Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626678573 - 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>2018-02- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-21 00:26:27 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626690883</tt>.<br> | ||
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So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5. | So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by "W". Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5. | ||
==Secondary splits== | ==Secondary splits*== | ||
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with: | Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with: | ||
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vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128) | vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128) | ||
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if | More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps. | ||
For a pergen (P8/m, | For a pergen (P8/m, P5), any interval (a,b) in which b mod m = 0 splits into at least m parts. For a pergen (P8, (a,b)/n), any interval (a',b') for which (a'·b - a·b') mod (n·|b|) = 0 splits into at least n parts. A double-split pergen (P8/m, M/n) contains all the secondary splits of both (P8/m, P5) and (P8, M/n), perhaps plus others. | ||
(a,b) = n·G | |||
(a',b') = i·(n·G) + j·(n·P8) = (i·a + j·n, i·b) | |||
a'·b - a·b' = i·a·b + j·b·n - i·a·b = j·b·n | |||
(a'·b - a·b') mod n·|b| = 0 | |||
|| | P8 = m·P | ||
(a',b') = i·(m·P) + j·(m·P12) = (i, j·m) | |||
|| ||= (P8/2, P5) ||= half-8ve ||= M2/2, M3/4, A4/6, m6/2, M10/2, d12/2 || | b' mod m = 0 | ||
M2/4 = (-3,2)/4 | |||
(2a +3b) mod 8 = 0 | |||
|| | b must be even | ||
(-6,4) = 81/64 = M3 | |||
mod nb/m? (-1,2) = 9/2 = WM9 = | |||
false double: (P8/m, (a,b)/n) | |||
gcd (m,n) = |b| | |||
(a'·b - a·b') mod | |||
The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occuring splits are listed too, under "all pergens". | |||
||||~ pergen ||~ secondary splits <= 12th || | |||
||||= all pergens ||= M3/2, d5/2, A4/3, A5/4, m7/2, M9/2, m10/3, A11/2 || | |||
||||~ half-splits ||~ || | |||
||= (P8/2, P5) ||= half-8ve ||= M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2 || | |||
||= (P8, P4/2) ||= half-4th ||= m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2 || | |||
||= (P8, P5/2) ||= half-5th ||= A1/2, m3/2, M7/2, m9/2, P11/2 || | |||
||= (P8/2, P4/2) ||= half-everything ||= (every 3-limit interval)/2 || | |||
||||~ third-splits ||~ || | |||
||= (P8/3, P5) ||= third-8ve ||= m3/3, M6/3, d5/6, A11/3, d12/3 || | |||
||= (P8, P4/3) ||= third-4th ||= A1/3, m7/6, M7/3, m10/9, M10/3 || | |||
||= (P8, P5/3) ||= third-5th ||= m2/3, m6/3, M9/6, A8/3, A12/3 || | |||
||= (P8, P11/3) ||= third-11th ||= M2/3, M3/6, A4/9, A5/12, m9/3, P12/3 || | |||
||= (P8/3, P4/2) ||= third-8ve half-4th ||= half-4th + third-8ve + M6/6, m10/6, A11/12 || | |||
||= (P8/3, P5/2) ||= third-8ve half-5th ||= half-5th + third-8ve + m3/6, d5/12 || | |||
||= (P8/2, P4/3) ||= half-8ve third-4th ||= half-8ve + third-4th + A4/6, M10/6 || | |||
||= (P8/2, P5/3) ||= half-8ve third-5th ||= half-8ve + third-5th + m6/6, M9/6, A12/6 || | |||
||= (P8/2, P11/3) ||= half-8ve third-11th ||= half-8ve + third-11th + M2/6, M3/12, A4/18, A5/24 || | |||
||= (P83, P4/3) ||= third-everything ||= (every 3-limit interval)/3 || | |||
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<!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div> | <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Further Discussion-Naming very large intervals">Naming very large intervals</a></div> | ||
<!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits">Secondary splits</a></div> | <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 2em;"><a href="#Further Discussion-Secondary splits*">Secondary splits*</a></div> | ||
<!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
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So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | So far, the largest multigen has been a 12th. As the multigen fractions get bigger, the multigen can get quite large. To avoid cumbersome degree names like 16th or 22nd, for degrees above 12, widening by an 8ve is indicated by &quot;W&quot;. Thus 10/3 = WM6 = wide major 6th, 9/2 = WWM2 or WM9, etc. For (P8, M/n), valid multigens are any voicing of the fifth that is less than n/2 octaves. For (P8, M/6), M can be P4, P5, P11, P12, WWP4 or WWP5.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Secondary splits"></a><!-- ws:end:WikiTextHeadingRule:71 -->Secondary splits</h2> | <!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc7"><a name="Further Discussion-Secondary splits*"></a><!-- ws:end:WikiTextHeadingRule:71 -->Secondary splits*</h2> | ||
<br /> | <br /> | ||
Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:<br /> | Besides the octave and/or multigen, a pergen splits many other 3-limit intervals as well. The composer can use these secondary splits to create melodies with equal-sized steps. For example, third-4th (porcupine) splits intervals other than the 4th into three parts. Of course, many 3-limit intervals split into three parts even when untempered, e.g. A4 = 3·M2. The interval's monzo (a,b) must have both a and b divisible by 3. The third-4th pergen furthermore splits any interval which is the sum or difference of the 4th and the triple 8ve WWP8. Stacking 4ths gives these intervals: P4, m7, m10, Wm6, Wm9... The last two are too large to be of any use melodically. Subtracting 4ths from the triple 8ve gives WWP8, WWP5, WM9, WM6, M10, M7, A4... The first four are too large, this leaves us with:<br /> | ||
| Line 2,523: | Line 2,542: | ||
vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)<br /> | vM7/2: C - F^ - Bv (vM7 = 15/8, probably more harmonious than M7 = 243/128)<br /> | ||
<br /> | <br /> | ||
More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if | More remote intervals include A1, d4, d7 and d10. These require a very long genchain. The most interesting melodically is A1: C - C^ - C#v - C#. From C to C# is 7 5ths, which equals 21 generators, so the genchain would contain 22 notes if it had no gaps.<br /> | ||
<br /> | |||
For a pergen (P8/m, P5), any interval (a,b) in which b mod m = 0 splits into at least m parts. For a pergen (P8, (a,b)/n), any interval (a',b') for which (a'·b - a·b') mod (n·|b|) = 0 splits into at least n parts. A double-split pergen (P8/m, M/n) contains all the secondary splits of both (P8/m, P5) and (P8, M/n), perhaps plus others. <br /> | |||
<br /> | |||
(a,b) = n·G<br /> | |||
(a',b') = i·(n·G) + j·(n·P8) = (i·a + j·n, i·b)<br /> | |||
a'·b - a·b' = i·a·b + j·b·n - i·a·b = j·b·n<br /> | |||
(a'·b - a·b') mod n·|b| = 0<br /> | |||
<br /> | |||
P8 = m·P<br /> | |||
(a',b') = i·(m·P) + j·(m·P12) = (i, j·m)<br /> | |||
b' mod m = 0<br /> | |||
<br /> | |||
M2/4 = (-3,2)/4<br /> | |||
(2a +3b) mod 8 = 0<br /> | |||
b must be even<br /> | |||
(-6,4) = 81/64 = M3<br /> | |||
<br /> | |||
mod nb/m? (-1,2) = 9/2 = WM9 =<br /> | |||
<br /> | |||
false double: (P8/m, (a,b)/n)<br /> | |||
gcd (m,n) = |b|<br /> | |||
(a'·b - a·b') mod<br /> | |||
<br /> | <br /> | ||
<br /> | <br /> | ||
<br /> | |||
The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occuring splits are listed too, under &quot;all pergens&quot;.<br /> | |||
<br /> | <br /> | ||
| Line 2,533: | Line 2,574: | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan="2">pergen<br /> | <th colspan="2">pergen<br /> | ||
</th> | </th> | ||
| Line 2,541: | Line 2,580: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td colspan="2" style="text-align: center;">all pergens<br /> | <td colspan="2" style="text-align: center;">all pergens<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">M3/2, d5/2, A5/4, m7/2, M9/2, A11/2<br /> | <td style="text-align: center;">M3/2, d5/2, A4/3, A5/4, m7/2, M9/2, m10/3, A11/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
< | <th colspan="2">half-splits<br /> | ||
</ | </th> | ||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">(P8/2, P5)<br /> | <td style="text-align: center;">(P8/2, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve<br /> | <td style="text-align: center;">half-8ve<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">M2/2, M3/4, A4/6, m6/2, M10/2, d12/2<br /> | <td style="text-align: center;">M2/2, M3/4, A4/6, A5/8, m6/2, M10/2, d12/2, A12/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P4/2)<br /> | <td style="text-align: center;">(P8, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-4th<br /> | <td style="text-align: center;">half-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">m2/2, M6/2, A8/2, m10/ | <td style="text-align: center;">m2/2, M6/2, m7/4, A8/2, m10/6, A11/4, P12/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P5/2)<br /> | <td style="text-align: center;">(P8, P5/2)<br /> | ||
</td> | </td> | ||
| Line 2,579: | Line 2,616: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P4/2)<br /> | <td style="text-align: center;">(P8/2, P4/2)<br /> | ||
</td> | </td> | ||
| Line 2,589: | Line 2,624: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th> | <th colspan="2">third-splits<br /> | ||
</th> | |||
<th><br /> | |||
</th> | </th> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P5)<br /> | <td style="text-align: center;">(P8/3, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-8ve<br /> | <td style="text-align: center;">third-8ve<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">m3/3, M6/3, A11/3, d12/3<br /> | <td style="text-align: center;">m3/3, M6/3, d5/6, A11/3, d12/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P4/3)<br /> | <td style="text-align: center;">(P8, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-4th<br /> | <td style="text-align: center;">third-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">A1/3, m7/6, M7/3, M10/3<br /> | <td style="text-align: center;">A1/3, m7/6, M7/3, m10/9, M10/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P5/3)<br /> | <td style="text-align: center;">(P8, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-5th<br /> | <td style="text-align: center;">third-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">m2/3, m6/3, M9/3, | <td style="text-align: center;">m2/3, m6/3, M9/6, A8/3, A12/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P11/3)<br /> | <td style="text-align: center;">(P8, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-11th<br /> | <td style="text-align: center;">third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">M2/3, M3/ | <td style="text-align: center;">M2/3, M3/6, A4/9, A5/12, m9/3, P12/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P4/2)<br /> | <td style="text-align: center;">(P8/3, P4/2)<br /> | ||
</td> | </td> | ||
| Line 2,647: | Line 2,670: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, P5/2)<br /> | <td style="text-align: center;">(P8/3, P5/2)<br /> | ||
</td> | </td> | ||
| Line 2,657: | Line 2,678: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P4/3)<br /> | <td style="text-align: center;">(P8/2, P4/3)<br /> | ||
</td> | </td> | ||
| Line 2,667: | Line 2,686: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P5/3)<br /> | <td style="text-align: center;">(P8/2, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve third-5th<br /> | <td style="text-align: center;">half-8ve third-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve + third-5th + m6/6, M9/6<br /> | <td style="text-align: center;">half-8ve + third-5th + m6/6, M9/6, A12/6<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, P11/3)<br /> | <td style="text-align: center;">(P8/2, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve third-11th<br /> | <td style="text-align: center;">half-8ve third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-8ve + third-11th + M2/6, M3/12, A4/18<br /> | <td style="text-align: center;">half-8ve + third-11th + M2/6, M3/12, A4/18, A5/24<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P83, P4/3)<br /> | <td style="text-align: center;">(P83, P4/3)<br /> | ||
</td> | </td> | ||
| Line 2,694: | Line 2,707: | ||
</td> | </td> | ||
<td style="text-align: center;">(every 3-limit interval)/3<br /> | <td style="text-align: center;">(every 3-limit interval)/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.<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. It includes alternate enharmonics for many pergens.<br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||