The circle of fifths doesn't actually close

Start on C. Go up in fifths — C, G, D, A — and twelve steps later you should be back on C, seven octaves up. You're not. You land about a quarter of a semitone high, and that gap has never gone away. Every tuning system in Western history is a different answer to one question: where do you hide it? This is what your piano is hiding, why every major chord you have ever heard has a flutter in it, and what it cost to make every key playable. CHAPTERS 0:00 Every piano is lying to you 0:28 The octave, and the fifth 1:28 Twelve fifths should bring you home 1:58 They don't — you land 23.46 cents high 2:49 So where do you put the error? 3:13 Four ways to hide it 4:06 The wolf 4:29 What it cost: the third 5:20 What a chord in tune sounds like 6:01 When keys had colours 6:34 The ones who escapedCharpentier (c. 1682) “cruel and hard”; Mattheson (1713) grave and pathetic; Schubart (1806) “the key of love, of devotion, of intimate conversation with God.” One key, three verdicts. Ward & Martin — J. Acoust. Soc. Am. 1954 — the listening test in which only 3 of 20 subjects told just from equal-tempered scales apart above chance. Who got there first — Zhu Zaiyu published the twelfth-root-of-two calculation in 1584; Simon Stevin independently in 1585. Broadwood was still retraining its tuners onto equal temperament in 1846. The instrument — The Clavemusicum Omnitonum — Vito Trasuntino, Venice 1606, thirty-one notes to the octave. International Museum and Library of Music, Bologna. The wolf story — Recorded by Edward John Hopkins in 1895. Tradition, not documentation — which is why it is flagged as such on screen. Deliberately left out — The Well-Tempered Clavier title-page squiggle (Bradley Lehman, Early Music 2005, and Lindley & Ortgies arguing against him in 2006). Genuinely unresolved, so it is not in an eight-minute video. Made with Blender, Python and a lot of measuring. Corrections are welcome and will be pinned. 7:09 Nothing you've heard was ever in tune SOURCES AND FURTHER READING The number — The Pythagorean comma is 531441/524288 — 23.460 cents. Ross W. Duffin — _How Equal Temperament Ruined Harmony (and Why You Should Care)_, Norton 2007. Source of the “invisible elephant” framing of the 14-cent third. His verdict is contested; the number is not. Rita Steblin — _A History of Key Characteristics in the Eighteenth and Early Nineteenth Centuries_. The standard work on the key-colour question, and the reason this video hedges where it does. The three writers on E-flat —