The octave that is not two to one

Two to one is the least controversial number in music. The octave is the interval every scale system in the world closes at, it is the one interval even the systems that reject the rest of the apparatus keep, and its ratio is exact by construction.

Ask a listener to set one by ear and they will not set it at two to one. They will set it wide, consistently, by an amount that grows with frequency, and they will do it with pure tones.

Why the pure-tone part is decisive #

The piano’s octave is stretched and the reason is thoroughly understood. A real string is stiff, so its partials run progressively sharp of whole-number multiples, and a tuner who eliminates the beating between the lower note’s second partial and the upper note’s first has necessarily set the fundamentals wider than 2:1. It is beat-nulling on inharmonic partials and it is computable from the wire’s diameter, length and tension.

That account is complete and it explains nothing in this essay, because a pure tone has exactly one partial. There is no second partial to be sharp, no coincidence to null, and no beating of any kind. The stretch survives anyway.

It also survives when the two tones are presented one after the other rather than together, which removes any possibility that beating between them is being used. And it survives dichotically — one tone to each ear, so the two never meet in the air at all, and any peripheral interaction is impossible.

Three independent ways of removing the inharmonicity explanation, and the effect is still there. Whatever is happening is happening to the pitch scale itself.

How big, and how reliably #

This is the point at which the essay has to be careful, because the published spread is wide.

The direction is not in doubt: every study since Ward’s in 1954 has found stretch rather than compression. The size is another matter. Reported values for the middle of the range run from about ten to about twenty-five cents, and the between-listener spread within a single study is comparable to the difference between studies. The figure’s curve is the middle of the published range, drawn as a line because a line is what a figure can draw, and it should be read as the centre of a broad band rather than as a measurement.

Two things are more robust than the absolute size.

It grows with frequency. The stretch is small in the bass and larger at the top, in every study.

It is well above the discrimination limen. Five cents is what a listener can detect, and the stretch is two to five times that. So this is not a measurement error and it is not a subtlety — it is a systematic departure several times larger than the resolution of the instrument measuring it.

Drawn against the discrimination limen, the stretch is comfortably above it across the whole range — 8 cents at 125 hertz against a limen near 5, and 30 cents at 4,000 against a limen near 5.5 — so it is not a measurement artefact and every listener has it.

The other stretched intervals #

The octave is not alone, and the pattern across intervals is what turns a curiosity into a claim about a scale.

Asked to set a fifth by ear, listeners set it slightly wide too, by a few cents — less than the octave’s stretch. Asked to set a major third, they set it wide as well, and by less again. The general finding is that melodic intervals set by ear come out larger than their just ratios, and the excess grows with the size of the interval.

That is precisely what would happen if the perceptual pitch scale were slightly expanded relative to the logarithm of frequency: every interval measured on it would be reported as smaller than it is, so a listener aiming for a target size would overshoot in frequency, and the overshoot would scale with the interval. One distortion of one scale accounts for the whole family.

The model has one parameter and it can be fitted from the octave alone, which makes the rest of the family a prediction rather than a description. Taking the octave’s stretch at A440 from the curve above — 13.4 cents on 1,200 — the expansion is 1.12 per cent, and every other interval follows:

A previous version of the sentence above had the major third stretched by more than the fifth, which cannot be right on this account and cannot be right beside the sentence that follows it: a third is smaller than a fifth, so an excess that grows with interval size has to make it smaller. The two statements were inconsistent and the model settles which one to keep.

One distortion is not enough for both jobs #

“One distortion of one scale accounts for the whole family” is the right shape of claim and it is one parameter short.

A constant expansion of the pitch scale predicts a stretch proportional to interval size, which is the family above. It also predicts a stretch that is the same number of cents at every frequency, because a constant fractional expansion applied to a constant interval gives a constant excess. The measured octave stretch is not constant: it is 8 cents at 125 hertz and 30 at 4,000, a factor of nearly four across the range, and every study agrees about that shape.

So the expansion factor must itself vary with frequency — 0.67 per cent in the bass and 2.5 per cent at the top. That is still one distortion of one scale, but it is a function rather than a number, and the difference matters for what the two physiological candidates have to explain. A place map that is slightly non-logarithmic would produce exactly a frequency-dependent expansion, because the departure from logarithmic is itself frequency-dependent. A fixed combination of two disagreeing estimates would not, unless the weighting also moves with frequency.

The frequency dependence is therefore the more diagnostic half of the finding, and it is the half the interval family cannot speak to at all. A study measuring fifths and thirds at one frequency tests the proportionality and says nothing about the shape; a study measuring octaves across the range tests the shape and says nothing about proportionality. Both have been done and this essay draws only the second.

There is also a sharp asymmetry between melodic and harmonic intervals which the essay should not gloss over. Everything above concerns intervals set melodically, one tone after another. Harmonic intervals, sounded together with complex tones, are set close to the just ratios, because the listener has beats to work with and beat-nulling is an entirely different and far more precise task. A performer with two references — the melodic sense of interval size and the harmonic evidence of beating — has two instruments that disagree by a few cents, and the tension between them is a real and recurring feature of ensemble intonation.

What might be doing it #

There is no settled explanation, and the honest state of the subject is that there are three candidates and no decisive experiment between them.

The neural place map is not logarithmic. The cochlea maps frequency to position, and the map is approximately but not exactly logarithmic. If pitch is read from position, a slight departure from logarithmic mapping produces exactly this kind of systematic stretch, and the sign works out. Against it: the effect appears in listeners across a range of conditions where the place code should be doing different things.

Two mechanisms are being combined and they disagree slightly. Pitch is available both from place — which region is excited — and from timing — the periodicity of the firing pattern. The two do not give identical answers, and a weighted combination of two slightly disagreeing estimates could easily land off the exact ratio. Against it: the effect persists above about 5 kHz, where the timing mechanism is thought to fail entirely.

It is learned from stretched instruments. Listeners in Western musical cultures have spent their lives listening to pianos, which are stretched. Perhaps the internal octave has been calibrated on them. Against it, decisively: the effect is found in listeners with little exposure to keyboard instruments, and it is found for tones far outside any piano’s range.

The third is the most testable and it fails, which is the most useful thing that can be said about the three. That leaves two physiological candidates and a preference for neither.

What it does to octave equivalence #

The result has a consequence for something this site has been treating as an axiom.

Octave equivalence — the claim that two notes an octave apart are in some sense the same note — underlies pitch-class notation, the circle of fifths, the whole apparatus of naming seven of the twelve, and every argument on this site that reduces a chord to a set of integers modulo twelve.

If the perceptual octave is not 2:1, then equivalence is not a relation between frequencies at all. It is a relation between categories, and the category has a preferred size that is a little wider than the ratio and a tolerance that is a good deal wider still. Two notes at exactly 2:1 are in the equivalence relation; so are two notes fifteen cents wider; so, probably, are two notes fifteen cents narrower. What is being asserted is membership, not identity.

That reframing is not a demolition of anything. Every argument this site makes about pitch classes survives, because they are arguments about which category a note is in and the categories are robust. But it changes what the octave is from a fact about frequency into a fact about a listener with a very good but not exact sense of one particular relation — which is the same move the categorical-hearing essays make about every other interval, arriving at the one interval that was supposed to be exempt.

That distinction is worth holding onto, because the two effects are constantly conflated. A piano’s deviation from equal temperament at its top note is large — thirty cents or more — and almost all of it is accumulation.

The listener’s stretch was previously described here as not compounding, on the reasoning that what is distorted is a relation rather than a coordinate. Nothing on this site supports that. The only quantity here is a per-octave stretch that rises with frequency, so two octaves taken as two steps must compound and must compound by more than double: at A440 the first octave is 13.4 cents and the second, starting higher, is 16.5, for a stacked total of 29.9 — 2.2 times the single-octave figure, not one.

Whether a listener asked to set a direct two-octave interval produces less than that is a real and separate question, it is the question the “does not compound” claim was reaching for, and this essay has no data that speaks to it. The curve above is a table of one-octave judgements. It can say what stacking predicts and it cannot say what a two-octave judgement gives, and the difference between those two would be the sharpest available test of whether the distortion is in the scale or in the relation.

Which computation produced the numbers #

Two lines in the hero figure, arrived at two different ways, and the difference is the essay.

The piano’s octave is computed. The inharmonicity coefficient of a wire follows from its diameter, length, tension and Young’s modulus; the stretch a beat-nulling tuner produces follows from the coefficient. At A440 the site’s own wire model gives about three cents; near the top of the keyboard it gives twenty-four. Nothing is fitted, and the same computation produces the Railsback curve as an accumulation of these octaves outward from the middle.

The listener’s octave is quoted. It is a small table of published values — 8 cents at 125 hertz rising to 30 at 4,000 — interpolated in log frequency, and it could not be computed here because nothing derives it. That asymmetry is the honest state of the subject.

Both of the derived quantities in this essay come out of that same table and nothing else. The expansion factor is the stretch at a stated frequency over 1,200; the stacked two-octave figure is the table read twice, at the base frequency and at the stretched octave above it. Neither is a new measurement and both are the published curve rearranged, which is worth saying because a rearrangement can look like corroboration. What they establish is what the table does and does not commit its user to, not whether the table is right.

The comparison the figure exists to make is therefore between a derivation and a measurement, and the finding is that the measurement is several times the derivation in the middle of the keyboard. A piano tuned purely for its own inharmonicity would still have octaves narrower than a listener’s preferred one, and in practice tuners stretch more than the wire requires — which is a small, well-known piece of craft knowledge that this comparison explains.

Whose octave, and what performers do with it #

The stretch shows up in practice, and where it shows up is informative.

Piano tuners stretch more than inharmonicity requires, as noted above, and they say so — the practice is described in the trade as “stretching to taste” and the amount is a matter of style, larger for solo repertoire and smaller for accompaniment. That variability is exactly what would be expected of a preference rather than a physical requirement.

Orchestral players tune their instruments’ octaves wide. String players setting the octaves between adjacent strings, and wind players adjusting their registers, both tend towards slightly wide octaves. This is documented in intonation studies and has no acoustic justification for instruments whose partials are effectively harmonic.

Singers do not, reliably. The measurement is much harder and the data much noisier, and it would be a mistake to claim more than that the effect has not been clearly established for unaccompanied singing.

Whose practice. All three observations come from Western art-music performance in the twentieth century, which is where the intonation measurement literature lives. Whether the same stretch appears in traditions with different instruments and different training is, as far as this essay’s sources go, unstudied — and the perceptual effect being found in listeners with little keyboard exposure is a reason to expect it would, not evidence that it does.

What the picture cannot show #

The spread, which is the most important missing thing. Drawing a single curve for a quantity whose between-listener range is a factor of two is the least defensible thing in this essay, and the figure would be better as a band. It is drawn as a line because the published studies do not report compatible enough error terms to draw a band honestly, and a fabricated band would be worse than a line with a caveat.

Which task was used. “Set an octave” is at least three experiments: adjust the upper tone until it sounds an octave above; adjust until it sounds best; and judge whether a given pair is an octave. They give different answers, all stretched, and the figure collapses them.

And nothing about melodic context. All of these are isolated pairs. What a listener does with an octave inside a phrase, where a key and a scale are supplying expectations, has been studied much less and there is no reason to assume the answer is the same.

And the octave’s own arithmetic is the cleanest ratio there is: one wave fits exactly twice into the other, every partial of the upper note lands on a partial of the lower, and the sum repeats at the lower note’s own period. That is why the octave is the strongest consonance and why a harmonic octave is set to 2:1 by anybody with beats to listen to. The stretch is a melodic phenomenon, and the two tasks are measuring different things with the same name.

Where the ladder goes next #

This ladder — how finely pitch is resolved, and whether the scale it is resolved on is straight — is complete in the sense that both questions now have numbers. What it opens is a question about what a listener remembers rather than what they can discriminate: a listener with absolute pitch is not detecting a difference, they are comparing an incoming tone against a stored one, and that is a different faculty with different limits and a different failure mode. It is also, unlike anything in this essay, dated — a memory for a convention drifts when the convention does.

Sideways, the octave’s non-exactness meets the one scale system built without an octave at all. The Bohlen–Pierce scale divides a 3:1 on the grounds that octave equivalence is a consequence of spectra rather than an axiom. This essay adds a second reason to be suspicious of the axiom, arriving from the listener rather than from the spectrum, and the two arguments are independent.