Field

Pitch and tuning

Frequency ratios, the comma that will not close, and every compromise ever made about it.

Twelve fifths do not make seven octaves. Pitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round 7 times and a little further, finishing 23.46 cents past seven octaves — the Pythagorean comma. Nothing in the chain closes that gap; it is what a tuning has to absorb, hide or spread.

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

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.

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.

The syntonic comma on the lattice. Pitch classes reached by multiplying by three — a fifth, one step east — and by five — a major third, one step north. Two routes to the same note name arrive at different pitches, and the syntonic comma is the size of that disagreement: 21.51 cents.

A second comma, arriving by a different road

Four pure fifths ought to land on a pure major third, two octaves up. They miss by 21.5 cents — a different gap from the one twelve fifths leave, produced by a different route, and the two are not the same size.

Key character in Werckmeister III. The twelve major keys in circle-of-fifths order, each with a bar as long as its major third is sharp of a pure 5:4. The bars run from 3.9 to 21.5 cents, so the keys genuinely differ.

Keys that had characters, and could be measured

Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.

Every equal division from 5 to 60, and how wrong it is. For each number of equal steps in the octave, how far its best fifth and its best major third fall from the pure ratios, in cents. The divisions people have actually used are the ones with small errors in both, and no other criterion was applied to pick them out.

Nineteen, thirty-one and fifty-three

Divide the octave into a different number of equal parts and different problems disappear. Which ones vanish is not a matter of taste — it can be computed from the prime factors of the intervals, and each division makes a different decision.

Beat rates for laying equal temperament. Each link in the chain of fifths, inside one bearing octave, with the rate at which its coincident partials beat — the third partial of the lower note against the second of the upper for a fifth, the fourth against the third for a fourth. Rates run from 0.59 to 1.12 beats per second, a spread of 0.52. This temperament narrows every fifth equally, and the rates still differ, because a beat rate is a difference in hertz and scales with the register.

A tuner counts beats, and that is the whole method

Laying a temperament sounds like the most subjective job in music, and it is arithmetic. Every interval in the bearing octave has a target rate in beats per second, the rate follows from the ratio and the register, and a tuner who hits the numbers has produced equal temperament without ever thinking about a cent.

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.

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.

A guitar cannot be in tune

Three errors land on the same instrument — equal temperament's own thirds, the sharpening that comes from pressing a string down to a fret, and the fact that the only correction available is one length per string. Searching over every setting a luthier could choose leaves a worst-case error of about fifteen cents, and most of what is left is the temperament, which no saddle can reach.

Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.

A fraction of a comma

The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.

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.

Nine commas, and the line under which none of them matters. Each named comma at its true size in cents, against the difference limen of 3.87 cents at 500 Hz — the smallest change of frequency a listener can detect, computed from the same formula the perception essays use. One comma falls below it, and it is the only gap in the subject that no tuning system has to do anything about.

A comma under the threshold

Eight pure fifths taken downwards arrive at a major third 1.95 cents flat of a perfect 5:4. That gap has a name and a history, and it is the only one in the subject nobody has ever had to hide — it is under the smallest difference a listener can detect, which makes a chain of untempered fifths a source of almost-just thirds and makes one eighteenth-century tuning nearly free.

How far a chain of fifths is from closing, after each number of fifths. The distance from a whole number of octaves after each number of pure fifths, in cents, out to 60. It is never zero, because a fifth is 3/2 and no power of 3 is a power of 2. The running minima are 1, 2, 5, 12, 41, 53 fifths, and the residue at each is that division's comma: 23.46 cents at 12, 19.84 cents at 41, 3.62 cents at 53. A 3-against-2 polyrhythm asked the same question closes exactly and at once — 3 pulses of the 3 are 2 of the 2, to the last microsecond — because 3/2 is a ratio of whole numbers and log2(3/2) is not.

A comma is a polyrhythm that never closes

Seven earlier essays have rested on a common grid. Two cycles have one when their ratio is a ratio of whole numbers, and the fifth against the octave is not — provably, from unique factorisation. So there is no grid, no composite and no least common multiple, and what is left over instead is the sequence 2, 5, 12, 41, 53 with a comma attached to each.

The same census, in every universe from four to thirty. How many sets have all four properties, in a universe of n equal steps. None at all when n is two more than a multiple of four — 6, 10, 14, 18, 22, 26, 30 — exactly one when n is a multiple of four, and exactly two when n is odd. The multiples of four each have their survivor at n/2 + 1 notes generated by n/2 − 1 steps, so twelve's seven notes generated by the fifth is the general answer with n put at twelve rather than a fact about twelve.

Every universe has one, or none

Run the census that found the diatonic set in a universe of nineteen equal steps, or twenty-four, or fifty-three. The answer is completely regular and nobody appears to have written it down — none at all when the universe is two more than a multiple of four, exactly one when it is a multiple of four, and exactly two when it is odd.

Two ways to detune a pair, and they disagree by an octave. A pair of instruments tuned deliberately apart, drawn across 4 octaves. Holding the detuning at 10 cents gives a beat rate that rises from 0.64 to 10.20 beats a second — a factor of 16, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at 3 beats a second instead gives a flat rate and an interval that shrinks from 46.6 cents at the bottom to 2.9 at the top. A tuner has to choose, the choice is audible across the range, and a table of cents can only write down the first of the two.

A tuning is not a table of cents

Two instruments of a pair are tuned deliberately apart, and the tuner has to choose between a fixed number of cents and a fixed number of beats — which differ by a factor of sixteen across four octaves and cannot both be held. A list of pitches records the first, cannot record the second, and cannot record at all that there are two instruments.

A note sung at 440 Hz, drawn in cents. Deviation from the notated pitch against time, for a vibrato of ±71 cents at 6.0 cycles a second — Seashore's and Prame's measured values, which agree. The note is 142 cents wide, and the shaded bands across the middle are the syntonic comma at 21.5 cents, the Pythagorean comma at 23.5 cents, the difference limen at 440 Hz at 4.0 cents. The pitch a listener reports is near the middle of the excursion rather than at either edge — and not exactly at the middle either: because cents are logarithmic and frequency is not, the mean frequency of this trace sits 0.73 cents above the centre line.

A note that is never at its pitch

Eleven essays in this field argue about differences of one to twenty-four cents. An ordinary operatic vibrato is a hundred and forty cents wide and completes six excursions a second, so every one of those distinctions fits inside a single sung note several times over — and the beat rate a tuner would null passes through zero eleven times a second.

A wind instrument is a thermometer

Pitch goes as the square root of absolute temperature, so a warming clarinet sharpens by about three cents a degree and passes a Pythagorean comma after eight. The strings beside it go flat as they warm. Nobody chose either number, no temperament addresses either of them, and together they are twice the size of the discrepancy this whole field is named after.

The note that is sharp because of where it goes

Eleven essays here have tuned notes against other notes sounding at the same time. A melodic line makes the opposite demand of the same note. In C major the leading note wants to be 1088 cents if the argument is vertical — a pure third above the dominant — and 1110 if it is horizontal, because a leading note is a note that is about to arrive somewhere and performers narrow the step. The two answers differ by 21.5 cents, which is a syntonic comma exactly, and no keyboard has ever been able to hold both.

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.

$Where the two spellings meet, and where they cross. G♯ minus A♭ against the fraction of a comma each fifth is narrowed by — twelve fifths against seven octaves, and nothing else in the calculation. In Pythagorean tuning G♯ is 23.46 cents ABOVE A♭; in quarter-comma meantone it is 41.06 cents BELOW it; the two spellings coincide at 0.09090 of a comma, which is what equal temperament is. A page that distinguishes the two names is exact in every tuning on this line except one point on it, and at that point it is wrong by 0.0014 cents rather than by nothing, because one eleventh is not quite the crossing.$

Two names for one key

A keyboard has one key between G and A and the page has two names for it. That looks like redundancy and it is not: the two names are twelve fifths apart on a chain, and in every tuning anybody played before the nineteenth century they are two different pitches. The size of the difference is twelve fifths against seven octaves and nothing else — twenty-three cents one way in Pythagorean, forty-one the other way in meantone, and zero at exactly one point in between.

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

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.

Blowing harder is playing sharper

Every frequency computed so far for an air column is a resonance of the tube, and no wind instrument plays at its bore's resonance. It plays between the bore and whatever is driving it, weighted by how sharply each is tuned — and a flute's driver is an air jet whose own preferred frequency goes as the square root of the blowing pressure. Across a tenfold pressure range the jet's preference rises by a factor of 3.16 and the bore holds it to 112 cents of that. A wind player's dynamics and their intonation are one control, and the size of the coupling between them is the valve's Q.

The pair tuned apart on purpose

A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.

Which partials of a natural horn can be lipped into tune. Every partial of the natural series against the nearest note of twelve equal, in cents, with the band the player's lips can actually move it drawn around each one. The band is ±23.1 cents, computed from the Q-weighted mean of a bore at Q 40 and lips at Q 12 rather than chosen. 4 of the first 16 partials fall outside it: 7, 11, 13, 14. The worst is the 11th at -48.7 cents, which would need lips of Q 38 to reach — comparable with the bore's own, at which point the bore has stopped deciding the pitch at all.

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.

One pair, 12 beat rates. Two notes at 220 hertz, 15 cents apart, drawn as the beat rate between each pair of partials. The fundamentals beat at 1.91 hertz and the k-th partials at k times that, so the series of rates is a straight line and it crosses 15 hertz at the 8th partial. Below the line the fluctuation is counted; above it the same physical fluctuation is heard as roughness. 1 per cent of this spectrum's energy is on the roughness side. The bars are drawn at each partial's own amplitude, so a spectrum with little energy high up crosses the line where nothing is listening.

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 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.

cylinder and Bessel flare: the length each mode behaves as though it has. Each mode's own acoustic length, m·c over twice its frequency, for a bore 148 cm long. A cylinder would give one number repeated. This gives 164 cm at the second mode and 154 at the 8th — a spread of 10.1 centimetres, or 110 cents, because a flare's end correction is a length that shrinks as the note rises. The first mode is off the top of this axis and is not a mode a player uses.

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 closing, and the note it is holding. The fourth resonance of a 370 cm horn against how much of the bore the hand occludes, at 97 per cent of the way along where the radius is 36 mm. Nothing much happens for the first ninety per cent. The note then falls to -401 cents — most of a fourth — over the next few, and between 99.5 and 99.90 per cent it jumps to 96 cents SHARP, because the lowest resonance has left the bottom of the range and every mode has taken the place of the one below it. The series' ratios are unchanged across the jump.

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.

The first five peaks, followed as the hand closes. Each line is one member of the series, tracked by its rank rather than by its frequency, and each dot's size is that peak's height. The lowest peak falls from 38 hertz to 34 as the hand closes and then jumps to 45, which is the renumbering computed earlier: past the wall the series is one member shorter at the bottom and every peak has taken the place of the one below it. The dots shrink through the middle of the travel and grow again at the far end, so the transition costs the player support as well as pitch — and the cost is temporary, which is why a fully stopped horn is a usable instrument and a nearly stopped one is not.

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.

Up a trumpet, the two clocks disagree. Every impedance peak of a trumpet, with the settling time each implies. The Q rises up the ladder and so does the frequency, and the settling time is their ratio — so in milliseconds the wait falls from 153 at the C♯2 to 24 at the B5, while in periods it rises from 11 to 24. Both curves are monotone and they point opposite ways, which means a player going up the instrument gets notes that arrive sooner and take longer in their own terms. Nothing here is a measurement of an instrument: it is the transmission-line solve of a trumpet-shaped bore, whose peak Qs are sensitive to how finely the sweep is sampled at about five per cent.

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.

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.

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.

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.

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 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.

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.

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.

The channels a fluctuation is analysed into, and how wide they are. A modulation filterbank of quality factor 1, drawn every half octave, with two of the beats a mistuned octave at middle C produces marked at 0.89 and 1.26 a second. A channel of quality one has its half-power points at 0.618 and 1.618 of its centre, so two fluctuations closer than a factor of 1.618 never end up in different channels. The two marked rates differ by a factor of 1.42, which is inside that. They are one fluctuation and not two — which is the question an earlier essay asked and left open, answered by a bank rather than by a convention.

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.

A struck octave becomes countable a second after the strike, or never. The number of separable beats a mistuned octave on A3 delivers, second by second after both notes are struck at 80 decibels, with every partial dying at its own rate (a 12-second fundamental, losses rising as frequency to the power 0.7). Read with the filter broadened by the level of the whole note, which is how the level-dependent count was first computed, the count is zero at every instant: the partials fall below audibility before the filter has narrowed enough to separate them. Read with the filter broadened by the level inside itself, which is what the published parameterisation was fitted against, the count is 0 at the strike, reaches 2 at 1.0 s and falls to nothing at 3.3 s.

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.

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.

A tempered fifth is a beat a second on the violin and one in four and a half seconds on the cello. How fast each open fifth of a string quartet beats when it is narrowed by 1.955 cents, the narrowing that meets an equal-tempered keyboard. The beat is the lower string's third partial against the upper string's second, so it is proportional to the lower string's frequency. Cello C2–G2: 0.22 a second, one beat every 4.5 seconds; cello G2–D3: 0.33 a second, one beat every 3.0 seconds; cello D3–A3: 0.50 a second, one beat every 2.0 seconds; viola C3–G3: 0.44 a second, one beat every 2.3 seconds; viola G3–D4: 0.66 a second, one beat every 1.5 seconds; violin G3–D4: 0.66 a second, one beat every 1.5 seconds; violin D4–A4: 1.00 a second, one beat every 1.0 seconds; violin A4–E5: 1.49 a second, one beat every 0.7 seconds. The slowest, the cello's C2–G2, is 6.7 times slower than the violin's A4–E5.

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.

Four pure fourths and a pure third leave a comma on the top strings. Each of a guitar's six open strings, in cents from the note its own frets make, when the strings are tuned against each other three ways. fretted unisons: E2 +0.00, A2 +0.00, D3 +0.00, G3 +0.00, B3 +0.00, E4 +0.00. harmonics, pure third on G–B: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 −19.55, E4 −21.51. harmonics, B from the low E's twelfth: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 +1.96, E4 +0.00. Tuned with pure fourths and a pure third the high E closes two octaves 21.51 cents flat, a syntonic comma; taking the B from the low E's twelfth lands the high E exactly and moves the error into the third between the G and B strings.

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.

A string that decays twice opens its count at once and shuts it early. The count of separable beats a mistuned octave on A3 delivers after both notes are struck at 80 decibels, with each filter read at the level inside it, under one exponential decay of 12 seconds and under two stages — a prompt sound of 1.5 seconds carrying all but the last 20 decibels, and an aftersound of 12 seconds. One exponential: open from 1.00 s to 3.30 s, 11.8 beats. Two stages: open from 0.15 s to 1.77 s, 8.8 beats — and the single exponential struck 20 decibels softer closes at 1.77 s.

A string that decays twice is counted early

A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.

One of these tunings has no comma to place. Five guitar tunings, each with every adjacent interval set to the pure ratio nearest the interval it spans, drawn as how far each string then sits from where the frets put it. standard, spanning 5–5–5–4–5 semitones, closes 21.5 cents short and spreads its strings over 21.5 cents; drop D, spanning 7–5–5–4–5 semitones, closes 17.6 cents short and spreads its strings over 19.6 cents; D A D G A D, spanning 7–5–5–2–5 semitones, closes exactly and spreads its strings over 3.9 cents; open G, spanning 5–7–5–4–3 semitones, closes exactly and spreads its strings over 15.6 cents; open D, spanning 7–5–4–3–5 semitones, closes exactly and spreads its strings over 15.6 cents. Standard tuning's four fourths and a third fall a syntonic comma short of two octaves, which is what was found earlier. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four — two octaves — so nothing has to be put anywhere.

One tuning has no comma to place

Standard guitar tuning is four fourths and a third, and tuned pure it closes two octaves exactly a syntonic comma short — so tuning by ear decides where the comma goes rather than whether. An open tuning replaces the interval list. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four, so its chain closes and there is nothing to place: its six strings sit within four cents of the frets. The open-chord tunings close too, and pay for it with one string fifteen cents flat — which every chord barred straight across inherits exactly, and which is why every such chord is precisely as pure as the open one.

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.

A tuning is right for some chords and wrong for the rest

An earlier essay asked which tuning minimises a piece's total departure from just, and guessed the answer would be a few cents off a named one. Searched over all six strings, two of three pieces get back the tuning they were already in — because their chains close and there is nothing to improve. The third lands on a tuning nobody names, three and nine tenths of a cent better than anything a player would have tried, and it is not a compromise: two of its three chords are exactly just and the third is abandoned by thirteen cents.

Counted in beats, the abandoned chord comes back sooner and by steps. The piece in standard tuning with its G major, open chord given more and more of the piece, searched twice: once minimising the mean departure from just in cents, once minimising the mean beat rate. The vertical axis is how far that chord sits from just at each optimum. Counted in cents it stays at 13.29 cents until it holds exactly 44 beats — the same as the rest of the piece — and then drops to zero in one step. Counted in beats it drops at 30.4 beats to 4.96, then at 47.4 beats to 3.91, then at 73.1 beats to 0.00 cents. At the piece's own 8 beats the two counts agree exactly: the corner does not move, and what the beats change is where the steps are.

Counting beats moves the price of a chord, not the tuning

The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.

Two shapes asking fifteen pairs of strings for different things. Every pair of the six strings, and the difference in cents between the two strings' offsets that would make the interval each shape puts on that pair exactly just. The piece's chords fall into two families that never disagree with themselves: E major, open and A major barred at 5 (44 beats), and G major, open (8 beats). They disagree on 13 of the 15 pairs: E2–A2 is asked for 1.96 and -13.69; E2–D3 is asked for 0.00 and 1.96; E2–G3 is asked for -13.69 and 0.00; E2–B3 is asked for 1.96 and -13.69; A2–D3 is asked for -1.96 and 15.64; A2–G3 is asked for -15.64 and 13.69; A2–E4 is asked for -1.96 and 13.69; D3–G3 is asked for -13.69 and -1.96; D3–B3 is asked for 1.96 and -15.64; D3–E4 is asked for 0.00 and -1.96; G3–B3 is asked for 15.64 and -13.69; G3–E4 is asked for 13.69 and 0.00; B3–E4 is asked for -1.96 and 13.69. The ring on each row is where the piece's cheapest tuning actually puts the pair. It sits on the first family's demand every time, which is what abandoning the other chord means: no weighting of the error can put a ring on two different places.

The chord a tuning gives up is a fingering

A guitar piece's best tuning makes two of its chords exactly just and abandons the third by thirteen cents, and no way of counting error — cents, beats, squared cents, or only the beats slow enough to hear — brings the third chord back. Fingered as a barre chord in the shape the other two already use, it comes back completely, and the whole piece is exactly just. The conflict was never between chords. It was between hands.

Tuned for the E shape, a cent of intonation is worth about five frets of hand movement. Every way through a ground on E major — I, IV, I, V, IV, I, for 16, 8, 8, 4, 4 and 8 beats — that is the cheapest route at some exchange rate between the piece's mean departure from just and the frets the hand moves, choosing each chord's fingering from its five moveable shapes; once with the guitar tuned to make the E shape exactly just and once on equal-tempered frets. Tuned for the E shape: 12.09 cents for 0 frets; 4.30 cents for 2 frets (worth it above 0.3 frets a cent); 2.09 cents for 10 frets (worth it above 3.6 frets a cent); 0.00 cents for 20 frets (worth it above 4.8 frets a cent). On the frets: 6.85 cents for 0 frets; 6.15 cents for 2 frets (worth it above 2.8 frets a cent); 5.95 cents for 4 frets (worth it above 10.2 frets a cent); 5.78 cents for 10 frets (worth it above 34.5 frets a cent); 5.60 cents for 20 frets (worth it above 57.5 frets a cent). Tuned for the shape, the route that keeps every chord in it is the cheapest once a cent is worth 4.8 frets; on the frets it needs 57.5.

Tuned for one shape, a cent is worth five frets

A guitar can be tuned so that one chord shape is exactly just wherever it is moved on the neck, and every other shape is then between twelve and fifteen cents out — worse than on the frets. So a player of such a tuning chooses, chord by chord, between intonation and the distance the hand travels. Searched exactly over every fingering of a I–IV–V ground, the route that keeps every chord in the one shape is the cheapest as soon as a cent of the piece's mean error is worth 4.8 frets of movement. On equal-tempered frets the same route needs a cent to be worth fifty-seven.

In every key with an open chord, the shape to tune for is that chord's. For a I–IV–I–V–IV–I ground in each of seven major keys, how far the hand moves on the route that keeps every chord exactly just, with the guitar tuned for each of the five moveable shapes, in frets. C: C 20, A 24, G 26, E 28, D 28; D: C 24, A 24, G 20, E 28, D 16; E: C 24, A 20, G 28, E 20, D 24; G: C 24, A 28, G 16, E 24, D 26; A: C 28, A 16, G 24, E 24, D 20; F: C 28, A 28, G 28, E 24, D 24; B♭: C 28, A 24, G 24, E 30, D 28. The keys whose tonic is an open chord — C, D, E, G and A — are served best by the shape of that chord in 5 of 5 cases; F and B♭ have no open tonic, and every shape leaves the hand further from the nut.

Tune a guitar for the shape of its key's open chord

A guitar can be tuned for any of five moveable chord shapes, and a player who then keeps every chord in that shape plays exactly in tune while the hand travels. Which shape should the tuning serve? Searched over seven keys and all five shapes, in every key whose tonic is an open chord the answer is that chord's own shape — C for C, D for D, G for G, A for A, and a tie between the E and A shapes in E. The reason is not the key and not the progression. It is that an open chord needs no barre: its hand sits one or two frets above the nut, and that saving is the whole of the difference.

On a guitar tuned for any major shape, no minor chord is within eight and a half cents of just. Each moveable minor shape — the open E minor, A minor and D minor chords — with its mean departure from just when the guitar is tuned to make one of the five major shapes exactly just, and on equal-tempered frets. E minor shape: 10.30 tuned for C, 10.95 tuned for A, 10.04 tuned for G, 9.78 tuned for E, 10.92 tuned for D; 5.74 on the frets. A minor shape: 10.17 tuned for C, 11.73 tuned for A, 14.47 tuned for G, 8.60 tuned for E, 12.20 tuned for D; 6.65 on the frets. D minor shape: 16.94 tuned for C, 9.78 tuned for A, 10.75 tuned for G, 11.41 tuned for E, 14.66 tuned for D; 8.15 on the frets. The least-bad is the A minor shape on a guitar tuned for the E shape, at 8.60 cents, and every one is worse than the frets.

A tuned guitar has no minor chord in tune

A guitar tuned to make one major chord shape exactly just can play every major chord in tune by moving that shape. It cannot play a minor chord in tune anywhere: every minor shape disagrees with every major shape about some pair of strings by eleven cents or more, and on any of the five tunings the least-bad minor chord — the A minor shape on a guitar tuned for the E shape — is 8.60 cents out, worse than on the frets. Put one minor chord into a I–vi–IV–V ground and the most-in-tune tuning in every key becomes the E shape's, because it is the one that hosts that chord least badly.

With its silent string placed for the minor chord, the C shape's tuning is the most in tune in every key. The I–vi–IV–V ground twice round in five major keys: the mean departure from just of the most-in-tune route under each major shape's tuning, with the strings that shape does not sound left where the one-shape tuning put them and placed for the minor chord. C: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. D: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. E: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. G: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. A: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. Under the C shape's tuning every major chord is exact and the minor chord is an E minor shape 6.26 cents out, for a quarter of the piece.

A silent string tunes the minor chord

A continuous search over a guitar's six strings, run on a piece with one minor chord in it, never settles between two chord shapes: at any share of the piece, the best tuning is exact for one chord and makes the other pay. But the tunings the earlier essays called exact for a shape were exact only on the strings that shape sounds. The C shape never sounds its low E string, and tuned to the C chord's own E it hosts an E minor chord at 6.26 cents — not the 8.60 that was said to be the best any tuned guitar could do. The pop ground's most in-tune tuning is then the C shape's, in every key, at 1.57 cents.

Tuned for the whole route, the I–vi–ii–V ground wants no shape at all until the hand is allowed to move. The best tuning for the I–vi–ii–V ground twice round, found by routing and retuning in turn from seventeen starts, in five keys and at eight exchange rates between a cent of mean error and a fret of hand movement: what chord shape it makes exact, its cents from just and the frets its route moves. C major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 30 frets); 50, C (3.13 cents, 30 frets). D major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (6.52 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 33 frets); 50, C (3.13 cents, 33 frets). E major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, E (4.60 cents, 15 frets); 12, A minor (4.11 cents, 20 frets); 20, A minor (3.78 cents, 26 frets); 50, C (3.13 cents, 49 frets). G major: 1, nothing (7.71 cents, 0 frets); 2, nothing (7.71 cents, 0 frets); 3, nothing (7.71 cents, 0 frets); 5, E minor (4.71 cents, 9 frets); 8, E minor (4.27 cents, 12 frets); 12, D (3.45 cents, 21 frets); 20, C (3.13 cents, 27 frets); 50, C (3.13 cents, 27 frets). A major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (5.90 cents, 3 frets); 8, C (4.11 cents, 15 frets); 12, C (4.11 cents, 15 frets); 20, C (4.11 cents, 15 frets); 50, C (3.13 cents, 35 frets). At the bottom of the range the best tuning is exact for no chord and the hand does not move; at the top it is one major shape with its silent string placed.

A guitar tuned for a still hand is tuned for nothing

Tune a guitar for a whole route through a ground rather than for one chord shape — routing, retuning and routing again until neither changes — and the answer depends on how much the player is willing to move. Valuing intonation highly, the best tuning is one major shape with its silent string placed, as before. In the middle it can be exact for a minor shape instead, because the route that keeps the hand near the nut plays its minor chords open. And for a player who will not move at all, the best tuning is exact for no chord whatever: every string within two cents of its fret, beating equal temperament by a few tenths of a cent. It is the compromise the earlier search went looking for, and it lives where nobody looked.

The set's tuning is the D shape's on the strings it sounds, and the set's own on the two it does not. The six strings' offsets from equal temperament, low E pinned, of the single tuning found for the I–vi–IV–V ground in five keys at two, five and fifty frets a cent, beside the D shape's placed tuning, dashed. 2: low E +0.0, A +2.0, D +2.0, G +3.9, B +2.0, high E −11.7, exact for D; 5: low E +0.0, A +2.0, D +2.0, G +4.0, B +2.0, high E −11.7, exact for D; 50: low E +0.0, A +13.7, D +0.0, G +15.6, B +13.7, high E +0.0, exact for C. The D shape's placed tuning: low E +0.0, A +2.0, D +8.5, G +10.5, B +8.5, high E −5.2. On the D, G, B and high E strings, measured from the D string, the five-frets tuning and the placed D tuning agree; they differ only on the two strings the D shape leaves silent.

A guitar tuned for a set is tuned for the D shape

Tune a guitar once for an evening of songs in five keys rather than once per song, and the tuning the set wants is not the compromise the single-song search predicted. At the valuation a typical player puts on a cent, it is exact for the D shape — the one fingering every key's route uses for some chord — with that shape's two silent strings placed for the set rather than for any song. Not retuning between songs costs a fifth of a cent. Only a set that mixes two grounds, played by a player who will not move, wants a tuning exact for nothing.

However many keys the set spans, its tuning is exact for a shape — the D shape from four keys on. The I–vi–IV–V ground in a set of songs whose keys are added one at a time round the circle of fifths from C — C, G, D, A, E, B, F♯, D♭, A♭, E♭, B♭, F — at five frets a cent: the set's mean cost with each song given its own tuning, with one tuning for the set, and on equal-tempered frets, with the shape the set's tuning is exact for above each point. 1 key: own 5.17, set 5.17 (C), frets 7.60; 2 keys: own 5.27, set 5.27 (C), frets 7.67; 3 keys: own 5.27, set 5.43 (C), frets 7.54; 4 keys: own 5.01, set 5.33 (D), frets 7.39; 5 keys: own 5.14, set 5.36 (D), frets 7.43; 6 keys: own 5.23, set 5.41 (D), frets 7.46; 7 keys: own 5.29, set 5.47 (D), frets 7.48; 8 keys: own 5.34, set 5.49 (D), frets 7.49; 9 keys: own 5.38, set 5.51 (D), frets 7.51; 10 keys: own 5.41, set 5.54 (D), frets 7.52; 11 keys: own 5.43, set 5.55 (D), frets 7.53; 12 keys: own 5.45, set 5.57 (D), frets 7.51. The set never stops sharing a shape: up to three keys it is the C shape's, and from four to all twelve the D shape's.

One guitar tuning serves every key, and it is the D shape's

A guitar tuned once for a set of pop songs in five keys is tuned exact for the D shape. The open question was the converse: which songs a given tuning serves, and how many keys a set can span before its best tuning stops being any shape's. Grown one key at a time round the circle of fifths, the set never stops sharing a shape: up to three keys it is the C shape's tuning, and from four keys to all twelve it is the D shape's, with the same offsets to a fifth of a cent. The five-key tuning serves the seven keys outside the set better than the two open keys inside it, and not retuning between songs costs less the more keys the evening spans.

A set in twelve keys keeps its shape through any number of rock songs and loses it to two minor ones. The I–vi–IV–V ground in all twelve keys, with songs outside it added one key at a time — minor-key songs on A, E, B, F♯ and on round the circle, or rock songs on G, D, A, E and on — at five frets a cent: the set's mean cost under its one best tuning, with the shape that tuning is exact for above each point. minor-key songs, i–iv–V–i: 0 added 5.57 (D), 1 added 5.77 (D), 2 added 5.97 (C), 3 added 6.01 (C), 4 added 6.04 (C), 6 added 6.08 (C), 8 added 6.12 (C), 12 added 6.03 (E minor). rock songs, I–♭VII–IV–I: 0 added 5.57 (D), 1 added 5.49 (D), 2 added 5.42 (D), 3 added 5.36 (D), 4 added 5.22 (D), 6 added 5.14 (D), 8 added 5.09 (D), 12 added 4.96 (D).

Two minor songs retune a guitar set for their dominant

A guitar tuned once for an evening of pop songs in every key is tuned exact for the D shape, because every key's route through the ground passes through it. Add songs that are not that ground and the tuning holds or breaks by a rule nobody would guess. Rock songs on I, ♭VII and IV never move it off the D shape, however many are added: they retune only the two strings the D shape leaves silent. One minor-key song leaves it alone too. Two move it to the C shape's tuning, and the chord they move it for is not one of their minor chords but their one major chord — the dominant, fingered in the C shape. Twelve move it again, to the E-minor shape's.

What each minor-key song gains from the C shape's tuning. For a minor-key song, i–iv–V–i, in each of the twelve keys, the saving in cents plus frets over five from playing it under the C shape's tuning rather than the twelve-key ground's own, the D shape's: C minor 1.69, C♯ minor 1.69, D minor 1.69, E♭ minor 1.69, E minor 1.69, F minor 1.69, F♯ minor 1.69, G minor 0.33, A♭ minor 1.69, A minor 1.69, B♭ minor 1.69, B minor 1.61. Switching the tuning costs the twelve ground songs 3.90 between them, so the switch pays when the added songs save more than that: the lines mark a half and a third of it, what each of two or three songs would have to save. Ten keys save 1.69 each — less than half, more than a third. G minor saves 0.33, because under the ground's tuning its dominant is the open D chord, already exact.

A guitar set keeps its tuning until the third minor song

Two minor-key songs were found to move a pop set's guitar tuning from the D shape to the C shape. Priced as a vote — what the move costs the twelve ground songs, against what it saves each minor song — two songs do not pay for it and three do. The earlier search had never tried keeping the tuning the set already had. And the key a minor song is in matters only at the nut: ten keys vote alike, and G minor, whose dominant is the open D chord, hardly votes at all.

The minor songs' tonic takes the tuning only at the ninth minor song. The twelve-key ground with minor-key songs added on A, E, B, F♯ and on round the circle: the set's mean cost, in cents plus frets over five, under three fixed tunings — the ground's own (the D shape's), the C shape's (the minor songs' dominant exact) and the E-minor shape's (their tonic exact). 0: 5.567, 5.892, 6.499; 1: 5.767, 5.937, 6.436; 2: 5.938, 5.975, 6.327; 3: 6.081, 6.008, 6.285; 4: 6.211, 6.037, 6.247; 5: 6.326, 6.062, 6.214; 6: 6.428, 6.085, 6.185; 7: 6.520, 6.105, 6.159; 8: 6.602, 6.123, 6.135; 9: 6.638, 6.102, 6.076; 10: 6.707, 6.118, 6.058; 11: 6.711, 6.134, 6.042; 12: 6.772, 6.148, 6.028. The ground's tuning is cheapest to two songs, the C shape's from three to eight, and the E-minor shape's from nine — when minor songs are nine of twenty-one.

A guitar set tunes for its minor chords only when nearly half its songs are minor

Add a third candidate to the vote — the tuning that makes the minor songs' own tonic chord exact — and a pop set's guitar passes through three tunings as minor songs are added: the ground's to the second, the dominant's from the third, and the tonic's only from the tenth, when ten songs in twenty-two are minor. Nothing in between is ever chosen: the two minor-song tunings differ on two strings, and every compromise between them costs the ground more than either end. Two keys break the pattern at the nut, and one earlier figure turns out to be wrong by a song.

One retuning is worth what the pop songs pay for the minor tuning

A guitarist allowed to retune once in an evening plays the pop songs in their own tuning and the minor songs in theirs. What that saves rises with every minor song until the tenth, and then stops: from there on it is exactly what the twelve pop songs would have paid to be played in the minor songs' tuning, and adding minor songs cannot raise it. The C shape's tuning, which a one-tuning guitar spends most of the range in, is never used. And the whole saving belongs to the running order. Put the same ten minor songs in two runs instead of one and a single retuning is worth three-fifths as much; spread them through the evening and it is worth a fifth.

The minor half's own tuning belongs to one key

A guitarist who retunes once between the pop songs and the minor songs has been choosing among three tunings found for whole evenings. Searched for the minor songs alone, the best tuning is none of them — but only in A, B and D minor, where it gives the tonic and subdominant one shape at the nut and both are in tune. In A it saves six-tenths a song, and the refund a retuning buys stops having a ceiling. In the other nine keys, and in any minor half that mixes keys, the E-minor shape's tuning was already the answer.

With a capo up to fret 7, 8 of the twelve minor keys are played as B minor in one tuning. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five, under the tuning searched for the song on A (the B string raised 13.7 cents): at the nut, and at the capo position up to fret 7 that suits it best, with the E-minor shape's tuning at its best capo for comparison. C minor 5.72 → 4.53 (capo 1); G minor 5.72 → 5.72; D minor 5.72 → 4.53 (capo 3); A minor 5.09 → 5.09; E minor 5.72 → 4.53 (capo 5); B minor 4.53 → 4.53; F♯ minor 5.72 → 4.53 (capo 7); C♯ minor 5.72 → 4.53 (capo 2); A♭ minor 5.72 → 5.72; E♭ minor 5.72 → 4.53 (capo 4); B♭ minor 5.72 → 5.09 (capo 1); F minor 5.72 → 4.53 (capo 6). 8 keys reach 4.53, B minor's price, by putting the capo where B minor's open-string chords become theirs. E minor costs 4.53 at capo 5, against 4.91 in the E-minor shape's tuning.

A capo is a transposition the tuning cannot see

Tempered frets make a chord shape's error the same wherever it sits, so a song with a capo at the fifth fret is, to the tuning, the song five semitones lower on a shorter neck. That lets every minor key borrow B minor's open-string fingerings, and eight of the twelve take the raised-B tuning's best price with a capo no higher than the seventh fret. A minor half in mixed keys, which gave a retuning nothing, now nearly doubles its refund — but the capo pays the single-tuned evening as well, and a minor half all in A gets less from its retuning than before.

With the capo left at fret 5, E minor becomes B minor and the other keys pay for it. Twelve minor-key songs, i–iv–V–i, one in each key round the circle, in the raised-B tuning, in cents plus frets over five: at the nut, with the capo left at fret 5 for every song, and at each song's own best capo up to fret 7. C minor 5.72, 8.62 at fret 5, 4.53 at its own; G minor 5.72, 5.72 at fret 5, 5.72 at its own; D minor 5.72, 5.09 at fret 5, 4.53 at its own; A minor 5.09, 6.13 at fret 5, 5.09 at its own; E minor 5.72, 4.53 at fret 5, 4.53 at its own; B minor 4.53, 8.62 at fret 5, 4.53 at its own; F♯ minor 5.72, 5.72 at fret 5, 4.53 at its own; C♯ minor 5.72, 8.62 at fret 5, 4.53 at its own; A♭ minor 5.72, 5.72 at fret 5, 5.72 at its own; E♭ minor 5.72, 5.72 at fret 5, 4.53 at its own; B♭ minor 5.72, 7.82 at fret 5, 5.09 at its own; F minor 5.72, 5.72 at fret 5, 4.53 at its own. The twelve cost 66.82 at the nut, 78.04 with the capo at fret 5 and 57.87 with it moved for every song. At fret 5 the song on E takes B minor's fingerings and the song on D A minor's; every other key is moved to a key the tuning does not favour, on a neck 5 frets shorter.

A capo left in place serves one key

Moved for every song, a capo lets nearly every minor key borrow B minor's open-string fingerings. Left at one fret for a whole run of minor songs, as most guitarists leave it, it does something much weaker: it renames the keys. Over a run in every key that is a reshuffle on a shorter neck and can never beat the nut; over the four keys guitarists use most it gains nothing either, because two of them are already the tuning's best. A fixed capo pays only a run in very few keys, spaced the way the tuning's cheapest keys are.

A set ordered by key relation moves the capo most

Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.

A set steps by fifths where the capo moves

A set of twelve minor songs in E, A, D and B can be ordered for the capo or for key relation, and the two orders pull apart. Searched over every one of its 41,304 orders, the trade between them is exact and simple: a set can put a fifth between neighbouring songs at each capo change for nothing, because a fifth always crosses frets and a capo change is where crossing is paid for. Beyond that each extra fifth costs about seven points of the capo's saving. The tone steps the earlier essay suspected of carrying the compromise are where the capo stays still.

A set free to change key becomes a set in two

A set list orders its songs for the music and a capo wants them ordered by fret, and the trade between the two looked as if it might disappear if a singer would move a song a tone. It does not. Searched exactly, moving songs adds five to twenty points of the capo's saving at any count of fifths and stops improving at six moved; at its limit the best set rocks between B minor and E minor, because a count of fifths rewards a pendulum. A set that really walks the chain of fifths has three to spend, not eleven.

A meantone keyboard is a stretch of the chain of fifths, and every split key lengthens it by one place at an end. 14 keys hold the positions from A♭ to D♯. That is 10 major and 10 minor triads; major keys E♭, B♭, F, C, G, D, A, E; minor keys with a raised leading note C, G, D, A, E.

A split key buys one of everything, and two buy C and E

An organ builder with money for two extra keys could have put them anywhere, and the arithmetic of triads cannot say where. On a meantone keyboard every added key buys exactly one major triad, one minor triad, one major key and one minor key, wherever it goes. So the count is no guide, and the choice was about which keys. The two accidentals most often divided, G♯/A♭ and D♯/E♭, are one on each side of the chain, and they are exactly what completes the minor keys on C and E. What they buy by the roughness sum is seven per cent.

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