Stack pure fifths and the last pitch misses seven octaves by 23.46 cents. Seven steps show how tuners hid that leftover.
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Hear the shared partial
Two thirds of a string sounds a fifth.
Stop a string at two thirds of its length. The short part sounds a fifth above the full string. This 3:2 ratio has a shared partial at 392.44 Hz. The third partial of C3 meets the second partial of G3 there, which helps the pair fuse.
Play C3 and G3 separately, then use the main control to hold the 3:2 pair. Listen for two pitches to fuse without losing their identities. The third partial of C3 and the second partial of G3 meet at the same frequency, so part of each tone reinforces the other. That shared component helps explain why a pure fifth can sound settled.
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