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Keenan's comma pump page: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
Why do we temper intervals? The only truly essential reasons are [[Why_Microtonality?#Why Microtonality?-5. For "puns"|puns]] and [[comma_pump|comma pump]]s.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-25 06:52:13 UTC</tt>.<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.
: The original revision id was <tt>248380063</tt>.<br>
 
: The revision comment was: <tt></tt><br>
If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
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.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Candidate 5-limit commas:
 
81/80 (meantone)
==Candidate 5-limit commas==
128/125 (augmented)
<ul><li>81/80 (meantone)</li><li>128/125 (augmented)</li><li>135/128 (mavila)</li><li>250/243 (porcupine)</li><li>256/243 (blackwood)</li><li>648/625 (diminished)</li><li>2048/2025 (srutal)</li><li>3125/3072 (magic)</li><li>6561/6250 (ripple)</li><li>15625/15552 (hanson)</li><li>16875/16384 (negri)</li><li>20000/19683 (tetracot)</li><li>20480/19683 (superpyth)</li><li>32805/32768 (helmholtz)</li></ul>
135/128 (mavila)
 
250/243 (porcupine)
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.
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)


5-limit commas NOT tempered out by 12-equal:
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)


Candidate 7-limit commas:
<ul><li>135/128 (mavila)</li><li>250/243 (porcupine)</li><li>256/243 (blackwood)</li><li>3125/3072 (magic)</li><li>15625/15552 (hanson)</li><li>16875/16384 (negri)</li><li>20000/19683 (tetracot)</li><li>20480/19683 (superpyth)</li></ul>This seems like a really good list.
49/48 (2.3.7 semaphore)
 
50/49 (jubilisma)
==Mavila==
64/63 (archytas)
135/128 = 2/1 * 5/4 / (3/4)^3
126/125 (starling)
 
225/224 (marvel)
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.
245/243 (octarod)
 
525/512
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/mikebattagliaexperiments/sets/the-mavila-experiments http://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments]).
686/675 (senga)
 
875/864 (supermagic)
==Porcupine==
1029/1000 (liese-related)
250/243 = (4/3)^2 / (6/5)^3
1029/1024 (gamelisma)
 
1728/1715 (orwellian)
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.
2240/2187
 
2401/2400 (breedsma)
Works in hedgehog too! (Also nautilus, ammonite...)
2430/2401
 
3125/3087
==Blackwood==
3136/3125 (parahemwuer)
256/243 = (4/3)^5 / (2/1)^2
3645/3584
 
4000/3969 (octagari)
Blackwood comma pumps involve 5 motions by 4/3 around a "circle of fifths" that's about as small as it can reasonably get.
4375/4374 (ragisma)
 
5103/5000
==Magic==
5120/5103 (hemifamity)
3125/3072 = (5/4)^5 / (3/2) / (2/1)
5625/5488
 
6144/6125 (hewuermity)
The simplest magic comma pumps involve 5 motions by 5/4 and one by 3/2. Bringing 6/5 into the picture only makes things more complicated.
8748/8575
 
9604/9375
Basically, moving the root up by 5/4 three times leads you to a chord one diesis (128/125) lower than what you started with. But in magic the diesis is the same as 25/24, so next you get to chords on the roots 6/5 and 3/2 (if that's the direction you're going).
10976/10935 (parahemfi)
 
12288/12005
==Candidate 7-limit commas==
15625/15309
<ul><li>49/48 (semaphore)</li><li>50/49 (jubilisma)</li><li>64/63 (archytas)</li><li>126/125 (starling)</li><li>225/224 (marvel)</li><li>245/243 (sensamagic)</li><li>256/245 (bapbo)</li><li>405/392</li><li>525/512 (avicennma)</li><li>686/675 (senga)</li><li>729/700</li><li>875/864 (keema)</li><li>1029/1000 (keega)</li><li>1029/1024 (gamelisma)</li><li>1323/1280</li><li>1728/1715 (orwellian)</li><li>2240/2187</li><li>2401/2400 (breedsma)</li><li>2430/2401 (nuwell)</li><li>2500/2401</li><li>3125/3024</li><li>3125/3087 (gariboh)</li><li>3136/3125 (hemimean)</li><li>3200/3087</li><li>3645/3584</li><li>4000/3969 (octagari)</li><li>4096/3969</li><li>4375/4374 (ragisma)</li><li>5103/5000</li><li>5120/5103 (hemifamity)</li><li>5625/5488</li><li>6144/6125 (porwell)</li><li>6272/6075</li><li>8192/7875</li><li>8505/8192</li><li>8748/8575</li><li>9604/9375</li><li>10976/10935 (hemimage)</li><li>12005/11664</li><li>12288/12005</li><li>15625/15309</li><li>16128/15625</li><li>16807/16200</li><li>16807/16384 (blacksmith-related)</li><li>16875/16807 (mirkwai)</li><li>17280/16807</li><li>17496/16807</li><li>19683/19208 (squares-related)</li><li>19683/19600 (cataharry)</li><li>28672/28125</li><li>31104/30625</li><li>33075/32768</li><li>33614/32805</li></ul>
16807/16384 (blacksmith-related)
16875/16807 (mirkwai)
17280/16807
19683/19208 (squares-related)
19683/19600 (cataharry)
28672/28125
31104/30625
33075/32768
33614/32805


7-limit commas NOT tempered out by 12-equal:
7-limit commas NOT tempered out by 12-equal:
49/48 (2.3.7 semaphore)
 
245/243 (octarod)
<ul><li>49/48 (semaphore) - 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.</li><li>245/243 (sensamagic) = 5/3 / (9/7)^2 - works in: magic, sensi, godzilla, superpyth, hedgehog.</li><li>525/512 (avicennma) = (5/4)^2 / (4/3 * 8/7) - works in: negri, flattone, armodue, muggles (magic in 19edo), lemba</li><li>686/675 (senga)</li><li>875/864 (keema)</li><li>1029/1000 (keega)</li><li>1029/1024 (gamelisma)</li><li>1323/1280</li><li>1728/1715 (orwellian)</li><li>2240/2187</li><li>2401/2400 (breedsma)</li><li>2430/2401 (nuwell)</li><li>3125/3024</li><li>4375/4374 (ragisma)</li><li>6144/6125 (porwell)</li><li>6272/6075</li><li>8505/8192</li><li>8748/8575</li><li>9604/9375</li><li>10976/10935 (hemimage)</li><li>12005/11664</li><li>12288/12005</li><li>15625/15309</li><li>16807/16200</li><li>16807/16384 (blacksmith-related)</li><li>16875/16807 (mirkwai)</li><li>17280/16807</li><li>17496/16807</li><li>19683/19208 (squares-related)</li><li>19683/19600 (cataharry)</li><li>31104/30625</li><li>33075/32768</li><li>33614/32805</li></ul>    
525/512
[[Category:comma_pump]]
686/675 (senga)
[[Category:Lists]]
875/864 (supermagic)
 
1029/1000 (liese-related)
{{todo|improve synopsis}}
1029/1024 (gamelisma)
1728/1715 (orwellian)
2240/2187
2401/2400 (breedsma)
2430/2401
4375/4374 (ragisma)
6144/6125 (hewuermity)
8748/8575
9604/9375
10976/10935 (parahemfi)
12288/12005
15625/15309
16807/16384 (blacksmith-related)
16875/16807 (mirkwai)
17280/16807
19683/19208 (squares-related)
19683/19600 (cataharry)
31104/30625
33075/32768
33614/32805
</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Keenan's comma pump page&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Candidate 5-limit commas:&lt;br /&gt;
81/80 (meantone)&lt;br /&gt;
128/125 (augmented)&lt;br /&gt;
135/128 (mavila)&lt;br /&gt;
250/243 (porcupine)&lt;br /&gt;
256/243 (blackwood)&lt;br /&gt;
648/625 (diminished)&lt;br /&gt;
2048/2025 (srutal)&lt;br /&gt;
3125/3072 (magic)&lt;br /&gt;
6561/6250 (ripple)&lt;br /&gt;
15625/15552 (hanson)&lt;br /&gt;
16875/16384 (negri)&lt;br /&gt;
20000/19683 (tetracot)&lt;br /&gt;
20480/19683 (superpyth)&lt;br /&gt;
32805/32768 (helmholtz)&lt;br /&gt;
&lt;br /&gt;
5-limit commas NOT tempered out by 12-equal:&lt;br /&gt;
135/128 (mavila)&lt;br /&gt;
250/243 (porcupine)&lt;br /&gt;
256/243 (blackwood)&lt;br /&gt;
3125/3072 (magic)&lt;br /&gt;
15625/15552 (hanson)&lt;br /&gt;
16875/16384 (negri)&lt;br /&gt;
20000/19683 (tetracot)&lt;br /&gt;
20480/19683 (superpyth)&lt;br /&gt;
&lt;br /&gt;
Candidate 7-limit commas:&lt;br /&gt;
49/48 (2.3.7 semaphore)&lt;br /&gt;
50/49 (jubilisma)&lt;br /&gt;
64/63 (archytas)&lt;br /&gt;
126/125 (starling)&lt;br /&gt;
225/224 (marvel)&lt;br /&gt;
245/243 (octarod)&lt;br /&gt;
525/512&lt;br /&gt;
686/675 (senga)&lt;br /&gt;
875/864 (supermagic)&lt;br /&gt;
1029/1000 (liese-related)&lt;br /&gt;
1029/1024 (gamelisma)&lt;br /&gt;
1728/1715 (orwellian)&lt;br /&gt;
2240/2187&lt;br /&gt;
2401/2400 (breedsma)&lt;br /&gt;
2430/2401&lt;br /&gt;
3125/3087&lt;br /&gt;
3136/3125 (parahemwuer)&lt;br /&gt;
3645/3584&lt;br /&gt;
4000/3969 (octagari)&lt;br /&gt;
4375/4374 (ragisma)&lt;br /&gt;
5103/5000&lt;br /&gt;
5120/5103 (hemifamity)&lt;br /&gt;
5625/5488&lt;br /&gt;
6144/6125 (hewuermity)&lt;br /&gt;
8748/8575&lt;br /&gt;
9604/9375&lt;br /&gt;
10976/10935 (parahemfi)&lt;br /&gt;
12288/12005&lt;br /&gt;
15625/15309&lt;br /&gt;
16807/16384 (blacksmith-related)&lt;br /&gt;
16875/16807 (mirkwai)&lt;br /&gt;
17280/16807&lt;br /&gt;
19683/19208 (squares-related)&lt;br /&gt;
19683/19600 (cataharry)&lt;br /&gt;
28672/28125&lt;br /&gt;
31104/30625&lt;br /&gt;
33075/32768&lt;br /&gt;
33614/32805&lt;br /&gt;
&lt;br /&gt;
7-limit commas NOT tempered out by 12-equal:&lt;br /&gt;
49/48 (2.3.7 semaphore)&lt;br /&gt;
245/243 (octarod)&lt;br /&gt;
525/512&lt;br /&gt;
686/675 (senga)&lt;br /&gt;
875/864 (supermagic)&lt;br /&gt;
1029/1000 (liese-related)&lt;br /&gt;
1029/1024 (gamelisma)&lt;br /&gt;
1728/1715 (orwellian)&lt;br /&gt;
2240/2187&lt;br /&gt;
2401/2400 (breedsma)&lt;br /&gt;
2430/2401&lt;br /&gt;
4375/4374 (ragisma)&lt;br /&gt;
6144/6125 (hewuermity)&lt;br /&gt;
8748/8575&lt;br /&gt;
9604/9375&lt;br /&gt;
10976/10935 (parahemfi)&lt;br /&gt;
12288/12005&lt;br /&gt;
15625/15309&lt;br /&gt;
16807/16384 (blacksmith-related)&lt;br /&gt;
16875/16807 (mirkwai)&lt;br /&gt;
17280/16807&lt;br /&gt;
19683/19208 (squares-related)&lt;br /&gt;
19683/19600 (cataharry)&lt;br /&gt;
31104/30625&lt;br /&gt;
33075/32768&lt;br /&gt;
33614/32805&lt;/body&gt;&lt;/html&gt;</pre></div>
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