five ways to be wrong about the same note
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twelve pure fifths overshoot seven octaves by 23.46 cents. the circle of fifths is a spiral. it never closes. every tuning system is a different way of forcing it shut.
equal temperament flattens every fifth by 1.955 cents. democratic. characterless. every key sounds the same — which means no key sounds like itself.
just intonation tunes to the harmonic series: small integer ratios where harmonics align and the chord rings. a major triad is 4:5:6 — three consecutive harmonics. but move to another key and the ratios break. you can be perfect here or perfect there. not both.
the wolf is what happens when you tune eleven fifths sweet and the twelfth absorbs all the error. 737 cents instead of 702. it howls. meantone hid it between G♯ and E♭. composers avoided those keys for three hundred years.
a barbershop quartet singing in just intonation generates a phantom fifth voice. combination tones reinforce at the root — the chord makes its own bass. this cannot happen on a piano. the 13.7-cent error in the equal-tempered major third prevents the alignment.
every tuning system is a theology. pythagorean: the mystical perfection of three. meantone: the renaissance wanting sweetness. well temperament: the enlightenment compromise. equal temperament: the industrial revolution. standardize everything. sacrifice character for interchangeability.