A tempered third cannot be one sound with its own ghost

The played notes already name the ghost bass closed on a prediction, and the prediction is wrong.

Its finding was that a just third’s two played notes and the products their partials make in the ear are, together, a set of whole multiples of one absent fundamental, so a single harmonic template explains all of them and names the same bass at every dynamic. Its closing section named the condition that made that work — everything above depends on a just interval — and guessed what a tempered one would do:

The prediction is that a tempered third behaves like the figure-and-ground reading after all, because its products and its notes disagree about the fundamental, and that the note named softly and loudly will differ by the tuning’s error multiplied up the series.

The first half of that sentence is right and the second does not follow from it. A tempered third’s notes and products do disagree about the fundamental. What they do then is not name two notes in turn. They name nothing. No fundamental within the template’s tolerance explains the set, at any dynamic, and the reading that produced a number for every just interval simply returns no candidate.

Three families, three displacements #

The reason is arithmetic and it is visible on one axis.

Mistune the upper note by fourteen cents and three things happen at once, each of them uniform within its own family.

The lower note’s partials do not move. They sit at harmonics 4, 8, 12, 16 and 20 of the implied fundamental and they are exactly on them, because nothing was done to the lower note.

The upper note’s partials all move by fourteen cents, every one of them by the same amount. A partial at five times the fundamental goes to five times it plus fourteen cents; one at fifteen times goes to fifteen times it plus fourteen. That is what a ratio does: multiplying a frequency by a constant shifts every one of its harmonics by the same number of cents.

The products move by twenty-three cents, the other way. A cubic product of the two notes’ fundamentals sits at 2 × 4 − 5 = 3 times the implied fundamental, and moving the upper note moves it by −5/3 of that motion — the magnification the third sound magnifies cents, not hertz measured for the notes, written for a pair of partials. Fourteen times five thirds is twenty-three.

So the set a template is asked to cover spans thirty-seven cents of displacement, and a template is allowed thirty. There is no fundamental it can be fitted to. Each family on its own is a perfect harmonic series — the products alone are the fit the ghost bass drops a twelfth at a forte made, and it survives temperament exactly as just, the crowd is one series would predict — and the three of them together are three series a few cents apart, which is the one thing a harmonic template cannot represent.

The tempered crowd is not a ruin. It is a perfectly good harmonic series on the wrong note, with a scatter around it. A listener attending to the products alone is offered a bass seventy cents flat of the one the notes imply; a listener taking everything as one sound is offered no bass at all. That is a sharper statement of the choice the played notes already name the ghost bass named than that essay could make, because for a just interval the two readings agreed about the note and differed only about how complete its series was.

The window, and which intervals fall inside it #

The width of the window is the question, and it is the same computation run at every mistuning until the fit fails.

The windows are all of the same order and all narrow. Softly they run from about four and a half cents to nine on either side of just; loudly from one to three and a half. And equal temperament’s errors are not of the same order at all: two cents on the fifth and the fourth, and thirteen and a half to fifteen and a half on the thirds and the sixths.

So the four intervals equal temperament tempers heavily fall outside their windows and the two it barely touches fall inside. That is not a coincidence in need of explaining; it is what a temperament is. Twelve fifths very nearly close the circle and twelve thirds do not come near it, so a system of twelve equal steps can afford to leave the fifth alone and has to spend the whole discrepancy on the thirds. What is new is that the discrepancy has a threshold attached, and that the threshold falls between the two.

The consequence is a statement about what equal temperament costs that is not the usual one. The usual complaint about a tempered third is that it beats, which is true and is what the roughness model prices. This is a different cost and a categorical one: a just third is one sound, with a bass implied by everything in it at once, and an equal-tempered third is a chord of two notes with a foreign series underneath.

The sixths, which are the worst of it #

The thirds are where the historical argument about temperament happens, and they are not the intervals with the narrowest windows.

Softly, the widest window in the set belongs to the major third — eight and a half cents sharp and nine flat, against the fifth’s five and a half either side. That is the reverse of what the beat-rate argument would suggest, and it is also the reverse of what the spread of magnifications suggests: the third’s spread at that dynamic is 6.7 and the fifth’s 3.9, so the rule of the last section gets the ordering of these two exactly wrong. They are the two points furthest from the line in the figure above, at nearly twice the predicted window and at three-quarters of it.

The likeliest reason is the one the rule leaves out, and it is worth naming rather than claiming. A template is not obliged to sit on the just fundamental; it fits the best one it can find, so a set whose families are displaced by nought, plus one and minus five-thirds of the mistuning can be met halfway by a fundamental placed between them, while a set displaced by nought, plus one and minus two and a third cannot be met as efficiently. How much of the spread a shift of the fundamental can absorb depends on where in the series the displaced components sit, which is exactly the thing the spread throws away. Checking it would mean reading the fitted fundamental’s own displacement out of each fit, which is one number the search already computes and does not report.

Loudly the order changes again. The major third’s window closes to four cents, the minor third’s to four and a half, and the fifth’s to four. The interval with the least room at a forte is the minor sixth at two and a half, whose worst component sits at the fourth harmonic of the implied fundamental and moves six cents for every cent of mistuning.

None of that changes the verdict on equal temperament, because equal temperament’s error on a third or a sixth is three to six times any of these windows at any dynamic. What it changes is the advice a player could take from it. An ensemble that has the thirds and sixths under its own control, as a string quartet or a consort of viols does, is not choosing between in tune and out of tune on a continuum: it is choosing whether the interval is a single sound, and the boundary is five to nine cents wide at a piano and one to three at a forte. That is narrower than the tuning scatter a quartet’s own open strings were measured to carry, which is a real claim about what an ensemble can achieve and a reason to expect the effect to be something players hear on their best nights rather than on all of them.

Louder is stricter #

The two dynamics in the window figure are far apart, and the fall between them is not gradual.

Below about sixty-five decibels the only products above threshold are the cubic ones, and every cubic product of a pair of partials is displaced in the same direction by a factor of the same sign. The set is then two families — the played partials and one crowd — and two families are easier to cover than three.

Above about seventy the difference tones arrive, and a difference tone’s magnification has the opposite sign and a much larger magnitude. The component at the implied fundamental itself, made from the sixth partial of the lower note and the fifth of the upper, moves twenty-five cents for every cent the upper note moves. One component like that is enough to put the set out of reach on its own.

So a loud third has to be tuned better than a soft one to stay a single sound, by a factor of four. That is a claim with an odd shape for this collection, because every instrument that plays thirds is tuned once and played at many dynamics. A keyboard tuner sets an interval that will be right at one dynamic and wrong at another, and no tuning can be right at both unless it is right at the strictest — which is to say, unless it is just.

Why the window is the width it is #

The magnification is not only the explanation of the tempered third’s failure; it predicts the width of every window drawn above, without searching for it.

For each interval and each dynamic, take every component the ear receives, work out how many cents it moves for one cent of mistuning of the upper note, and take the difference between the largest and the smallest. That number — the spread — is how fast the set pulls apart, and the window ought to be the tolerance divided by it.

It is, to within a factor of three at every one of the twelve points, and the logarithms correlate at −0.60. A major third at eighty decibels has a spread of forty and a window of four cents; a minor third at fifty-five has a spread of 3.8 and a window of eleven and a half. The factor of three of slop is not noise: a template can shift its own fundamental to split the difference between families, which buys some of it, and it has to assign whole harmonic numbers, which costs some.

The useful form of the rule is that the window belongs to the interval’s worst-magnified audible component and to nothing else. A fifth’s products are mildly magnified because its ratio has small numbers close together; a third’s difference tone at the fundamental is magnified twenty-five-fold because 5 and 4 differ by one. The same arithmetic that made the third sound a fine measuring instrument makes it a fragile one to build a single percept out of.

Six systems against the window #

The window is a tuning tolerance, and tuning systems are things that spend tolerances.

Read down the third’s column and the systems sort themselves into the order the historical argument put them in, for a reason nobody in that argument could have stated. Just intonation and quarter-comma meantone put the third exactly where it is, and both fuse. A well temperament’s best key puts it six cents sharp, which is inside the nine-cent window, and its worst key puts it twenty-one and a half sharp, which is not. Equal temperament puts every third fourteen sharp and Pythagorean puts four of them twenty-one and a half sharp.

The fifth’s column is the other half of the trade and it barely moves. Quarter-comma meantone’s fifth is five and a half cents narrow, which sits a tenth of a cent inside the window’s edge; Vallotti’s best-key fifth is five and nine tenths narrow, half a cent outside it. A temperament that buys pure thirds by narrowing its fifths is spending the fifth’s fusion down to the last tenth of a cent, and did not know it. That is a coincidence rather than a design, but it is the kind that is worth recording: the amount a fifth can be narrowed before it stops being one sound and the amount it has to be narrowed to make a third pure are the same number to within the width of this measurement.

Which computation produced the numbers #

The interval is two complex tones of six partials each falling as one over their number, the lower one fixed and the upper one mistuned by the stated number of cents. The implied fundamental is 65.4 hertz and the played notes are C4 and the interval above it, which is the object fit’s own arrangement. The products are that essay’s: every difference tone and every cubic product of every pair of partials, kept when it clears its own threshold under the masking the primaries cast, with the loudest kept when several pairs land together.

The object fit takes the lowest eight members of the union of the played partials and the audible products, assigns rising harmonic numbers, fits a fundamental by least squares and requires every component to sit within thirty cents of its assigned harmonic. Components within fifteen cents of each other are treated as one, which matters only when the interval is tempered. The window is found by doubling outward from just and then bisecting, to half a cent.

The magnification of a component is worked out from the pair that makes it: a difference tone of partial j of the lower note and k of the upper sits at kq − jp times the fundamental and moves by kq/(kq − jp) times the mistuning; a cubic product at 2jp − kq moves by −kq/(2jp − kq) times it.

Where the model stops #

One template, one tolerance. Thirty cents is the figure the object fit was built with and it decides the answer. A mechanism that allowed sixty would admit a template for a tempered third, but what it would admit is not the fundamental — it is a note a third of an octave below, supported by two components and contradicted by nine.

Only the upper note is moved. A real temperament moves both notes of an interval relative to some reference, and the arithmetic here is about their ratio, so that is equivalent for the ratio’s sake. What it is not equivalent to is a real ensemble, where both notes drift.

A six-partial spectrum falling as one over n. A bowed string has more partials and a clarinet has a quite different set; which products are audible, and so which magnifications are in the set, depends on both.

Nothing here is about beats. The beat rate of a tempered third is a separate quantity with a separate history — beats are arithmetic is where it is priced — and a listener hearing a tempered third hears both things at once. The two even point the same way, which makes them hard to separate by ear and is a reason to be careful about attributing anything a listener reports to one of them.

The mistuning is fixed for the length of the note. A real performance’s intonation moves inside a note, by vibrato most obviously. A vibrato of the ordinary depth sweeps an interval through several times the width of the windows measured here, and what that does to a template fitted to a steady set is a question this arithmetic cannot ask.

What the numbers cannot say about a listener #

Whether anything fuses. The computation says that a single harmonic template covers a just third’s components and not a tempered third’s. That a listener runs such a template, and that running it is what hearing one sound consists in, is the assumption all of these essays have been built on and is not tested by anything here.

What a listener hears instead. Two notes and a foreign bass is one possibility; two notes and nothing is another; and the third possibility — that the failure is heard as roughness rather than as a division into objects — is the one the essays here have most evidence for and least ability to separate from the others.

Whether the loud-and-strict result is why intonation is argued about at a forte. It predicts that it should be. Predicting a practice is not measuring one.

Still open: whether the line survives the tempo #

Everything here and in the four essays before it is a steady state — two notes held, the products in equilibrium, a template fitted to what is audible. The bass line under a passage in thirds drew a scale harmonised in thirds whose difference tones spell a diatonic bass, and a scale at a walking tempo gives each of its intervals two or three tenths of a second.

That is the same order as the time a residue pitch takes to establish, and the crowd changes wholesale at every step, because the implied fundamental moves by a whole tone or a fourth. Running the fit over a window that slides across the passage — with the window’s length as the free parameter and the tempo as the dial — would say what fraction of each note a listener has a bass under, and at what tempo the answer reaches zero. It would decide whether the Tartini bass is a laboratory object that exists only under held intervals, or the thing eighteenth-century players said they listened for when they tuned by it.