Keenan's comma pump page
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author keenanpepper and made on 2011-08-28 20:05:01 UTC.
- The original revision id was 248975337.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
Why do we temper intervals? The only truly essential reasons are [[Why Microtonality?#Why Microtonality?-5. For "puns"|puns]] and [[comma pump]]s. Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune "I've Got Rhythm" to just intonation. If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps. Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. [[81_80|81/80]] is an ideal comma pump comma in many ways. ==Candidate 5-limit commas== 81/80 (meantone) 128/125 (augmented) 135/128 (mavila) 250/243 (porcupine) 256/243 (blackwood) 648/625 (diminished) 2048/2025 (srutal) 3125/3072 (magic) 6561/6250 (ripple) 15625/15552 (hanson) 16875/16384 (negri) 20000/19683 (tetracot) 20480/19683 (superpyth) 32805/32768 (helmholtz) Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift. 5-limit commas NOT tempered out by 12-equal: 135/128 (mavila) 250/243 (porcupine) 256/243 (blackwood) 3125/3072 (magic) 15625/15552 (hanson) 16875/16384 (negri) 20000/19683 (tetracot) 20480/19683 (superpyth) This seems like a really good list. ==Mavila== 135/128 = 2/1 * 5/4 / (3/4)^3 The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated. In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the "wrong" kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see [[http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments]]). ==Porcupine== 250/243 = (4/3)^2 / (6/5)^3 The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated. ==Blackwood== 256/243 = (4/3)^5 / (2/1)^2 Blackwood comma pumps involve 5 motions by 4/3 around a "circle of fifths" that's about as small as it can reasonably get. ==Candidate 7-limit commas== 49/48 (semiphore) 50/49 (jubilisma) 64/63 (archytas) 126/125 (starling) 225/224 (marvel) 245/243 (octarod) 256/245 405/392 525/512 686/675 (senga) 729/700 875/864 (supermagic) 1029/1000 (liese-related) 1029/1024 (gamelisma) 1323/1280 1728/1715 (orwellian) 2240/2187 2401/2400 (breedsma) 2430/2401 2500/2401 3125/3024 3125/3087 3136/3125 (parahemwuer) 3200/3087 3645/3584 4000/3969 (octagari) 4096/3969 4375/4374 (ragisma) 5103/5000 5120/5103 (hemifamity) 5625/5488 6144/6125 (hewuermity) 6272/6075 8192/7875 8505/8192 8748/8575 9604/9375 10976/10935 (parahemfi) 12005/11664 12288/12005 15625/15309 16128/15625 16807/16200 16807/16384 (blacksmith-related) 16875/16807 (mirkwai) 17280/16807 17496/16807 19683/19208 (squares-related) 19683/19600 (cataharry) 28672/28125 31104/30625 33075/32768 33614/32805 7-limit commas NOT tempered out by 12-equal: 49/48 (semiphore) - This doesn't really produce good comma pumps. It basically just modifies the 7-limit tonality diamond so that 8/7~7/6; there's no cycle of consonant shifts that wouldn't also be consonant in JI. 245/243 (octarod) = 5/3 / (9/7)^2 - works in many temperaments including magic, sensi, godzilla, superpyth 525/512 = (5/4)^2 / (4/3 * 8/7) - basically a negri thing 686/675 (senga) 875/864 (supermagic) 1029/1000 (liese-related) 1029/1024 (gamelisma) 1323/1280 1728/1715 (orwellian) 2240/2187 2401/2400 (breedsma) 2430/2401 3125/3024 4375/4374 (ragisma) 6144/6125 (hewuermity) 6272/6075 8505/8192 8748/8575 9604/9375 10976/10935 (parahemfi) 12005/11664 12288/12005 15625/15309 16807/16200 16807/16384 (blacksmith-related) 16875/16807 (mirkwai) 17280/16807 17496/16807 19683/19208 (squares-related) 19683/19600 (cataharry) 31104/30625 33075/32768 33614/32805
Original HTML content:
<html><head><title>Keenan's comma pump page</title></head><body>Why do we temper intervals? The only truly essential reasons are <a class="wiki_link" href="/Why%20Microtonality%3F#Why Microtonality?-5. For "puns"">puns</a> and <a class="wiki_link" href="/comma%20pump">comma pump</a>s.<br /> <br /> Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune "I've Got Rhythm" to just intonation.<br /> <br /> If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.<br /> <br /> Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. <a class="wiki_link" href="/81_80">81/80</a> is an ideal comma pump comma in many ways.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Candidate 5-limit commas"></a><!-- ws:end:WikiTextHeadingRule:0 -->Candidate 5-limit commas</h2> 81/80 (meantone)<br /> 128/125 (augmented)<br /> 135/128 (mavila)<br /> 250/243 (porcupine)<br /> 256/243 (blackwood)<br /> 648/625 (diminished)<br /> 2048/2025 (srutal)<br /> 3125/3072 (magic)<br /> 6561/6250 (ripple)<br /> 15625/15552 (hanson)<br /> 16875/16384 (negri)<br /> 20000/19683 (tetracot)<br /> 20480/19683 (superpyth)<br /> 32805/32768 (helmholtz)<br /> <br /> Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift.<br /> 5-limit commas NOT tempered out by 12-equal:<br /> 135/128 (mavila)<br /> 250/243 (porcupine)<br /> 256/243 (blackwood)<br /> 3125/3072 (magic)<br /> 15625/15552 (hanson)<br /> 16875/16384 (negri)<br /> 20000/19683 (tetracot)<br /> 20480/19683 (superpyth)<br /> This seems like a really good list.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Mavila"></a><!-- ws:end:WikiTextHeadingRule:2 -->Mavila</h2> 135/128 = 2/1 * 5/4 / (3/4)^3<br /> The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated.<br /> <br /> In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the "wrong" kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see <a class="wiki_link_ext" href="http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments" rel="nofollow">http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments</a>).<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Porcupine"></a><!-- ws:end:WikiTextHeadingRule:4 -->Porcupine</h2> 250/243 = (4/3)^2 / (6/5)^3<br /> The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Blackwood"></a><!-- ws:end:WikiTextHeadingRule:6 -->Blackwood</h2> 256/243 = (4/3)^5 / (2/1)^2<br /> Blackwood comma pumps involve 5 motions by 4/3 around a "circle of fifths" that's about as small as it can reasonably get.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-Candidate 7-limit commas"></a><!-- ws:end:WikiTextHeadingRule:8 -->Candidate 7-limit commas</h2> 49/48 (semiphore)<br /> 50/49 (jubilisma)<br /> 64/63 (archytas)<br /> 126/125 (starling)<br /> 225/224 (marvel)<br /> 245/243 (octarod)<br /> 256/245<br /> 405/392<br /> 525/512<br /> 686/675 (senga)<br /> 729/700<br /> 875/864 (supermagic)<br /> 1029/1000 (liese-related)<br /> 1029/1024 (gamelisma)<br /> 1323/1280<br /> 1728/1715 (orwellian)<br /> 2240/2187<br /> 2401/2400 (breedsma)<br /> 2430/2401<br /> 2500/2401<br /> 3125/3024<br /> 3125/3087<br /> 3136/3125 (parahemwuer)<br /> 3200/3087<br /> 3645/3584<br /> 4000/3969 (octagari)<br /> 4096/3969<br /> 4375/4374 (ragisma)<br /> 5103/5000<br /> 5120/5103 (hemifamity)<br /> 5625/5488<br /> 6144/6125 (hewuermity)<br /> 6272/6075<br /> 8192/7875<br /> 8505/8192<br /> 8748/8575<br /> 9604/9375<br /> 10976/10935 (parahemfi)<br /> 12005/11664<br /> 12288/12005<br /> 15625/15309<br /> 16128/15625<br /> 16807/16200<br /> 16807/16384 (blacksmith-related)<br /> 16875/16807 (mirkwai)<br /> 17280/16807<br /> 17496/16807<br /> 19683/19208 (squares-related)<br /> 19683/19600 (cataharry)<br /> 28672/28125<br /> 31104/30625<br /> 33075/32768<br /> 33614/32805<br /> <br /> 7-limit commas NOT tempered out by 12-equal:<br /> 49/48 (semiphore) - This doesn't really produce good comma pumps. It basically just modifies the 7-limit tonality diamond so that 8/7~7/6; there's no cycle of consonant shifts that wouldn't also be consonant in JI.<br /> 245/243 (octarod) = 5/3 / (9/7)^2 - works in many temperaments including magic, sensi, godzilla, superpyth<br /> 525/512 = (5/4)^2 / (4/3 * 8/7) - basically a negri thing<br /> 686/675 (senga)<br /> 875/864 (supermagic)<br /> 1029/1000 (liese-related)<br /> 1029/1024 (gamelisma)<br /> 1323/1280<br /> 1728/1715 (orwellian)<br /> 2240/2187<br /> 2401/2400 (breedsma)<br /> 2430/2401<br /> 3125/3024<br /> 4375/4374 (ragisma)<br /> 6144/6125 (hewuermity)<br /> 6272/6075<br /> 8505/8192<br /> 8748/8575<br /> 9604/9375<br /> 10976/10935 (parahemfi)<br /> 12005/11664<br /> 12288/12005<br /> 15625/15309<br /> 16807/16200<br /> 16807/16384 (blacksmith-related)<br /> 16875/16807 (mirkwai)<br /> 17280/16807<br /> 17496/16807<br /> 19683/19208 (squares-related)<br /> 19683/19600 (cataharry)<br /> 31104/30625<br /> 33075/32768<br /> 33614/32805</body></html>