Magic Music Clacton — musical sound explained | magicmusicclacton.com
Scales are built by stacking intervals, but pure ratios never close the circle — so every tuning system is a deliberate compromise.
Build a scale from first principles and the problem appears quickly. Stack twelve pure fifths (3:2) and the chain lands seven octaves above the starting note — but sharp by about 23 cents. That gap, the Pythagorean comma, proves that pure fifths and pure octaves cannot coexist in a closed system.
Pythagorean tuning accepts the comma and keeps all fifths pure except one, the wolf, left howling and unusable. It suits medieval melody, where fifths dominate, but its major thirds — a sharp 408 cents — sound restless to ears trained on later harmony.
Just intonation retunes the thirds to their pure 5:4 and 6:5 forms, producing chords of extraordinary stillness. The cost is mobility: intervals that are pure in one key turn sour in another, which is why just intonation thrives in unaccompanied singing, where pitch can adjust chord by chord.
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Meantone temperaments, dominant in the Renaissance and Baroque eras, narrow the fifths slightly so the major thirds come out pure or nearly so. Several keys sound glorious; distant keys expose the wolf. Much early keyboard music was written with exactly this palette in mind.
Equal temperament takes the radical option: divide the octave into twelve identical semitones of 100 cents each. Every fifth is 2 cents narrow, every major third 14 cents wide — and every key is equally usable. The compromise, described mathematically by Zhu Zaiyu in 1584, became the near-universal default of Western instruments in the nineteenth century.
Reference pitch is a separate convention layered on top. Modern instruments are built around A4 = 440 Hz (ISO 16), but historical standards ran lower, period ensembles often use 415 Hz, and some orchestras sharpen to 442 or 443 Hz. The ratios define the system; the reference pitch merely anchors it.
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