The spiral that never closes

Stack twelve perfect fifths and you almost — but not quite — land seven octaves up. That tiny leftover, the Pythagorean comma, is why the piano has exactly twelve keys per octave.

twelve perfect fifths

(3/2)¹²

= 129.746

seven octaves

2⁷

= 128

the leftover

the comma

≈ 23.5 ¢

Twelve fifths almost equal seven octaves — but land 23.5 cents too high. Nature won't give us a whole number, so the piano splits the octave into twelve equal steps and shares the tiny error out.

stack a fifth at a time — ×3/2, over and over

fifths stacked0

total multiple (3/2)^0×1.0000

that's how many octaves?0.0000

Watch the wheel: each fifth hops five seats clockwise. Twelve hops bring you all the way around the twelve notes and back home — except you overshoot the start by a hair. That never-quite-closing spiral is why twelve is the magic number.

For the classroom

Learning goals

Stacking a musical interval means multiplying its ratio repeatedly: n fifths = (3/2)ⁿ.
Twelve fifths give (3/2)¹² ≈ 129.75, just past seven octaves (2⁷ = 128) — the ratio 129.75/128 is the Pythagorean comma (~23.5 cents).
Because no power of 3/2 ever equals a power of 2, the octave can't be filled by pure fifths — equal temperament divides it into 12 identical steps instead.

Try this

1.Stack fifths one at a time and read the 'how many octaves' number. Ask students to predict when it first passes a whole number of octaves, and by how much it misses at 12.
2.Compute (3/2)¹² and 2⁷ by hand or calculator and subtract — connect that tiny gap to the wobble they'll hear in the next lab.