Field

Timbre and acoustics

What makes a clarinet a clarinet, and what a room does to it before it reaches an ear.

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.

Four spectra of the same note. The amplitude of each partial for 4 timbres at the same pitch — pure, string, clarinet, bell. These are the exact lists the sound buttons here synthesise from, so the picture and the sound are the same data.

The ear hears the list, not the shape

Two sounds with the same partials and different phases have completely different waveforms and sound identical. What the ear extracts is a list of frequencies and strengths, and everything else is discarded.

Three envelopes. How loudness changes over the life of a note, for plucked, bowed and struck. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum.

The shape of a note, which is most of what an instrument is

Cut the first fifty milliseconds off a recorded piano and listeners stop calling it a piano. The attack carries more identity than the steady tone it leads into, and it is the part every spectrum plot leaves out.

A small room's lowest modes. The first few axial standing waves of a room, drawn in plan, with the frequency of every mode below 160 hertz listed underneath. The low modes are far apart in frequency, so some bass notes are loud in one corner and absent in another. The sound buttons play these two octaves above their real pitch, because a room's lowest modes are below what most speakers reproduce.

The room is part of the instrument

A room has frequencies it supports and frequencies it will not. In a small one those frequencies are far apart, so some bass notes are loud in one corner and absent in another — and no equipment fixes it.

The vowel in "hod", sung at 110 Hz. The partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung.

A vowel is two resonances

The vowel in "heed" is the same vowel sung high or low, and nothing about it is a property of the note. It is two peaks in the response of the mouth, sitting at fixed frequencies while the partials of the voice slide underneath them.

Where a tuned piano actually sits. The departure from equal temperament of every key of a small upright, computed from the stiffness of its strings and the fact that a tuner sets octaves without beats rather than at a ratio of two to one. The treble ends up 52 cents sharp and the bass 18 cents flat, and neither is an error.

The piano is tuned wrong on purpose

Every well-tuned piano has a sharp treble and a flat bass, by up to a third of a semitone at the extremes. It is not an error, it is not a compromise about keys, and it follows from one property of a steel wire that can be computed from its diameter and its length.

Reverberation time, by two formulas. Sixty-decibel decay time against average absorption, divided by the room's volume-to-surface ratio so that every room sits on the same pair of curves. Sabine's equation, which is the one every textbook gives, and Eyring's correction to it. They agree in the reflective rooms Sabine measured and separate above ᾱ ≈ 0.18: at 0.6 Sabine reads 53% high, and at ᾱ = 1 — a room whose walls absorb everything, which is the outdoors — it still returns a positive time for a space with no reverberation at all. Marked: a concert hall 1.59 s, a stone church 3.37 s, a studio live room 0.45 s, a carpeted bedroom 0.13 s, an anechoic chamber 0.03 s.

How long a room rings, and where the formula stops

Sabine's reverberation time is one line of arithmetic — volume over absorption — and it built the modern concert hall. It also predicts that a room whose walls absorb everything still rings, which is a room with no reverberation at all, and the error is largest in exactly the rooms most music is now made in.

Three attacks, the first 50 ms. How loudness changes over the life of a note, for plucked, bowed and struck, drawn over the first 50 milliseconds. By the right-hand edge the plucked note is at 91%, the bowed note is at 36%, the struck note is at 97% — attack times of 4 ms, 140 ms, 2 ms, a spread of 70 to one, and the part a listener uses to tell them apart. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum.

The first fifty milliseconds

A spectrum is supposed to be what makes a trumpet a trumpet. Cut the first fifty milliseconds off a recorded note and listeners stop being able to name the instrument — while the spectrum they are hearing is unchanged. Identity is in the part of the sound that ends before the note has properly started.

Partial 3, mistuned by 3%. A 10-partial tone on 220 Hz with one partial treated differently from the rest. Mistuning it moves it off the harmonic grid by 3.0 per cent, which is 19.8 Hz — slow enough to be heard as a beat rather than as a separate pitch, and enough for the partial to be heard out of the note as a whistle of its own. An onset difference does the same to a partial that is exactly in tune.

What makes two partials one note

A note is a stack of ten or twenty simultaneous tones and is heard as one thing. The obvious explanation is that they are whole-number multiples of a fundamental — and the obvious explanation is not sufficient. Mistune one partial by three per cent and it leaves the note; give a perfectly harmonic partial a thirty-millisecond head start and it leaves too. Shared behaviour beats arithmetic.

A bowed string on 196 Hz, through a violin body. The source is a sawtooth at one over n; the filter is the body's measured response, with A0 at 275 Hz, B1− at 460 Hz, B1+ at 540 Hz, bridge hill at 2500 Hz. What is radiated is their product, drawn as the bars. The resonances stay where they are when the note changes, exactly as a vowel's formants do — which is why an instrument has a voice rather than a tone, and why the same argument that identifies a vowel identifies a violin. The body frequencies are measured means over full-size instruments rather than computed from a plate.

The body is the filter

A violin string radiates almost nothing. What reaches a room is the string's sawtooth multiplied by the body's response, and that response is a comb of measured resonances that stays put while the note moves. It is the same arithmetic that identifies a vowel, on wood instead of a mouth — which is why an instrument has a voice rather than a tone.

A string mode swept through a body resonance at 460 Hz. What the string plays against what comes out. Away from the resonance the two are the same and the line is the diagonal. Near it the mode splits into a pair, and the note warbles at the difference between them — 12.9 Hz at the centre, which is slow enough to be counted and far too fast to be a tremolo. The splitting is a coupled oscillator and has nothing to do with the wolf fifth of a tuning system, which is a twenty-three cent arithmetic residue and shares only the word.

The other wolf

A cellist's wolf note is a string mode landing on a body resonance, at which point the two stop being separable and start exchanging energy — the mode splits in two and the note warbles at the difference. It is a coupled oscillator. The tuning system's wolf is twelve fifths failing to close by 23.5 cents. They share a word and nothing else.

Where each frequency goes, from a source 18 cm across. Polar response of a circular radiator of radius 9 cm at 200 Hz (ka = 0.3), 800 Hz (ka = 1.3), 2000 Hz (ka = 3.3), 5000 Hz (ka = 8.2). Zero degrees is straight ahead. The low frequency is a circle — it goes everywhere — and the high one is a narrow lobe with nulls either side of it, so a listener off to the side hears the same note with its top missing.

An instrument points

A source radiates evenly while it is small compared with the wavelength and beams once it is not, and the crossover is one number. So the same instrument is omnidirectional in its bottom octave and a searchlight in its top one — which means its spectrum depends on where the listener is standing, and a microphone position is a choice about what the instrument sounds like.

6 chords in a gothic cathedral. Each chord's reverberant decay in a room with a 8 second reverberation time, at 1 chord a second. Decay is linear in decibels, so each line is straight with a slope of -7.5 dB a second. When a chord arrives, 2 earlier ones are still above 20 dB down.

The room chooses the harmonic rhythm

A chord in a cathedral is still sounding, seven decibels down, when the next one arrives — and the one after that, and the one after that. Reverberation is linear in decibels, so the number of chords audible at once is one number divided by another, and it puts a hard ceiling on how fast a composer writing for that building can change harmony. The ceiling is computable, and the music written for those rooms sits under it.

Where each room stops being a set of resonances. The Schroeder frequency of 6 rooms — a carpeted bedroom at 217 Hz, a domestic living room at 183 Hz, a rehearsal room at 120 Hz, a jazz club at 68 Hz, a shoebox concert hall at 21 Hz, a gothic cathedral at 36 Hz — marked on a logarithmic frequency axis with the ranges of 4 instruments underneath. Below the mark a room is a handful of separable modes and a note's loudness depends on where the listener is standing; above it the modes overlap and the room is described by one decay time.

Where a room stops being a room

A small room has frequencies it supports and frequencies it will not, and a hall has a reverberation time. Those are two separate accounts and they are descriptions of the same building at different frequencies. The crossover is one formula, and in a bedroom it lands at about two hundred hertz — in the middle of the bass register, and above nothing at all in a concert hall.

Six reverberation times for one room. Sabine's arithmetic evaluated in each octave band from the published absorption coefficients of the surfaces. a large stone church runs from 6.3 seconds at 125 Hz to 2.4 at 4 kHz — a bass ratio of 1.38, where concert halls are specified between 1.1 and 1.25.

A room does not decay evenly

Sabine's arithmetic gives one number and absorption is a strong function of frequency, so a room has six reverberation times rather than one. A stone church rings for 6.3 seconds at 125 hertz and 2.4 at 4 kilohertz, which means a chord left in it does not fade — it changes shape, losing its top before it loses its bottom, and arriving at the listener as a different sonority from the one played.

The first eighty milliseconds are a different room

Draw a line across a room's decay and the energy on either side is two opposite verdicts about one building — clarity before it, reverberation after. The line is a property of the ear, not of the room, and putting it at 50 milliseconds and at 80 makes the two published design targets fall out — a room for speech at one second, a room for music at 1.6.

The roughness curve for ideal bar. The partials are at 1 : 2.756 : 5.404 : 8.933 : 13.340 times the fundamental, which is not a harmonic series, so nothing about where the wells fall can be read off the small whole numbers. Sensory dissonance between two complex tones as the upper one is swept through 13.00 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit 7 cents below 3/2, 14 cents below 5/3, 13 cents above 7/4, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

The spectrum that was supposed to explain the gamelan

Run the model that designed the Bohlen–Pierce scale on a bar's partials and it asks for a compressed pseudo-octave at 1166 cents and wells at 694 and 870. A measured slendro's degrees are at 231, 474, 717 and 955, and only one of the four is near anything the model wants. Sweep the partials and a spectrum wanting a 240-cent step can be found — sitting sixty per cent of the way up its own roughness curve, which is what a scale-level test says about the whole comparison.

Long-term average spectra: an orchestra, playing forte against a trained operatic soloist. Each source's mean spectrum over a long passage, in decibels below its own strongest region, on a logarithmic frequency axis. An orchestra, playing forte peaks at 250 Hz and is 30 dB down by 3,150 Hz; a trained operatic soloist peaks at 250 Hz and is 11 dB down by 3,150 Hz. The shapes are the same until about 1 kHz and separate above it: at 3153 Hz the difference is 19.0 decibels, which is the largest anywhere in the range. Nothing here is about level. Both curves are drawn against their own peaks, so what is being compared is shape.

One voice over ninety players

A soloist heard over a full orchestra is not louder than it and could not be. What the trained voice does instead is put a peak of energy at three kilohertz, which is where the orchestra's spectrum has already fallen away and where the ear's own threshold happens to be lowest. Nineteen decibels of advantage, in a place nobody is competing for.

$What gets out of an opening, for 4 openings. The fraction of the wave's energy radiated at an open end against frequency, in the baffled-piston model — the radiation resistance of a circular piston, normalised to the tube's own impedance. Each curve runs from nothing at the bottom, where the opening is far smaller than a wavelength and the wave simply turns round, to everything above ka ≈ 2. Half the energy leaves at 6364 Hz for a flute's embouchure end (radius 10 mm), 2015 Hz for a clarinet's bell (radius 30 mm), 975 Hz for a trumpet's bell (radius 62 mm), 403 Hz for a horn's bell (radius 150 mm). The crossover goes as one over the radius, so the widest and narrowest here are 15.8 times apart in frequency. The same number decides how strongly the tube resonates and how much sound it makes, which is why a bell cannot brighten an instrument without also weakening its own resonances.$

The bell decides what gets out

A tube resonates because the wave turns round at the open end, and it is audible because some of the wave does not. Those are the same number with opposite signs. One quantity — the size of the opening against a wavelength — decides how loud an instrument is, how bright it is and how directional it is, and a bell moves the boundary rather than removing it.

The sound a listener knows best

A voice is recognisable across every vowel it says, across two octaves of pitch, down a bad telephone line and in a whisper where there is no pitch at all. Nothing that survives all of that can be a frequency. What survives is a ratio: the resonances of a vocal tract are set by its length, so a shorter tract multiplies every formant by the same factor, and identity is a scale on the spectral envelope rather than a position within it. Between an adult man and a child the whole pattern moves by a fifth, and the vowel does not change at all.

$The fluctuation stays; the rate goes. A unison of n voices with a spread of 15 cents, averaged over 5 draws. The depth of the amplitude fluctuation does not fall as voices are added — a choir is no steadier than a duet — but the fraction of that fluctuation in any single modulation component falls from 77 per cent at two voices to 24 at 32. Two voices make one beat and it can be counted; 16 make 120 and none of them is a rate. That is why a choir cannot be tuned by nulling anything.$

What a choir does that a soloist cannot

Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift this site measured describes only ensembles that hold their pitch still.

A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there.

The mark that is not a level

There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes.

A note gets duller as it dies. Each partial of a string note against time, with the loss rising as the partial number to the power 1 — so the fundamental takes 6 seconds to fall sixty decibels and the 8th takes 0.75. The heavy line is the power-weighted centroid, falling from partial 1.77 toward the fundamental; it is halfway there after 0.15 seconds. A single-rate envelope would draw all of these as parallel lines and the centroid as a horizontal one, and a struck string does neither: what is left at the end of a long note is very nearly a sine.

The note that gets duller as it dies

Every envelope drawn so far is one curve applied to a whole sound, and no struck string behaves that way. A string loses energy to air, to internal friction and to the bridge, and all three losses rise with frequency — so a note with a six-second fundamental has a sixteenth partial that is gone in under half a second, and the sound moving toward the listener is a spectrum collapsing toward its own fundamental. Which means an instrument is identified twice: once by the fifty milliseconds of its attack, which the earlier essays measured, and again by how fast its colour drains, which they did not.

One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string.

An instrument is not one timbre

Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

How many intervals a duet is smooth at. Every ordered pairing of 5 radiators, with the number of wells in its dissonance curve over an octave from 262 hertz. Rows are the instrument underneath and columns the one above, so the grid is not symmetric about its diagonal and that asymmetry is the result. The count runs from 1 to 7 across the grid, and a cell and its mirror need not agree: a violin under a clarinet has 1 and a clarinet under a violin has 2. The fifth is a well in every one of the 25 pairings and no other interval is.

Which instrument is underneath

Every roughness curve so far compares two tones of the same timbre, which is a duet nobody plays. Give the two notes different instruments and the sum stops being symmetric: the same written interval, on the same two players, is up to five times rougher depending on which of them takes the lower note. From G3 upward the fifth is a well in all twenty-five pairings and no other interval is; below it, three pairings lose even that, and all three have a clarinet on top.

Six ways to put three players on three notes. The same chord — G3, B♭3, D4 — played by clarinet, oboe, voice in all 6 possible assignments, scored by the roughness each produces. Every bar is the same pitches and the same instruments; only who is on which note changes. The worst is 1.42 times the best, which is a factor a score can control and a chord symbol cannot express at all. Each row is labelled from the bottom note upward.

Which player on which note

An interval's roughness depends on which instrument is underneath, so the pair does not commute. Three players over three notes is the smallest thing that asymmetry has anywhere to go: six assignments, all of them the same chord, and across 450 of them the roughest averages half again the smoothest and reaches six times it. It is orchestration in the only form that can be computed here — not which chord, and not which voicing, but who is on which note.

How far equal temperament puts each interval's coincidence out. For each interval, the pair of partials it brings together and how many cents equal temperament mistunes that coincidence by. A fifth's third-against-second is out by 2 cents and a major third's fifth-against-fourth by 14 — so the same temperament that is inaudible on a fifth produces, at 220 hertz, a beat of 8.7 per second between two sections singing a third, with every singer in both of them perfectly in tune.

A section against another section

The choir has been treated as a unison, and no choir sings only unisons. Two sections an interval apart beat between partials rather than between fundamentals — the third brings the fifth partial of one against the fourth of the other — and equal temperament puts that coincidence fourteen cents out. So two sections singing a tempered third beat at nearly nine per second with every singer in both of them perfectly in tune, and the same temperament is inaudible on a fifth.

C4: the pulse computed and the pulse assumed. Above, the force the hammer delivers to the string at C4, integrated forward against the felt's nonlinear force and the string's returning corner, drawn against the half-sine of 1.60 milliseconds that every earlier figure assumed. The computed contact lasts 2.21 milliseconds and the corner comes home 4.6 times inside it. Below, the excitation each pulse gives to each partial. They agree at the bottom and part company higher up — worst at partial 6, by 34 decibels — because the assumed pulse has nulls the computed one does not.

The pulse that was assumed

Every figure until now low-passes the string's excitation with the spectrum of a half-sine, which is what a hammer would deliver against a rigid wall. An earlier essay said so and declined to do better. Doing better takes forty lines and refuses the prediction that came with it: the corner's round trips govern the spectrum as expected, and the contact time is governed by something else entirely — the mass ratio discovered one essay earlier.

What a trumpet mouthpiece does to the series. Each bore's modes before a mouthpiece is fitted and after, on the axis that says which harmonic each mode is: zero means the m-th mode sits at m times the spacing and is the m-th harmonic. Every shape starts near -0.14 and ends near 0.05. The cup holds 3.0 millilitres against a throat 3.7 mm across and pops at 642 hertz. What does not improve is the regularity: the spacing stays as uneven as the flare left it.

What the mouthpiece is actually for

A cup and a throat look like a comfort fitting and are a component. Put a real one on the front of a real bore in the horn equation and the whole mode series slides sideways by about a seventh of its own spacing — which is exactly the distance between a series whose second resonance is 1.85 times the spacing and one whose second resonance is the second harmonic. The flare decides whether the modes are evenly spaced; the mouthpiece decides which harmonic each one is.

The partial above which the returning corner has lost its phase. Up a piano, the lowest partial whose extra travel after one round trip to the near end exceeds a quarter of a cycle — above it, the component comes home somewhere other than where a corner needs it. The curve follows the inharmonicity coefficient, which is smallest in the tenor and rises both ways, so the corner survives best at A2 — to the 24th partial — and worst at A7, where only 5 partials are still in step. The earlier coupled model assumes a clean corner at every partial it uses.

The corner does not come back a corner

Stepping a hammer forward against the string's own returning wave showed that the coupling matters most in the bass. It assumed the wave that came back was the shape that left. On a stiff string it is not: partials travel at different speeds, and by the top octave of a piano the sixth partial is the first to arrive out of place. But a periodic wave can only see phase modulo a cycle, and counted that way twenty-one of forty-eight partials are still in step — which is why the returning wave is degraded rather than destroyed.

Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.

$Which of the four terms is doing the removing, note by note. Each of three terms has a corner — a partial number above which it is the one taking the energy away — and the lowest corner at each pitch is the term that binds. The comb is flat at 8, because where the hammer lands is a fraction of the length and does not change with the note. The contact time's corner falls from 34 at A0 to 0.5 at A7. They cross at about E3, 165 hertz: below it the strike point decides the spectrum and above it the contact time does. The dispersion's corner is above both at every pitch on the instrument — 5.8 at the top against a contact corner of 0.5 — so that earlier term is real and is nowhere the binding one. That is a statement no one of them could make on its own.$

Four terms, and only one of them binds

Six earlier essays each computed one term of a piano's excitation spectrum and handed it to the next, and nobody had multiplied them. Multiplied, three of the four have a corner — a partial above which they are the term doing the removing — and putting the three corners on one axis says which is in charge at each pitch. Below E3 the strike point decides; above it the contact time does; and the dispersion, which is the newest and most laborious term, is nowhere the binding one.

The flare that makes a series harmonic

Every figure until now treats a partial as n times a fundamental, and the whole of brass instrument design is the business of making that true. A plain cylinder is 127 cents from a harmonic series and a plain cone is 21; the best flare found by sweeping is 4.6, and only two per cent of the swept surface comes within five cents of it. The shape is forced rather than chosen.

$Where each exciter's contact corner sits against its comb. The merger's four corners, drawn for three exciters. The comb is the strike point and is a horizontal line, because it is a fraction of the string and has no pitch in it. The contact corner is the exciter's, and it falls up the compass because the contact time is a fixed number of milliseconds against a shrinking period. Where they cross, the binding term changes. piano hammer crosses at E3; harpsichord plectrum crosses nowhere in the compass; dulcimer beater crosses at A5. So the crossing found earlier is not a fact about pianos. It is a fact about compliance: a felt hammer stays on the string for about 1.6 milliseconds and a plectrum for 0.05, and a difference of that size is a corner in a different place.$

The crossing belongs to the felt

On a piano the strike point decides the top of the spectrum below E3 and the hammer's contact decides it above. The merger's own bookkeeping asked whether that crossing is a fact about pianos or about hammers. A plectrum never crosses at all and a hard beater crosses two octaves higher, and the boundary is a contact time of about an eighth of a millisecond — ten times shorter than felt.

The bell and the cup, on one surface. Every combination of a flare exponent and a cup depth, shaded by how far the resonance series is from a harmonic series over partials 2 to 8, in cents. The pale line is the best flare for each cup — the ridge — and it runs diagonally: the exponent that suits the shallowest cup here is 0.93 and the one that suits the deepest is 0.70, a spread of 0.22. The ring marks the bell found earlier by sweeping the flare alone, at an exponent of 1.00 — which reaches 4.6 cents with nothing in front of it, and sits where the ring is, at 24.9 cents, once a catalogue mouthpiece is on it. It is on none of the ridge, and that is the whole finding: the two choices are not separable, and a bell optimised alone is not the bell an instrument wants.

The bell is tuned for the cup

Sweeping a brass bell's two flare parameters found a shape that makes the resonance series harmonic to four and a half cents. Bolt the mouthpiece that instrument is actually played with onto it and the series is twenty-five cents out — worse than a plain cone. The flare and the cup are not two independent choices, and the best bell for a maker to build is one that is deliberately wrong.

Intonation is a unison problem and nothing else. The roughness between two instruments on one note, against how far apart they are in cents, drawn for a unison and for the intervals beside it. A perfect unison is 0.0007 — the partials coincide and there is nothing to beat. Five cents apart it is 0.0465, 65 times as rough, and ten cents apart it is rougher than a major third played exactly. The mechanism is that partial n of a note mistuned by c cents is mistuned by c cents as well, which is n times as many hertz — so the top of the spectrum enters the critical band long before the fundamental does. The other curves are flat, because a third's roughness is set by which partials nearly coincide and a few cents does not change which.

Two players on one note

Six essays have put one instrument on each note of a chord, and the commonest thing an orchestrator actually does is put two on the same note. Two independent sources add in power, so the composite is neither of them — except that it nearly always is one of them, because the level at which ownership changes hands is rarely at zero. And a unison ten cents out is rougher than a major third dead in tune.

Who owns clarinet, oboe, voice at every balance. The composite of three players at 392 hertz belongs to whichever of them it is nearest in log-spectral distance, and here that is drawn over the whole plane of balances a conductor could set — the second and third players from 24 decibels below the first to 24 above. voice owns 79 per cent of the square. The three regions meet where all three distances are equal, which is the only balance at which the composite belongs to nobody: it is at -0.3 and -19.1 decibels, inside the square and therefore a balance an ensemble could actually be asked for. A trio has a colour of its own at one point, not over a region.

A section has a loudest member, not a colour

Two players on one note have a balance at which the composite belongs to neither, and that is what blending means. Three should have three such balances and no reason for them to agree — a trio with a rock-paper-scissors ownership would have no strongest member at all. Twenty trios, sixty pairwise comparisons, and not one disagreement: the possibility is real, arbitrary spectra do it once in twenty, and instruments never do.

Which pairs blend is a question about the note. The level at which a doubled pair's composite changes owner, drawn for all 15 pairs of 6 radiators over 2.6 octaves from 131 to 784 hertz. A pair blends when that level is inside the shaded band, which is the twenty-four decibels either way two players can manage; a curve outside it, or absent, is a pair one instrument owns at every balance. 6 of 15 pairs blend at the bottom of the range and 12 at the top. Every filter in this collection is fixed in frequency and the fundamental is not, so a radiator's shape is a function of pitch and so is everything computed from two of them — the blend ranking at the bottom and at the top disagree on 70 of 105 comparisons, which is more than half, so the order has turned over rather than merely shuffled.

The blend table has a row for every note

Eight earlier essays sound their instruments at one note, and one of them says why that cannot be innocent: every filter here is fixed in frequency and the fundamental is not. Swept over four octaves, the number of pairs that blend doubles from six to twelve, the ranking turns over rather than shuffles — seventy of a hundred and five comparisons swap — and a clarinet with an oboe goes from the best pair in the collection to the eleventh.

The bare bore does not care which valve is down. The distance in cents of partials 2 to 8 from the harmonic series that fits them best, at each valve combination of a B♭ trumpet, computed with the mouthpiece in place and again with the bore alone. The bare bore is flat — 0.78 cents from best to worst across the whole set — so the length changes nothing. With one cup serving all of them the same bore runs 19.53 to 24.83 cents, a spread of 5.30, because the cup's popping frequency stays at 642 hertz while the series underneath it drops.

One cup and seven lengths

A trumpet is seven tubes with one mouthpiece serving all of them, and the harmonicity of its resonance series is a different number in every position — 19.5 cents open, 24.8 with all three valves down. The bare bore is flat across the same seven lengths to within eight-tenths of a cent, so none of it is a length problem. It is the cup, standing still while the series walks past it, and pressing all three valves does to the series exactly what fitting a mouthpiece 56 per cent of the catalogue depth would do.

The top that falls while the note lasts

Four earlier essays have asked where the harmonic series stops, and all four answered with a number computed for a tone that never ends. A struck C3 has 152 audible partials at the strike and eight after two-thirds of a second, so the ear's own resolution limit governs the first ten per cent of the note and the decay governs the rest. Playing ten decibels louder buys the ear a further tenth of a second, and doubling its reign would take fifty-eight.

A struck note's two ends are the same for every loss law. The partial levels of a string spectrum struck at 80 decibels on 130.8 hertz, and what is left of it when the fundamental itself falls under the threshold of hearing, for three laws relating a partial's decay rate to its number. The left panel is every one of them: a loss law cannot change the spectrum at the instant of the strike, because no time has passed. The other three are every one of them too: whatever the law, the note ends with nothing above the threshold. So both ends of the slide are shared, and everything that distinguishes an exponent of 0.5 from an exponent of 1 from an exponent of 2 is in the middle.

The middle nobody could have guessed

A struck note has no steady state, only a slide from one spectrum to another — so the question is what the middle carries that the ends do not. The answer is exact rather than statistical: every loss law in the family leaves the strike with the same spectrum and ends in the same silence, so both endpoints carry precisely nothing about which of them it is. The whole difference is 41.3 decibels, and it peaks 0.38 seconds in, seven per cent of the way through the note.

The collapse belongs to the bass

Every envelope drawn until now has no pitch in it: the decay model counts partials rather than measuring them in hertz. Put a fundamental in and the collapse it describes shrinks monotonically up the keyboard, from 25.2 semitones of colour drain at A0 to 6.6 at C8, because a partial has to fit under twenty kilohertz to exist and the top note has four of them. Above C6 a struck note has fewer partials than the ear could have resolved, and there is nothing left for a collapse to take away.

Both notes of a fifth, and the coincidences going out. The highest partial still above the threshold of hearing for each note of a fifth struck at 80 decibels on 130.8 hertz, against time, with a 1/n spectrum and a loss rising as the partial number to the power 1. Both curves come down from off the top of the frame — the lower note starts with 152 partials and the upper with 102, because twenty kilohertz is a ceiling in frequency and not in partial number. The rings are the interval's partial coincidences at the moment they stop existing — successive multiples of one ratio, which go out from the top down: 12:8 at 0.47 s, then 9:6 at 0.64 s, then 6:4 at 1.04 s, then 3:2 at 2.14 s. The lowest, 3:2, is the last, and after it the two notes have no partial in common that either of them can still supply.

A fifth on a piano is not a fifth a second later

Nine essays draw one note in silence, and a listener is given a texture. Put two struck notes an interval apart and consonance comes apart into two quantities that had agreed while nothing moved: the partial coincidence that names the interval outlives the strike in exactly the order common-practice theory ranks its intervals, and the roughness that scores it reorders itself inside a third of a second, with the fifth overtaken by four intervals every treatise calls harsher.

The blend arrives before the note does

Nine essays on spectrum draw a steady state, and the strongest cue that two instruments are two instruments is that they do not start together. Two envelopes rising at different rates turn out to be a balance — the same dial an earlier essay swept — so the attack is that dial moved by the clock, and its whole travel is fixed at twenty times the log of the two attack times. It is six decibels for a clarinet with a violin against a crossing twelve to twenty-two decibels out, so one pair in ten changes hands during its own attack, and which one depends on a convention rather than on the instruments.

An inversion lasts as long as its outer sixth

A struck interval keeps the partial coincidence that names it for a time set by its ratio, and a chord is three intervals at once. Voiced over one bass and struck on a piano, a triad keeps the evidence of all three only as long as its weakest one lasts, and for an inversion that is the sixth on the outside: a major sixth lasts as long as a major third, a minor sixth dies first. So the major six-four and the minor sixth chord are the most durable voicings of their triads and the major sixth chord and the minor six-four the least — and unlike a dyad, a triad's inversions keep their order by roughness through almost the whole decay.

A doubled pizzicato gives its note away while it is still the louder. The power of a violin plucked, against a flue pipe holding the same note at 392 hertz, through the first 600 milliseconds of the pluck, with the pluck starting 12 decibels up and its fundamental decaying over 1 second. With each partial losing level in proportion to its number, the composite stops resembling the pluck at 70 ms, when the pluck is still 5.2 decibels the louder. With every partial fading together it would keep the note until 543 ms. The dashed line is the balance at which the steady-state doubling changes owner, minus 20.6 decibels: the release crosses the owner long before its balance gets there, because what hands the note over is the pluck's upper partials going, not its level.

A doubled pizzicato gives its note away early

The attack turns the balance between two players on one note by a few decibels and stops. A pluck does not stop — every partial of it decays, so a pizzicato doubled by a held instrument walks the balance for the whole note, and the expectation was a handover as slow as the decay. It is fast. A one-second pizzicato over a flute loses its note in 70 milliseconds, while it is still five decibels the louder, because what hands the note over is its upper partials going first. A uniform fade would have kept it eight times as long.

How long a doubled pizzicato keeps its note, seat by seat, in two rooms. How long a doubled violin pizzicato on 392 hertz keeps its note against the metres from the players, the pluck starting 12 dB up and decaying over 1 s with a loss exponent of 1. a concert hall, a flue pipe: 1 → 86 ms, 1.5 → 123 ms, 2 → 226 ms, 3 → 359 ms, 5 → 445 ms, 7 → 481 ms, 10 → 506 ms, 15 → 522 ms, 20 → 528 ms, 30 → 532 ms; a concert hall, an oboe: 1 → 52 ms, 1.5 → 55 ms, 2 → 59 ms, 3 → 77 ms, 5 → 149 ms, 7 → 195 ms, 10 → 224 ms, 15 → 242 ms, 20 → 248 ms, 30 → 254 ms; a concert hall, a clarinet: 1 → 44 ms, 1.5 → 45 ms, 2 → 45 ms, 3 → 47 ms, 5 → 53 ms, 7 → 65 ms, 10 → 86 ms, 15 → 105 ms, 20 → 112 ms, 30 → 118 ms; a large stone church, a flue pipe: 1 → 440 ms, 1.5 → 578 ms, 2 → 651 ms, 3 → 728 ms, 5 → 784 ms, 7 → 803 ms, 10 → 814 ms, 15 → 820 ms, 20 → 822 ms, 30 → 824 ms; a large stone church, an oboe: 1 → 56 ms, 1.5 → 65 ms, 2 → 89 ms, 3 → 207 ms, 5 → 281 ms, 7 → 303 ms, 10 → 315 ms, 15 → 322 ms, 20 → 325 ms, 30 → 326 ms; a large stone church, a clarinet: 1 → 44 ms, 1.5 → 43 ms, 2 → 43 ms, 3 → 44 ms, 5 → 53 ms, 7 → 67 ms, 10 → 80 ms, 15 → 89 ms, 20 → 92 ms, 30 → 94 ms. The mid-band critical distance is 5.3 m in a concert hall and 2.3 m in a large stone church. In none of the 60 cases does the note return to the pluck once it has left.

A room keeps a pizzicato from giving its note away

Doubled by a flute, a one-second pizzicato loses its note in 70 milliseconds dry, because its upper partials go first. The question left open was whether a room, whose reverberation keeps those partials alive, gives the note back afterwards. It does not give it back. It stops the note going: ten metres into a concert hall the pluck keeps it for 506 milliseconds, in a stone church for 814, and the room's own uneven decay takes back between a quarter and two fifths of that. In a room the loss law that decided everything dry matters a tenth as much, because the room's decay has become the clock.

A bow shows the loss law a blow conceals. The same string on 130.8 hertz under three loss laws, drawn twice each: struck, and held by a continuous drive. The pale marks are the spectrum a blow produces, and they are identical in all three rows — a strike is the source spectrum and has no loss in it yet, which is why both endpoints of a struck note were found to carry nothing about the law. The solid marks are where each partial settles when a drive balances its own loss, at drive over loss, so the steady spectrum rolls off as the source's roll-off plus the exponent. At an exponent of 0.5 the held spectrum's centroid sits at 167 hertz, 5.7 semitones under the strike's 232; At an exponent of 1 the held spectrum's centroid sits at 144 hertz, 8.2 semitones under the strike's 232; At an exponent of 2 the held spectrum's centroid sits at 133 hertz, 9.6 semitones under the strike's 232. The quantity that is invisible at both ends of a struck note is the slope of a bowed one, for as long as the bow moves.

A bow holds the number a blow hides

Two struck notes with different loss laws are identical at the strike and identical at the end, which is why separating them at all meant looking in the middle. Drive the same two strings continuously and the loss law stops being a rate and becomes a slope: each partial settles at its drive over its own loss, so the exponent adds to the source's roll-off and sits in the spectrum for as long as the bow moves. It is 18.7 decibels of separation available from the first instant, against 41.3 that a blow delivers after four tenths of a second and then takes away.

The room is the slower of the two

A reverberant field is the source convolved with the room, so a partial's tail falls at the slower of the two rates rather than at their sum — and the room is slower for exactly the partials the string is losing fastest. Half a note's colour is gone in 0.163 seconds in no room at all, 0.313 in a concert hall and 1.441 in a stone church. The destination is identical in all three, because a room cannot hold a partial up above the fundamental it is also holding. What a hall takes away is the rate, and the rate was the whole of the identity cue.

The drain does not stop when the key does. The note does.. Semitones of colour gone, against time, for a note on 130.8 hertz left to ring and for the same note released after 0.4 seconds onto a damper of 0.15 seconds. The two curves lie on each other until the key comes up, and the damped one then ends: the note is inaudible at 0.52 seconds with 8.7 of the free note's 9.9 semitones delivered. A damper adds one loss to every partial alike, so it adds the same number to every decay rate and leaves every DIFFERENCE between rates exactly as it was — the spectrum at each instant is the ringing spectrum shifted bodily down by 400 decibels a second. The colour goes on draining at its own rate the whole time. What the damper takes away is not the drain but the seconds.

A damper changes the clock, not the colour

A damper is an extra loss on the string rather than a second decay, so it adds the same number of nepers a second to every partial — and adding a constant to every rate leaves every difference between rates exactly where it was. The damped spectrum at any instant is the ringing spectrum at that instant shifted bodily down, to machine precision. The colour goes on draining at its own rate; the note simply runs out of seconds, and how many it gets is written on the page as a note value and a tempo.

A room pulls the compass apart rather than evening it out. How long a pizzicato entering 6 decibels above a held note keeps the composite spectrum, at eight pitches across two and a half octaves, heard 15 metres from the stage. no room: 0.07, 0.05, 0.06, 0.06, 0.02, 0.05, 0.05, 0.04 seconds; a concert hall: 0.61, 0.27, 0.50, 0.38, 0.01, 0.25, 0.27, 0.19 seconds; a large stone church: 1.03, 0.39, 0.79, 0.56, never, 0.33, 0.35, 0.23 seconds. Dry the figures barely move — a spread of 3.0 across the whole compass — because a room is the thing that varies with frequency and there is none. In a hall the spread is 44. The room does not scale the dry answer by a constant: it multiplies it by between four and nine times depending on the note, and at C6 it makes the pluck's position worse rather than better, because the two instruments' spectra nearly coincide there and the pluck starts only 4.0 decibels ahead instead of twelve.

One note in the compass loses its pizzicato

Dry, how long a pluck keeps the composite spectrum barely depends on which note it plays: three-hundredths of a second at the worst pitch and seven at the best, a spread of three. In a concert hall the same eight notes spread by a factor of forty-three, and in a stone church one of them never gets the note at all. The room does not scale the dry answer by a constant — it multiplies it by between four and nine times depending on the pitch, and at the one note where the two instruments' spectra nearly coincide it makes the pluck's position worse instead of better.

A damper is a loss on the string, so a room can overrule it. Decay rate in nepers a second against partial number, for a note on 130.8 hertz in a room of 2 seconds. The rising line is the string's own loss, 1.15 nepers a second at the fundamental and growing as the partial number to the power 1. The line above it is that plus the damper's 46.1, which is what the string does once the key comes up. The flat line is the room. What a listener receives is the SLOWER of the damped string and the room, because a hall goes on radiating what the string has already given it — and here the room is slower on 8 of 8 partials, from the fundamental upward. The composition proposed earlier — take the slower of the string and the room, then add the damper to whichever won — would put the damper outside the minimum, where nothing can overrule it, and would predict a note 2.54 seconds shorter than ringing where the arithmetic here predicts 0.90.

A damper cannot reach into the room

The essay before this one proposed the arithmetic for a damped note in a hall: take the slower of the string's rate and the room's, then add the damper's to whichever won. The composition is wrong, and it is wrong in the one place that decides the answer. A damper is a loss on the string, so it belongs inside the minimum where a room can overrule it — and past about three seconds of reverberation it is overruled on every partial, so the damper removes no audible seconds of note at all.

No seventh chord can be spaced to last as long as a triad. Every inversion and spacing within 2 octaves over C3, struck at 80 dB, for two triads and five seventh chords: the bar is the longest any spacing keeps every pair's partial coincidence, and the tick is the bound set by the chord's worst pitch-class distance — the longest any presentation of that distance lasts. major triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; minor triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; dominant seventh: 0.66 s over 32 voicings, bound 0.64 set by the tone; major seventh: 0.42 s over 32 voicings, bound 0.38 set by the semitone; minor seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; half-diminished seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; diminished seventh: 0.86 s over 32 voicings, bound 0.87 set by the tritone. Every seventh chord contains a distance worse than any a triad contains, except the diminished seventh, whose distances are only minor thirds and tritones.

A seventh chord cannot be spaced to last like a triad

A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.

A staccato is the direct sound's, and the room takes it within a fifth of the critical distance. A note on 130.8 Hz held 0.4 s and damped, in a room of 2 s reverberation, heard at distances from 0.02 to 5 times the critical distance: how long after the release the note takes to fall 10 dB and 20 dB. To fall 10 dB: 24 ms at the source, 333 ms far away; 0.02: 24 ms, 0.05: 25 ms, 0.1: 25 ms, 0.15: 26 ms, 0.2: 28 ms, 0.3: 35 ms, 0.5: 103 ms, 0.75: 187 ms, 1: 234 ms, 1.5: 281 ms, 2: 302 ms, 3: 318 ms, 5: 328 ms; doubled by 0.38 of the critical distance. To fall 20 dB: 49 ms at the source, 667 ms far away; 0.02: 49 ms, 0.05: 51 ms, 0.1: 60 ms, 0.15: 117 ms, 0.2: 198 ms, 0.3: 308 ms, 0.5: 436 ms, 0.75: 520 ms, 1: 568 ms, 1.5: 614 ms, 2: 635 ms, 3: 652 ms, 5: 661 ms; doubled by 0.14 of the critical distance. Where the direct sound and the room are equal, the damper's work is already hidden: the room's copy is only 20 dB below the direct sound at a tenth of the critical distance, and a 20 dB fall reaches it there.

Only the player hears a staccato end

A damper stops a string in a seventh of a second, and in a hall the room goes on for two. A listener hears both, mixed in proportion to how close they sit, and the question was at what distance the short part stops mattering. The answer is closer than any seat. A damped note's twenty-decibel fall has doubled in length by a seventh of a hall's critical distance — 77 centimetres in a two-second concert hall — and by a quarter of it in a jazz club. The end of a staccato is something the pianist hears and the front row does not.

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