Keenan's comma pump page

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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