Intervals and chords

The series has three tops

How far up the harmonic series can an ear go? The question has three answers and they are an order of magnitude apart. Consecutive partials stop being separately resolvable somewhere around the eighth, and where exactly depends on the fundamental. They stop being a semitone apart at the seventeenth, at every fundamental, because the ratio does not know what it is measured in. And they stop being distinguishable in pitch at all between the thirty-fourth and the hundred and fortieth. Every claim about where the series runs out is a claim about which of the three was meant.

Assumes: A string does everything at once · The series is not a chord

The harmonic series is the most-used object on this site. It is the reason a string sounds like one note, the source of every just ratio, the thing a brass player selects from, and the object three of this collection’s ladders start on.

It is also infinite, and every argument that uses it stops somewhere — the series is not a chord partly because of where each argument stops. Six, for the derivation of the triad. Eight, for the register where a natural trumpet can play a tune. Sixteen, for the tables of partials. Those numbers are quoted in different essays for different reasons and none of them says why it is that number rather than another.

There is a reason, and there are three of them.

Why there is more than one answer

The interval between consecutive partials closes as the series climbs. Between the first and second it is an octave; between the second and third, a fifth; between the fifteenth and sixteenth, 105 cents.

Where the series stops being a chord and becomes a scale. The interval between each partial and the next, in semitones, against partial number. It is an octave, then a fifth, then a fourth, a major third, a minor third — and it crosses 2 semitones between partials 9 and 10. Each shaded band is one octave of pitch and holds twice as many partials as the band below it: 2 between 1 and 2, 3 between 2 and 4, 5 between 4 and 8, 9 between 8 and 16. Nothing about the series changes with height; what changes is the density, and a set of notes becomes usable as a melody when its neighbours are a step apart. — Fig. 1 The interval between each partial and the next, against partial number. It falls as 1200 log₂((n+1)/n), which is a smooth curve with no features in it at all: no knee, no shoulder, nothing that could be pointed at as the place the series ends. Any threshold has to be brought from outside.

That is the point. The series has no internal structure that says where to stop, so a stopping point is always a fact about the listener imported into a fact about the sound — and there is more than one thing a listener can fail to do.

Three of those failures matter and they happen in a fixed order.

The order is worth stating before the numbers, because it is what makes three separate measurements comparable at all. A listener stops being able to itemise the partials first, stops being able to use the step as a scale degree second, and stops being able to tell the two apart last — each a weaker demand than the one before it, and at the fundamentals this essay draws, each threshold duly comes out higher.

It is tempting to go further and say the order is guaranteed: that a weaker demand must be satisfied later, so a different order would be evidence of an error rather than of a discovery. That argument is wrong, and a section near the end of this essay is about where it fails and why. It fails because the three demands are not measured against the same variable, and two curves that are not functions of the same thing are free to cross.

The first top: hearing them out

The ear is a bank of filters, and two partials inside one filter are not heard as two — which is what makes two partials one note. The width of an auditory filter is roughly constant in hertz at low frequencies and grows above about five hundred, while the spacing between consecutive partials is exactly constant in hertz — so the two curves cross, and above the crossing the partials are not separately resolved.

That gives the first number. At a fundamental of 55 hertz the crossing is at the sixth partial; at 110, the eighth; at 220 and 440, the ninth.

Below the crossing a partial is an object. It can be heard out, it can be tuned, it can be matched. Above it, it exists only as a contribution to the timbre of the whole. That is a boundary this collection has already used twice: it is where the dominance region for pitch runs out and it is why the shift experiment that closes the residue ladder uses partials in the low teens rather than higher up.

The bandwidth behind the first answer, read as an interval, is more than an octave wide at the bottom of the bass and under three semitones at the top — which is why the resolution limit depends on the fundamental and the other two do not.

The second top: a semitone

The second question is different and its answer is the surprising one.

Ask not whether the partials can be separated, but whether the step between two of them is small enough to be a step of a scale. A semitone is a hundred cents. The interval between partials n and n+1 falls below a hundred cents at n = 17 — and it does so at every fundamental, because 1200 log₂((n+1)/n) contains no frequencies at all.

The series has three tops. Where the harmonic series stops, asked three ways, at four fundamentals. Consecutive partials stop being separately resolvable around partial 8 — that one depends on the fundamental, since a critical band is a fixed width in hertz and the spacing is not. They stop being a semitone apart at partial 17 at every fundamental, because the ratio (n+1)/n does not know what n is measured in. And they stop being distinguishable in pitch at all between partials 34 and 140. Every claim about how far up the series something happens is a claim about which of these three was meant. — Fig. 3 The three thresholds at four fundamentals. The resolvability limit moves — six, eight, nine, nine. The difference limen moves a great deal — 140, 90, 56, 34. The semitone sits at seventeen in every row, because it is the only one of the three that is a statement about a ratio rather than about the ear. The third button cannot be played at the point the third threshold happens: for a 55-hertz fundamental that is the hundred and fortieth partial at 7.7 kilohertz, where most speakers have rolled off, so it plays the pair just below instead — nineteen cents at 4.9 kilohertz, against a limen of 6.8 there.

That is the number behind the register where a natural instrument can play a melody: above the eighth partial the steps are two semitones or less, above the sixteenth they are one or less, and a bugle plays fanfares in the low register and tunes in the clarino because those are the same fact.

It is also, read one way, an answer to why anybody would build a scale from the top of the series rather than the bottom. Partials 16 to 32 are a chromatic scale, in the sense of a run of steps around a semitone; they are wildly out of tune with equal temperament — the seventeenth is 105 cents above the sixteenth and the nineteenth is 89 above the eighteenth — but they are a scale in shape.

The first 16 partials of a string. A string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning. — Fig. 4 The first sixteen partials with the notes they are nearest and the cents they miss by. The bottom of the series is a chord and the top of it is a scale, and the changeover is where the steps get small — which is the same crossing as the semitone limit, one octave lower.

The third top: telling them apart at all

The last question is the crudest and the answer is the largest. Two frequencies presented one after the other can be judged different only if they differ by more than the difference limen, which for a sustained tone in the middle of the range is a few cents.

Against the difference limen across the range, a comma is 21.5 cents and a semitone 100, and the limen is near 5 — so the third top, the point at which two partials stop being separable frequencies at all, is far above the other two and is a fact about the ear’s acuity rather than about the series.

Set the step between consecutive partials against that curve and it crosses far up: at a fundamental of 110 hertz, the two are equal around the ninetieth partial; at 440, around the thirty-fourth; at 55, around the hundred and fortieth.

Above the crossing, consecutive partials are not merely unresolved — they are, in the pitch sense, the same note. There is no ordering of them left.

That number is not musically useful and it is worth having anyway, because it is the one that makes the phrase “the harmonic series is infinite” concretely wrong for an ear. A series that has run out of distinguishable steps has run out, and where that happens depends on the fundamental by a factor of four.

There is a fourth boundary and it is not a threshold about the series at all: above about eighteen kilohertz there is nothing, so a fundamental of 110 hertz has 163 partials in the audible range and one of 440 has forty. That is an outer wall rather than a gradual failure, and at the fundamentals used above it arrives long after all three of the others.

Where the order does not hold

The claim made near the top of this essay — that the three thresholds come in a fixed order because each demand is weaker than the one before, so a different order would be an error rather than a discovery — is too strong, and the site’s own machinery says so when it is asked at fundamentals the figure does not draw.

The order breaks above a fundamental of about 1,208 hertz. At 1,320 the semitone limit is the seventeenth partial, as it is everywhere, and the difference-limen limit is the fifteenth: consecutive partials become indistinguishable in pitch before they come within a semitone of each other. At 3,520 the two are seventeen and seven.

The a priori argument fails because the three demands are not nested in the way it assumes. Two of them are measured in cents at a fixed partial number and one is measured in cents at that partial’s own frequency — and the difference limen in cents rises steeply toward the top of the audible range. A weaker demand does not have to be satisfied later when the two criteria are functions of different variables; the curves simply cross, and where they cross is a measurement rather than a deduction.

The fourth boundary gets there first, and that is the more useful correction. The eighteen-kilohertz wall is described above as arriving long after the other three, and at 110 or 220 hertz it does. It stops being true well inside the musical range:

	third top (difference limen)	second top (a semitone)
f₀ = 220	partial 56, at 12.3 kHz	partial 17, at 3.7 kHz
f₀ = 440	partial 34, at 15.0 kHz	partial 17, at 7.5 kHz
f₀ = 660	partial 26, at 17.2 kHz	partial 17, at 11.2 kHz
f₀ = 880	partial 21, at 18.5 kHz	partial 17, at 15.0 kHz

The third top passes above the wall at a fundamental of about 783 hertz, which is the G at the top of the treble staff, and the second top passes above it at about 1,059. So for a soprano’s high notes, a piccolo, or the top octave of a piano, the wall is not the last of the four boundaries — it is the first, and the two thresholds this essay spends most of its length on describe partials that are not there to be resolved, stepped through, or told apart.

That reorders the essay’s own conclusion for part of the range. Below about 700 hertz there are three tops and a wall behind them; above it there is a wall, and the tops are arithmetic about frequencies nobody can hear. The first top survives everywhere, because resolvability tops out around the ninth or tenth partial at every fundamental and that partial is always audible.

And the resolvability limit is more criterion-dependent than recorded. The computation note below says that requiring the spacing to fall below half a bandwidth rather than a whole one moves every figure down by two or three partials and changes no ordering. Running it moves them the other way and much further: the crossing goes from 6 to 15 at 55 hertz, 8 to 17 at 110, 9 to 18 at 220 and 9 to 19 at 440 — up by nine or ten in every case. At 220 and 440 that puts the first top above the second, so the ordering this essay presents as necessary is broken at ordinary musical fundamentals by nothing more than a change in where the resolvability criterion is drawn.

None of that undoes the essay’s main point, which is that a stopping place on the series is always imported from outside it. It sharpens it: the boundaries are not merely imported, they are not reliably ordered, so quoting one without saying which — and at what fundamental, and under which criterion — is worse than the essay had already argued.

There is something worth pausing on there. Everything low in the series is small whole numbers and everything high in it is not, and the transition is smooth — but the use changes discontinuously. Below the crossing the ratios matter because they are simple; above it the ratios do not matter at all and only the spacing does. The same object is a source of consonances at the bottom and a source of steps at the top, and it is the ear’s two failures that separate the two regimes rather than anything in the series.

Which top each argument in this collection meant

The reason to separate the three is that this site has quoted all of them, in different essays, without saying which.

The triad stops at six. The third rung of this ladder found that roughness does not justify that stopping point — roughness per pair is lowest at the first four partials and rises monotonically from there. The three tops do not justify it either: six is below every one of them. So the derivation’s stopping point is not a perceptual boundary, and the honest account remains the one that rung gave, which is that six is where the answer is.

The clarino register begins around eight. That is the semitone top, one octave down: partials 8 to 16 are within two semitones of each other.

The dominance region ends around six to eight. That is the resolvability top, and it is the one boundary of the three with an experimental literature attached to it.

The tables of partials stop at sixteen. That is a convention about how much fits on a page, and it is worth noticing that sixteen is one short of the semitone limit — so the standard table is exactly the part of the series in which every step is bigger than a semitone.

Each prefix of the series sounded as a chord has its own roughness, and the smoothest is the first four — which is a fourth criterion again, and the one that selects six is none of these three.

The first 24 partials of a string. A string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning. — Fig. 6 And the partials against the notes a keyboard has, which is where the three answers meet a fourth question. The seventh partial is 31 cents flat of anything on the instrument, the eleventh 49, the thirteenth 41 — so **the series stops being usable as a source of notes long before it stops being resolvable, audible or a semitone apart.** That limit is not on this essay’s list because it is not a fact about the series at all: it is a fact about the twelve-note division the series is being read against.

That distinction is the one worth carrying out of this essay. Two of the three tops are facts about a listener and one is a fact about arithmetic, and the number everybody actually quotes — six, or seven, or eight — is a fact about a tuning system and belongs to a fourth category entirely.

The one place the three disagree usefully

Between the resolvability limit and the semitone limit there is a band — roughly partials 8 to 17 — where consecutive partials are not separately resolvable and are more than a semitone apart. That band is peculiar and it has a name in practice even though the theory rarely marks it.

It is the band where a spectrum has structure the ear cannot itemise but can hear the consequences of. Two partials a whole tone apart at 2 kHz beat and roughen; they cannot be heard out; and the result is a texture rather than a chord. Everything this site says about roughness lives there, and everything it says about pitch lives below it.

How many partials two notes share at each interval, computed with a twelve-cent tolerance, is the count every consonance argument in this collection runs on — and it uses the resolution top rather than either of the others, because a partial the ear cannot separate is a partial that cannot coincide with anything in particular.

Which computation produced the numbers

The step in cents is 1200 log₂((n+1)/n), evaluated directly. The semitone crossing is where that falls below 100, found by scanning n rather than solved, and the exact solution — 1/(2^(1/12) − 1) = 16.82 — confirms the seventeenth partial is the first.

The resolvability crossing uses the equivalent rectangular bandwidth, 24.7(0.00437 f + 1) hertz, and marks the first partial whose neighbours are closer together than that width. The criterion is that the spacing, which is f₀, is less than the bandwidth at that partial’s frequency.

That criterion was previously recorded here as insensitive — a half-bandwidth threshold was said to move every figure down by two or three partials and to change no ordering. Run against the same function it moves them the other way and by nine or ten: 6 to 15 at a fundamental of 55 hertz, 8 to 17 at 110, 9 to 18 at 220 and 9 to 19 at 440. It also changes the ordering, at 220 and at 440. The full-bandwidth criterion is the one every figure here uses and it is the conventional one; what is no longer claimed is that the choice does not matter.

The difference limen uses this collection’s own fitted curve for the frequency difference limen, converted to cents. That curve is a fit to published data across the audible range and is the same one the pitch acuity ladder is built on.

Every number in the figures is computed at the stated fundamental. Nothing is interpolated between rows.

What the picture cannot show

The resolvability limit is not a line. It is a gradual failure, it depends on level, on duration, on whether the listener has been told what to listen for, and on whether the partial has been made to stand out by mistuning or by onset. “Around the eighth” is a summary of a soft transition and the sharp number in the figure is an artefact of having to draw one.

A partial’s own audibility is not modelled. In a real tone the upper partials are far weaker than the lower ones, so many of them are below threshold long before any of these limits applies. All three tops here are computed for partials of equal strength.

Nothing here is measured. The bandwidth formula, the difference limen curve and the audible upper limit are all quoted models fitted by other people to other people’s listeners.

Two of the thresholds are not independent. The resolvability limit and the roughness curve come from the same critical-band formula, so an essay that uses one to explain the other is going in a circle; what saves the argument here is that the semitone limit and the difference limen come from somewhere else entirely.

And the whole essay assumes a harmonic series. For an inharmonic partial set — which includes every piano string above the middle — the spacing is not f₀ and the three crossings move. On a real piano the stiffness stretch is enough to matter by the twentieth partial.

The ladder from here

This rung took a single object and found three boundaries in it, and the useful consequence is a rule for reading the rest of this collection: when an argument stops the series somewhere, ask which of the three it means. What the ladder still owes is the fourth question nobody asks — how far up a player can go, which is a fact about a lip and a bore rather than about an ear, and which is the real limit on every instrument that selects partials for a living.