An interval is two errors

The previous rung found that every difference limen on this site is quoted for a tone that lasts as long as the listener needs, that no note in music does, and that a tone of duration T cannot have its frequency specified more finely than about 1/2T however good the ear is. It ended by naming the obvious next question: the same bound applied to an interval rather than to a note.

The step is one square root and the consequence is larger than it looks.

Two estimates, and the thing judged is their difference #

A melodic interval is not heard. What is heard is two notes, one after the other, and the interval is a quantity the listener computes from them.

Each of those notes arrives with an error. The previous rung gave that error its two bounds: the steady-tone limen of the listener, about 4 cents at A440, and the Fourier bound imposed by the note’s own length, whichever is larger. The interval is the difference between two such estimates, and independent errors add in quadrature — the variance of a difference is the sum of the variances.

For two notes of the same length at nearby pitches that is a factor of √2. It is not a large factor and it is not the interesting part of the result. The interesting part is what it does to a number that was already, at speed, very large.

At a crotchet at 120 beats a minute the interval limen is 5.4 cents. At a quaver, 9.4. At a semiquaver, 18.8. At a semiquaver at 200 beats a minute, 31.3.

A semiquaver passage is intonated to about a fifth of a semitone, and that is a bound on the listener rather than a claim about the player.

The number that was already being quoted #

Here is the part that makes this a rung rather than an arithmetic exercise.

This site has carried, since its first essays about the listener, a figure with three resolutions on it: pitch discrimination at about 4 cents, the beat-counting resolution of a held fifth at 1.3, and — between them and an order of magnitude coarser than discrimination — melodic interval judgement at 25 to 50 cents. That last range is a published measurement, quoted from the literature, and nothing in this collection had ever produced it from anything.

The quadrature sum lands in that band, and it does so without having been aimed at it. The two ingredients — a steady-tone limen fitted to discrimination data, and a Fourier bound that is arithmetic — were both on this site for other reasons. Put a note length of 75 to 150 milliseconds into them, which is the length of the notes an intonation judgement is usually made on, and out comes 15.7 to 31.3 cents.

That is not a derivation of the published range and should not be read as one: the published figure is for a specific task with specific stimuli, and the agreement is to within a factor of about two at the edges. What it is, is a mechanism.

The honest way to state the agreement is to run it backwards, since the note length was chosen and the band was not. Inverting the curve asks what duration the published band implies rather than what band an assumed duration produces, and the answer is narrower and more useful:

At A440 the band implies notes of 47 to 94 milliseconds, which is about sixty per cent of the 75 to 150 assumed above — so the computed range and the published one overlap over their upper half rather than coinciding, and the arithmetic in the paragraph before this one lands at 15.7 to 31.3 against a published 25 to 50. The overlap is real and it is a third of each range, not a match.

The register column is the part that matters more. The implied duration spans a factor of sixteen across three octaves, because both terms of the limen worsen toward the bottom of the range. So the mechanism does not predict a duration; it predicts a relation between duration and register, and a published figure quoted as “25 to 50 cents” with no frequency attached cannot be compared with it except at whatever register the experiment used. If those experiments were run around A440 the agreement stands as stated; if they were run in the bass it is out by a factor of four in the other direction.

That is the test this rung actually offers, and it is sharper than the coincidence it was written around. The prediction is that melodic interval judgement should get worse toward the bass at a rate the arithmetic fixes, roughly quadrupling between A440 and A110 at a fixed note length, and that is a measurement somebody could make with the stimuli the published figure already used.

One assumption is worth pricing beside it. Everything above adds the two errors as independent, and the model’s own alternative — a listener holding a reference and judging both notes against it — shares part of the error and shrinks the sum. At a hundred-millisecond note at A440 the limen falls from 23.5 cents at no correlation to 17.3 at a half and 9.7 at nine tenths. So the choice between the two models is worth a factor of more than two, which is the same size as the agreement being claimed. The reason a melodic interval is judged an order of magnitude more coarsely than two pitches are compared is not that interval perception is a separate and worse faculty. It is that the notes are short and there are two of them.

The bass is worse, and by a lot #

The steady-tone limen rises toward the bottom of the range and the Fourier bound rises much faster, because 1/2T in cents is larger when T is fixed and the frequency is low. Both effects push the same way.

Ninety-three cents at a tenth of a second at A110. A semitone is a hundred. A walking bass line at any ordinary tempo is played at a precision the listener has no way of assessing melodically — which does not mean the notes are not in tune, only that whatever is keeping them in tune, it is not a listener hearing the intervals between them.

Something is, of course, and this collection knows what. The bass line’s notes are simultaneous with the harmony above them, and a simultaneity is a completely different measurement.

The simultaneity, and why it is a different instrument #

Two notes sounding together do not present the ear with two pitch estimates to subtract. Their partials coincide — the third partial of the lower note of a fifth against the second of the upper — and a mistuning turns that coincidence into a beat, which is a difference frequency and is counted rather than estimated.

1.31 cents against 23.5. The same interval, the same two frequencies, the same listener, and a factor of eighteen — decided entirely by whether the notes arrive together or one after the other.

The factor is smaller for long notes, because the melodic limen has a floor: 5.4 cents against 1.31 is a factor of four. It is much larger at speed and in the bass. Across the range of ordinary musical notes it runs from four to seventy.

Which intervals are protected, and which are not #

The simultaneity limen is not the same for every interval, and the pattern in it is the one the whole subject is built on.

A mistuning has to produce a countable beat, and the beat happens at whichever pair of partials coincides. For an octave that is the lower note’s second against the upper’s first, and the coincidence is low in the spectrum where a given mistuning in cents is a small number of hertz. For a minor third it is the sixth against the fifth, far higher up, where the same mistuning in cents is a larger number of hertz and therefore a faster beat.

Run the numbers and the ordering is: octave 0.98 cents, fifth 1.31, fourth 1.47, major third 1.56, minor third 1.65. The intervals with the simplest coincidences are the ones a listener can tune most precisely, and they are the same intervals every tuning tradition treats as fixed while leaving the thirds to be argued about. That is the chain of fifths getting its authority from a limen rather than from a ratio.

Both halves of that figure move in the same direction the ordering predicts, and they move by very different amounts. The simultaneity limen spans a factor of 1.7 from the octave to the minor third; the melodic limen barely distinguishes them at all, because it is set by how long the notes are and not by which notes they are. So the whole of the ordering lives in the simultaneous case, which is another way of saying that a tradition which fixes its octaves and fifths and argues about its thirds is a tradition tuning things it can hear and leaving alone the ones it cannot.

That is a small result and it is worth stating because it runs the usual argument backwards. The simple ratios are not treated as fixed because they are consonant; they are treated as fixed because a mistuning of one of them is audible, and a mistuning of a complicated one is not.

What it bounds in this collection #

This is the second consecutive rung of this ladder whose main product is a boundary drawn across other essays, and the boundary this one draws is sharper.

Temperament arguments survive. Every one of them is about intervals sounded together — a third in a chord, a fifth in a bearing plan, a comma accumulated over a progression whose chords are held. The simultaneity limen of about 1.3 cents clears the syntonic comma by a factor of sixteen and the schisma by half. Nothing in the tuning ladder is threatened.

Melodic intonation arguments do not survive at speed. A leading note played sharp because of where it goes is sharpened by 10 to 20 cents in the measurements. That is audible as a melodic interval only if the note lasts longer than about 160 milliseconds. In a fast passage the sharpening is real, measurable in the performance, and below the threshold at which a listener could tell it from an unsharpened one — which does not make it pointless, but does mean its effect is not being heard as pitch. A melody is a walk rather than a set, and the walk’s steps are measured with a coarser ruler than the collection has been assuming.

And the vibrato result gets a second reading. An operatic vibrato is 140 cents wide and completes six excursions a second, so each excursion is about 80 milliseconds long. The essay’s finding was that every distinction the tuning ladder argues about fits inside a single sung note. The arithmetic here adds the reason the singer gets away with it: 80 milliseconds is not long enough for the interval to be judged to better than about 30 cents anyway.

And the vibrato result gets its second reading from the same curve. One second of an ordinary operatic vibrato at A440 is six excursions of about eighty milliseconds each, spanning 140 cents peak to peak — wider than the syntonic comma by a factor of six, so on the face of it a listener should hear the note wandering. Read through the arithmetic above, eighty milliseconds is a note whose interval to its neighbour cannot be judged to better than about thirty cents in any case, so the excursion is not merely tolerated but is below the resolution at which it could be assessed. The two facts are usually offered as a puzzle and its excuse; they are one measurement read at two durations.

Which computation produced the numbers #

Each note’s limen is durationLimen, unchanged from the previous rung: the larger of the steady-tone difference limen, which is a published fit to discrimination data across frequency, and the Fourier bound of 1/2T converted to cents at that frequency.

The interval’s limen is the square root of the sum of the two notes’ squared limens. That is the variance of a difference of two independent random variables and nothing more.

The independence is the assumption, and it is the one place the model can be attacked. If a listener holds a reference — a tonic, a drone, the memory of the first note as an absolute rather than as an estimate — then the two errors share a component and partly cancel. With a correlation of 0.5 the floor at A440 falls from 5.4 cents to 3.8; with 0.8 it falls to 2.5, below the single-note limen, which is the model announcing that it has been pushed past where it means anything.

So the independent model is drawn and the correlated one is not, and the reason is not that independence is known to be right. It is that the correlated model has a free parameter with no measurement behind it, and a curve with a dial on it that can be turned to reach any answer is not evidence. What can be said is the direction: any correlation between the two notes’ errors makes the interval limen smaller than drawn, so every figure here is an upper bound on how badly a melodic interval is heard.

The simultaneity limen is the site’s own beatLimitCents with no change: the mistuning that produces one beat every two seconds at the coinciding partial. Which partials coincide is searched rather than assumed, because a spectrum without the relevant partial has no coincidence and therefore no beat to count — a pair of flutes at a fifth has a much weaker coincidence than a pair of strings, and the number quoted is for a spectrum that has one.

Whose music, and when #

The melodic result is about listeners rather than repertoires, but its consequences are not evenly distributed.

A tradition whose intonation is carried by held simultaneities — four-part harmony, a barbershop chord, an organ registration — is operating in the regime where the ear is precise to about a cent, and its theory can afford to argue about commas. A tradition whose intonation is carried melodically over a drone is in an intermediate regime: the drone supplies the simultaneity, so each note is a held interval against it, and the fine judgement is available for the notes that are sustained and not for the ones that pass. A tradition of fast unaccompanied monophony is in the coarse regime throughout.

That is a prediction rather than an observation, and it is the kind that could be checked: the traditions with the most elaborate microtonal theories should be the ones whose practice sustains notes against a reference, and the ones with the least should be the fast monophonic ones. This collection has entered enough tuning systems to notice that the prediction is not obviously wrong — a maqam against a drone, a raga against a tanpura — and not enough to call it tested.

What the picture cannot show #

Quadrature assumes Gaussian, independent, unbiased errors and none of the three is established. The limen is a threshold from a discrimination task, converted to a standard deviation by an assumption that is standard and is still an assumption.

Nothing here is a judgement of an interval’s identity. The categories are wide — a major third can be seventeen cents wrong and still be a major third — so a listener asked which interval this is has a much easier task than one asked whether it is in tune. The limen computed here is for the second question.

The two notes are the same length and the same loudness. Real melodic intervals are between notes of different durations, and the shorter note dominates the sum, so an interval between a long note and a short one is judged nearly as badly as one between two short ones.

And the simultaneity number assumes somebody is counting beats. A tuner does; a listener at a concert does not, and whether an untrained ear extracts anything like that precision from a held chord is a listening question this collection cannot settle. What the number bounds is what is available in the signal, not what is taken from it.

Where this ladder goes next #

Four rungs. The first put a number on how finely two pitches can be told apart; the second found the octave is not where it should be; the third found the whole family is a function of note length. This one applies that to the thing music actually asks a listener to judge and finds that a published range the collection had been quoting comes out of it.

The rung after it is the one the correlation question makes unavoidable. Everything above treats each note as an independent estimate, and a listener in a key does not: a note is heard as a scale degree, which is a category with a centre, and a categorical anchor is exactly the correlated reference the model above refused to fit. The measurement that would settle the correlation is an interval-discrimination run inside an established key against the same run with no key at all, and the difference between them is the size of the anchor. That is an experiment. What can be computed first is the size of the effect it would have to find to matter — and from the numbers above, it is a factor of two.