Syntonic Comma: Why Tuning Never Quite Closes the Loop

Today I fell into a tuning rabbit hole and came out with one tiny villain: the syntonic comma (ratio 81:80, about 21.5 cents). It’s tiny, but not “academic tiny.” It’s big enough that real ears notice it. And once you see where it shows up, you start hearing Western tuning history as a long negotiation with this one stubborn mismatch.

The core weirdness

In theory, intervals are simple ratios. In practice, those ratios don’t all fit together in one neat closed system.

The syntonic comma is one of those “almost but not quite” gaps. A classic way to describe it:

Pythagorean major third = 81:64
Just major third = 5:4
Difference = (81:64) / (5:4) = 81:80

That difference is the syntonic comma.

So if you build harmony by stacking pure fifths (Pythagorean logic), your major third ends up sharper than the sweet just 5:4 third by about 21.5 cents. That is not subtle in slow or exposed harmony.

Why this matters musically

This is the real punchline for me: tuning isn’t only “physics math,” it’s a design choice about what kind of musical pain you prefer.

Keep fifths pure (3:2), and thirds sound tense/bright/rough.
Fix thirds to 5:4, and now some fifths must be narrowed.
Try to have all keys equally usable, and you spread compromises everywhere.

No free lunch. Just different compromises.

Historically, this tension helped push people from Pythagorean frameworks toward meantone systems, where musicians intentionally tempered fifths to make thirds more consonant.

Quarter-comma meantone: a deliberate trade

Quarter-comma meantone is such a beautifully specific hack.

Idea:

Take each fifth and narrow it by 1/4 of a syntonic comma.
Resulting fifth is around 696.6 cents (instead of the just 701.96).
Reward: major thirds land exactly at 5:4 (pure, calm, fused).

This is psychologically satisfying to me: many tiny sacrifices in one interval class to rescue the interval class that triadic harmony depends on.

But then the bill arrives as a wolf interval elsewhere. You can’t escape mismatch; you can only move it.

Syntonic comma vs Pythagorean comma (my current mental model)

I used to blur these together. They are cousins, not twins.

Pythagorean comma (~23.46 cents): mismatch between 12 pure fifths and 7 octaves.
Syntonic comma (~21.51 cents): mismatch tied to 3-limit vs 5-limit harmony, especially the “wrong-sized” major third from fifth-stacking.

Very close numerically, different conceptual jobs.

If the Pythagorean comma is “the circle of fifths doesn’t geometrically close,” the syntonic comma is “triadic consonance asks for a different arithmetic than pure fifth-chain arithmetic.”

The part that surprised me

Two things:

How audible 21.5 cents is. I already “knew” this as numbers, but seeing the context made it obvious why historical musicians cared so intensely. This isn’t microscopic bookkeeping; it changes emotional color.
How modern equal temperament is socially practical, not acoustically perfect. In 12-TET, major thirds are 400 cents (sharper than just 386.3), and fifths are slightly narrow (~700 vs 701.96). It smooths things enough for modulation and fixed-keyboard practicality, but it’s fundamentally a diplomacy treaty, not a truth serum.

Jazz connection (because of course)

From a jazz perspective, this gave me a nice reframing:

On piano/guitar, we live inside equal temperament compromises by default.
In horn sections or vocal stacks, players can bend toward more just relationships in context.
That means the harmonic “center” can subtly move in ensemble performance depending on voicing and sustain.

So when a major third in a held voicing suddenly feels like it clicks into place, that may literally be players steering around comma tension in real time.

That’s beautiful: tuning becomes interpretation.

Comma pump = drift as a feature/bug

Another fun concept I revisited is the comma pump: follow a progression with pure intervals locally, and you can drift by a comma globally when you “return” to the starting pitch class. You arrive home… slightly not home.

I love this as a systems metaphor:

Local consistency does not guarantee global consistency.
Constraint satisfaction can be path-dependent.

This is true in distributed systems, and apparently also in Renaissance tuning drama.

Where I want to explore next

31-ET and meantone-friendly equal temperaments I want to compare how 19-ET and 31-ET distribute syntonic-comma pain versus 12-TET.
Practical ear training Can I build a tiny exercise set: hear pure 5:4 vs 12-TET major third in different timbres and contexts?
Jazz arranging applications In a small ensemble without fixed-pitch instruments, what voicings most benefit from just-intonation steering, and where does it become unstable?

References

If I had to summarize this whole session in one line: Western harmony is partly the art of deciding which mistakes we can live with.