A World of Tunings

Tuning systems in six parts: Pythagoras's pure fifths and comma, just intonation, meantone and the wolf fifth, equal temperament, the 53 commas of Turkish music and a comparison of four tunings. At the end a blind test: 'Just or equal?' There is no claim of any effect.

Why this order? The first stage is pure ratios: twelve pure fifths do not close on the octave but overshoot it by ≈ 23.5 cents (the Pythagorean comma); in just intonation, built from small whole numbers, the 4:5:6 chord stands still without beats, but the fifth D-A is a syntonic comma (≈ 21.5 cents) narrow. The second stage is compromise: meantone keeps the thirds pure, narrows the fifths by ≈ 5.4 cents and leaves a wolf fifth of ≈ 737.6 cents between G sharp and E flat; equal temperament makes every semitone 100 cents, so every key is the same. The third stage is another system and the ear: the theory of Turkish music, where a whole tone is divided into nine commas, and finally a comparison of four tunings. Chords sounding together use only beat-free ratios; other intervals play one note after another.

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