Just intonation — where it appears
Named by 66 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
Three is the largest agreeable number
Take every chord of every size that can be built from the twelve, score each one for roughness and for the size of the whole numbers it needs, and ask how many are good on both counts. At two notes the answer is one. At three notes it is still one. At four it is eight, and the question stops having an answer.
The progression that never comes home
Just intonation is usually said to fail because a keyboard has only twelve keys. That is not the reason. Take one of the commonest chord sequences in tonal music, play every chord in exact whole-number ratios, and it arrives back on its starting chord a syntonic comma flat — on any instrument, including a voice.
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.
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.
Somebody has to pay the comma
A choir has no frets and no keys, so it can tune every chord exactly — and a common progression sung that way arrives a comma flat, every circuit. The freedom a keyboard lacks turns out to be a freedom about where the error goes rather than whether there is one, and the two costs always add to the same number.
The third the model has no opinion about
A neutral third — halfway between major and minor, and a scale degree in most of the music between Morocco and Iran — is nothing at all on an ordinary roughness curve. It becomes a well at 347 cents, which is exactly 11:9, only for a spectrum whose ninth and eleventh partials are at least three times their natural strength. No instrument has that spectrum.
The root an ear supplies
A major triad's notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass. A minor triad's fit two different series with two different answers a major sixth apart, and the model cannot choose between them. The ambiguity theorists argued about for two centuries is a computable quantity, and the spectrum invented to remove it is one no object produces.
A scale built downward from a fourth
A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.
The register where the series becomes a scale
A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.
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.
What a choir does that a soloist cannot
Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift measured for tuned singing describes only ensembles that hold their pitch still.
The series is not a chord
The major triad is derived from the first six partials of the harmonic series in every textbook that derives it from anything, and the derivation works: the first six contain a major triad and nothing else. Take the first four instead and there is no third at all; take seven and there is a dominant seventh with a flat seventh thirty-one cents below any keyboard note; take sixteen and there are eight pitch classes. Six is the last stopping point that gives the answer, roughness prefers three, four or ten depending on the register, and no criterion in the arithmetic selects six over any of them.
A section against another section
The choir has been treated as a unison, and no choir sings only unisons. Two sections an interval apart beat between partials rather than between fundamentals — the third brings the fifth partial of one against the fourth of the other — and equal temperament puts that coincidence fourteen cents out. So two sections singing a tempered third beat at nearly nine per second with every singer in both of them perfectly in tune, and the same temperament is inaudible on a fifth.
The scale least committed to its own instrument
The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.
The setting is not the preference
Every published number for an interval listeners prefer — the pure third a quartet is said to find, the raised leading note, the harmonic seventh — is a value somebody adjusted until it sounded right. A listener adjusting inside a key is adjusting through the posterior computed just before, so the value they stopped at is not the one they preferred: it is the one whose heard size equals it. On quarter-second notes the correction for a pure major third is 12.8 cents, which is nearly a syntonic comma and is larger than the 13.7-cent preference it corrects.
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.
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.
A boundary beside a fifth
A closed form established earlier says an n-category division costs n·σ·√(2/π) whatever the widths are, so a boundary costs the same wherever it is put and the step pattern cannot matter. Its own last section named the assumption that produces the null: one σ, applied to every boundary in the octave. Let σ follow how securely each interval is held and the formula becomes a sum over boundaries rather than n copies of one — the diatonic's error rises from 5.1 per cent to 11.8, and where the degrees are put matters again.
The best seven of the twelve
Once the noise is allowed to differ from boundary to boundary, a scale can be chosen to minimise identification error — and the choice is a search over four hundred and sixty-two sets rather than an argument. Run, it returns a cluster of semitones around the tonic and around the fifth, and puts the diatonic major at rank 376 of 462, in the worse fifth of the ranking. A criterion whose optimum is a scale nobody has ever played is a criterion that is not what scales are chosen for, and the reason it fails is legible in the model rather than in the music.
The unequal scale that is easier to name
The whole of the earlier result was that a scale's step pattern cannot matter, and every tradition it drew agreed with the equal division of its own size to three decimal places. With one noise per boundary the comparison is live again, and the tempered diatonic beats seven equal steps by eleven per cent — while the pentatonics gain nothing and the maqam scales gain two tenths of one per cent. Only one of the two security models produces the effect, and it is the one that was not measured on listeners raised inside the tradition it is being used to explain.
The bass line under a passage in thirds
A major scale harmonised in parallel thirds gives the ear a difference tone under every pair, and in five-limit just intonation those tones are a diatonic bass line — C, A, C, F, G, F, G, C — made of the scale's own notes. Tempered, the same line moves only by whole tones, a neutral third and a fourth stretched to 650 cents, and wobbles by up to 84 cents from note to note. In sixths the bass is drawn by the other product, because the cubic product of a pair is the difference tone of the same pair inverted.
A combination-tone bass needs a forte
A scale in just thirds draws a diatonic bass line through its difference tones, and in sixths the cubic product draws one. Given the two published level laws, with their constants swept, the thirds' bass is not heard at all below primaries of about 66 dB and is heard whole only from 71 to 81. The cubic products are a different kind of object: the primaries mask them decibel for decibel as they rise, so no dynamic changes whether they are heard. Most of the thirds' inner line never is, and the sixths' bass needs a forte and a gentle law.
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.
Three harmonics of the bass arrive before the bass
Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.
The ghost bass drops a twelfth at a forte
Both crowds arrive at once and every member of both is a multiple of the same absent fundamental, so a listener is never given a choice between them — only a different subset of one harmonic series at every dynamic. Softly, the subset is an exact gapless series on three times the fundamental. Loudly, the difference tones fill in the low harmonics and no template on the higher note survives them. Between 62 and 72 decibels, depending on the interval, the note the crowd names falls by an octave or a twelfth, and the two qualities of third cross at different levels.
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.
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.
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.
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.
The played notes already name the ghost bass
The products of a just third's partials, fitted on their own, name a note a twelfth above the bass when the interval is soft and drop to the bass when it is loud. Put the two played notes back beside them and the drop disappears: the notes and their products name the bass at every dynamic, because the notes' own partials are harmonics of it already. What the dynamic changes is not which note is implied but how complete its harmonic series is — nine holes from the notes alone, four when soft, none when loud.
A tempered third cannot be one sound with its own ghost
The essay before this one fitted one harmonic template to a just third's played partials and the products of those partials together, and it worked at every dynamic. Tempered, it does not work at all — and not by naming a different note. There is no template. The lower note's partials sit exactly on their harmonics, the upper note's are every one fourteen cents out, and the products are twenty-three cents out the other way, a span of thirty-seven cents across a tolerance of thirty. The window is nine cents wide when the third is soft and two when it is loud, so equal temperament's fourteen falls outside it — while the fifth's two cents fall inside. The consonances a temperament leaves nearly alone are exactly the ones that stay single sounds.
The seventh has two places to be
Tune a dominant seventh's seventh as the seventh partial and its coincidences last 0.87 seconds instead of 0.66 — exactly the diminished seventh's figure, which is the highest any four-note chord in this census reaches. It gets there by levelling rather than by improving: the root's pair with the seventh gains a third of a second and the fifth's loses one, and all three pairs that contain the seventh arrive at the same number. The route is a valley. A seventh tuned six cents flat has lost the ninth-partial coincidence and not yet found the seventh, and lasts 0.46 seconds, which is worse than leaving it alone. And the same retuning halves the minor and half-diminished sevenths.
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.
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.
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.
A ghost bass an octave lower needs notes twice as long
A scale played in just thirds has a bass underneath it that nobody plays — the difference tones of its thirds spell a diatonic line. Every essay on that bass has held its intervals still. A moving passage gives the bass a new pitch every note, and a low pitch is not there until it has run a few of its own cycles. With four cycles, a passage in thirds from C4 keeps its bass for 83 per cent of each note at three notes a second and loses it completely below 73 milliseconds a note. An octave down, every number doubles: the passage from C3 has no bass at all in its lowest note at semiquavers, and it is the scale's one Pythagorean third that sets the limit.
Four notes cannot get below the seventh partial
Tune a dominant seventh as 4 : 5 : 6 : 7 — every note moved, not only the seventh — and its coincidences last 0.868 seconds, which is the one-note retuning's figure to the millisecond. Tuning the third and fifth buys nothing at the top and costs seven voicings a fifth of their life. Every other chord of four notes a harmonic series holds does worse, and the reason is not temperament: no four pitch classes avoid a semitone, a tone and a tritone together, and none of those three can be tuned to coincide below the seventh partial. The half-diminished seventh, which the one-note retuning ruined, reaches the same ceiling read down the series.
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.
The notes name the ghost bass before it has a cycle
A ghost bass that had to wait for four of its own cycles needed 61 milliseconds under a third from C4, and so vanished from fast passages. A pitch mechanism fitting the whole object — both notes' partials, which run their cycles many times faster, and the products — names the same bass in five. Seven of the eight thirds of a just scale have their bass for most of every note at any tempo a player reaches. The eighth, the scale's Pythagorean D–F, never has its own: the object hears it as a 6 : 5 and supplies the bass a D a comma lower would give, a semitone above the difference tone, until the difference tone arrives and breaks the fit.
The dynamic chooses which D the scale has
A just C major scale has one third that is not 5 : 4 or 6 : 5 — D–F, a Pythagorean 32 : 27 — and a pitch mechanism fitting the notes' partials hears it as the just minor third a comma away, with that third's bass, B♭1, beneath it. Only the Pythagorean third's own difference tone can contradict that, and it is audible only from 76 dB. So below a forte the ghost line under a scale in just thirds is C, B♭, C, F, G, F, G, C for the whole of every note, and the syntonic comma is resolved by the listener rather than the players; above it, and only in notes longer than 73 milliseconds, the B♭ is taken away.
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.
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.
A tuned seventh chord does not beat; it buzzes
Three four-note chords share a ceiling of 0.87 seconds on how long their partial coincidences last, and nothing in that measure could tell them apart. Their beating can. A chord tuned exactly to one harmonic series is one periodic sound, so no two of its partials can differ by less than the series' fundamental: the tuned dominant seventh has no slow beating at all, where the tempered diminished seventh has 125 beating pairs at the strike. Its roughness is another matter. Tuning removes the beats and changes the roughness by a few per cent, in either direction, and a seventh on its way to 7 : 4 beats more, not less, until it arrives.
A tempered seventh chord already has the tuned one's low note
A dominant seventh tuned 4 : 5 : 6 : 7 is one periodic sound, and its period is a note below the chord that nobody plays. The question left open was whether tuning adds that note. Run through the temporal account of residue pitch, the tempered chord turns out to repeat at the same place, a few cents sharp, at 0.82 of the tuned chord's exactness at the strike and 0.96 a second later. Tuning does not create the low note. It makes it exact, and it gets there by a route on which every step helps, which is not true of the beating.
A tuned close seventh chord low on the keyboard flutters at one rate
A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.
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.
The scale's ii chord borrows the dominant's bass
Heard quietly, the pair D–F of a just C major scale brings the bass of the other D with it — B♭, the bass the 10 : 9 D implies — so the comma is resolved by the listener rather than by the players. Add the A that makes the pair a ii chord and that stops being true. The whole-object fit now hears the chord over G, as the top of a dominant ninth, and it does so because the template prefers simple harmonics to accurate ones; a different bass is heard in each inversion, and at a forte none at all. Only a ii built on the 10 : 9 D has one bass in every voicing and at every dynamic. Played quietly from a fixed scale, I–vi–ii–V–I is heard with the bass line C, F, G, G, C: the ii merged into the chord after it, and the progression home.
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.
A tuned chord's flutter binds it only if its notes start together
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.
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.
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.
A tuned chord stays in step for half a cent-second
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it.
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.
A shared vibrato cannot hold a tuned chord together
A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.
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.
A chord comes back into step too briefly to bind
A vibrato that is shared imperfectly does not carry a tuned chord's notes ever further apart: each note's time is warped back and forth, and the chord passes back through step twice in every swing. Locked ninety-seven or ninety-eight per cent, it does so about ten times a second. But each return lasts on average less than one cycle of the chord's own flutter, which is the rhythm the ear's filters would have to share to show the binding — fewer than one return a second lasts that long. The chord is in step often and never for long enough.
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.
Two octaves up, a sung chord can bind at every swing
A vibrato's swing between two notes is the same number of milliseconds in every register; the tolerance for being in step halves each octave, and so does the period of the chord's own flutter. The first two make a scaling law — an octave higher is the same chord with its vibratos disagreeing half as much — and the third breaks the tie. Matched for time in step, the higher chord's returns carry twice as many flutter cycles: over C2 fewer than one return a second is long enough to bind, over C4 with the vibratos locked ninety-nine and a half per cent nearly every return is.
A narrow vibrato needs no lock
A tuned close seventh binds and releases at the vibrato's swings only where its notes' timing swings by about the width of its onset tolerance, and the swing is the vibrato's width times how far the singers' vibratos disagree. Down to a ninety-five per cent lock the two trade one for one: a 71-cent vibrato locked ninety-nine and a half per cent is the same chord as a 7-cent one locked ninety-five. Below that the trade fails, in the singers' favour. With no lock at all the swing stops growing with the disagreement, and a vibrato of three cents over middle C — twelve over the bass — binds the chord at most of its swings with each singer's vibrato its own.
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 straight tone drifts out of step and stays there
A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Equal temperamentHarmonic seriesIntonationOpen stringOptimisationCentsSyntonic commaVoicingDominant seventhBeatingCombination toneResidue pitch