Intervals and chords

The fourth top is the maker's

The series has three tops and all three are the ear's: where consecutive partials stop being resolved, where their spacing falls below a semitone, and where it falls below what the ear can hear at all. The fourth is how far up a player can go, and it is the only one somebody chose. Every modern brass bell puts the boundary at about the ninth partial across a family whose tubes differ by a factor of four — and the baroque natural trumpet, whose bell is small, puts it at the seventeenth, which is exactly the top of the clarino register.

Assumes: The series has three tops · The bell decides what gets out

The series has three tops took a single object and found three boundaries in it, and ended by naming a fourth question nobody asks: how far up a player can go.

That question is different in kind from the other three, and the difference is the point of this rung. Resolvability, semitone spacing and the difference limen are all facts about a listener; they are the same on every instrument and were the same before instruments existed. The fourth is a fact about a tube with a hole in the end of it, and somebody decided how big the hole is.

Every modern bell puts the boundary at the same partial. For each brass instrument, the partial at which its bell stops turning the wave round — the frequency at which half the energy escapes at the mouth, divided by the tube's own fundamental. The five modern instruments land between partial 8.4 and partial 10.2 despite tube lengths differing by a factor of four, because a bell is scaled with its instrument. The natural trumpet of the baroque, whose bell is small and whose tube is long, puts it at partial 17.2. — Fig. 1 For each brass instrument, the partial at which its bell stops turning the wave round — the frequency at which half the energy escapes at the mouth, divided by the tube’s own fundamental. Five modern instruments whose tubes differ by a factor of four land between the eighth partial and the tenth. The baroque natural trumpet lands at the seventeenth. The buttons play a natural trumpet’s partials 8 to 12 and then 13 to 18.

Every modern brass bell puts the boundary at about the ninth partial. A tuba’s tube is nearly four times a trumpet’s and its bell is three times as wide, and the ratio that decides this comes out very nearly the same on both.

Why there is a boundary at all

Selecting a partial requires that there be a partial to select. A brass player finds a note by putting the lips near one of the bore’s resonances and letting the bore capture them — the thing players call slotting — and a resonance exists because the wave arriving at the open end mostly turns round and comes back.

The bell decides what gets out computed the share that does not. The whole of it is ka: the opening’s radius against the wavelength. Below the crossover almost everything reflects, which is what makes the tube a resonator; above it almost everything leaves, which is what makes the instrument audible, and there is very little left inside to resonate with.

What gets out of an opening, for 3 openings. The fraction of the wave's energy radiated at an open end against frequency, in the baffled-piston model — the radiation resistance of a circular piston, normalised to the tube's own impedance. Each curve runs from nothing at the bottom, where the opening is far smaller than a wavelength and the wave simply turns round, to everything above ka ≈ 2. Half the energy leaves at 975 Hz for a trumpet's bell (radius 62 mm), 1344 Hz for a natural trumpet's bell (radius 45 mm), 403 Hz for a horn's bell (radius 150 mm). The crossover goes as one over the radius, so the widest and narrowest here are 3.3 times apart in frequency. The same number decides how strongly the tube resonates and how much sound it makes, which is why a bell cannot brighten an instrument without also weakening its own resonances. — Fig. 2 The radiated share against frequency for three bells. Every curve has the same shape and the radius slides it: a horn’s fifteen-centimetre mouth puts the crossover at 403 hertz and a natural trumpet’s four-and-a-half at 1344. What turns those into partial numbers is that each instrument’s tube length is also different, and the two nearly cancel.

So the fourth top is where the crossover falls in the series, and that is a ratio of two lengths: the bell’s radius and the tube’s length.

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 16 semitones between partials 1 and 2. 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, 17 between 16 and 32. 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. 3 Why the question has an answer at all: the interval between each partial and the next, in semitones, against partial number. Down at the eighth partial the neighbours are a whole tone apart and a player cannot miss; by the thirty-seventh they are less than a quarter-tone apart and a player cannot aim. Somewhere between those two the instrument stops offering a choice, and the whole of this essay is which mechanism gets there first.

The invariant, and the one instrument that breaks it

The ratio is very nearly constant across the modern family, and the constancy is not obvious. Nothing forces a maker to scale the bell with the tube; a trumpet with a horn’s bell is physically possible and would be a different instrument.

	tube	bell	boundary
B♭ tuba	5.50 m	38 cm	partial 10.2
F horn	3.70 m	30 cm	partial 8.7
tenor trombone	2.75 m	20 cm	partial 9.7
B♭ trumpet	1.48 m	12 cm	partial 8.4
cornet	1.48 m	12 cm	partial 9.0
baroque natural trumpet	2.20 m	9 cm	partial 17.2

The boundary is not an approximate constancy — it is exactly one number. The crossover frequency goes as one over the bell radius and the fundamental as one over twice the tube length, so their ratio is exactly the tube length over the bell radius. The boundary partial is the reciprocal of the bell’s radius-to-length ratio, and nothing else enters:

	bell radius ÷ tube length	boundary
tuba	0.0345	10.2
trombone	0.0364	9.7
horn, trumpet, cornet	0.0405	8.7
natural trumpet	0.0205	17.2

The five modern instruments span 0.0345 to 0.0405, a spread of 1.17, and their boundaries span 8.7 to 10.2 — the same 1.17, necessarily. So the finding is not that five different instruments happen to agree; it is that makers scale bells with tubes to within seventeen per cent across a family whose lengths differ by 3.7 times, and the boundary is that decision restated. There is one design constant here and the table above is showing it twice.

That also means the trumpet and cornet rows, which give identical dimensions, cannot give different boundaries. Both are 8.7.

Both counterfactuals are worth pricing, because they say how far the choice reaches. Give the natural trumpet a bell on the modern line and it wants a radius of 8.9 centimetres — an eighteen-centimetre mouth on a two-metre tube, wider than a trombone’s, and it would land at the eighth partial with the rest of them. Go the other way and ask what bell a B♭ trumpet needs to reach the seventeenth partial, and the answer is a radius of three centimetres: a six-centimetre mouth, five and a half times the bore, which is a tube that barely flares.

Neither instrument is impossible to build and neither is anywhere near the other’s dimensions. So the constancy across the modern five is not a consequence of scaling instruments up and down — a family scaled by any one dimension would hold this ratio automatically, but a family that departed from it by a factor of two, as the natural trumpet does, is evidently buildable. The modern makers converged on 0.035 to 0.041 and the baroque maker did not, and the two decisions are separated by an audible register. Which is the sense in which this top is the maker’s and the other three are not: the ear’s three boundaries can be measured, and this one had to be settled by somebody choosing a number and then holding it across a whole family of instruments for two centuries.

The natural trumpet is the outlier and it is the one that had to be. A valveless instrument has only its harmonic series to play with, and everything below the eighth partial is a bugle call — the notes are a fifth and a fourth apart down there and no melody fits — the register where the series becomes a scale is the essay about exactly that threshold. Between the eighth and the sixteenth partials the steps are a tone or a semitone and a diatonic scale appears, which is the register baroque trumpeters played in and which is called clarino.

The arithmetic says the instrument’s bell is what made that register available. Put a modern trumpet’s bell on it and the boundary drops from the seventeenth partial to the ninth, which is exactly where the melodic register begins.

The first 18 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 series a two-and-a-fifth-metre natural trumpet gives, out to the eighteenth partial. Below the eighth the intervals are a fifth and a fourth; from the eighth up they are tones and semitones and a scale is available, with the eleventh and thirteenth partials out of tune and the ones between them not — the four partials the lips cannot correct. The clarino register is the top half of this picture and it is the half a small bell keeps.

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. 5 The three tops found earlier, for comparison: where consecutive partials stop being resolved by the ear, where their spacing falls below a semitone, and where it falls below the difference limen. All three move with the fundamental and none of them moves with the instrument — a listener hearing a 55-hertz tone resolves eight partials whether it comes from a string, a pipe or a loudspeaker. The fourth top does not appear on this figure at all, because it is not about hearing.

That is the sharpest way to put the distinction. Ask where the series stops and there are two entirely different kinds of answer: three about what can be heard, and one about what can be made. The first three have been the same for as long as there have been ears, and the fourth changed in the eighteenth century because somebody wanted a louder instrument.

The other candidate, which does not bind

There is a second thing that could stop a player, and it is worth computing because the previous rung of the air-column ladder supplies the number.

The partial the lips cannot reach established that a brass player’s embouchure can move a note ±23.1 cents, from the Q-weighted mean of a bore at Q 40 and lips at Q 12. Slotting requires the player to be able to aim at one partial rather than another, and that requires the partials to be further apart than the lips can wander.

The lips' reach against the gap to the next partial. The interval in cents between consecutive partials, against partial number, with the band the lips can move a note drawn on it — ±23.1 cents, which is the number an earlier essay on air columns gives for a bore at Q 40 and lips at Q 12. Slotting fails when the whole reach spans the gap to the neighbour, because then any embouchure setting is within reach of two partials at once and the note is no longer found. That happens at the 37th partial, which is far above anything anybody plays. — Fig. 6 The interval between consecutive partials against partial number, with the lips’ whole reach of 46 cents drawn across it. The two meet at the thirty-seventh partial: above it any embouchure setting is within reach of two partials at once and there is no note to find. Nobody has ever been near it — the clarino register tops out at the eighteenth, where the partials are 94 cents apart and the lips reach half of that.

The lips would allow the thirty-seventh partial and the bell stops it at the ninth. So the fourth top is not the player’s at all, in the sense that the question was asked; it is the instrument’s, and the player is nowhere near their own limit.

That is a satisfying answer to a question that sounded like it was going to be about embouchure and turned out to be about a hole.

It also settles something about the eleventh partial that has been hanging over two ladders. The partial the lips cannot reach found the eleventh 48.7 cents flat and beyond the lips’ correction, and the hand in the bell found the remaining 26 cents available at a cost of four decibels. The eleventh partial sits at 858 hertz on a natural trumpet, which on that instrument’s small bell is comfortably below the boundary and on a modern one is right at it. The partial with the reputation is the partial where two independent problems meet, and only one of them is about tuning.

Which computation produced the numbers

Three quantities and one division.

The tube’s fundamental is the speed of sound over twice its acoustic length, which is the ladder’s first rung unchanged and which ignores the flare — a real brass instrument’s low modes are not exactly harmonic, and the bell is part of why. Using the plain half-wavelength length is an approximation that is worst at the bottom of the series and best in the register this rung is about.

The bell’s crossover is the site’s own radiationCrossover: the frequency at which the piston radiation function reaches a half, solved by bisection. It is the same function every figure in the bell rung is drawn from.

The boundary partial is the second divided by the first. Nothing is fitted and the only inputs are two lengths per instrument, both of them measurable with a ruler.

The lip limit is the previous rung’s ±23.1 cents against seriesStepCents, which is 1200·log₂((n+1)/n) and is arithmetic.

Where the model stops

A bell is a flare and this treats it as an aperture. The radiation crossover is computed for a flat piston of the mouth’s radius, and a real bell converts the wave gradually along its length. The proper quantity is the horn’s own cutoff frequency, which depends on how fast the flare opens rather than on where it ends, and which for a trumpet is usually quoted near 1500 hertz against the 975 this model gives. The model is low by about half an octave, and the consequence is a multiplication rather than an offset — the boundary is a ratio, so a crossover 1.54 times higher gives a boundary 1.54 times higher. That is +4.7 partials on a trumpet and +9.3 on a natural trumpet, not the partial and a half previously recorded here. It changes none of the comparisons, because it is the same factor on every row.

It does change how the historical argument reads, and in its favour. Corrected, a natural trumpet’s boundary sits at partial 26 and the clarino register tops out at 18, so the small bell leaves real headroom above the register it makes available. A modern trumpet’s sits at 13, which is inside the clarino register rather than at its bottom edge. The uncorrected numbers put the modern boundary at the register’s foot and the corrected ones put it a third of the way up, and both say the same thing about which instrument can play there.

The crossover is a half and not a wall. Nothing stops resonating at the crossover; the modes get weaker over about an octave either side of it. So a boundary at “partial 8.4” means the modes are getting hard to hold from about the sixth and are gone by about the twelfth, which is a fair description of where a modern trumpet’s upper register gets difficult.

And the player is not in the model at all. Everything above says where the instrument stops offering resonances. Whether a player can produce the air speed, the embouchure tension and the endurance to use the ones that are there is a different question with a different literature, and the historical evidence — that the clarino register required specialists and was lost for a century — says it is not a small one.

Whose music, and the register that disappeared

The claim about geometry is general. The claim about repertoire is specific and dated.

Baroque trumpet writing sits in the clarino register because that is where the instrument has a scale, and the parts are famously high and famously exposed. The practice is documented from the middle of the seventeenth century, it was the work of a guild of specialists, and it declined through the second half of the eighteenth.

The usual explanation is social — the guilds broke up, the players stopped being trained — and the arithmetic here adds a mechanical one that runs alongside it. Trumpet bells got wider through the same period, for reasons that had nothing to do with the clarino register and everything to do with wanting more sound in a larger orchestra in a larger room. A wider bell moves the boundary down the series, and the register the specialists played in is the register that a wider bell takes away.

That is a claim about a direction rather than a date, and it is the honest form of it. What the model gives is that a modern trumpet’s bell puts the boundary at the eighth partial and a baroque one’s at the seventeenth. What it does not give is the year in which the change crossed the line, because bells changed gradually and the practice declined gradually and nobody was measuring either.

It is worth adding what did not happen. Valves arrived in the 1820s and made the whole chromatic scale available in the comfortable middle register, which removed the reason to play high at all. So the clarino register was lost twice over: the instrument stopped supporting it and the music stopped needing it, and only the first of those is in this arithmetic.

The one number that would test it

The reading above is a hypothesis about history dressed as a calculation, and it is worth saying what would falsify it.

If the clarino register was lost because bells widened, then surviving instruments should show it: a trumpet dated 1650 and one dated 1800 should differ in the ratio of bell radius to tube length, in the direction and by roughly the amount the table needs. That is a measurement anybody with access to two museum instruments and a tape measure could make, and this collection has neither.

If instead the ratio was stable across the period and the register was lost socially, the arithmetic here is still correct about the modern instrument and says nothing about the history — which would be a real result too, and a duller one.

The prediction has the useful property of being about a ratio rather than about a size. Instruments were built at many pitches and many lengths; what the model says should have changed is bell radius divided by tube length, which is dimensionless and comparable across all of them.

What the picture cannot show

Whether the boundary is where players actually stop. Everything here is a property of the bore’s impedance and nothing is a measurement of a performance. A player can and does force notes above the point where the instrument stops helping, at a cost in security and endurance that no figure here contains.

Nor whether the fourth top belongs to the series at all. The other three are properties of a harmonic series and would be the same for any source producing one. This one is a property of a particular way of producing it, and a plucked string or a struck bar has no equivalent — a string does everything at once and its partials are all present together whether or not anybody could select them. The question “how far up can a player go” only means anything for an instrument that plays one partial at a time, which is brass, the overtone flutes, and the human whistle.

And the table’s dimensions are nominal. Bell diameters vary between makers by a couple of centimetres and acoustic lengths are not tube lengths — a brass instrument’s effective length includes the mouthpiece and the flare, both of which are frequency-dependent. The ratios are robust to a few per cent; they are not robust to twenty.

Where this ladder goes next

Five rungs. A string does everything at once; a stiff string’s partials are not quite the series; the series is not a chord; the series has three tops, all of them the ear’s; and now a fourth that is not, and that turns out to be a design decision made about the size of a hole.

What the ladder still owes is the thing the flare limitation names. Every figure in it treats a partial as n times a fundamental, and the whole of brass instrument design is the business of making that true — a plain cone or a plain cylinder does not give a harmonic series with a mouthpiece on one end and a bell on the other, and the shapes that do were arrived at empirically over three centuries. Computing which flare makes a series harmonic is a horn-equation problem this collection has never set up, and it is the same object the hand in the bell needed and did without.