Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 628122853 - Original comment: ** |
Wikispaces>TallKite **Imported revision 628136231 - 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>2018-03-29 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-29 14:07:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>628136231</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians. | Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a __huge__ number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians. | ||
Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). | Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk. | ||
||~ __pergen number__ ||||||||||||~ __prime subgroup__ || | ||~ __pergen number__ ||||||||||||~ __prime subgroup__ || | ||
||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) || | ||~ unsplit ||~ 2.3 ||~ 2.5 (M3 = 5/4) ||~ 2.7 (M2 = 8/7) ||~ 3.5 (M6 = 5/3) ||~ 3.7 (M3 = 9/7) ||~ 5.7 (WWM3 = 5/1, d5 = 7/5) || | ||
| Line 682: | Line 682: | ||
||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (M9, d5)* || | ||= 2 ||= (P8/2, P5) ||= (P8/2, M3) ||= (P8/2, M2) ||= (P12/2, M6) ||= (P12/2, M3) ||= (M9, d5)* || | ||
||= 3 ||= (P8, P4/2) ||= (P8, M2)* ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M2)* ||= (WWM3, m3)* || | ||= 3 ||= (P8, P4/2) ||= (P8, M2)* ||= (P8, M2/2) ||= (P12, M6/2) ||= (P12, M2)* ||= (WWM3, m3)* || | ||
||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, P5)* ||= (P12, | ||= 4 ||= (P8, P5/2) ||= (P8, m6/2) ||= (P8, P5)* ||= (P12, P4)* ||= (P12, m10/2) ||= (WWM3, M7)* || | ||
||= 5 ||= (P8/2, P4/2) ||= (P8/2, M2)* ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (M9, m3)* || | ||= 5 ||= (P8/2, P4/2) ||= (P8/2, M2)* ||= (P8/2, M2/2) ||= (P12/2, M6/2) ||= (P12/2, M3/2) ||= (M9, m3)* || | ||
||~ third-splits ||~ ||~ ||~ ||~ ||~ ||~ || | ||~ third-splits ||~ ||~ ||~ ||~ ||~ ||~ || | ||
| Line 689: | Line 689: | ||
||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, P4)* ||= (WWM3, WA6/3) || | ||= 8 ||= (P8, P5/3) ||= (P8, m6/3) ||= (P8, m7/3) ||= (P12, m7/3) ||= (P12, P4)* ||= (WWM3, WA6/3) || | ||
||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) || | ||= 9 ||= (P8, P11/3) ||= (P8, M10/3) ||= (P8, M9/3) ||= (P12, WWM3/3) ||= (P12, WM7/3) ||= (WWM3, WWm7/3) || | ||
||= 10 ||= (P8/3, P4/2) ||= (P8/3, M2)* ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= | ||= 10 ||= (P8/3, P4/2) ||= (P8/3, M2)* ||= (P8/3, M2/2) ||= (P12/3, M6/2) ||= (P12/3, M2)* ||= (WWM3/3, m3)* || | ||
||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, P5)* ||= (P12/3, P4)* ||= | ||= 11 ||= (P8/3, P5/2) ||= (P8/3. m6/2) ||= (P8/3, P5)* ||= (P12/3, P4)* ||= (P12/3, m10/2) ||= (WWM3/3, M7)* || | ||
||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||= | ||= 12 ||= (P8/2, P4/3) ||= (P8/2, M3/3) ||= (P8/2, M2/3) ||= (P12/2, M6/3) ||= (P12/2, M3/3) ||= (M9, d5/3)* || | ||
||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||= | ||= 13 ||= (P8/2, P5/3) ||= (P8/2, m6/3) ||= (P8/2, m7/3) ||= (P12/2, m7/3) ||= (P12/2, P4)* ||= (M9, WA6/3)* || | ||
||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||= | ||= 14 ||= (P8/2, P11/3) ||= (P8/2, M10/3) ||= (P8/2, M9/3) ||= (P12/2, WWM3/3) ||= (P12/2, WM7/3) ||= (M9, WWm7/3)* || | ||
||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||= | ||= 15 ||= (P8/3, P4/3) ||= (P8/3, M3/3) ||= (P8/3, M2/3) ||= (P12/3, M6/3) ||= (P12/3, P4)* ||= (WWM3/3, d5/3) || | ||
For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1. | For prime subgroup p.q, the unsplit pergen has period p/1. The generator is found by dividing q by p until it's less than p/1, and inverting if it's more than half of p/1. | ||
| Line 720: | Line 720: | ||
C1 --- G1 --- D2 --- A2 | C1 --- G1 --- D2 --- A2 | ||
Splitting the 5th adds notes to the horizontal edges of the square: | Splitting the 5th adds notes to the horizontal edges of the square: | ||
C2 Ev2 G2 | C2 Ev2 G2 | ||
| . . . . . . | | | . . . . . . | | ||
| Line 732: | Line 732: | ||
C1 Ev1 G1 | C1 Ev1 G1 | ||
From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square. | From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square. | ||
C2 ---- G2 | C2 ---- G2 | ||
| . A^1 . | | | . A^1 . | | ||
| Line 4,152: | Line 4,152: | ||
Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.<br /> | Conventional notation assumes the 2.3 prime subgroup. Non-8ve and non-5th pergens can be notated in a backwards compatible way as a subset of a larger prime subgroup which contains 2 and 3. Thus 5/4 = M3, 7/4 = m7, etc. The advantage of this approach is that conventional staff notation can be used. The disadvantage is that there is a <u>huge</u> number of missing notes and intervals. The composer may want to use a notation that isn't backwards compatible for composing, but use one that is for communicating with other musicians.<br /> | ||
<br /> | <br /> | ||
Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). | Just as all rank-2 pergens in which 2 and 3 are present and independent can be numbered, so can all 2.5 pergens, all 2.7 pergens, all 3.5 pergens, etc. Every rank-2 pergen except Blackwood-like ones can be identified by its prime subgroup and its pergen number. The pergens are grouped into blocks and sections as before. Within each section, the pergens are ordered by cents size of the generator. Sometimes the pergen can be simplified. For example, 2.7 pergen #4 is (P8, m7/2) which is (P8, P4), which is equivalent to (P8, P5). P5 represents not 3/2 but half of 16/7, which is 3/2 sharpened by half of 64/63. Simplified pergens are marked with an asterisk.<br /> | ||
| Line 4,251: | Line 4,251: | ||
<td style="text-align: center;">(P8, P5)*<br /> | <td style="text-align: center;">(P8, P5)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P12, | <td style="text-align: center;">(P12, P4)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">(P12, m10/2)<br /> | <td style="text-align: center;">(P12, m10/2)<br /> | ||
| Line 4,365: | Line 4,365: | ||
<td style="text-align: center;">(P12/3, M6/2)<br /> | <td style="text-align: center;">(P12/3, M6/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(P12/3, M2)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">(WWM3/3, m3)*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,381: | Line 4,381: | ||
<td style="text-align: center;">(P12/3, P4)*<br /> | <td style="text-align: center;">(P12/3, P4)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P12/3, m10/2)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(WWM3/3, M7)*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,397: | Line 4,397: | ||
<td style="text-align: center;">(P12/2, M6/3)<br /> | <td style="text-align: center;">(P12/2, M6/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P12/2, M3/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(M9, d5/3)*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,413: | Line 4,413: | ||
<td style="text-align: center;">(P12/2, m7/3)<br /> | <td style="text-align: center;">(P12/2, m7/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P12/2, P4)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(M9, WA6/3)*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,429: | Line 4,429: | ||
<td style="text-align: center;">(P12/2, WWM3/3)<br /> | <td style="text-align: center;">(P12/2, WWM3/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P12/2, WM7/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(M9, WWm7/3)*<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,445: | Line 4,445: | ||
<td style="text-align: center;">(P12/3, M6/3)<br /> | <td style="text-align: center;">(P12/3, M6/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P12/3, P4)*<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(WWM3/3, d5/3)<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 4,477: | Line 4,477: | ||
C1 --- G1 --- D2 --- A2<br /> | C1 --- G1 --- D2 --- A2<br /> | ||
<br /> | <br /> | ||
Splitting the 5th adds notes to the horizontal edges of the square: <br /> | Splitting the 5th adds notes to the horizontal edges of the square:<br /> | ||
C2 Ev2 G2<br /> | C2 Ev2 G2<br /> | ||
| . . . . . . |<br /> | | . . . . . . |<br /> | ||
| Line 4,489: | Line 4,489: | ||
C1 Ev1 G1<br /> | C1 Ev1 G1<br /> | ||
<br /> | <br /> | ||
From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square. <br /> | From G1 to C2 is a 4th, so splitting the 4th adds a note halfway between them, in the center of the square.<br /> | ||
C2 ---- G2<br /> | C2 ---- G2<br /> | ||
| . A^1 . |<br /> | | . A^1 . |<br /> | ||