Acoustics & Music Theory
why it sounds like that
Series

Pitch standard — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

    The pitch nobody agreed on, for four hundred years

    A440 is a committee decision from 1955. Before it, the same written note was played anywhere between about 392 and 465 hertz depending on the town, the building and the decade — a spread of very nearly a minor third, and every piece of theory here is untouched by it.

    part 1 · tuning
  2. Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

    A memory for the note itself, and it is dated

    Absolute pitch is usually described as a rare perceptual gift. It is better described as a memory for a convention — and conventions have dates. Possessors trained on A=440 mis-name Baroque pitch by a semitone, their own labels drift sharp with age, and meanwhile most listeners without it start familiar songs within a semitone of the record.

    part 2 · tuning
  3. Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A415. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

    The note that moved by a minor third

    A has been anywhere between 392 and 465 hertz in the surviving record, which is 296 cents — a minor third short of six. The usual conclusion is that absolute pitch level is a convention and nothing musical depends on it. That is true of everything written on paper and false of everything that happens in a throat or a body: a singer's register break sits at a fixed frequency, so across the historical range it lands three semitones further down the written page. Transposing a piece is not a uniform operation, because the performer does not transpose.

    part 3 · tuning
  4. How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string.

    A standard is a specification

    Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string.

    part 4 · tuning
  5. What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series.

    What the climb was a search for

    The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose.

    part 5 · tuning
  6. Re-gauging at a fixed tension: how close sheep gut comes to breaking. Holding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 1.79 times the stress sheep gut will take; at A = 466 it has 1.27. The margin reaches one at A = 525 hertz, which is far above anything the four hundred years of climb reached.

    It was never the strings that stopped the climb

    Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.

    part 6 · tuning
  7. A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year.

    A woodwind cannot be pulled to a new standard

    Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

    part 7 · tuning
  8. 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.

    part 8 · tuning
  9. Where the register break falls on a tenor's page. The two measured laryngeal crossings — 330 hertz going up and 294 coming down — read as WRITTEN notes, against the pitch standard the part is performed at. The crossings are frequencies and do not move; the notation does, so the seam slides down the stave by exactly the interval the standard rises. At A392 the upward crossing is written F♯4, at A415 it is F4, at A440 E4 and at A465 E♭4 — a minor third of movement across four centuries, on a part nobody rewrote. Across the range drawn the seam passes 4 written semitones. The shaded horizontal band is the tenor's written compass, C3 to A4; the seam is inside it at 7 of the 7 documented standards drawn.

    A standard moves the page, and not the seam

    Every earlier essay has priced a pitch standard against something with a fixed length in it. A voice has none, so nothing about it changes at all — what changes is where the written note falls against a break in the larynx that is a frequency and stays put. At A415 that break is written F4, at A440 it is E4 and at Chorton it is E♭4: a minor third of movement across four centuries, on a part nobody rewrote.

    part 9 · tuning
  10. Every pitch standard, given the width 8 degrees gives it. Each documented standard drawn not as a point but as the band an ensemble occupies while the room warms by 8 degrees: the air columns sharpen by 23.3 cents, the steel strings flatten by 20.1, and 13.8 cents of spread inside each wind instrument's own register cannot be pulled out because it is a gradient along the bore rather than an offset. The band is 57 cents wide, and 5 of the 6 adjacent steps in the whole record are narrower than it — which is to say that 5 of the distinctions four centuries of committees argued about are smaller than the pitch spread inside one orchestra on one evening.

    A standard is a point, and a performance is a band

    Nine earlier essays draw every pitch standard as a single number, because none of them has a temperature in it. An air column sharpens as the room warms and a steel string flattens, at 2.95 and 2.49 cents a degree; add the 13.8 cents of spread inside one wind instrument's own register and eight degrees makes an orchestra 57 cents wide. Five of the six steps in four hundred years of pitch standards are narrower than that.

    part 10 · tuning
  11. Two ways to match a tuning note, and they are not the same size. How finely one player can put their A on another's, at 440 hertz on a note of 2 seconds, by each of the two criteria available. Judging one pitch is good to 4.0 cents; comparing two of them adds two errors in quadrature and is good to 5.7. Nulling the beat between them is a different operation altogether — a mistuning of 2.0 cents makes a beat of 0.50 hertz, which shows one full cycle inside the note — and it is 2.9 times finer. The beat criterion is available only when the two tones sound together and share a partial. A player tuning to a note that has already stopped has the coarse one, and so does a singer with nothing to beat against.

    An orchestra is given a note

    Every earlier essay on pitch standards draws a standard as a number an ensemble is at, and no ensemble is at a pitch. It is handed one, by one player, on one note, and everything else is matched to it by ear — so a standard reaches an orchestra through a limen nobody quotes. Matching by comparing two pitches is good to 5.7 cents at A; nulling the beat between them is good to 2.0, and which of the two is available depends on how long the oboe holds the note. The crossover is at about seven tenths of a second, which is shorter than an oboe's A and longer than a plucked one.

    part 11 · tuning
  12. Who listens to whom decides how far apart an orchestra ends up. The spread an ensemble of 60 arrives at, for four ways of passing the tuning note around, with one match good to 2.0 cents. Matching the giver directly leaves every player one match away and a spread of 2.0 cents whatever the size; passing it along a line leaves the last player 59 matches away and a spread of 15.1. Orchestral practice is the middle one — principals to the oboe, sections to their principals — which is two matches and 2.8 cents, and is within half a cent of the best arrangement available at any ensemble size. A convention nobody derived sits one step off the optimum of an arithmetic nobody wrote down.

    Who listens to whom when an orchestra tunes

    One match is good to two cents and an orchestra is sixty of them, arranged in an order that nobody chose deliberately. Passed along a line, the errors accumulate and the last player is fifteen cents from the first; given to everybody at once, nobody is more than two. The convention every orchestra uses — principals to the oboe, sections to their principals — is two matches deep, costs 2.8 cents whether there are four players or a hundred, and sits within a cent of the best arrangement that exists.

    part 12 · tuning
  13. The interval an orchestra tunes on is the least sensitive one it plays. The smallest mistuning each interval betrays, on notes of 2 seconds with the lower note at 440 hertz, taking one full beat cycle as the criterion. A unison shows 1.97 cents; a fifth shows 0.66, a major third 0.39, a minor third 0.33. The ratio is exact and it is the interval's own upper term: the lowest coincidence of a p:q interval sits at p times the lower note's frequency, so the beat runs p times faster. The dashed line is what the same players manage by comparing two pitches instead, at 5.7 cents — coarser than every interval on the axis by between three and seventeen times.

    The unison is the coarsest thing in the room

    An orchestra tunes on a unison and then plays intervals, and the two are not the same test. The lowest coincidence of a p:q interval sits at p times the lower note's frequency, so a mistuning of a given number of cents makes a beat p times faster — a fifth betrays it three times sooner than a unison, a minor third six. The ritual that opens a rehearsal is therefore the least sensitive measurement anybody will make all evening, and every chord afterwards is a finer one.

    part 13 · tuning
  14. The ensemble agrees with itself more and more, about a pitch that is moving. Runs of 16 players each correcting toward the mean of their neighbours, with no term anywhere pulling them back to the note they were given, over 480 corrections. The shaded band is the root-mean-square displacement across all eight runs — the envelope a random walk has — and it grows from 0.84 cents a quarter of the way through to 2.01 at the end, which is the square-root growth a random walk has. Four individual runs are drawn inside it and the furthest of the eight over the top, ending at 4.16 cents. Meanwhile the spread AMONG the players falls from 2.8 cents to 0.5. A consensus with no anchor cannot hold a pitch, and it also cannot lose one quickly: a movement's worth of corrections is a few cents rather than the semitone unaccompanied choirs are said to fall by.

    A consensus with nothing to hold it

    Once the oboe has stopped, no reference is left in the room. Each player corrects toward what they hear around them, which is other players correcting toward them — and a consensus dynamic has a fixed point at every common value, so it pulls the ensemble together and nothing pulls it anywhere in particular. Simulated, the players' spread falls from 2.8 cents to 0.5 while the ensemble as a whole random-walks. The size is the result and it is small: two or three cents over a movement, which is a tenth of what unaccompanied choirs are said to lose.

    part 14 · tuning

All series