. Further study is needed.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:53:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:53 -->Extremely large multigens</h2> | <!-- ws:start:WikiTextHeadingRule:53:&lt;h2&gt; --><h2 id="toc6"><a name="Further Discussion-Extremely large multigens"></a><!-- ws:end:WikiTextHeadingRule:53 -->Extremely large multigens</h2> | ||
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If the pergen is not explicitly false, put the pergen in its <strong>unreduced</strong> form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.<br /> | If the pergen is not explicitly false, put the pergen in its <strong>unreduced</strong> form, which is always explicitly false if the pergen is a false double. The unreduced form replaces the generator with the difference between the period and the generator: (P8/m, M/n) becomes (P8/m, P8/m - M/n) = (P8/m, (n<span class="nowrap">⋅</span>P8 - m<span class="nowrap">⋅</span>M)/nm). The new multigen M' is the product of the original pergen's outer elements (P8 and n) minus the product of the inner elements (m and M), divided by the product of the fractions (m and n). Invert M' if descending (if P &lt; G), and simplify if m and n aren't coprime. M' will have a larger fraction and/or a larger size in cents, hence the name unreduced.<br /> | ||
<br /> | <br /> | ||
For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5)/(3<span class="nowrap">⋅</span>2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6<span class="nowrap">⋅</span>G - m3. The comma splits both the octave and the fifth.<br /> | For example, (P8/3, P5/2) is a false double that isn't explicitly false. Its unreduced generator is (2<span class="nowrap">⋅</span>P8 - 3<span class="nowrap">⋅</span>P5) / (3<span class="nowrap">⋅</span>2) = m3/6, and the unreduced pergen is (P8/3, m3/6). This <u>is</u> explicitly false, thus the comma can be found from m3/6 alone. G is about 50¢, and the comma is 6<span class="nowrap">⋅</span>G - m3. The comma splits both the octave and the fifth.<br /> | ||
<br /> | <br /> | ||
This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4)/(2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>M2)/8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.<br /> | This suggests an alternate true/false test: if neither the pergen nor the unreduced pergen is explicitly false, the pergen is a true double. For example, (P8/4, P4/2) isn't explicitly false. Its unreduced form has (2<span class="nowrap">⋅</span>P8 - 4<span class="nowrap">⋅</span>P4) / (2<span class="nowrap">⋅</span>4) = (2<span class="nowrap">⋅</span>M2) / 8, which simplifies to M2/4. The unreduced pergen is (P8/4, M2/4), which also isn't explicitly false, thus (P8/4, P4/2) is a true double. It requires two commas, one for each fraction. The two commas must use different higher primes, e.g. 648/625 and 49/48. Thus true doubles require commas of at least 7-limit, whereas false doubles require only 5-limit.<br /> | ||
<br /> | <br /> | ||
A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br /> | A false double pergen's temperament can also be constructed from two commas, as if it were a true double. For example, (P8/3, P4/2) results from 128/125 and 49/48, which split the octave and the 4th respectively.<br /> | ||
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<br /> | <br /> | ||
<span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | <span style="display: block; text-align: center;">P1 — \M3 — \\A5=/m6 — P8</span><span style="display: block; text-align: center;">C — E\ — Ab/ — C</span><span style="display: block; text-align: center;">P1 — ^\M3 — ^^\\A5=^^/m6=vv\M6 — ^<span style="vertical-align: super;">3</span>8=v/m9 — F</span><span style="display: block; text-align: center;">C — E^\ — Ab^^/=Avv\ — Dbv/ — F</span><br /> | ||
This is a lot of math, but it only needs to be done once for each pergen | This is a lot of math, but it only needs to be done once for each pergen!<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:63:&lt;h2&gt; --><h2 id="toc11"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:63 -->Alternate enharmonics</h2> | <!-- ws:start:WikiTextHeadingRule:63:&lt;h2&gt; --><h2 id="toc11"><a name="Further Discussion-Alternate enharmonics"></a><!-- ws:end:WikiTextHeadingRule:63 -->Alternate enharmonics</h2> | ||
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Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | ||
<br /> | <br /> | ||
For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as ^ | For example, small triple amber tempers out (12,-1,0,0,-3) from 2.3.11, making a third-11th pergen. The generator is 11/8. If 11/8 is notated as an ^4, the enharmonic is v<span style="vertical-align: super;">3</span>M2, but if 11/8 is notated as a vA4, the enharmonic is ^<span style="vertical-align: super;">3</span>dd2.<br /> | ||
<br /> | <br /> | ||
This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | ||
<br /> | <br /> | ||
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 /> | 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 /> | ||
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<br /> | <br /> | ||
General rules for combining pergens:<br /> | General rules for combining pergens:<br /> | ||
<ul><li>(P8/m, | <ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li></ul><br /> | ||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:71 -->Pergens and EDOs</h2> | <!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:71 -->Pergens and EDOs</h2> | ||
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<br /> | <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 /> | 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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:3873:http://www.tallkite.com/misc_files/pergens.pdf --><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><!-- ws:end:WikiTextUrlRule:3873 --><br /> | ||
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This app 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 /> | This app 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:75 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:75 -->Misc notes</h2> | ||
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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 <!-- ws:start:WikiTextUrlRule: | 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 <!-- ws:start:WikiTextUrlRule:3874:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3874 --><br /> | ||
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