Acoustics & Music Theory
why it sounds like that
Concept

Key colour — where it appears

The characteristic quality attributed to a particular key, which in an unequal temperament corresponds to that key's own interval sizes. In equal temperament there is nothing left for it to correspond to, and the attributions outlived the tunings.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

How far each system sits from equal temperament. Deviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp.

Where to hide the comma, which is the only real question

Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.

tuning · The comma The chain of fifths in quarter-comma meantone. The fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and one of them — G♯ to D♯, the 12th link, where the chain is forced to close — is the wolf, at 35.7 cents.

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

tuning · The comma The same total, distributed differently. The departure of each of the twelve major thirds from a pure 5:4, one row per temperament, keys ordered round the circle of fifths. The totals are equal — every closing temperament's twelve thirds add to 4800 cents — so what a temperament chooses is not how much error there is but which keys carry it.

What a temperament cannot do

Add up the twelve major thirds of any keyboard tuning whose chain of fifths closes and the answer is 4,800 cents. Every time, in every system, before a single fifth has been chosen. So no temperament has ever made the average third less wrong than any other one, and four hundred years of argument were about a distribution rather than about accuracy.

tuning · The comma How sure the reading is, bar by bar. Every earlier essay reports one best reading. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed and has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07.

The margin the dynamic program already had

Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.

scales · Key-relations Four of the five are a whole number of semitones, and one is exactly half of one. Each mismatch in cents, against the ticks at whole semitones — which are the only places a transposing keyboard can put a player. 4 of the 5 land within six cents of a tick: the Chorton–Kammerton gap is 197 cents against a whole tone's 200, and Chorton against French pitch is 296 against a minor third's 300. The exception is an English organ against Handel's fork, at 50 cents — 50 cents from the nearest tick, which is as far as it is possible to be. So the small mismatches are the unsolvable ones, and the large ones were solved by shifting the keys.

The instrument that cannot be moved

A string is regauged and a woodwind is scaled. An organ's pitch is the length of its pipes, and metal can be cut off and cannot be put back — so an organ is a ratchet that only goes sharp. The mechanical answer was to shift the keyboard against the pipes, and its cost is not the transposition. It is that the temperament's key colours rotate out from under the notation, by an amount measured in fifths rather than in semitones.

tuning · Pitch standard A string quartet's open strings are five keys of a Pythagorean keyboard. The five pitch classes a string quartet's open strings sound — C, G, D, A and E — laid out as the chain of fifths they are tuned along, outward from the A the ensemble is given, with each fifth pure. The bars give each string's departure from the same note on an equal-tempered keyboard: C −5.87 cents, G −3.91 cents, D −1.96 cents, A 0.00 cents, E +1.96 cents. Above, the strings each instrument owns: the violin G, D, A, E; the viola C, G, D, A; the cello the same four an octave lower. The cello's C2 is 0.221 hertz below the keyboard's, and the widest span of the chain, from the cello's C to the violin's E, is a Pythagorean third and two octaves, 21.5 cents wider than a just one.

The tuning a string quartet cannot change

A string quartet can put every stopped note wherever it likes, and it has five pitches it cannot move once the pegs are turned: the open strings C, G, D, A and E, tuned in pure fifths from the A it was given. That makes its only fixed tuning a five-key Pythagorean keyboard — the cello's C nearly six cents below a piano's, and every third two open strings can make a syntonic comma from just. Counted key by key, the clash is worst in G, C and F, the keys that use every open string, and absent from A and E.

tuning · Open strings An unaccompanied quartet settles where its open strings put it. The average pitch of a quartet correcting toward itself over 480 corrections, in cents from the note it was given, averaged over 24 runs. With no pull from the open strings the ensemble random-walks, and the shaded band is how far: 3.7 cents root-mean-square by the end. With each open string pulling the notes that share its pitch class at a weight of 0.05, the ensemble settles at −0.97 cents in A major, against −1.01 from the open strings' weighted mean; −2.46 cents in C major, against −2.42 from the open strings' weighted mean; −2.53 cents in E♭ major, against −2.54 from the open strings' weighted mean.

An open string pulls the quartet flat

Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.

tuning · Open strings Three G strings, and they are not one pitch. Each instrument's four open strings, at the Pythagorean position its own chain of fifths puts them, with the uncertainty its own tuning leaves drawn as a band. The A is given and carries no error; every other string is reached from it one fifth at a time, and a fifth set by ear is set by nulling a beat whose rate falls with frequency — so the error accumulates down the chain and is worst at the bottom. violin: G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00, E5 2.0 ± 0.44; viola: C3 -5.9 ± 2.83, G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00; cello: C2 -5.9 ± 2.83, G2 -3.9 ± 1.77, D3 -2.0 ± 0.98, A3 0.0 ± 0.00. The three G strings share a pitch class and are expected to sit 2.5 cents apart; the two C strings 4.0.

The quartet settles at two pitches, not four

The quartet's open strings have been treated as five fixed pitches on one chain, and they are not: the violin, the viola and the cello each tuned a G string by ear and the three are expected to sit two and a half cents apart. Giving each player their own strings, with their own scatter, and pulling each toward only their own, changes the ensemble's settled pitch by a hundredth of a cent. What it does change is systematic rather than random: a violin has an E string and no C, the lower instruments have a C and no E, so the quartet splits by section by a tenth of a cent in every key.

tuning · Open strings

Named alongside it

The objects these essays reach for when they reach for this one.

Chain of fifthsEqual temperamentIntonationOpen stringTemperamentResonanceTuning by earAmbiguityBeatingCentsCircle of fifthsCirculating temperament

All concepts