Stack pure fifths and the last pitch misses seven octaves by 23.46 cents. Seven steps show how tuners hid that leftover.
Step 4 of 7
1584, Zhu Zaiyu
Twelve equal steps of 1.0594631.
Zhu Zaiyu calculated the equal semitone in 1584. Call its frequency ratio r. Twelve equal steps multiply a starting frequency by r^12, and an octave doubles that frequency. Therefore r^12 = 2, which makes r = 2^(1/12), or 1.0594631. Seven steps make a fifth of exactly 700.00 cents. A pure 3:2 fifth is 701.96 cents, so the equal fifth is 1.96 cents narrower. Simon Stevin described this twelfth-root method in the Low Countries around 1605. That even compromise makes every step interchangeable.
Press Climb twelve equal steps before comparing Pure fifth with Equal fifth. Listen for the climb to land exactly at C5. Use Two fifth tones to overlap them and hear the beat that equal spacing adds.
The upper tones differ by 0.4 Hz, so the Two fifth tones control swells about once every two seconds. Those two are bare sine tones, with nothing above the fundamental to add a faster beat.
Two commas
Cents measure each gap.
A comma is the leftover distance between two tunings that nearly agree. The Pythagorean one is 23.46 cents wide, about a quarter of a 100-cent step. The 21.51-cent syntonic comma lies between the 81:64 Pythagorean third and the pure 5:4 third.
Drag the fifth narrower, watch the open circle close, then play the gap. Twelve pure fifths land 23.46 cents beyond seven octaves because the powers of 3 and 2 never meet. Equal steps distribute that miss by narrowing every fifth about two cents. Closing the picture reveals the compromise that lets a fixed keyboard use one tuning in every key.
Drag the fifth narrower, watch the circle close, then play the gap. Twelve pure fifths pass seven octaves by 23.46 cents, so equal steps narrow each fifth about two cents. The closed picture reveals the tuning compromise that lets one fixed keyboard play in every key.
Size the fifth701.96 ¢ fifth · twelve of them land 23.46 ¢ over