Exploring the Structure of Comma Pumps in 72-EDO

This record documents an experimental attempt to construct a comma pump within the 72-tone equal temperament (72-TET) system. Based on pure intervals such as 3/2, 4/3, and 5/4, the piece explores how small discrepancies—known as commas—accumulate when these ratios are successively stacked and mapped onto an equal-tempered framework.
While the notation is not strictly accurate in conventional terms, the design reflects a high level of structural thinking, wherein tuning theory, auditory perception, and fractional arithmetic interact dynamically. The piece progresses through incremental pitch shifts, gradually generating divergence from just intonation.
By investigating how pure ratios distort over time in a microtonal grid, this experiment offers insight into the structural behavior of tuning systems, and raises questions about how harmonic identity can be preserved or transformed under alternate temperaments.

Reiji's Observations

Note: The following is an English translation of a spoken log in Japanese, recorded while Reiji was looking at a score he had notated himself using MuseScore.

The notation and playback are limited by the application, which supports a maximum resolution of 72-tone equal temperament.
As a result, this score isn’t entirely accurate—the pitches deviate by about 5 cents from the intended values.
Ideally, the pitch would go up by 14 cents per step, but since 72-TET divides the octave into 16.666... cents, each step ends up about 2 cents off from just intonation.
I’ve currently written 32 bars of this, and by that point, the accumulated drift is probably close to the 41-cent comma that results from stacking 12 fifths in a meantone temperament.
So what does a “comma pump in 72-TET” mean? …Well, I guess I should explain a bit more before getting into that.

A comma pump is a structure built up from simple intervals—meaning it uses pure ratios like octaves, 3/2, 5/4, etc., as in 5-limit tuning.
Let me show you what I mean: You start on G, then multiply by 3/2. After that, go up an octave (×2).
The next pitch is roughly like an A, but if you don’t know the exact ratio, you can just multiply the previous one again.

So, 3/2 × 3/2 = 9/4.
Then you go back to the root G and raise it by a perfect fourth.
That means multiplying G by 4/3 (which is like moving a perfect fifth down from an octave). That gives you something close to C.
Then, raise it by a major third: 4/3 × 5/5 = 5/3.
Next, you’d go up a minor third—but at this point, you might think, “Wait, doesn’t that just bring us back to the octave?”
But actually, this part is being carried over from the A before, and if you hear it from E, the fourth gets a bit muddy.
So yeah, I realize now that was wrong—better to start from 9/4 instead.

From 9/4, if you halve it, you get 9/8.
Then going up a fifth gives 27/16.
Earlier I said 27/20, but that was a mistake—27/16 is correct, and a bit higher in pitch.

Wait a sec… (wandering around thinking)
Hold on… (doing fraction calculations on paper)
Ah, it’s 27/16. Continuing on—sorry, 27/20 was wrong.

Now, if you multiply that by 4/5, you get a pitch kind of like C.
Hang on, I’m doing more fraction math.
Hmm... the numbers are getting pretty big, and I’m not sure how to simplify them easily.
Okay, let me just say the fraction: multiply by 27/100—
Wait, no! That drops the octave too much. It should be 27/20. Okay, 27/20 it is.

This interval—multiplying the root by 27/20—actually has a name. I’m not sure how to pronounce it in English, but maybe “sharp fourth”? “Sur-fourth”? Maybe “ei-fourth”? Not sure.
Then, if you multiply that by 3/2, you get 81/40. That’s an interval just 1/40 higher than an octave.
And as you keep stacking like this, you can see how the pitch slowly keeps drifting upward.

Notation: Comma Pump in 72-TET (scored in MuseScore)

Each measure ascends by a step based on a 72-TET interpretation of 14-cent increments. Although the score uses standard notation, the actual sound includes finer microtonal details.