An instrument points

Every figure in this field so far has computed what an instrument produces. None of them has asked where it goes, and the answer is that it does not go everywhere.

One number decides it #

A source radiates evenly in all directions while it is small compared with the wavelength it is producing, and becomes directional when it is not. The parameter is the ratio of the two, conventionally written $ka$ — the wavenumber times the radiating radius, which is the circumference of the source measured in wavelengths.

Below $ka = 1$ the source is effectively a point and the radiation is omnidirectional. Above it, contributions from different parts of the source arrive at an off-axis listener with different delays, they interfere, and the pattern develops a main lobe and nulls.

The crossover frequency is therefore $c/2\pi a$: for a radiator 9 cm in radius, about 600 Hz. Below that the instrument fills the room; above it, it points.

The consequence: an instrument has no single spectrum #

Here is what makes this an essay rather than a footnote about loudspeakers.

A note’s partials are spread across frequency. Its fundamental may be well below the crossover, radiating in all directions; its upper partials are above it, beaming. So the spectrum an instrument radiates is direction-dependent: on-axis a listener gets everything, off-axis they get the low partials and progressively less of the high ones.

An instrument therefore does not have a spectrum. It has a family of spectra indexed by direction, and every figure in the body-filter essay, every spectrum in the tone-hole essay and every partial list in the harmonic-series essay is one member of such a family — the one measured wherever the measurement was made.

A microphone position is a choice about what the instrument sounds like, not a choice about how well it is captured. So is a seat in a hall, and so is where a player stands relative to a section.

Which instruments point, and where #

The radiating area differs enormously across the orchestra and so, therefore, does the crossover.

A brass bell is 12 to 30 cm across and radiates from one place. Its crossover is a few hundred hertz, and above about 2 kHz a trumpet is strongly directional straight out of the bell. That is why brass are placed at the back facing forward, why a trumpet turned aside is a completely different sound, and why the instruction to play into a stand is a real change and not a stage direction.

A flute radiates from two places — the embouchure hole and the first open tone hole — each of them small. Both are omnidirectional over most of the range, and the interference between them produces a pattern that is complicated and not strongly beamed.

A violin radiates from a large, irregular, multiply-resonant surface. Its pattern above about 1 kHz is a mess: not a clean lobe but a set of sharply direction-dependent peaks and nulls that changes with frequency. This is why moving a microphone 20 cm around a violin changes the recorded tone substantially, and why violin recording is regarded as difficult.

A singer radiates from a mouth a few centimetres across, and is among the least directional sources in an ensemble below about 4 kHz. The singer’s formant near 3 kHz is right about where directionality is starting, which is one of the reasons a soloist facing the audience carries over an orchestra.

What is lost off-axis, in partials rather than in decibels #

It is worth converting the directivity index into something musical, because decibels of on-axis gain are not what a listener notices.

Take a note whose fundamental is 300 Hz on a source with a 600 Hz crossover. Its first partial is below the crossover and goes everywhere. Its second is at the crossover. Its fourth, at 1,200 Hz, has $ka = 2$; its eighth, at 2,400 Hz, has $ka = 4$; its sixteenth has $ka = 8$ and a main lobe only a few degrees wide.

Evaluating the piston response at each partial rather than reasoning about ka gives the actual shape, and it is not the shape the reasoning suggests:

Off-axis is not a low-pass filter. It is a notch followed by a plateau: the response falls smoothly to a deep null — partial 11 at forty-five degrees, partial 7.7 at ninety — and then the side lobes bring it back to eighteen or twenty-four decibels down and hold it there. A listener at forty-five degrees loses the eleventh partial almost entirely and gets the sixteenth back at thirty times the eleventh’s level.

That corrects the comparison this paragraph was making. A note played softly rolls off smoothly and monotonically; a note heard off-axis has one partial missing and the rest of its top shelved. Those are different timbres and the second has no acoustic analogue in playing quietly at all — it is closer to a comb filter, which is what an interference pattern is.

And the notch moves with the listener. Its partial number is inversely proportional to the sine of the angle, so it sits at the eleventh partial at forty-five degrees, the ninth at sixty and the eighth at ninety. A listener walking round a directional source hears a null sweeping down through the spectrum, and an ensemble spread across a stage is a set of listeners each with the null in a different place.

What survives the correction is the essay’s main claim, and more cleanly. Below the crossover everything is within a decibel at any angle, so the low partials genuinely do go everywhere; the direction-dependence is entirely in the top of the spectrum, and it is more violent than a roll-off rather than less.

The perceptual consequence is one this site has already established. The upper partials are what identify an instrument, and the first fifty milliseconds carry most of the identification — and an attack transient is broadband, which means it is precisely the part of a note most affected by a source that beams its high frequencies. An instrument heard off-axis is harder to identify, and that is a testable claim rather than an impression. The notch sharpens it into a prediction with a location: identification should fail worst at whatever angle puts the null on a partial the instrument’s identity depends on, which is a different angle for every instrument and every note. That is a stranger prediction than further off-axis is worse and it is the one the arithmetic actually makes.

What a room does to all of it #

A directional source in a dead room is heard as its on-axis pattern from wherever the listener sits — they get whatever direction they are in. A directional source in a live room is heard twice: once directly, and once as reverberation, which arrives from everywhere and has been averaged over all directions.

So a room undoes directivity, partially, by scrambling it. The direct sound carries the on-axis spectrum and the reverberant field carries the power-averaged one, and what a listener gets is a mixture whose proportions depend on their distance from the source.

The notch is the part a room undoes most completely, and for a reason worth stating. A null is an interference minimum in one direction, and the reverberant field is an average over every direction — so a partial that is fifty decibels down on the direct path is at its ordinary level in the reverberation, and the mixture has no null in it at all. A directional source in a live room loses its notch and keeps its shelf, because the shelf is a genuine reduction in radiated power at high frequencies and the notch is only a redistribution of it. That is the difference between a directivity index, which is about power, and a polar pattern, which is about direction, and the room separates them.

That is the acoustic content of a fact every recording engineer knows: a close microphone sounds brighter and more direct, and a distant one sounds duller and more blended. It is not only a matter of the room adding decay. It is that the two positions are receiving different spectra from the same source.

A number that decides three things at once #

The parameter $ka$ has now appeared three times in this phase, doing a different job each time, and the coincidence is not one.

It sets the end correction. A tube behaves as though it were longer than it is because the wave carries on past the opening before reflecting, and how far it carries on depends on the opening’s size against the wavelength.

It sets how much escapes. The reason the reflection is imperfect is that some of the wave leaves, and how much leaves is the same comparison. A perfectly reflecting opening would be silent.

And it sets where what escapes goes, which is this essay.

So an opening’s size is not one design parameter among several: it is a single knob wired to intonation, loudness and directionality simultaneously, and it cannot be turned for one of them without moving the other two. A loud instrument reflects poorly, is harder to keep in tune across its range, and beams its high frequencies. That is not a list of separate compromises; it is one compromise seen from three sides.

The instruments of the orchestra are a set of settings of that knob, and reading them off is a fair summary of the whole field. A flute’s small openings: quiet, well behaved in tune, omnidirectional. A trumpet’s large bell: loud, awkward across registers, sharply directional. A violin’s radiating plates: in between and irregular, which is why it is the hardest of the three to describe and the most interesting to measure.

Directivity is one half of what an opening does. The other half is how much gets out of it at all.

So an opening that is too small to point is also too small to radiate, and the two limits arrive together because they are the same ratio of size to wavelength. An instrument that beams its high partials is an instrument that is only getting its high partials out, which is the same fact told twice.

Why this is a room-acoustics essay #

This sits on the room ladder rather than in the instruments field, and the reason is a distinction worth naming.

The instruments field asks what an object produces. This essay is about what happens between producing and hearing, which is where a room lives — and it turns out that the source’s own directionality is one of the terms in that transfer, alongside the room’s absorption, its modes and its decay time.

A room’s treatment cannot be designed without it. An orchestral platform’s reflectors are placed with the sources’ patterns in mind: a reflector above the brass is doing something quite different from one above the strings, because the brass are sending most of their high-frequency energy in one direction and the strings are not.

The player is inside the pattern too #

One consequence that is easy to miss: the person producing the sound is in the worst possible place to hear it.

A violinist’s ears are directly above and behind the instrument, which is not on any axis a listener occupies. A flautist’s are a few centimetres from the embouchure hole and nowhere near the first open tone hole. A brass player is behind the bell. In every case the player is receiving a spectrum that no member of the audience receives, and receiving it at a level dominated by the direct sound with almost no room in it.

This is the acoustic content of a very old piece of teaching, which is that a player cannot judge their own tone and needs either a room, a recording or another listener. It is usually explained by bone conduction, which is real and is a much larger effect for singers than for anybody else. For instrumentalists the larger term is geometric: they are standing off-axis, close, in the direct field.

It also explains the specific complaint that a hall “does not give anything back”. A player relies on early reflections returning some of the on-axis sound to them, and a platform designed without that in mind leaves them hearing only their own off-axis near field — which is why stage reflectors exist, and why they are aimed at the players rather than at the audience.

What the picture cannot show #

The piston model is an idealisation, and a violin is not a piston. The polar patterns drawn here are for a flat circular radiator in a baffle, which is a good model for a brass bell and a poor one for a violin’s top plate or a flute’s two openings. What transfers is the scaling — round below $ka = 1$, beamed above it, with the crossover set by the size — and not the detailed lobe structure of any particular instrument.

A real instrument’s pattern is not axially symmetric. A violin’s differs above and below, and in front and behind, in ways that need a full spherical measurement to describe. Those measurements exist and are large data sets, not a curve.

Nothing here is about the room’s own effect on frequency. Absorption is frequency-dependent — a room absorbs high frequencies far more than low ones — so the reverberant field is duller than the direct sound for a second reason that has nothing to do with directivity.

And no listener has one ear. Direction is computed from the difference between two of them, and a listener off-axis from a directional source is receiving a spectrum that differs between their ears as well as from the on-axis one.

The measurement problem this creates #

One practical note, because it bears on every figure elsewhere on this site.

Measuring an instrument’s spectrum requires choosing a microphone position, and the choice is not neutral. The two defensible answers are the on-axis spectrum, which is what a listener directly in front receives, and the power-averaged spectrum, which integrates over every direction and is what a reverberant field carries. They differ by many decibels at high frequencies on a directional source, and the literature reports both, sometimes without saying which.

The site’s own figures compute spectra from an instrument’s parameters rather than measuring them, which sidesteps the choice and replaces it with a different assumption: that every partial the source produces is equally available. That is the on-axis idealisation, and it is stated here rather than left implicit.

Whose instruments, and when #

The piston-radiator theory is Rayleigh’s and is nineteenth-century. Systematic measurement of orchestral instruments’ directivity is twentieth: Jürgen Meyer’s Acoustics and the Performance of Music, first published in German in 1972, is the standard reference and contains polar patterns for most of the orchestra measured in third-octave bands. Modern spherical-array measurements from the 2000s onward have refined them and have not overturned them.

The claims about seating and reflectors are claims about Western concert-hall practice as it developed in the nineteenth and twentieth centuries. They are not universal: a gamelan, a Javanese pendopo and a Balinese courtyard are an ensemble and a space designed together under quite different assumptions, and a mosque or a cathedral is a room whose acoustics were chosen for a voice and not for an orchestra.

Where this goes #

This closes the instruments phase and the room ladder’s third rung, and what it leaves is a gap this site should be explicit about. Every spectrum drawn anywhere on the site is an on-axis spectrum, computed from an instrument’s own parameters, with no direction attached. That is the right thing to draw — the alternative is a family of curves nobody can read — and it is an idealisation that this essay is the record of.

The other direction is back into the field this phase opened. A tube’s end correction and its radiation efficiency are the same physics seen twice: what escapes is what fails to reflect, and $ka$ governs both. An instrument that is loud is one that reflects poorly, is hard to keep in tune across its range, and points. Three properties, one parameter, and every instrument in the orchestra is a decision about where to set it.