Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626629395 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626631169 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2018-02-19 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-19 20:02:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626631169</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> | ||
| Line 319: | Line 319: | ||
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: | ||
P4: C - Dv - Eb^ - F | P4/3: C - Dv - Eb^ - F | ||
A4: C - D - E - F# (the lack of ups and downs indicates that this interval was already split) | A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split) | ||
m7: C - Eb^ - Gv - Bb | m7/3: C - Eb^ - Gv - Bb or m7/6: C - Dv - Eb^ - F - Gv - Ab^ - Bb | ||
M7: C - Ev - G^ - B | M7/3: C - Ev - G^ - B | ||
m10: C - F - Bb - Eb (also already split) | m10/3: C - F - Bb - Eb (also already split) (m10/9 also occurs) | ||
M10: C - F^ - Bv - E | M10/3: C - F^ - Bv - E | ||
Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies: | Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies: | ||
^m3: C - Dv - Eb^ (^m3 = 6/5) | ^m3/2: C - Dv - Eb^ (^m3 = 6/5) | ||
^m6: C - Dv - Eb^ - F - Gv - Ab^ (^m6 = 8/5) | ^m6/5: C - Dv - Eb^ - F - Gv - Ab^ (^m6 = 8/5) | ||
vm9: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15) | vm9/4: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15) | ||
vM7: C - F^ - Bv (vM7 = 15/8, 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 | 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 unbroken. | ||
For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer). | For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer). | ||
A double-split perhgen like (P8/3. P4/2) contains all the secondary splits of both (P8/3, P5) and (P8, P4/2), plus others. The following table shows the secondary splits for all pergens up to the third-splits. The split intervals range from d5 to A5 on the genchain of 5ths. For convenience, naturally occuring splits are listed too, under "all pergens". | |||
||~ block ||||~ pergen ||~ secondary splits <= 12th || | ||~ block ||||~ pergen ||~ secondary splits <= 12th || | ||
| Line 2,507: | Line 2,509: | ||
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 /> | ||
<br /> | <br /> | ||
P4: C - Dv - Eb^ - F<br /> | P4/3: C - Dv - Eb^ - F<br /> | ||
A4: C - D - E - F# (the lack of ups and downs indicates that this interval was already split)<br /> | A4:/3 C - D - E - F# (the lack of ups and downs indicates that this interval was already split)<br /> | ||
m7: C - Eb^ - Gv - Bb<br /> | m7/3: C - Eb^ - Gv - Bb or m7/6: C - Dv - Eb^ - F - Gv - Ab^ - Bb<br /> | ||
M7: C - Ev - G^ - B<br /> | M7/3: C - Ev - G^ - B<br /> | ||
m10: C - F - Bb - Eb (also already split)<br /> | m10/3: C - F - Bb - Eb (also already split) (m10/9 also occurs)<br /> | ||
M10: C - F^ - Bv - E<br /> | M10/3: C - F^ - Bv - E<br /> | ||
<br /> | <br /> | ||
Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:<br /> | Of course, an equal-step melody can span other intervals besides 3-limit ones. Simply extend or cut short the earlier examples to find other melodies:<br /> | ||
<br /> | <br /> | ||
^m3: C - Dv - Eb^ (^m3 = 6/5)<br /> | ^m3/2: C - Dv - Eb^ (^m3 = 6/5)<br /> | ||
^m6: C - Dv - Eb^ - F - Gv - Ab^ (^m6 = 8/5)<br /> | ^m6/5: C - Dv - Eb^ - F - Gv - Ab^ (^m6 = 8/5)<br /> | ||
vm9: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15)<br /> | vm9/4: C - Eb^ - Gv - Bb - Db^ (vm9 = 32/15)<br /> | ||
vM7: C - F^ - Bv (vM7 = 15/8, 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 | 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 unbroken.<br /> | ||
<br /> | <br /> | ||
For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).<br /> | For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).<br /> | ||
<br /> | |||
A double-split perhgen like (P8/3. P4/2) contains all the secondary splits of both (P8/3, P5) and (P8, P4/2), plus others. The following table shows the secondary splits for all pergens up to the third-splits. The split intervals range from d5 to A5 on the genchain of 5ths. For convenience, naturally occuring splits are listed too, under &quot;all pergens&quot;.<br /> | |||
<br /> | <br /> | ||
| Line 5,654: | Line 5,658: | ||
<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. 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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7108: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:7108 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
<br /> | <br /> | ||
| Line 5,664: | Line 5,668: | ||
<br /> | <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 /> | 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 /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7109:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:7109 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||