These are musical examples which demonstrate the modern harmonic technique of "comma pumping."

Music often consists of chord progressions which first depart from the tonic, and then cyclically return you to it. Such a chord progression is called a "comma pump" if it can only exist in a certain regular temperament, and would not work in just intonation or in other temperaments that don't eliminate the same comma. A canonical example is the chord progression ||: Cmaj | Am | Dm | G7 :|| in meantone temperament, which would deposit you the tiny interval of 81/80 lower from the root if it were performed in just intonation. This chord progression sounds perfectly natural to most western listeners, as it is very common in western music. One way to think of it is that it takes advantage of the relationship that V/V/V/V/V = III, or alternatively that V/V = IV/vi.

Analogous chord progressions exist in other temperaments: for example, in Porcupine temperament, the minor thirds are tuned slightly sharp and the minor sevenths slightly flat, so that the relationship bIII/bIII/bIII = IV/IV holds true. In Mavila temperament, displayed on this page under the "Mavila Experiments," the fifths are flat, so that V/V/V/V/V = bIII and IV/IV/IV = III. In Blackwood temperament, the fifths are so sharp that five of them closes at the octave, so that V/V/V/V/V = I; other intervals such as the major third need to be placed "outside" of the circle of fifths in this temperament.

These are some examples of comma pumps, chosen to demonstrate their unique character and somewhat hypnotic feel. To a western listener, these may seem somewhat analogous to "auditory illusions," as they are not consistent with and violate any internal 12-EDO based map that one may have.

Comma Pumps

This is a chord progression in Hanson temperament, tuned to 34-equal. In this system, rather than 4 minor thirds landing you at the octave, 6 minor thirds lands you at an octave + a fifth. The minor thirds in this system are even closer to the harmonic series than the familiar ones in 12-equal, which contributes to the "bright" feel of this tuning. Don't think too much about the name "Hanson."

Comma pump in Hanson/Kleismic temperament, 34-equal

This is a chord progression in Blackwood temperament, tuned to 15-equal. In this system, the circle of fifths is truncated at 5 steps, meaning that for C-G-D-A-E-B all of the fifths are a little bit sharp so that the B on top ends up being another C again. This means, by definition, that the E on top is more like an F.

Since this chain by itself doesn't really give us a usable major or minor third, we have to add them manually and in a way that's not accessible by the base circle of fifths that proceeds from the root. This chord progression exploits this property.

Comma pump in Blackwood temperament, 15-equal

This is a comma pump in the unnamed 17496/16807 temperament, tuned to 34-equal. Although the basic root movement is by 7/6, the chords are tuned to 5-limit minor 7 chords, to make them brighter.

17496/16807 Comma Pump, Minor Chords, 34-equal

This is a comma pump in the unnamed 17496/16807 temperament, tuned to 34-equal. The basic root movement is by 7/6, and the chords are tuned to 7-limit subminor 7 chords, to make the voice leading smoother.

17496/16807 Comma Pump, Subminor Chords, 34-equal

This is a comma pump in porcupine which goes up by three 6/5's and then down two 4/3's back to 1/1. I designed it as an example of something that I thought sounded perfectly "natural," but which doesn't work at all in 12-EDO. I'll use this in a piece somewhere down the road.

The tuning here is 22-EDO.

A Porcupine Comma Pump - 22-EDO version

This is mostly goofy experimental music theory stuff to see how different microtonal scales sound - no real musical compositions here (yet)

Comma pump in Hanson/Kleismic temperament, 34-equal

Mike Battaglia