The note that is sharp because of where it goes

Every tuning argument in this collection has been vertical.

The comma is the gap between twelve fifths and seven octaves, and it matters because the fifths have to sound together. The syntonic comma is the gap between four fifths and a pure third, and it matters for the same reason. Meantone flattens the fifth to buy a pure third; a temperament is a policy for where to put an error between simultaneous notes; a choir that tunes each chord pure walks downhill. Eleven essays, one question: what should this note be, given the note beside it.

There is a second question about the same note and it has a different answer. What should this note be, given the note after it?

The two questions have been asked separately for six hundred years and the collection has only ever asked the first, which is a gap worth naming before filling: a melody is a path, and a path makes demands on a pitch that a simultaneity does not.

Two arguments about one note #

Take the leading note of C major — the B — and ask where it should sit.

The vertical argument. B is the third of the dominant triad, G–B–D. A pure major third is the ratio 5:4, which is 386.3 cents; a pure fifth above C is 702.0; so B, tuned as a pure third above a pure fifth, is at 1088.3 cents above the tonic. That is the just intonation answer, it is what makes the dominant chord beatless, and it is what every harmonic argument on this site would produce.

The horizontal argument. B is a leading note. What it does is go to C, and a leading note that is close to its destination arrives more strongly; performers with continuous pitch control raise it, and have been recorded doing so for as long as anyone has measured. Tuning B by iterating pure fifths — five of them up from C — gives 1109.8 cents. That is the Pythagorean answer, and it is what a melodic argument produces.

The two answers are 1088.3 and 1109.8. The gap is 21.5 cents, and the number is not a coincidence.

The gap is a syntonic comma, and it has to be #

The two constructions differ in exactly one respect: whether B is reached by a third or by fifths.

On the lattice of pure fifths and pure thirds the two answers are two different arrival points for one note name: four fifths up lands 21.5 cents above one third up plus two octaves, and that gap is the syntonic comma. Against the difference limen at 500 hertz it is four times over — so the disagreement is not a rounding, it is an interval a listener can hear.

So the melodic and harmonic answers for the leading note are the two ends of the syntonic comma. This is not a new comma and it is not a new problem — it is the comma the whole of Renaissance tuning theory was about — but it is arriving from a direction none of those essays used. The comma is normally introduced as a conflict between two simultaneities: a pure third and a stack of pure fifths cannot both be right in one chord. Here it is a conflict between a simultaneity and a succession.

The size is worth putting in perspective.

Read as a semitone, it is starker #

The comma is easier to feel when it is put where a performer would feel it: in the size of the step from B to C.

A performer sharpening the leading note is narrowing the semitone. The just leading note gives a semitone of 111.7 cents, which is wider than equal temperament’s, and to a string player or a singer it sounds slack — the note fails to lean. The Pythagorean leading note gives 90.2, a step so narrow it is nearly a fifth of a tone smaller, and it is what expressive intonation has been measured doing. Nineteenth-century violin treatises say so in as many words, instructing that the leading note be taken high; the instruction is old enough to predate any measurement of it, which makes the later measurements a confirmation rather than a discovery.

Measurements of string quartets, solo violinists and unaccompanied singers reproduce this consistently: leading notes are played sharp of equal temperament, not flat of it, and by more than the difference limen. Those measurements are quoted here, not computed — this site has no corpus of performances — and they are the reason for taking the horizontal argument seriously rather than treating it as folklore.

Both are right, and they are right about different things #

The temptation is to declare one of them correct. Neither is.

The vertical argument is right about a chord and its criterion is beats. Tune B as a pure third above G and the dominant triad has no beating between its third and its root’s fifth partial; move it 21.5 cents and it beats at a rate the tuning ladder can compute exactly.

The horizontal argument is right about a step and its criterion is arrival. There is no beating in a melodic interval — the notes are not sounding together — so the criterion that decides the vertical case has nothing at all to say about the horizontal one. What is left is expectation, and a narrower step is a stronger implication.

These are not two opinions about one quantity. They are two quantities that happen to share a note name. A keyboard has to choose because a keyboard has one pitch per key; nothing else does.

The same conflict, one interval down #

The leading note is the sharpest case but it is not a special case, and the pattern across the whole scale is worth seeing at once.

That is the general statement of the conflict. The vertical argument wants the five-limit values because those are the ones that stop the beating; the horizontal argument wants the three-limit ones because a chain of fifths gives narrow diatonic semitones and wide whole tones, which is what a melodic step wants. Each degree with a five in it is a place where a performer has a decision to make, and there are three of them in a major scale.

The whole scale, and the second melodic complaint #

The leading note is the case the argument is usually made about, and the same comparison over all seven degrees says two things the single note does not.

The disagreement is exactly one syntonic comma at three degrees and exactly zero at the other three. It is not a gradient with the leading note at its steep end; the third and the sixth are out by precisely the same 21.51 cents, and the second, fourth and fifth are not out at all, because they are built from twos and threes and have no five in them. What makes the leading note the clearest case is not the size of the gap but that the melodic argument for taking the sharp side is strongest where the note is about to resolve by a semitone.

The second thing is a melodic complaint the vertical argument has no way to raise, and it is arguably the stronger one. Read the same two tunings as a list of steps rather than a list of positions:

Just intonation has three sizes of step in a major scale and Pythagorean has two. Two of just’s whole tones are 203.9 and two are 182.4 — a difference of a syntonic comma between two steps a melody treats as the same interval. A performer walking up a just scale plays two different whole tones and has to know which is which; walking up a Pythagorean one, every whole tone is the same and every semitone is the same.

That is a melodic objection to just intonation which has nothing to do with leading notes and does not appear in the vertical argument at all, because a chord never asks two whole tones to match. A scale that is smooth vertically is uneven horizontally, and the unevenness is the same comma, distributed differently — which is the whole-scale version of the conflict and the reason a single tuning cannot answer both questions.

What performers actually do, which is neither #

The interesting resolution is not a compromise but a switch.

What performers actually do is neither: measurements put the leading note 10 to 20 cents sharp of equal temperament in melodic contexts and close to just in sustained chords, which is a continuous adjustment rather than a choice of system. Placed against the pitch-resolution figure, both of those quantities are comfortably above the melodic band and comfortably below a semitone.

The reported behaviour of good ensembles is that the same note is tuned differently depending on what it is doing at that moment. A note held as part of a sustained chord is tuned to the chord. A note passing through on the way somewhere is tuned to its destination. A leading note in a final cadence, sustained under a fermata, is pulled back toward the chord because it is now a chord tone; the same leading note in a rapid passage is sharpened because it is now a step.

This is exactly what a wind player’s tuning does with temperature and what a singer’s vibrato does with pitch: the note is not a value, it is a behaviour. What is new here is that the direction of the adjustment is decided by whether the note has a neighbour above it or after it.

And it explains something about the keyboard repertoire that is otherwise odd. Keyboard temperaments are argued about entirely on vertical grounds — which keys get the good thirds — and a keyboard’s melodic intonation is never discussed, because there is nothing to discuss. A pianist has no access to this variable at all.

Why a keyboard cannot split the difference #

Equal temperament looks like the obvious compromise: 1100 cents is almost exactly halfway between the two answers, 11.7 cents from one and 9.8 from the other.

It is a compromise in the arithmetic and not in the effect, and the reason is that the two criteria are not commensurable. Being 11.7 cents sharp of the just third means the dominant triad beats — audibly, at a countable rate on a sustained chord. Being 9.8 cents flat of the Pythagorean leading note means the semitone is a tenth of a semitone too wide, which is not audible as a defect at all in a rapid passage and is faintly slack in a slow one. Splitting a difference between a criterion measured in beats per second and one measured in rhetorical strength does not produce a solution half as bad as each; it produces something that fails the first plainly and the second mildly.

Which is roughly what equal temperament is usually accused of, from the harmonic side only. The melodic side of the complaint has no constituency because the instruments that would make it are the ones that were never in the temperament argument to begin with.

Where the argument stops #

Three limits, and the first is the largest.

Twenty-one cents is not what performers are measured to do; it is the size of the disagreement. Actual measured leading notes sit above equal temperament by amounts that vary enormously between players, passages and recordings — some of them beyond Pythagorean, many between equal and Pythagorean, and some flat of equal. The arithmetic here bounds the argument; it does not predict a performance.

The uneven-whole-tone objection is an objection to one construction, not to vertical tuning. Just intonation as tabulated here puts the sixth at 5:3 and the third at 5:4, which is one of several five-limit layouts of a major scale, and a different arrangement moves which whole tone is the narrow one without removing it. What cannot be removed is that a scale with two pure thirds in it has two sizes of whole tone somewhere, because the difference between them is the comma the thirds were bought with — so the complaint survives any relabelling and the specific row it lands on does not.

A great deal of intonation is not about either axis. Ensemble intonation drifts, instruments are imperfect, and much of what looks like a tuning decision is a consequence of temperature, fatigue or a fingering. Attributing a measured sharpness to a melodic intention requires ruling those out and this essay rules out none of them.

And the horizontal argument has no model behind it comparable to the vertical one. Beating is a computation with a formula. “A narrower step implies its destination more strongly” is a description of a practice, supported by measurement, with no mechanism attached — the closest thing to one in this collection is the tonal hierarchy, which is a frequency count. That asymmetry is real and the two halves of this essay are not equally well founded.

And across four temperaments’ twelve thirds the totals are identical, so what a temperament chooses is not how much error there is but which keys carry it — a fact that settles nothing about this argument, because the disagreement here is inside one key.

Which computation produced the numbers #

The just leading note is 15:8, which is a 5:4 above a 3:2, and its size in cents is 1200 log₂(15/8) = 1088.27. The Pythagorean one is five pure fifths reduced into the octave: 1200 log₂((3/2)⁵/4) = 1109.78. Their difference is 21.51 cents, and the syntonic comma — 81:80 — is 21.51 cents. The two are the same number because 15/8 divided into (3/2)⁵/4 is exactly 81/80.

The semitone sizes are 1200 minus each of those. The difference limen is the same function the perception essays use, evaluated at 500 Hz.

Nothing in this essay is a measurement of a performance. Every performance figure is quoted and attributed in the text as quoted.

Whose music, and when #

The vertical argument is European and its period is roughly 1450 to 1750 — the era in which thirds became consonances and the syntonic comma became the central problem of tuning theory. Before it, Pythagorean tuning was standard precisely because the third was not a consonance to be tuned pure, and the leading note being sharp was not a conflict with anything.

That is the historically startling part and it deserves stating: the melodic answer came first. Medieval European practice used the Pythagorean leading note, deliberately and by theory, and the conflict this essay describes did not exist because one side of it had not yet arrived.

The horizontal argument is not confined to Europe. Traditions with continuous pitch and strong melodic grammar — Hindustani and Carnatic classical music, Arabic maqam practice, Persian dastgah — all describe degrees whose intonation depends on the direction of approach and on what follows. A degree defined by where it goes next is the normal case outside keyboard cultures, and the tuning consequence is the same one.

What the picture cannot show #

It cannot show a performance. Every figure here is a tuning system evaluated, which is a table of pitches. A performance is a trajectory through pitch and the interesting quantity is where it is at each instant, which no static diagram carries.

It cannot show vibrato. An operatic vibrato is 142 cents peak to peak, which is six times the disagreement this whole essay is about. For a voice with vibrato the question of whether the leading note is at 1088 or 1110 is very nearly meaningless, and the essay applies to instruments and voices that hold a pitch still.

It cannot show a chord changing under a held note. The most difficult real case is a note that is sustained while the harmony moves under it — beginning as a chord tone and becoming a dissonance, or the reverse. Its correct tuning changes during the note, and no static picture of a scale has anywhere to put that.

And it has one note in it, though the whole-scale version is computed in a section above and is not what the phrase “in weaker form” would have suggested: the arithmetic gap at the third and the sixth is the same syntonic comma, to the hundredth of a cent, and only the melodic case for taking the sharp side is weaker.

The ladder from here #

This closes the first five rungs of the melody ladder, and the shape of them is one argument: a tune is a path, and nearly everything interesting about it follows from the path rather than from the material it walks on. Steps, reversals, the arch, the register a valveless instrument can play in, and now the tuning of a single degree — five properties, each of which reads as a preference and none of which is one.

What is owed is the two constraints this ladder has not touched. Rhythm is one: every figure in these five essays plots a note as a column and none of them knows how long it lasts, and melodic identity is at least as much rhythmic as it is intervallic. Memory is the other: the whole account of contour rests on what a listener retains, and this collection has measured how long a repeat can wait without ever asking what is being held in the interval.