Commas Everywhere

Most musicians learn the circle of fifths as a practical tool and leave it there: stack 12 perfect fifths, drop 7 octaves, and you come almost back to where you started. That “almost” is the Pythagorean comma — the tiny mismatch between (3/2)^12 and 2^7.

Numerically, that mismatch is about:

C = (3/2)^12 / 2^7 ≈ 1.0136433.

Years ago, I ran into an odd little hieroglyphic-style diagram in an obscure musicology journal: a circle from ancient Egypt simply split into 360 wedges. No explanation, just sitting there in the middle of an article about tuning. I started treating that 360-fold division as a hint: if you extend the same fifths-and-octaves logic out to a full 360-step cycle, does anything interesting happen?

It does: the comma reaches its smallest value at the 360th iteration — the tightest “almost-closure” in the entire extended spiral. That’s what first made the diagram feel intentional.

And then, when you compare the size of the Pythagorean comma to the correction between a 360-day “perfect circle” year and our actual 365-day solar year, you get this:

C ≈ 365 / 360.

More explicitly:

Pythagorean comma ≈ 1.0136433

Year correction (365/360) ≈ 1.0138889

They differ by about 0.024%. In other words, the little sliver by which 12 fifths fail to close on 7 octaves is almost exactly the same sliver by which a 360-day schematic year fails to match the real sky.

You can read that as coincidence if you want. But structurally, it’s the same pattern showing up twice:

• a 12-step, 5-based spiral in pitch space

• a 360-step angular cycle in orbital space

Both forced to “almost close,” both leaving behind the same tiny excess.

For the ancients who talked about the harmony of the spheres, this isn’t a metaphor so much as a design principle: the cosmos isn’t tuned perfectly to our neat circles, and the circles aren’t tuned perfectly to themselves. The same little comma that haunts our tuning systems also sits between the geometric year (360) and the lived year (365-ish). Somewhere between those numbers, music and planets are solving the same geometric problem.