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>2011-08-28 20:05:01 UTC</tt>.<br>
: The original revision id was <tt>248975337</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>
<h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">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.
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.
Line 12: Line 5:
If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.
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.
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==  
==Candidate 5-limit commas==
81/80 (meantone)
<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>
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.
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:
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==  
<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.
 
==Mavila==
135/128 = 2/1 * 5/4 / (3/4)^3
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.
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]]).
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]).


==Porcupine==  
==Porcupine==
250/243 = (4/3)^2 / (6/5)^3
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.
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==  
Works in hedgehog too! (Also nautilus, ammonite...)
 
==Blackwood==
256/243 = (4/3)^5 / (2/1)^2
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.
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==  
==Magic==
49/48 (semiphore)
3125/3072 = (5/4)^5 / (3/2) / (2/1)
50/49 (jubilisma)
 
64/63 (archytas)
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.
126/125 (starling)
 
225/224 (marvel)
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).
245/243 (octarod)
 
256/245
==Candidate 7-limit commas==
405/392
<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>
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:
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
<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 = (5/4)^2 / (4/3 * 8/7) - basically a negri thing
[[Category:comma_pump]]
686/675 (senga)
[[Category:Lists]]
875/864 (supermagic)
 
1029/1000 (liese-related)
{{todo|improve synopsis}}
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</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;Why do we temper intervals? The only truly essential reasons are &lt;a class="wiki_link" href="/Why%20Microtonality%3F#Why Microtonality?-5. For &amp;quot;puns&amp;quot;"&gt;puns&lt;/a&gt; and &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pump&lt;/a&gt;s.&lt;br /&gt;
&lt;br /&gt;
Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune &amp;quot;I've Got Rhythm&amp;quot; to just intonation.&lt;br /&gt;
&lt;br /&gt;
If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.&lt;br /&gt;
&lt;br /&gt;
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. &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt; is an ideal comma pump comma in many ways.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Candidate 5-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Candidate 5-limit commas&lt;/h2&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;
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.&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;
This seems like a really good list.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Mavila"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Mavila&lt;/h2&gt;
135/128 = 2/1 * 5/4 / (3/4)^3&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the &amp;quot;wrong&amp;quot; kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see &lt;a class="wiki_link_ext" href="http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments" rel="nofollow"&gt;http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments&lt;/a&gt;).&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Porcupine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Porcupine&lt;/h2&gt;
250/243 = (4/3)^2 / (6/5)^3&lt;br /&gt;
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.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Blackwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Blackwood&lt;/h2&gt;
256/243 = (4/3)^5 / (2/1)^2&lt;br /&gt;
Blackwood comma pumps involve 5 motions by 4/3 around a &amp;quot;circle of fifths&amp;quot; that's about as small as it can reasonably get.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Candidate 7-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Candidate 7-limit commas&lt;/h2&gt;
49/48 (semiphore)&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;
256/245&lt;br /&gt;
405/392&lt;br /&gt;
525/512&lt;br /&gt;
686/675 (senga)&lt;br /&gt;
729/700&lt;br /&gt;
875/864 (supermagic)&lt;br /&gt;
1029/1000 (liese-related)&lt;br /&gt;
1029/1024 (gamelisma)&lt;br /&gt;
1323/1280&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;
2500/2401&lt;br /&gt;
3125/3024&lt;br /&gt;
3125/3087&lt;br /&gt;
3136/3125 (parahemwuer)&lt;br /&gt;
3200/3087&lt;br /&gt;
3645/3584&lt;br /&gt;
4000/3969 (octagari)&lt;br /&gt;
4096/3969&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;
6272/6075&lt;br /&gt;
8192/7875&lt;br /&gt;
8505/8192&lt;br /&gt;
8748/8575&lt;br /&gt;
9604/9375&lt;br /&gt;
10976/10935 (parahemfi)&lt;br /&gt;
12005/11664&lt;br /&gt;
12288/12005&lt;br /&gt;
15625/15309&lt;br /&gt;
16128/15625&lt;br /&gt;
16807/16200&lt;br /&gt;
16807/16384 (blacksmith-related)&lt;br /&gt;
16875/16807 (mirkwai)&lt;br /&gt;
17280/16807&lt;br /&gt;
17496/16807&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 (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.&lt;br /&gt;
245/243 (octarod) = 5/3 / (9/7)^2 - works in many temperaments including magic, sensi, godzilla, superpyth&lt;br /&gt;
525/512 = (5/4)^2 / (4/3 * 8/7) - basically a negri thing&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;
1323/1280&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;
3125/3024&lt;br /&gt;
4375/4374 (ragisma)&lt;br /&gt;
6144/6125 (hewuermity)&lt;br /&gt;
6272/6075&lt;br /&gt;
8505/8192&lt;br /&gt;
8748/8575&lt;br /&gt;
9604/9375&lt;br /&gt;
10976/10935 (parahemfi)&lt;br /&gt;
12005/11664&lt;br /&gt;
12288/12005&lt;br /&gt;
15625/15309&lt;br /&gt;
16807/16200&lt;br /&gt;
16807/16384 (blacksmith-related)&lt;br /&gt;
16875/16807 (mirkwai)&lt;br /&gt;
17280/16807&lt;br /&gt;
17496/16807&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>

Latest revision as of 04:14, 16 December 2024

Why do we temper intervals? The only truly essential reasons are puns and comma pumps.

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 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/mikebattagliaexperiments/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.

Works in hedgehog too! (Also nautilus, ammonite...)

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.

Magic

3125/3072 = (5/4)^5 / (3/2) / (2/1)

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.

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).

Candidate 7-limit commas

  • 49/48 (semaphore)
  • 50/49 (jubilisma)
  • 64/63 (archytas)
  • 126/125 (starling)
  • 225/224 (marvel)
  • 245/243 (sensamagic)
  • 256/245 (bapbo)
  • 405/392
  • 525/512 (avicennma)
  • 686/675 (senga)
  • 729/700
  • 875/864 (keema)
  • 1029/1000 (keega)
  • 1029/1024 (gamelisma)
  • 1323/1280
  • 1728/1715 (orwellian)
  • 2240/2187
  • 2401/2400 (breedsma)
  • 2430/2401 (nuwell)
  • 2500/2401
  • 3125/3024
  • 3125/3087 (gariboh)
  • 3136/3125 (hemimean)
  • 3200/3087
  • 3645/3584
  • 4000/3969 (octagari)
  • 4096/3969
  • 4375/4374 (ragisma)
  • 5103/5000
  • 5120/5103 (hemifamity)
  • 5625/5488
  • 6144/6125 (porwell)
  • 6272/6075
  • 8192/7875
  • 8505/8192
  • 8748/8575
  • 9604/9375
  • 10976/10935 (hemimage)
  • 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 (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.
  • 245/243 (sensamagic) = 5/3 / (9/7)^2 - works in: magic, sensi, godzilla, superpyth, hedgehog.
  • 525/512 (avicennma) = (5/4)^2 / (4/3 * 8/7) - works in: negri, flattone, armodue, muggles (magic in 19edo), lemba
  • 686/675 (senga)
  • 875/864 (keema)
  • 1029/1000 (keega)
  • 1029/1024 (gamelisma)
  • 1323/1280
  • 1728/1715 (orwellian)
  • 2240/2187
  • 2401/2400 (breedsma)
  • 2430/2401 (nuwell)
  • 3125/3024
  • 4375/4374 (ragisma)
  • 6144/6125 (porwell)
  • 6272/6075
  • 8505/8192
  • 8748/8575
  • 9604/9375
  • 10976/10935 (hemimage)
  • 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
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