Kite's thoughts on pergens: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-19 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-19 21:18:13 UTC</tt>.<br> | ||
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==Pergens and MOS scales== | ==Pergens and MOS scales== | ||
MOS scales | Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk. | ||
||||||~ | ||||~ pergen ||~ MOS scales ||~ of 5-12 notes ||~ ||~ ||~ ||~ || | ||
||= | ||= (P8, P5) ||= unsplit ||= 5 = 2L 3s ||= 7 = 5L 2s |||| 12 = 7L 5s (or 5L 7s) || || || | ||
||= 2L 2s ||= (P8/2, P5) | ||||~ halves ||~ ||~ ||~ ||~ ||~ ||~ || | ||
||= 3L 1s ||= (P8, P5/2) | ||= (P8/2, P5) ||= half-8ve ||= 6 = 2L 4s ||= 8 = 2L 6s ||= 10 = 2L 8s |||||| 12 = 2L 10s (or 10L 2s) || | ||
||||||~ Pentatonic | ||= (P8, P4/2) ||= half-4th ||= 5 = 4L 1s ||= 9 = 5L 4s ||= ||= || || || | ||
||= (P8, P5/2) ||= half-5th ||= 7 = 3L 4s ||= 10 = 7L 3s ||= ||= || || || | |||
||= (P8/2, P4/2) ||= half-everything ||= 6 = 4L 2s ||= 10 = 4L 6s ||= ||= || || || | |||
||||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ || | |||
||= (P8/3, P5) ||= third-8ve ||= 6 = 3L 3s ||= 9 = 3L 6s |||| 12 = 3L 9s (or 9L 3s) || || || | |||
||= (P8, P4/3) ||= third-4th ||= 5 = 1L 4s ||= 6 = 1L 5s ||= 7 = 1L 6s ||= 8 = 7L 1s || || || | |||
||= (P8, P5/3) ||= third-5th ||= 5 = 1L 4s ||= 6 = 5L 1s ||= 11 = 5L 6s ||= || || || | |||
||= (P8, P11/3) ||= third-11th ||= 5 = 2L 3s ||= 7 = 2L 5s ||= 9 = 2L 7s ||= 11 = 2L 9s || || || | |||
||= (P8/3, P4/2) ||= third-8ve, half-4th ||= 6 = 3L 3s * ||= 9 = 6L 3s ||= ||= || || || | |||
||= (P8/3, P5/2) ||= third-8ve, half-5th ||= 6 = 3L 3s * ||= 9 = 3L 6s ||= 12 = 3L 9s ||= || || || | |||
||= (P8/2, P4/3) ||= half-8ve, third-4th ||= 6 = 2L 4s * ||= 8 = 6L 2s ||= ||= || || || | |||
||= (P8/2, P5/3) ||= half-8ve, third-5th ||= 6 = 4L 2s * ||= 10 = 6L 4s ||= ||= || || || | |||
||= (P8/2, P11/3) ||= half-8ve, third-11th ||= 6 = 2L 4s * ||= 8 = 2L 6s ||= 10 = 2L 8s ||= 12 = 2L 10s || || || | |||
||= (P8/3, P4/3) ||= third-everything ||= 6 = 3L 3s * ||= 9 = 6L 3s * ||= ||= || || || | |||
||||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ || | |||
||= (P8/4, P5) ||= quarter-8ve ||= 8 = 4L 4s |||| 12 = 4L 8s (or 8L 4s) ||= || || || | |||
||= (P8, P4/4) ||= quarter-4th ||= 5 = 1L 4s ||= 6 = 1L 5s ||= 7 = 1L 6s ||= 8 = 1L 7s || 9 = 1L 8s || 10 = 9L 1s || | |||
||= (P8, P5/4) ||= quarter-5th ||= 5 = 1L 4s ||= 6 = 1L 5s ||= 7 = 6L 1s ||= || || || | |||
||= (P8, P11/4) ||= quarter-11th ||= 5 = 3L 2s ||= 8 = 3L 5s ||= 11 = 3L 8s ||= || || || | |||
||= (P8, P12/4) ||= quarter-12th ||= 5 = 3L 2s ||= 8 = 5L 3s ||= ||= || || || | |||
||= (P8/4, P4/2) ||= quarter-8ve half-4th ||= 8 = 4L 4s * ||= 12 = 4L 8s ||= ||= || || || | |||
||= (P8/2, M2/4) ||= half-8ve quarter-tone ||= 6 = 2L 4s * ||= 8 = 2L 6s * ||= 10 = 2L 8s ||= 12 = 2L 10s || || || | |||
||= (P8/2, P4/4) ||= half-8ve quarter-4th ||= 6 = 2L 4s * ||= 8 = 2L 6s * ||= 10 = 8L 2s ||= || || || | |||
||= (P8/2, P5/4) ||= half-8ve quarter-5th ||= 6 = 2L 4s * ||= 8 = 6L 2s * ||= ||= || || || | |||
||= (P8/4, P4/3) ||= quarter-8ve third-4th ||= 8 = 4L 4s * ||= 12 = 8L 4s * ||= ||= || || || | |||
||= (P8/4, P5/3) ||= quarter-8ve third-5th ||= 8 = 4L 4s * ||= 12 = 4L 8s * ||= ||= || || || | |||
||= (P8/4, P11/3) ||= quarter-8ve third-11th ||= 8 = 4L 4s * ||= 12 = 4L 8s * ||= ||= || || || | |||
||= (P8/3, P4/4) ||= third-8ve quarter-4th ||= 6 = 3L 3s * ||= 9 = 3L 6s * ||= 12 = 9L 3s * ||= || || || | |||
||= (P8/3, P5/4) ||= third-8ve quarter-5th ||= 6 = 3L 3s * ||= 9 = 6L 3s * ||= ||= || || || | |||
||= (P8/3, P11/4) ||= third-8ve quarter-11th ||= 6 = 3L 3s * ||= 9 = 3L 6s * ||= 12 = 3L 9s * ||= || || || | |||
||= (P8/3, P12/4) ||= third-8ve quarter-12th ||= 6 = 3L 3s * ||= 9 = 3L 6s * ||= 12 = 3L 9s * ||= || || || | |||
||= (P8/4, P4/4) ||= quarter-everything ||= 8 = 4L 4s * ||= 12 = 8L 4s * ||= ||= || || || | |||
About half of the quarter-split pergens have nothing but starred scales. The next table continues with fifth-splits, etc., showing only unstarred scales. It also excludes those MOS scales with L/s > 5. | |||
||||~ pergen ||||||||||~ MOS scales of 5-12 notes (unstarred scales only, L/s < 5 only) ||~ ||~ || | |||
||||~ fifths ||~ ||~ ||~ ||~ ||~ ||~ ||~ || | |||
||= (P8, P4/5) ||= fifth-4th ||= 8 = 1L 7s ||= 9 = 1L 8s ||= 10 = 1L 9s ||= 11 = 1L 10s ||= 12 = 1L 11s (or 11L 1s) ||= ||= || | |||
||= (P8, P5/5) ||= fifth-5th ||= 6 = 1L 5s ||= 7 = 1L 6s ||= 8 = 1L 7s ||= 9 = 8L 1s ||= ||= ||= || | |||
||= (P8, P11/5) ||= fifth-11th ||= 7 = 4L 3s ||= 11 = 7L 4s ||= ||= ||= ||= ||= || | |||
||= (P8, P12/5) ||= fifth-12th ||= 7 = 3L 4s ||= 10 = 3L 7s ||= ||= ||= ||= ||= || | |||
||= (P8, WW4/5) ||= fifth-WW4th ||= 7 = 2L 5s ||= 9 = 2L 7s ||= 11 = 2L 9s ||= ||= ||= ||= || | |||
||= (P8/2, P4/5) ||= half-8ve fifth-4th |||||| 12 = 2L 10s (or 10L 2s) ||= ||= ||= ||= || | |||
||||~ sixths ||~ ||~ ||~ ||~ ||~ ||~ ||~ || | |||
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A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | |||
||~ MOS scale ||||~ primary example ||~ secondary examples || | |||
||~ Pentatonic ||~ ||~ ||~ || | |||
||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic ||< third-4th, quarter-4th, quarter-5th || | ||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic ||< third-4th, quarter-4th, quarter-5th || | ||
||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic ||< third-11th || | ||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic ||< third-11th || | ||
||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic ||< quarter-11th || | ||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic ||< quarter-11th || | ||
||= 4L 1s ||= (P8, P4/2) [5] ||= half-4th pentatonic ||< || | ||= 4L 1s ||= (P8, P4/2) [5] ||= half-4th pentatonic ||< || | ||
||||||~ | ||~ Hexatonic ||~ ||~ ||~ || | ||
||= 1L 5s ||= (P8, P4/3) [6] ||= third-4th hexatonic ||< quarter-4th, quarter-5th, fifth-4th, fifth-5th || | ||= 1L 5s ||= (P8, P4/3) [6] ||= third-4th hexatonic ||< quarter-4th, quarter-5th, fifth-4th, fifth-5th || | ||
||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic ||< || | ||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic ||< || | ||
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||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic ||< || | ||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic ||< || | ||
||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic ||< || | ||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic ||< || | ||
||||||~ | ||~ Heptatonic ||~ ||~ ||~ || | ||
||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic ||< quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th || | ||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic ||< quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th || | ||
||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic ||< fifth-WW4th, sixth-WW5th || | ||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic ||< fifth-WW4th, sixth-WW5th || | ||
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||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic ||< sixth-WW4th || | ||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic ||< sixth-WW4th || | ||
||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< || | ||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic ||< || | ||
||||||~ | ||~ Octotonic ||~ ||~ ||~ || | ||
||= 1L 7s ||= (P8, P4/4) [8] ||= quarter-4th octotonic ||< fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th || | ||= 1L 7s ||= (P8, P4/4) [8] ||= quarter-4th octotonic ||< fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th || | ||
||= 2L 6s ||= (P8/2, P5) [8] ||= half-8ve octotonic ||< || | ||= 2L 6s ||= (P8/2, P5) [8] ||= half-8ve octotonic ||< || | ||
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||= 6L 2s ||= (P8/2, P4/3) [8] ||= half-8ve third-4th octotonic ||< || | ||= 6L 2s ||= (P8/2, P4/3) [8] ||= half-8ve third-4th octotonic ||< || | ||
||= 7L 1s ||= (P8, P4/3) [8] ||= third-4th octotonic ||< || | ||= 7L 1s ||= (P8, P4/3) [8] ||= third-4th octotonic ||< || | ||
||||||~ | ||~ Nonatonic ||~ ||~ ||~ || | ||
||= 1L 8s ||= (P8, P4/4) [9] ||= quarter-4th nonatonic ||< fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th || | ||= 1L 8s ||= (P8, P4/4) [9] ||= quarter-4th nonatonic ||< fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th || | ||
||= 2L 7s ||= (P8, W<span style="vertical-align: super;">3</span>P5/8) [9] ||= eighth-W<span style="vertical-align: super;">3</span>5th nonatonic ||< third-11th, fifth-WW4th || | ||= 2L 7s ||= (P8, W<span style="vertical-align: super;">3</span>P5/8) [9] ||= eighth-W<span style="vertical-align: super;">3</span>5th nonatonic ||< third-11th, fifth-WW4th || | ||
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||= 7L 2s ||= (P8, WWP5/6)[9] ||= sixth-WW5th nonatonic ||< (lopsided unless 5th is sharp) || | ||= 7L 2s ||= (P8, WWP5/6)[9] ||= sixth-WW5th nonatonic ||< (lopsided unless 5th is sharp) || | ||
||= 8L 1s ||= (P8, P5/5) [9] ||= fifth-5th nonatonic ||< || | ||= 8L 1s ||= (P8, P5/5) [9] ||= fifth-5th nonatonic ||< || | ||
||||||~ | ||~ Decatonic ||~ ||~ ||~ || | ||
||= 1L 9s ||= (P8, P5/6) [10] ||= sixth-5th decatonic ||< fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th || | ||= 1L 9s ||= (P8, P5/6) [10] ||= sixth-5th decatonic ||< fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th || | ||
||= 2L 8s ||= (P8/2, P5) [10] ||= half-8ve decatonic ||< half-8ve quartertone, half-8ve third-11th || | ||= 2L 8s ||= (P8/2, P5) [10] ||= half-8ve decatonic ||< half-8ve quartertone, half-8ve third-11th || | ||
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||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||
The | The pentatonic MOS scales don't include fifthj-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen. | ||
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | ||
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How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7. | How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7. | ||
How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime | How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. | ||
Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen. | Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). | ||
For N-edo, if P = p\N and G = g\N, ... | |||
It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen. | |||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | ||
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||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||
==Combining pergens== | ==Combining pergens== | ||
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<!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:82 -->Pergens and MOS scales</h2> | <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:82 -->Pergens and MOS scales</h2> | ||
<br /> | <br /> | ||
MOS scales | Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br /> | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan=" | <th colspan="2">pergen<br /> | ||
</th> | </th> | ||
<th> | <th>MOS scales<br /> | ||
</th> | |||
<th>of 5-12 notes<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | </th> | ||
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<td style="text-align: center;"> | <td style="text-align: center;">(P8, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">unsplit<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 2L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">7 = 5L 2s<br /> | ||
</td> | </td> | ||
<td colspan="2">12 = 7L 5s (or 5L 7s)<br /> | |||
<td | |||
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< | <th colspan="2">halves<br /> | ||
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< | <th><br /> | ||
</ | </th> | ||
<th><br /> | |||
</th> | |||
<th | <th><br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
| Line 2,529: | Line 2,580: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/2, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-8ve<br /> | ||
</td> | |||
<td style="text-align: center;">6 = 2L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">8 = 2L 6s<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">10 = 2L 8s<br /> | ||
</td> | </td> | ||
<td | <td colspan="3">12 = 2L 10s (or 10L 2s)<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 4L 1s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">9 = 5L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P5/2)<br /> | ||
</td> | |||
<td style="text-align: center;">half-5th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 = 3L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">10 = 7L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/2, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-everything<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 4L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">10 = 4L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
< | <th colspan="2">thirds<br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</ | </th> | ||
< | <th><br /> | ||
</ | </th> | ||
<th><br /> | |||
</th> | |||
< | <th><br /> | ||
</ | |||
< | |||
</ | |||
< | |||
</ | |||
<th | |||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
| Line 2,631: | Line 2,664: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/3, P5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-8ve<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 3L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">9 = 3L 6s<br /> | ||
</td> | </td> | ||
<td colspan="2">12 = 3L 9s (or 9L 3s)<br /> | |||
<td | |||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 1L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">6 = 1L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 = 1L 6s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 7L 1s<br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 1L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">6 = 5L 1s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11 = 5L 6s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P11/3)<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 2L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">7 = 2L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9 = 2L 7s<br /> | |||
<td style="text-align: center;">2L | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">11 = 2L 9s<br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/3, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-8ve, half-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">9 = 6L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/3, P5/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">third-8ve, half-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">9 = 3L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">12 = 3L 9s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/2, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-8ve, third-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 2L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">8 = 6L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/2, P5/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-8ve, third-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 4L 2s *<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">10 = 6L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/2, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-8ve, third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 2L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">8 = 2L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">10 = 2L 8s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 2L 10s<br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/3, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-everything<br /> | |||
<td style="text-align: center;">third- | |||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9 = 6L 3s *<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td><br /> | |||
<td | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th colspan=" | <th colspan="2">quarters<br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
| Line 2,859: | Line 2,858: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8/4, P5)<br /> | ||
</td> | |||
<td style="text-align: center;">quarter-8ve<br /> | |||
</td> | |||
<td style="text-align: center;">8 = 4L 4s<br /> | |||
</td> | |||
<td colspan="2">12 = 4L 8s (or 8L 4s)<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P4/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 1L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">6 = 1L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 = 1L 6s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 1L 7s<br /> | ||
</td> | </td> | ||
<td | <td>9 = 1L 8s<br /> | ||
</td> | </td> | ||
<td | <td>10 = 9L 1s<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P5/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 1L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">6 = 1L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">7 = 6L 1s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P11/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 3L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">8 = 3L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">11 = 3L 8s<br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P8, P12/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-12th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">5 = 3L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: | <td style="text-align: center;">8 = 5L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
<td | <td><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8/4, P4/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-8ve half-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 4L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 4L 8s<br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 2,993: | Line 2,964: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/2, M2/4)<br /> | |||
<td style="text-align: center;">(P8/2, | |||
</td> | </td> | ||
<td style="text-align: center;">half-8ve<br /> | <td style="text-align: center;">half-8ve quarter-tone<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6 = 2L 4s<br /> | <td style="text-align: center;">6 = 2L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8 = 2L 6s<br /> | <td style="text-align: center;">8 = 2L 6s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">10 = 2L 8s<br /> | <td style="text-align: center;">10 = 2L 8s<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;">12 = 2L 10s<br /> | ||
</td> | |||
<td><br /> | |||
</td> | |||
<td><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P4/ | <td style="text-align: center;">(P8/2, P4/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-4th<br /> | <td style="text-align: center;">half-8ve quarter-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 2L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 2L 6s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">10 = 8L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,043: | Line 3,000: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, P5/ | <td style="text-align: center;">(P8/2, P5/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">half-5th<br /> | <td style="text-align: center;">half-8ve quarter-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 2L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 6L 2s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,061: | Line 3,018: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/ | <td style="text-align: center;">(P8/4, P4/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-8ve third-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 4L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 8L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,079: | Line 3,036: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
< | <td style="text-align: center;">(P8/4, P5/3)<br /> | ||
</td> | |||
<td style="text-align: center;">quarter-8ve third-5th<br /> | |||
</ | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 4L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 4L 8s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 3,113: | Line 3,054: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8/4, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third- | <td style="text-align: center;">quarter-8ve third-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 4L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 4L 8s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 3,131: | Line 3,072: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8/3, P4/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third- | <td style="text-align: center;">third-8ve quarter-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">9 = 3L 6s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 9L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,149: | Line 3,090: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8/3, P5/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third- | <td style="text-align: center;">third-8ve quarter-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">9 = 6L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
| Line 3,167: | Line 3,108: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, | <td style="text-align: center;">(P8/3, P11/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-8ve | <td style="text-align: center;">third-8ve quarter-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6 = 3L 3s<br /> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9 = | <td style="text-align: center;">9 = 3L 6s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">12 = 3L 9s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,185: | Line 3,126: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/3, | <td style="text-align: center;">(P8/3, P12/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-8ve | <td style="text-align: center;">third-8ve quarter-12th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">6 = 3L 3s<br /> | <td style="text-align: center;">6 = 3L 3s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">9 = 3L 6s<br /> | <td style="text-align: center;">9 = 3L 6s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">12 = 3L 9s<br /> | <td style="text-align: center;">12 = 3L 9s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,203: | Line 3,144: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8/ | <td style="text-align: center;">(P8/4, P4/4)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">quarter-everything<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 4L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12 = 8L 4s *<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,219: | Line 3,160: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
</tr> | |||
</table> | |||
<br /> | |||
About half of the quarter-split pergens have nothing but starred scales. The next table continues with fifth-splits, etc., showing only unstarred scales. It also excludes those MOS scales with L/s &gt; 5.<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th colspan="2">pergen<br /> | |||
</th> | |||
<th colspan="5">MOS scales of 5-12 notes (unstarred scales only, L/s &lt; 5 only)<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<th colspan="2">fifths<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8 | <td style="text-align: center;">(P8, P4/5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">fifth-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">8 = 1L 7s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">10 = | <td style="text-align: center;">9 = 1L 8s<br /> | ||
</td> | |||
<td style="text-align: center;">10 = 1L 9s<br /> | |||
</td> | |||
<td style="text-align: center;">11 = 1L 10s<br /> | |||
</td> | |||
<td style="text-align: center;">12 = 1L 11s (or 11L 1s)<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8 | <td style="text-align: center;">(P8, P5/5)<br /> | ||
</td> | |||
<td style="text-align: center;">fifth-5th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">6 = 1L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 = 1L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">8 = | <td style="text-align: center;">8 = 1L 7s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">9 = 8L 1s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8 | <td style="text-align: center;">(P8, P11/5)<br /> | ||
</td> | |||
<td style="text-align: center;">fifth-11th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 = 4L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">11 = 7L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,269: | Line 3,252: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
< | <td style="text-align: center;">(P8, P12/5)<br /> | ||
</td> | |||
<td style="text-align: center;">fifth-12th<br /> | |||
</ | </td> | ||
< | <td style="text-align: center;">7 = 3L 4s<br /> | ||
</ | |||
<td style="text-align: center;"> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">10 = 3L 7s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8, WW4/5)<br /> | ||
</td> | |||
<td style="text-align: center;">fifth-WW4th<br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">7 = 2L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">9 = 2L 7s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">11 = 2L 9s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">(P8, | <td style="text-align: center;">(P8/2, P4/5)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">half-8ve fifth-4th<br /> | ||
</td> | </td> | ||
<td | <td colspan="3">12 = 2L 10s (or 10L 2s)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <th colspan="2">sixths<br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;"><br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,375: | Line 3,366: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,393: | Line 3,386: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,447: | Line 3,446: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,465: | Line 3,464: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,483: | Line 3,486: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 3,501: | Line 3,506: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
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<br /> | |||
A MOS scale tends to be generated by just a few pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | |||
<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<th>MOS scale<br /> | |||
</th> | |||
<th colspan="2">primary example<br /> | |||
</th> | |||
<th>secondary examples<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<th>Pentatonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/3) [5]<br /> | |||
</td> | |||
<td style="text-align: center;">third-5th pentatonic<br /> | |||
</td> | |||
<td style="text-align: left;">third-4th, quarter-4th, quarter-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5) [5]<br /> | |||
</td> | |||
<td style="text-align: center;">unsplit pentatonic<br /> | |||
</td> | |||
<td style="text-align: left;">third-11th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P12/5) [5]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-12th pentatonic<br /> | |||
</td> | |||
<td style="text-align: left;">quarter-11th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/2) [5]<br /> | |||
</td> | |||
<td style="text-align: center;">half-4th pentatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>Hexatonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 5s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/3) [6]<br /> | |||
</td> | |||
<td style="text-align: center;">third-4th hexatonic<br /> | |||
</td> | |||
<td style="text-align: left;">quarter-4th, quarter-5th, fifth-4th, fifth-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5) [6]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve hexatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/3, P5) [6]<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve hexatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P4/2) [6]<br /> | |||
</td> | |||
<td style="text-align: center;">half-everything hexatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/3) [6]<br /> | |||
</td> | |||
<td style="text-align: center;">third-5th hexatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>Heptatonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 6s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/3) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">third-4th heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;">quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 5s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P11/3) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">third-11th heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;">fifth-WW4th, sixth-WW5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/2) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">half-5th heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;">fifth-12th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P11/5) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">fifth-11th heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;">sixth-12th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">unsplit heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;">sixth-WW4th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/4) [7]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-5th heptatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>Octotonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 7s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/4) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-4th octotonic<br /> | |||
</td> | |||
<td style="text-align: left;">fifth-4th, fifth-5th, sixth-4th, sixth-5th, seventh-4th, seventh-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 6s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve octotonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 5s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P11/4) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-11th octotonic<br /> | |||
</td> | |||
<td style="text-align: left;">seventh-WW4th, seventh-WW5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/4, P5) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-8ve octotonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P12/4) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-12th octotonic<br /> | |||
</td> | |||
<td style="text-align: left;">(very lopsided, unless 5th is quite flat)<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P4/3) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve third-4th octotonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/3) [8]<br /> | |||
</td> | |||
<td style="text-align: center;">third-4th octotonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>Nonatonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 8s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/4) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-4th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">fifth-4th, sixth-4th, sixth-5th, seventh-4th/5th, eighth-4th/5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 7s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, W<span style="vertical-align: super;">3</span>P5/8) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">eighth-W<span style="vertical-align: super;">3</span>5th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">third-11th, fifth-WW4th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 6s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/3, P5) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">third-8ve half-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 5s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P12/7) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">seventh-12th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">sixth-11th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/2) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">half-4th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">(lopsided unless 4th is sharp), seventh-11th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/3, P4/2) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve half-4th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, WWP5/6)[9]<br /> | |||
</td> | |||
<td style="text-align: center;">sixth-WW5th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;">(lopsided unless 5th is sharp)<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">8L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/5) [9]<br /> | |||
</td> | |||
<td style="text-align: center;">fifth-5th nonatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th>Decatonic<br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
<th><br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 9s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/6) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">sixth-5th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">fifth-4th, sixth-4th, seventh-4th/5th, eighth-4th/5th, ninth-4th/5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 8s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">half-8ve quartertone, half-8ve third-11th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 7s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P12/5) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">fifth-12th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">eighth-WW4th, eighth-WW5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">4L 6s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P4/2) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-everything decatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">5L 5s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">(lopsided unless 5th is quite flat)<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">6L 4s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5/3) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve third-5th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">7L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/2) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-5th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">ninth-WW5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">8L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P4/4) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve quarter-4th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;">half-8ve quarter-12th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">9L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/2) [10]<br /> | |||
</td> | |||
<td style="text-align: center;">quarter-4th decatonic<br /> | |||
</td> | |||
<td style="text-align: left;"><br /> | |||
</td> | |||
</tr> | |||
</table> | |||
<br /> | |||
The pentatonic MOS scales don't include fifthj-split pergens, because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<br /> | |||
<br /> | |||
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br /> | |||
<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:84 -->Pergens and EDOs</h2> | |||
<br /> | |||
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | |||
<br /> | |||
Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.<br /> | |||
<br /> | |||
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.<br /> | |||
<br /> | |||
How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br /> | |||
<br /> | |||
Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). <br /> | |||
<br /> | |||
For N-edo, if P = p\N and G = g\N, ...<br /> | |||
<br /> | |||
It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br /> | |||
<br /> | |||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | ||
<br /> | <br /> | ||
| Line 3,907: | Line 4,698: | ||
</table> | </table> | ||
<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:86 -->Combining pergens</h2> | <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:86 -->Combining pergens</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:5974: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:5974 --><br /> | ||
<br /> | <br /> | ||
Alt-pergenLister 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 /> | Alt-pergenLister 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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:5975:http://www.tallkite.com/misc_files/alt-pergensLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><!-- ws:end:WikiTextUrlRule:5975 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:3616:&lt;img src=&quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 460px; width: 704px;&quot; /&gt; --><img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><!-- ws:end:WikiTextLocalImageRule:3616 --><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:90 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc19"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:90 -->Misc notes</h2> | ||