Just intonation — where it appears
Named by 32 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 this site measured 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.
Named alongside it
The objects these essays reach for when they reach for this one.
CentsSyntonic commaEqual temperamentBeatingDifference limenHarmonic seriesTuning by earCombination toneIntonationOpen stringPartialDifference tone