Consonance is half learned, and this is the half

The oldest claim in this subject is that consonance is small whole numbers, and this site has spent four phases computing consequences of it. Roughness can be computed from a spectrum and a model of the critical band; the triad falls out of a double criterion; the dissonance curve wells on the fourth, the fifth and the octave for any harmonic spectrum.

All of that is computation and none of it is in question. What is in question is what it is a computation of.

Two models, one word #

That two models exist is not a scandal. That they are routinely treated as the same model is.

Roughness is a peripheral, mechanical account. Two partials within a critical band beat; beating at a few tens of hertz is heard as roughness; sum the roughness over every pair of partials in a chord and the total is a prediction. It is entirely a fact about the cochlea and it would be true of any listener with a cochlea.

Harmonicity is a central, pattern-matching account. The ear is looking for a harmonic series, because harmonic series are what physical objects produce; a chord that fits one fundamental well is easy to interpret, and one that fits nothing is not. It is a fact about a template rather than about a membrane, and it is the same machinery that produces the missing fundamental.

They make different predictions, and this site has already found one place where they part company sharply. Scored on roughness alone, the smoothest three-note chord available is not the triad — it is stacked fourths, 0–5–10, because roughness rewards wide spacing and every pair in that chord is wide. Scored on integer simplicity alone, 8:9:10 does well and is one of the roughest chords there is. It took both criteria together to select the triad, and the fact that two are needed is the fact this essay is about.

The evidence from a long way away #

If consonance were purely the roughness model, it would be the same everywhere, because cochleas are the same everywhere. Testing that requires listeners who have not spent their lives inside the Western repertoire, and such listeners are hard to reach — which is why the decisive work is recent.

The clearest result comes from studies with the Tsimane’, a society in the Bolivian Amazon with very limited exposure to Western music. What was found, in outline:

They discriminate roughness normally. Asked which of two sounds is rougher, they answer as Western listeners do. The peripheral machinery is intact and identical.

They prefer harmonic sounds to inharmonic ones, and they match octaves in the standard way. So it is not that no acoustic property is available to them.

And they rate consonant and dissonant chords as equally pleasant. The preference — the thing the word consonance is usually taken to name — is absent, while every perceptual capacity it is supposed to rest on is present.

The same studies find the preference increasing with exposure to Western music across populations, from rural to town-dwelling to city-dwelling to Western listeners. It is graded, and it tracks exposure.

That combination is what makes the result decisive rather than merely interesting. If the Tsimane’ had failed to discriminate roughness, the finding would be about hearing. They discriminate it perfectly and do not mind it, which makes the finding about preference, and preference is the part that is learned.

The objection, and it is a good one #

There is a serious objection to reading that evidence as decisively as the section above does, and an essay that skipped it would be advocacy.

The objection is about what the question means across a language and a culture. Asking a listener to rate how pleasant a chord is presupposes that rating isolated chords for pleasantness is a coherent activity, and it is a very unusual activity even for Western listeners. A null result on a task that may not translate is weaker evidence than a null result on a task that does.

Three things make the objection less damaging than it first looks.

The same listeners give non-null answers to other questions in the same format. They rate rough sounds as rougher and inharmonic sounds as less pleasant than harmonic ones. So the task was intelligible and the scale was being used; what was flat was one specific dimension.

The graded result across populations is hard to explain away. If the task simply failed to translate, the answers should be noise. Instead they move monotonically with exposure to Western music, which is what a real learned preference looks like and not what a broken instrument looks like.

And the direction was predicted in advance. The hypothesis was stated before the data were collected, which is worth more than a post-hoc account of a surprise.

It remains true that the strongest form of the claim — that there is no innate component to consonance preference — is not established, and other work reports small preferences for harmonic intervals in infants and in non-human animals. The defensible statement is narrower and is the one this essay makes: whatever innate component exists is far too small to explain the preference Western listeners have, and the remainder is learned.

What survives, and it is most of the site #

It would be easy to over-read this, and the correction cuts in a specific place rather than everywhere.

The computations survive completely. The dissonance curve is a computation over a spectrum and a critical-band model, and its wells are where they are regardless of anybody’s culture. Nothing about the arithmetic of the comma, beating, or the critical band is touched.

The discriminations survive. A listener anywhere can hear that a tempered third beats and a just third does not. That is a perceptual fact and it is not in question.

What does not survive is the inference from smooth to preferred. The claim “the fifth is consonant because its partials coincide” contains a hidden step: from it does not beat to it is liked. The first half is acoustics and universal. The second half is a preference, and it is trained.

That is a real limit and it is worth stating in the site’s own terms. The dissonance curve predicts which intervals are smooth. It does not predict which intervals a tradition will build a scale on, and the several traditions that have built scales on intervals the curve rates as rough are not making an error.

The traditions that chose otherwise #

The essay’s claim is easier to accept with cases in front of it, and there are several. Each is a tradition whose central intervals are ones the roughness model rates badly.

Gamelan. Javanese and Balinese ensembles are built on metallophones and gongs whose partials are strongly inharmonic, and the tuning systems — sléndro and pélog — do not correspond to any small-integer ratios. Paired instruments are deliberately detuned by a few hertz so that the ensemble shimmers. On the Western curve that is a design brief for maximum roughness, and it is the desired sound.

Bulgarian and other close-harmony singing traditions. Two voices a major second apart, held, is the characteristic sound of several Balkan vocal traditions. A major second at those frequencies is deep inside one critical band and beats hard. It is not tolerated; it is the effect.

And the barbershop tradition, which went the other way and is the useful control. Barbershop singing pursues just intonation with unusual explicitness, tuning chords until the beating stops, and it is unmistakably a Western practice pursuing the roughness model to its conclusion. Two traditions with the same cochleas taking opposite decisions is the shape of an aesthetic choice, not of a perceptual difference.

That last observation is the most useful thing in this section, because it shows the model earning its keep on exactly the material that is supposed to embarrass it. Roughness is real and it is universal. What is not universal is whether roughness is a defect.

The reconciliation that does not work #

There is an attractive way to make the cross-cultural finding go away without giving anything up, and this essay raises it below as a limitation: a gamelan’s instruments are inharmonic, so scoring its music on a harmonic spectrum’s dissonance curve is scoring it against the wrong curve, and a large part of the apparent disagreement may be agreement about different instruments. That is the argument the Bohlen–Pierce scale applies deliberately, and the site has the machinery to test it rather than propose it.

It does not survive the test. Computing the dissonance curve for a bar spectrum — the ideal free–free bar, and filed bars with the second partial swept from 2.2 to 3.0, which is what filing one does — and asking how near slendro’s steps fall to its minima:

Slendro fits a string’s dissonance curve better than any bar’s, which is the opposite of what the reconciliation requires. Changing the spectrum to the one the instruments actually have does not move the tuning into the wells; it moves it further out.

And then the control that decides what any of those numbers mean. A curve with many minima is near any set of degrees by accident, so the 42 cents has to be scored against what a random five-note scale would get:

The Western scale does not beat chance either. A dissonance curve with eight minima across an octave is within about forty cents of almost any scale anyone writes down, and the aggregate statistic that seemed to be measuring a fit has no power to distinguish the just diatonic from a random draw. The chromatic scale scores worse than a random draw.

That is not a result against the curve, and reading it as one would be a second version of the same mistake. The individual identifications are exact and remain so: the just fourth at 498 cents sits on a well at 498, the fifth at 702 on a well at 702, the major sixth at 884 on a well at 884, and the just major third at 386 one cent from a well at 387. Four of the seven diatonic degrees are dead on a minimum.

What the baseline exposes is the other three, and the wells nobody quotes. The major second at 204 cents is 183 cents from the nearest well. The leading note at 1088 is 112 cents from one. And the curve has minima at 583, 814 and 969 cents — a tritone, a minor sixth and a septimal seventh — which no diatonic scale places a degree on. The curve explains four intervals precisely and is silent or contradicted on the rest, in the Western scale exactly as in slendro, and a summary that averages over both halves hides which is which.

So the essay’s own escape route is closed, and closed from an unexpected direction. Gamelan is not the wrong-spectrum case; it is the case where an argument that was never doing as much work as it looked like is asked to do the work alone.

Which computation produced the numbers #

Two scores, both computed here, and one large quotation.

Roughness is summed Plomp–Levelt over every pair of partials of every pair of notes, using the site’s own critical-band model — the same function every other consonance figure on the site uses, run on chords rather than on intervals.

Harmonicity is computed as the smallest integer set that expresses the chord’s frequency ratios within a stated tolerance: the largest of those integers is the score, so a small number means the chord sits low in one harmonic series. The tolerance is a parameter and the result is sensitive to it, which is a known weakness of every integer-fitting measure and one this site has recorded before — at sixteen cents of tolerance the tempered triad scores second of twelve and at twelve cents it scores thirtieth, a four-cent change nobody could hear producing a rank change of twenty-eight.

The cross-cultural findings are quoted, and this essay is the most citation-dependent on the site. It cannot compute a Tsimane’ listener’s ratings and it does not have the data. What it can do is state precisely which of its own claims the finding bounds, and that is what the section above does.

The site’s habit is that every claim gets a test it could fail, and the test here is the disagreement itself: the figure computes both scores independently and reports their correlation. If the two models agreed closely, the essay’s argument would collapse — there would be one model with two descriptions, and no room for a learned component to sit in. The correlation is moderate rather than high, which is what the argument needs and is a result the figure could have refused to produce.

What the picture cannot show #

The listeners. Every figure here is a computation over chords. The whole force of the essay comes from an experiment on people, and none of it is drawable from a spectrum.

Timbre, which changes the answer. Both models are evaluated on a harmonic spectrum. On a spectrum with no even partials the consonances move, and any tradition using inharmonic instruments is being scored against a spectrum its instruments do not have. That is a real limitation on the models; what the section above establishes is that it is not an explanation of the cross-cultural finding, because putting the right spectrum under slendro makes the fit worse rather than better.

Context. A chord’s effect depends on what precedes it. A dissonance that resolves is a different object from the same dissonance in isolation, and everything in this essay is isolated chords.

And the word itself. “Consonance” has been used for at least three distinct things — smoothness, fusion, and stability in a key — and the three come apart. The third is a matter of tonal expectation and is unambiguously learned; the first is unambiguously peripheral; and the second sits between them.

The critical-band picture is where the two accounts come apart — a band is a fixed fraction of a frequency, so it is more than an octave wide at the bottom of the bass and under three semitones at the top, and the roughness account is therefore register-dependent in a way the integer account is not.

Whose music, and what it does to the founding claim #

The claim under this site’s name is that consonance is frequency ratios of small integers, and that almost everything interesting follows from those integers refusing to line up. The second half is untouched — the comma is arithmetic and it does not care who is listening.

The first half now has a boundary drawn round it, and the boundary is worth writing out. Small whole numbers predict which intervals are smooth and which fuse, for harmonic spectra, for any listener with a human cochlea. That is a strong claim and it is supported. Small whole numbers predict which intervals a tradition will prefer is a weaker claim, it is culturally specific, and the evidence is now against it as a universal.

A large part of the Western tradition’s distinctiveness is that it built a harmonic practice on the first claim as though it implied the second. That was an enormously productive thing to do — most of what this site is about is the consequences — and it was a choice, not a discovery.

Where the ladder goes next #

This anchor has four rungs now and each one has narrowed the claim: the ratio, the roughness computation, the double criterion that selects the triad, and this one, which says what the whole apparatus is a theory of. What it opens is the question the boundary leaves standing — if preference is learned, what teaches it, and in what order? Statistical learning has an answer for the tonal hierarchy and no clear one for interval preference, which is the next rung and is not yet writable.

Sideways, the timbre caveat above is a whole argument on its own. A tradition whose instruments are inharmonic has a different consonance curve, and its scales sit on that curve’s wells rather than on this one’s — which is exactly the reasoning the Bohlen–Pierce scale applies deliberately, and which the gamelan arrived at without any theory at all.