Every essay — page 2
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Pitch and tuning
Frequency ratios, the comma that will not close, and every compromise ever made about it.
The partial the lips cannot reach
A wind instrument plays between its valve's preferred frequency and its bore's, which suggested that the natural trumpet's notorious eleventh partial would therefore turn out to be a statement about the lips. It is not. Run the same Q-weighted mean over the whole series and the lips can move that partial twenty-three cents, which is a quarter of what it needs — and the Q that would fix it is the Q at which the bore stops choosing the note.
Every partial beats at its own rate
Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.
The hand that changes the bore
A conjecture refuted here left a question with a number on it: the natural trumpet's eleventh partial is 48.7 cents flat, the lips can move it 23.1, and 25.6 cents are owed by something that is not an embouchure. There is exactly one thing a player can change about the bore while playing, and putting the hand in the bell buys those 25.6 cents at a cost of 3.8 decibels — because the aperture that tunes the instrument is the aperture that radiates it.
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.
A horn has one length per partial
Every tube until now has had an acoustic length: its physical length plus a correction for the wave carrying on past the opening. A flaring bore does not have one. Its second mode behaves as though the tube were 164 centimetres long and its eighth as though it were 154, and the ten centimetres between them are the same physical fact — a fixed correction against a shrinking wavelength — arriving as a hundred cents.
The hand goes in, and the note jumps
A horn player's hand closes the bell and the pitch falls — 19 cents, then 55, then 132, then four hundred, accelerating the whole way. Then, in the last half per cent of closure, it stops falling and lands a semitone above where it started. An earlier essay guessed the mechanism was the boundary condition changing kind and the series going odd-only. It is not. The series never changes at all.
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.
A resonance has a strength as well as a frequency
What eleven earlier essays drew is a row of frequencies, because the solver behind it has no losses and a lossless resonance has no width. Put the losses in and every one of them acquires a height and a Q — and the hand closing a horn's bell turns out to take away nine and a half per cent of the instrument's total support before giving all of it back, in a window a few per cent wide where the horn is genuinely hard to play.
The higher note speaks sooner and takes longer
Up a brass instrument the settling time in milliseconds falls by a factor of seven and the settling time in periods rises by a factor of three. Both curves are read off the same impedance sweep, both are monotone over most of the compass, and they point in opposite directions — so the slowest note of the instrument depends entirely on which clock is used to time it.
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.
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.
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.
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.
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.
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.
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.
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.
One fluctuation or two
Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.
How hard the note was struck
The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.
Counted in the decay, or not at all
A mistuned octave struck hard delivers no countable beat at the strike, and the reconciliation offered for that was that a tuner listens to the decay. Computed through a real decay it fails on its own terms: the partials fall silent before the filter has narrowed enough to separate them. It succeeds only when the filter is broadened by the level inside it, which is what the published parameterisation was fitted against — and then the window opens at a twelfth of the note's life and shuts at a quarter.
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.
The cello cannot hear its own tempering
Narrowing a quartet's fifths to meet a piano is one number, 1.96 cents a fifth, and it is a different beat on every string: once every two thirds of a second on the violin's A–E and once every four and a half seconds on the cello's C–G. Set by ear for two seconds a fifth, the violin's E lands within two thirds of a cent and the cello's C within 5.7 — which is as large as the Pythagorean error the tempering was meant to remove. The string whose tuning is most wrong is the string whose tuning is least certain, and a cellist tuning down the chain cannot tell pure from tempered.
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.
A guitar tuned by harmonics hides a comma
A quartet's stopped notes go wherever its players put them, so its open strings can sit on a pure chain. A guitar's stopped notes are fixed by frets that are already an equal temperament, and its six open strings — four fourths and a third — close two octaves exactly a syntonic comma short when tuned pure, since (4/3)⁴·(5/4) is 4 × 80/81. Tuning by harmonics decides where the comma goes, not whether: a pure third puts it in the octaves of an open E major chord, 17.6 cents narrow; a B from the low E's twelfth puts it in the G–B third, 21.5 cents wide.