Acoustics & Music Theory
why it sounds like that
Instruments and their design

The register where the series becomes a scale

A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.
16 min read 8 figures The design was forcedSmall whole numbersThe instrument is not the model

Assumes: A melody is a walk, not a set

The first rung of this ladder found that a melody is almost all small steps, and that the reason is not entirely a matter of taste: above about eight notes a second a line that leaps stops being one line.

That is a constraint on what a melody may do. There is a harder one on what an instrument may offer, and for two hundred years the most important melodic instrument in the orchestra could not offer steps at all except in one place.

An instrument with no notes in between

A trumpet or horn without valves — which is what every trumpet and horn was until the 1830s — is a tube of fixed length. Its resonances are the harmonic series of that length, and the player selects among them with the lips. There is nothing else. No holes, no slide, no valve; the available pitches are the partials and only the partials.

So the question of what such an instrument can play is the question of how far apart the partials are, and that is one line of arithmetic.

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 from each partial to the next, in semitones. Twelve, then seven, then five, four, three — the gap shrinks steadily because the ratio (n+1)/n approaches one. It crosses two semitones between the eighth partial and the ninth, and the shaded bands show why that is inevitable: each band is one octave of pitch, and doubling the partial number doubles how many partials are inside it. Two notes in the first octave, three in the next, five, and nine.

That last observation is the whole essay in a sentence. A harmonic series has a fixed number of notes per octave only in the sense that it has a doubling number of them. Low down it is a chord; high up it is a scale; and the crossing is not a matter of degree but of whether consecutive notes are close enough to be melodic steps.

Where the crossing is

Reading the numbers off:

From To Interval
2 3 a fifth
3 4 a fourth
4 5 a major third
5 6 a minor third
6 7 2.67 semitones
7 8 2.31
8 9 2.04 — a whole tone
9 10 1.82
12 13 1.39

Below the sixth partial the smallest available move is a minor third. An instrument with those notes and nothing else cannot play a tune; it can play a bugle call, and the reason every bugle call in every army sounds like the others is that they are all drawn from the same four notes — a constraint of the same kind as the one a drum’s inharmonic modes impose, where the available material decides the repertoire before anybody composes anything.

The first eight 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. 2 The first eight partials of a natural trumpet in B flat, drawn as the modes of the tube. The set is a major triad twice over with a flat seventh added at the top. Everything a valveless instrument could play below its eighth partial is inside this — which is exactly the material of a fanfare, and is why the instrument’s ceremonial role and its acoustic limit are the same fact.

From the eighth upward the situation is different in kind. The step from 8 to 9 is a whole tone, 9 to 10 is a large semitone, and by the twelfth the steps are semitones. Partials 8 to 16 span exactly one octave and there are nine of them — a diatonic scale with a passing note, available without moving anything.

Nine notes to the octave is more than a diatonic scale needs and fewer than a chromatic one has, and counting them that way is already misleading: two of the nine are the only source of two ordinary degrees and are badly out of tune, so the instrument has five reliable degrees and four problems rather than seven degrees and two spare. The surplus is not evenly distributed either — the extra note sits between the fourth and fifth degrees rather than anywhere a player would have chosen. That is the first hint that what the tube offers and what the music wanted were never quite the same object, and the rest of this essay is about the gap.

That register has a name. Baroque trumpeters called it the clarino, they specialised in it to the exclusion of everything below, and the entire melodic literature for the natural trumpet — the second Brandenburg, the B minor Mass, every high trumpet part that sounds impossible — lives inside those nine notes.

The scale nobody designed

It is worth writing the clarino set out as pitches, because the coincidence is uncomfortable.

The simple ratios, and 12 equal steps. One octave laid out in cents. Above the line, the frequency ratios of small whole numbers, where they actually fall; below it, 12 equal steps of 100.00 cents each. The two sets almost never coincide, and how nearly they do is the case for or against the division.
Fig. 3 Partials eight to sixteen of a B flat tube, in cents above the eighth, against the twelve equal steps. Six of the nine land within fifteen cents of a keyboard note. Three do not: the eleventh sits 48.7 cents below its nearest note, the thirteenth 40.5 above, and the fourteenth 31.2 below.

Which six they are is the thing to look at, and it is not the reading this essay first gave. Six notes cannot be a seven-note scale, and the six that land are do, re, mi, sol, ti and the octave — 0, 204, 386, 702, 1088 and 1200 cents. That is the major scale without its fourth and sixth degrees.

And the two degrees it is missing are exactly the two the bad partials supply. The eleventh is fa and it is 48.7 cents out; the thirteenth is la and it is 40.5 out. The fourteenth adds a flattened seventh nobody asked for, at 31.2 out.

So the tube does not give a major scale with three awkward notes in it. It gives five degrees of one dead on, and then offers the two that are left only in positions no listener will accept. That is a much sharper constraint, and it is the one the repertoire is shaped by: baroque clarino writing spends its time on the tonic, the second, the third, the fifth and the leading note, and treats the fourth and the sixth as passing notes to be got through — which is exactly what a player would do with a degree that is a quarter-tone out and cannot be avoided by choosing a different partial, because there is no other partial that supplies it.

So the natural trumpet’s melodic octave is a five-degree scale with two more degrees available badly. The eleventh partial is the notorious one — almost exactly halfway between the fourth and the sharpened fourth, which is why it is sometimes written as one and sometimes as the other and is right as neither.

This is the same set of deviations that the series ladder measures for a different purpose: partials 7, 11 and 13 are the members that no temperament has a name for. Reached vertically, as chord members, they are the reason the harmonic series cannot be stacked into a usable chord. Reached horizontally, as melodic neighbours, they are three notes in a scale of nine that a player has to bend.

The same three partials, from two directions, with opposite consequences. Vertically they are what makes the series unusable. Horizontally they are a nuisance inside something that is otherwise perfectly usable, because a melody visits a note and leaves, and a chord holds it against others.

The horizontal case is worse than that formulation allows, though, and for a reason the vertical case does not have. A chord can omit its seventh; a scale cannot omit its fourth and sixth degrees and remain a scale. So the three bad partials are not three notes out of nine that a player must handle carefully — two of them are the only route to two degrees, and the third is surplus. Their nuisance value is not proportional to their number.

The scale it produces, against the one the theory produces

There is a comparison worth making explicitly, because the two objects arrive from completely different places and land close together.

Notes on the keyboard. A piano keyboard with the notes under discussion marked. The keyboard is used throughout because it shows distance rather than name, and distance is what the theory is about.
Fig. 4 The nearest keyboard notes to partials 8 through 16, marked on one octave. Six of the nine are the major scale with its fourth and sixth degrees missing; the eleventh partial is the note marked differently, sitting between the fourth degree and the sharpened fourth and belonging to neither. What the clarino gives is not the diatonic set but a five-degree subset of it, with the two absentees available only from the two worst partials in the series.

This collection has reached the diatonic set twice already by construction: as a chain of seven fifths and as the answer to a four-property census over all 349 shapes in twelve. The clarino octave is not a third route to it, and saying so is the honest version of a comparison that is tempting to make.

What the tube produces in tune is do, re, mi, sol and ti — which is not the pentatonic set either, since the pentatonic omits the fourth and the seventh and this omits the fourth and the sixth. It is its own object: the five degrees of the major scale that lie within fifteen cents of a chain of low harmonics, and nothing else. A length of brass produces that whether or not anybody has an opinion about scales, and what it produces is not any set this collection’s other two constructions arrive at.

It is not quite the diatonic set, and the discrepancy is exactly the three awkward partials. That matters, because it makes the coincidence checkable rather than merely suggestive: if the harmonic series simply were the source of the diatonic scale, the eleventh partial would not have to be argued about for three hundred years.

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 Where the series stops, asked three ways, because the register this essay is about is bounded above as well as below. Consecutive partials stop being separately resolvable around the eighth — that limit depends on the fundamental, since a critical band is a fixed width in hertz and the spacing is not — and they stop being a semitone apart at the seventeenth, at every fundamental, because that is arithmetic rather than perception. The melodic register is the strip between those two answers: high enough for the steps to be small, low enough for the notes to be separable. The 9:8 whole tone from the eighth partial to the ninth is its widest step and its bottom edge.

What the players actually did

The fix was in the lips, and it is well documented. A brass player can pull a resonance a substantial distance — the lip and the air column form a coupled system, and the player’s embouchure can drive it away from the tube’s own maximum by something on the order of a semitone at the cost of tone and stability.

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 two properties, on one pair of axes. The gap to the next partial falls as the series rises — 204 cents from the eighth to the ninth, 105 from the sixteenth to the seventeenth — and the band drawn on it is how far the lips can move a note, ±23.1 cents, computed from a bore at Q 40 against lips at Q 12 rather than chosen. Slotting fails where the reach spans the whole gap, because then any embouchure sits between two answers. So the register is melodic because the gaps are small and treacherous because the gaps are small: one number, two consequences, and the reason clarino playing was a profession rather than a technique.

That combination is worth stating plainly, because it explains why clarino playing was a separate profession rather than a technique. High partials are dense, which is what makes melody possible. They are also weakly defined, which is what makes hitting the right one hard, and what makes bending the wrong ones into tune possible at all. The property that made the register melodic and the property that made it difficult are the same property.

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. 7 And there is a ceiling the lips cannot reach past, which is the bell rather than the player. Above some partial a bell stops turning the wave round — half the energy escapes at the mouth instead of returning — and the tube stops having a usable resonance there at all. Computed for five modern brass instruments that boundary lands 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. Which is the sharpest available statement of what was lost: the clarino register begins at about the eighth partial, and a modern bell’s boundary sits at the bottom of it. The natural trumpet’s bell was small and shallow by comparison, so its boundary was higher and the register above the eighth partial was available to be played. The instrument was not merely harder to play in that register; it was built to have one.

The horn’s different answer

The horn arrived at the same problem and solved it another way, and the difference is instructive.

A horn player’s right hand sits in the bell, and closing it lowers the pitch of whatever partial is sounding by up to a semitone or so while changing its tone. Hand-stopping turns the natural horn’s gapped series into something approaching a chromatic scale over much of its range — at the price of an audibly different timbre on every stopped note.

So the two instruments made opposite trades. The trumpet kept one timbre and went up until the notes were close enough. The horn stayed in a comfortable register and accepted that half its notes would sound different from the other half. Eighteenth-century horn writing uses that difference as a colour rather than hiding it, which is the only sensible response to a limitation that cannot be removed.

There is a third answer and it belongs to the trombone, which had a slide from the fifteenth century and therefore never had the problem at all. That the trombone spent three hundred years playing sustained church polyphony while the trumpet played fanfares is usually told as a story about social function, and the acoustic version is simpler: one of them could play stepwise lines in a comfortable register and the other could not.

Both were made obsolete within a decade of the valve, and the loss is real: the clarino register and the stopped horn are two distinct sounds that a modern instrument does not make.

The same arithmetic on a string

The constraint is not about brass. It is about anything whose available pitches are a harmonic series, and a bowed or plucked string produces the whole series at once as well.

The difference is that a string player can stop the string and get any pitch at all, so the series is a colour rather than a limit. Harmonics on a violin or a guitar are used exactly where the density argument says they can be: as isolated high notes and as glassy runs high on a string, never as a melody in the low partials, because the low partials of a string are as far apart as the low partials of a tube.

The exception that proves it is the Aeolian harp and the whole family of instruments where the player has no stopping mechanism at all — an overtone flute, a jew’s harp, a tube with no holes. Every one of them has a repertoire that sits high, and the reason is on the vertical axis of this essay’s first figure.

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 1 semitones between partials 17 and 18. 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. 8 The same curve carried to the twenty-fourth partial with the threshold set at one semitone instead of two. It crosses at the sixteenth, so a series read that high offers semitone motion — which is what the highest clarino writing and the overtone-singing traditions both use. Nothing changes about the series between the two figures; what changes is what counts as a step.

Which computation produced the numbers

Everything above is one function. The interval in semitones between partials n and n+1 is twelve times the base-two logarithm of (n+1) over n, evaluated for each n; the deviation of partial n from equal temperament is 1200 log₂(n) minus the nearest multiple of 100.

Neither is measured or quoted. The mode frequencies drawn for the tube are the same cylindrical-bore model the wind essays use, so the picture of the trumpet’s resonances and the picture of the abstract series are drawn from different code and agree — which is the check that the tube really does produce the series and is not merely said to.

The clarino boundary at the eighth partial is a reading rather than a discovery: the step crosses two semitones between 8 and 9, and two semitones is where the step distribution of actual melodies has most of its mass. Choosing a different melodic threshold moves the boundary by a partial or two and does not change anything about the argument.

Whose music, and when

The clarino repertoire is European and it is narrow in time: roughly 1650 to 1750, concentrated in Germany and Italy, and effectively finished by the middle of the eighteenth century — before the valve, which is the point. The technique was lost for a century and a half and reconstructed in the twentieth.

The natural horn’s hand-stopping is later, from about 1750, and the repertoire that assumes it is Classical rather than Baroque — Mozart’s concertos are written for an instrument on which a significant fraction of the notes are stopped, and modern performances on valved horns lose a colour the writing takes for granted.

The constraint itself has no period. Any instrument whose available pitches are a harmonic series faces it, and several outside Europe do: the alphorn, whose repertoire sits in the same high register for the same reason and whose eleventh partial has a name of its own; the overtone flutes of Scandinavia and eastern Europe; and the whole family of overtone singing traditions, where the melody is carried by partials selected with the tract over a fixed drone and is therefore confined to exactly the same band of the series.

That last case is the strongest evidence that the arithmetic rather than the culture is doing the work. Four unrelated traditions, four different objects, one register.

What the picture cannot show

It cannot show the player. The instrument’s resonances are a fixed list; what comes out is the result of a coupled lip-and-tube system that can be driven a long way from the list. Every number here describes where the tube would like to sound, and every note of the surviving repertoire was produced by somebody overriding that to a degree the model has no term for.

It cannot show tone. The nine partials of the clarino octave are drawn as nine positions on a ruler. In sound they are not equal — the higher ones are thinner, harder to attack and quicker to crack — and the difference between the bottom and the top of that octave is far more than a difference of pitch.

It cannot show attack. Selecting the twelfth partial of a tube requires the lips to be already oscillating near that frequency before the note begins; a fractional error puts the note on the eleventh or the thirteenth. That is a transient problem, it is the whole difficulty of the register, and the same class of problem on a bowed string has an essay of its own.

And it has no bell in it. The cylindrical model gives the series exactly; a real trumpet is not cylindrical, and the bell shifts every mode — which is what makes a real instrument’s low partials playable at all, since a strictly cylindrical tube’s second mode is a twelfth rather than an octave above its first.

The ladder from here

Four rungs, and each has taken a fact about melodic shape and found something other than preference underneath it: streaming for stepwise motion, a range boundary for post-skip reversal, a return for the arch, and now the density of the harmonic series for what a valveless instrument may play.

The next rung leaves shape and takes tuning, and it is the one where this ladder collides with the rest of the site. Every tuning argument in this collection so far has been vertical — a note tuned against another sounding at the same time. A melodic line makes the opposite demand of the same note, in the opposite direction, and the size of the disagreement is very nearly a syntonic comma.

Part 4 of 8

One essay in the series on melody. The essays either side of this one:

The arch is not a preference. Every sequence of 6 notes over 8 scale degrees — 262,144 of them, enumerated rather than sampled — classified by contour, under three constraints. With none, the nine classes are spread. Requiring the sequence to return to its starting degree leaves only the arch, the valley and the flat, at 42.2 per cent each for the first two. Requiring it to begin and end on the LOWEST degree leaves the arch alone, at 99.6 per cent. Nothing here prefers a rise followed by a fall; the constraint is that the melody comes home, and a melody that comes home from below has nowhere to go but up first. The shape that survives everything else Part 3 — Throw away a melody's key, its tuning, its instrument and the sizes of its intervals, and what is left is a string of pluses and minuses. That string is what a listener who cannot name a note still has, and it costs 37 per cent of the tune to keep. The arch that melodic shape is famous for is not in it as a preference: enumerate every six-note sequence that begins and ends on the lowest degree it uses and 99.6 per cent of them are arches, because a melody that comes home from below has nowhere to go first but up. How far each system sits from equal temperament. Deviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp. The note that is sharp because of where it goes Part 5 — Eleven essays here have tuned notes against other notes sounding at the same time. A melodic line makes the opposite demand of the same note. In C major the leading note wants to be 1088 cents if the argument is vertical — a pure third above the dominant — and 1110 if it is horizontal, because a leading note is a note that is about to arrive somewhere and performers narrow the step. The two answers differ by 21.5 cents, which is a syntonic comma exactly, and no keyboard has ever been able to hold both.

Where this goes next

Ode to Joy as a path. Ode to Joy plotted as 30 notes against the 8 scale degrees it uses, one column per note. Its largest melodic interval is 2 semitones and it spans 7; the mean absolute step is 1.24 semitones. Beethoven, Ninth Symphony, finale, 1824 — the theme as first stated, eight bars. A melody is a walk, not a set Nine essays here are about which seven of the twelve a scale takes, and every one of them describes a set. A tune is not a set; it is a path across one, and the path is nearly all small steps. That is not a matter of taste. Above about eight notes a second the ear stops being able to hold a large interval and a small one in the same line, and at sixteen the choice disappears altogether — so a fast passage is scalar because a fast passage that leaps is two pieces of music. The first eight 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. A string does everything at once A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact. What a 60 cm tube supports, by how its ends are closed. The first 6 modes of an open cylinder and a stopped cylinder, all of the same acoustic length. An open tube supports every whole multiple of the fundamental and reaches the second mode an octave up. A cylinder stopped at one end supports only the odd multiples and reaches its second mode 1902 cents up, which is a twelfth. Its fundamental is also an octave below the others, because it fits a quarter of a wavelength where they fit a half. A tube that skips every other partial Stop one end of a cylinder and half its modes vanish. That single fact about where the pressure has to be decides that a clarinet sounds hollow, that it plays an octave below its length suggests, and that it must cover nineteen semitones with fingers before it can overblow — while every other woodwind covers twelve.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down, the 8 sharing most with it of 9.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

BoreClarinoHarmonic seriesJust intonationMelodic intervalNatural trumpetPartial