Comma pump

An example of a comma pump: in 12edo, which tempers out 81/80, this progression returns to the starting note; in 15edo, which observes this comma, it does not.

A comma pump is a JI or tempered chord progression whose starting and ending points differ by a comma.

There is some ambiguity in this term depending on whether the comma in question is tempered out (thus returning to the starting pitch) or not. If the comma is tempered out, the chord progression may be called a comma loop.

Given a comma, a comma pump may be constructed by stacking root movements by basic intervals in the JI group in question to that comma. Reordering movements of a comma pump, or reversing the progression, creates another valid comma pump progression for that comma.

Examples

Meantone

The I-vi-ii-V-I diatonic progression in Meantone diatonic is a Meantone or 81/80 comma loop:

CEG
-> ACEA (C -> A: down by 6/5)
-> DFA (A -> D: up by 4/3)
-> GDGB (D -> G: down by 3/2)
-> CEGC (G -> C: up by 4/3)

In JI and other non-Meantone tunings, attempting this comma pump results in the end point being flatter by (possibly tempered) 81/80 relative to the starting point. In 21edo, since 81/80 is mapped to -1\21, this progression raises the pitch by 1\21.

Porcupine

Here's a Porcupine (250/243) comma loop, which only returns to the starting pitch in Porcupine tunings (^/v = 81/80 alteration). Note that 250/243 is the interval between 81/80 and 25/24.

C Ev G
-> Av C Ev Av (C -> Av: down 6/5)
-> Dv F Av (Av -> Dv: up 4/3)
-> Bb^ Db^^ F^ (Dv -> Bvv = Bb^: down 6/5, Porcupine entails Dv = Db^^)
-> G Bb^ D G (Bb^ -> G: down 6/5)
-> C Ev G (G -> C: up 4/3)

Slendric

Slendric equates three 8/7's with one 3/2. Below, ^/v indicates alteration by 64/63.

C G Bbv
-> D^ A^ C (C -> D^: up 8/7)
-> Fv Cv Ebvv (D^ -> E^^ = Fv: up 8/7)
-> G D Fv (Fv -> G: up 8/7)
-> C Bbv G (G -> C: down 3/2)

Negri

(19edo notation)
F C F A
A C E (F -> A: 5/4 up)
E B E G# (A -> E: 4/3 down)
G# B D# (E -> G#: 5/4 up)
D# A# D# Fx (G# -> D#: 4/3 down)
Fx A# Cx = Gb Bbb Db (D# -> Fx: 5/4 up)
Db Ab Db F (Gb -> Db: 4/3 down)
F C F A (Db -> F: 5/4 up)