Acoustics & Music Theory
why it sounds like that
Concept

Masking — where it appears

The raising of the threshold for one sound by the presence of another, which decides when an earlier chord stops being heard as harmony. It works forward in time as well as simultaneously, and it reaches further upward in frequency than downward.

Named by 47 essays across 8 fields — each of them below, with the objects they name alongside it.

What one tone hides, and in which direction. The masked threshold beside a tone masker: any probe below one of these curves is inaudible while the masker sounds. The frequency axis is in Bark, the scale on which the ear's filters are evenly spaced, so the pattern is a pair of straight lines. The upper slope is much shallower than the lower one and gets shallower still as the masker gets louder — masking spreads upward, not downward.

One sound hides another, and it hides upward

A tone can be made completely inaudible by a second tone that is nowhere near it in frequency, and the region it disappears into is lopsided. Masking spreads up the spectrum and barely down it, and the reach grows with the masker's level — so which line in an arrangement vanishes is a prediction, not a matter of taste.

perception · Masking 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.

timbre · Room acoustics The most a 6 cm cone can make of a low note. The maximum sound pressure level at one metre from a circular radiator of effective radius 3.2 cm, moving 1.5 mm at its limit, in a system resonating at 250 Hz. Below resonance the cone is already at that limit and the pressure a piston makes goes as the square of frequency, so the curve falls at twelve decibels an octave: 66 dB at 40 Hz, 78 dB at 80 Hz, 94 dB at 200 Hz. The 40 Hz figure is 28 decibels below the 200 Hz one, and that gap is arithmetic about a radius and a displacement rather than a property of any particular loudspeaker. The dots are the harmonics of a 41.2 Hz note with a one-over-n source spectrum; the loudest of them is the 6th.

The bass a small loudspeaker does not make

A three-inch cone at its excursion limit produces sixty-six decibels at forty hertz, which the ear converts to seventeen phons — barely above nothing. The note is heard anyway, because its harmonics are radiated and its fundamental is supplied by the listener. Computing what arrives turns the residue from a curiosity into a design decision, and finds that the fundamental of a low note is not the loudest part of it on any system a listener is likely to own.

instruments · Missing fundamental 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.

timbre · The voice What a spread costs a chord. a major triad of 3 notes lasting 600 ms each, with the onsets spread by up to 320 ms. The upper line is the share of each note's length during which every note is sounding; the lower is the chord's roughness weighted by that share, since roughness is a property of two partials sounding at the same time. At a spread of 30 ms — the asynchrony at which a mistimed partial stops belonging to its note — the chord is still 89 per cent simultaneous. It stops being simultaneous at all at 300 ms, which is where the last note arrives after the first has finished.

The chord that is not played at once

Every chord until now starts its notes at the same instant, and no figure ever set the asynchrony to anything else. A spread chord is not a defective simultaneity: at forty milliseconds a triad of half-second notes is still eighty-seven per cent simultaneous, so it carries almost all its roughness, and it stops being a chord at all only when the last note arrives after the first has finished. What none of this can explain is the one thing every keyboard player knows — that a chord is rolled upward. The masking asymmetry that ought to explain it is 2.4 decibels at a close voicing, which is not enough.

intervals · The triad 3 tones, one power, and the interval between them. 3 tones of fixed total power, spread symmetrically about 440 hertz, drawn against the interval between neighbours. Piled on one pitch they are one sound of that power; separated by more than a critical band — 4.5 semitones here — they are 3 sounds whose loudnesses add, and the same power reaches 2.08 times the loudness at 5 semitones. The two lines are two models of the same rule and they disagree about how abrupt the change is, not about where it goes.

A chord is not as loud as its notes

The first essay on loudness said what a tone's loudness is and recorded that it had said nothing about a chord's. Here is the missing rule, and it has a musical consequence nobody would predict from it: the same three notes, at the same power, are twice as loud in the treble as in the bass — because the critical band that makes a low triad five times rougher also makes it one sound instead of three.

perception · Loudness Adding parts adds power, and very little loudness. Each part is played at the same level, and the chord is realised every way its parts allow and averaged over them, so the quantity is a property of the texture rather than of one arrangement. Going from 3 parts to 8 adds 4.3 decibels of power and 0.1 decibels of loudness, because the extra parts land in bands that are already occupied — the count of occupied critical bands FALLS from 6.0 to 3.9 as the parts crowd into the same register.

The dynamics are in the score already

Count the parts in each bar, realise them in their ranges, put every partial in its critical band, sum the loudnesses and run the result through the two smoothers built earlier. What comes out is a dynamic curve for a piece with no performance in it anywhere — and it says that doubling the number of parts inside a fixed register adds three decibels of power and about one of loudness, because the extra parts land in bands that were already occupied. Let the register widen with the parts and the same arithmetic gives eight phon, which is what a tutti actually is.

form · Loudness A loud chord is a smaller chord. The share of a voicing's partials that stand above what the rest of it masks, and the share of its computed roughness that is between partials a listener actually has, from 30 decibels to 100. Both fall: 83 per cent of the partials survive at 30 decibels and 38 at 100, and the roughness share goes from 88 per cent to 67. The direction is the upward spread of masking, which grows faster than linearly with level: a loud partial masks a band above itself much wider than a quiet one does, so the chord's own top disappears into its own bottom. Two earlier essays are drawn at one level, and this is what they were holding.

A loud chord is a smaller chord

Two earlier essays hold the level fixed, and the level decides how much of a chord a listener is given. At thirty decibels twenty of a triad's twenty-four partials stand above what the rest of it masks; at a hundred, nine do. Every roughness figure until now counts partials that are in the score, and a partial the chord masks is not a partial the listener has.

harmony · Masking Nothing at all until fifteen decibels, and then it depends on the tempo. The fraction of a line's partials that stay above threshold, over how fast the line moves and how much louder everything before each note is. Darker is more lost. The whole left-hand side is white: at equal levels a note cannot be masked by its predecessor at any tempo, and that is a proof rather than a measurement — forward masking leaves a threshold at most ten decibels below the masker, and a note's own partials mask each other from the same components at full level. The boundary is between twelve and eighteen decibels, and beyond it the loss grows with the tempo: at 280 to the crotchet and 36 decibels of contrast, 26 per cent of the line's partials are gone. Fifteen decibels is about the gap between a forte and a piano.

An equal note cannot be masked

Three earlier essays are about one instant, and forward masking lasts two hundred milliseconds — longer than a note at any brisk tempo. So a fast line should be a sequence of events hiding each other, and it is not: a note masks itself ten decibels harder than its predecessor can, at any speed. What does hide a line is dynamic contrast, and the boundary is fifteen decibels.

perception · Masking The top voice arrives whole and the bottom one arrives as a sine. 4 parts sounding together, each of 8 partials, with every partial tested against the summed masked threshold of every component in the texture. A filled mark is a partial the listener receives and an open one is a partial the part would have had alone and does not have here. The bass at C3 keeps 1 of 8, the tenor at C4 keeps 2 of 8, the alto at E4 keeps 5 of 8, the soprano at G5 keeps 8 of 8, every one of them at 70 decibels. Every part is at the same level and the difference is entirely where each one sits: masking spreads upward, so the part at the top of the texture has nothing above it to be masked by and the part at the bottom has everything.

The listener is given the top voice, and the bass as a sine

Four earlier essays put the masker and the probe in the same voice. Put them in different voices — a four-part texture at one level — and the soprano arrives with all eight of its partials, the alto with five, the tenor with two and the bass with one. Balancing the loudness, which is the constraint a scoring is solved under, changes none of that: equal loudness is not equal spectrum and cannot be made so.

perception · Masking The roughest chord on the page is at C1 and the roughest one heard is at A♭2. One close triad at 70 decibels through 17 registers, with its roughness computed twice: over every partial in the score, and over only the partials that stand above what the chord itself masks. The written curve rises all the way down and its maximum is the lowest register drawn, C1, which is the low-interval rule as it has always been computed here. The delivered curve turns over at A♭2 and falls to nothing below A♭1: a close triad down there is not rough, because it is not arriving as a chord — 1 of its 24 partials survives at C1 and there is almost nothing left to beat against anything.

A low chord stops being rough by stopping being a chord

Every count of audible partials until now is of a chord at middle C, and the three registers it did compare span C3 to C5 — a third of the range a chord is written in. Move the same triad down and the count collapses: 79 per cent of its partials arrive at E3 and 4 per cent at C1. So the roughest chord on the page is the lowest one and the roughest chord a listener receives is at G2, and where that maximum sits moves nearly two octaves with the dynamic.

perception · Masking What the chord before takes out of the chord after. Five chords at 1.2 seconds each, with the roughness each one has on its own — its simultaneous masking and the threshold of hearing already applied — and the roughness it actually has once the chord in front of it has raised the threshold. Four of the five are untouched. The fifth, i, two parts, follows the only step in this passage that falls more than fifteen decibels, and it arrives into a hole: it is entirely below threshold for its first 13 milliseconds and takes 240 to get all of itself back. Masking can only remove partials, so it can only lower a roughness — and the chords it can reach are the ones a written dynamic has just made quiet, which are already the smooth ones. Across the passage the dissonance contrast goes from 3021 to 3113: the mask widens it by 3.0 per cent rather than eating it.

A soft chord has to fade in

Forward masking sits between the two integration times already in play — two hundred milliseconds against a thirty-seven millisecond roughness window and a two-second loudness release — and it was owed as the term that might eat the dissonance contrast. It does not. It widens it, by three per cent at a chorale's pace and fifty-nine at four chords a second, because it can only ever remove partials and it can only reach the chord a dynamic has already made quiet. What it does instead is stranger: one chord in the passage is entirely inaudible for its first twelve milliseconds and takes a quarter of a second to arrive whole.

form · Orchestration Partials 3, 4, 12 of "hod", over one vibrato cycle. The level of three partials of a 220 hertz note on the vowel in "hod", each about its own mean, over one cycle of a vibrato of ±71 cents at 6.0 hertz. The pale curve is the frequency deviation itself, for phase reference. Partial 3 at 660 hertz swings 4.22 decibels and peaks with the frequency; Partial 4 at 880 hertz swings 0.41 decibels and peaks twice a cycle; Partial 12 at 2640 hertz swings 8.62 decibels and peaks against it. The formants of this vowel are at 730, 1090, 2440 hertz and do not move; a partial below one rises as the frequency rises and one above it falls, so the modulations of a single note run in opposite directions at the same instant.

The partial that gets louder as it goes sharp

Eleven earlier essays sweep a set of partials and hold their amplitudes still, and a real tract does not move with the fundamental. Put the formant account and the sweeping account together and every partial acquires an amplitude modulation at the vibrato rate: 0.07 decibels on the fundamental and 8.6 on the twelfth partial of the same note. They are in phase below a formant and anti-phase above one — not ninety degrees apart — and the whole note swings 0.82 decibels, because they cancel.

instruments · The voice Scored the way these figures score it, a clarinet is the worst of the six. One close triad at 70 decibels through 17 registers, drawn once for each of the 6 spectra to hand. The score is the share of every partial written, which is the quantity the register figure published. pure 100 per cent at best, string 79 per cent at best, clarinet 54 per cent at best, reed 71 per cent at best, bell 52 per cent at best, organ 72 per cent at best. A clarinet's four even partials are twenty-eight decibels below its odd ones and are inaudible beside their own neighbours before any chord is built, so counting them in the denominator makes the spectrum that survives its own masking best look like the one that survives it worst.

A clarinet keeps what a string loses

Every masker, probe, chord, line and texture until now is eight partials falling as 1/n, and it was not even an option a placement could pass. Sweeping the six spectra to hand says the clarinet is the worst of them — 54 per cent of itself at best against a string's 79 — and that answer is an artefact of the score. Counted against what each note keeps on its own, the clarinet keeps 100 per cent where the string keeps 79, because its components stand a twelfth apart rather than an octave. The missing parameter was the spectrum; the second missing parameter was the denominator.

perception · Masking One doubling, held down a phrase. Where a single held arrangement of 4 players on 3 notes stands among the 36 at each chord of a 5-chord passage, best at the top, with what each chord would rather have named along the bottom. The held answer is flue pipe · trumpet · clarinet+violin, and it is the chord's own first choice at 4 of 5 of them. Holding it costs 16.2 per cent of the passage's roughness against re-scoring every chord — which is 2.4 per cent of the range the choice actually spans, since the arrangements at one chord differ by a factor of 7.8 on average. The cost is not spread over the passage: 1 chord carries nearly all of it.

An orchestrator doubles a line, not a chord

Three earlier essays made the objective a functional over a passage and a later one went back to holding one chord still. Put the doubling back into time and the retreat turns out to have been cheap: one arrangement held down a five-chord phrase is that phrase's own best answer at four of its five chords and costs 2.4 per cent of the range the choice spans — while the forward mask named earlier as the third temporal constant reaches for twenty milliseconds rather than two hundred, and cannot change the answer at any pace at all.

instruments · Orchestration An entrance stops being a loudness event and never stops being a colour one. The same oboe entering on the same note at the same level, against how many players were already sounding. Its contribution to the loudness falls from 15.5 phons to 0.32 — a factor of 48 — and crosses the one-phon difference limen at 5 players already playing. Its contribution to the roughness rises by a factor of 12.3 over the same range, because roughness is a sum over pairs and the entrant makes one new pair with everybody. Both curves are drawn as a share of their own largest value, since a phon and a squared pascal have no exchange rate. The claim is the two directions, not the crossing point of two units.

An entrance is a change of colour

Eight essays on orchestration move the assignment and hold the ensemble still, and a score does the opposite: it brings players in and takes them out. Loudness is a sum over parts and roughness is a sum over pairs, so the player who joins adds one term to the first and one to the second for everybody already there. What the entrance is worth in phons falls by a factor of forty-eight across the range an ensemble spans and crosses the difference limen at five players; what it is worth in roughness rises by twelve, and by a further factor of ten for every ten decibels the passage is played at.

form · Orchestration The same player, arriving and leaving, read as a share of the change. One oboe joining 5 players and the same oboe leaving them again, with both loudness readings drawn as the share of their own change that has arrived. The entrance is half received in 90 milliseconds and the exit in 1.43 seconds, a factor of 15.9. The roughness readings, drawn faintly, are 25 and 25 milliseconds and lie on top of each other. A score that writes a diminuendo under a departing part is not softening the exit. It is doing the smoother's release for it, on a clock the smoother would otherwise take two seconds over.

A part that leaves is not a part that arrives

The same player, the same note, the same level, and the only difference is which way round it happens. A listener's loudness reading takes 1.43 seconds to receive half of a departure and 90 milliseconds to receive half of an arrival — a factor of sixteen with nothing asymmetric in the sound at all, since both readings integrate the same two states in the same order. The colour reading receives the two identically, because a window has no direction, so a departure is a change whose grain arrives at once and whose level takes most of two seconds.

form · Orchestration Which chord of a passage has room for the part that is entering. An oboe entering on one note, tried at each chord of a five-chord passage, scored by how far its own partials sit above the threshold the ensemble already sounding puts over them. The best moment gives it 9.0 decibels of margin and the worst 0.3, a spread of 8.7 — and the best moment is not the quietest chord, which is vi, close below. Room for an entrance is spectral rather than dynamic. A chord with a hole in its written spacing need not have one in its spectrum, because the partials of its bass fill the middle whatever the notes above it do.

The chord that has room for an entrance

Three essays have made the ensemble something a score can change and none of them has asked when. The ensemble already sounding puts a masked threshold over whatever register an entering part takes, and that threshold is set by the voicing rather than by the dynamic — so the five chords of one passage differ by 8.7 decibels in how much of an entering oboe survives them, and the quietest chord of the five is the worst place in the passage to bring somebody in. Swept over the entrant's own pitch, the choice of moment is worth as much as the choice of register.

form · Orchestration A loudly struck note hides the beats it is being struck to reveal. The share of the fluctuation in its own auditory filter that belongs to each of the first five members of a mistuned octave's beat family, against how loud the note is. The filter's lower skirt shallows by about 38 per cent of its 51-decibel value every ten decibels, so more of the neighbouring partials get into the filter and the pedestal each member sits on grows. The mean share falls from 0.55 at 40 decibels to 0.07 at 100. Past about 90 decibels the curves are flat because the model's skirt is clamped rather than because anything stops changing — that clamp is the model's floor and not a measurement.

How hard the note was struck

The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.

tuning · Beating The schedule that hears every entrance best holds the high parts back. Six parts waiting to enter a five-chord passage over four sounding players, each entering once and staying: the schedule under which the least audible entrance is as audible as it can be made. brass on E3 enters at I, open with -1.4 decibels of mean margin over the mask; oboe on E4 enters at vi, close below with -2.6 decibels of mean margin over the mask; clarinet on G4 enters at I, open with -1.4 decibels of mean margin over the mask; voice on C5 enters at IV, close above with 2.1 decibels of mean margin over the mask; violin on G5 enters at V, bracketing with 6.9 decibels of mean margin over the mask; flue pipe on C6 enters at I, hollow with 4.0 decibels of mean margin over the mask. The least audible entrance is at -2.6 decibels and the margins sum to 7.6; of all 15625 schedules 0 have a better least audible entrance and 576 a larger sum.

Room is used up by whoever enters first

The chord with the most room for a part entering alone is a fact about that chord. It stops being a fact the moment two parts want it, because each part that comes in raises the mask over everybody after it. Given six parts waiting to enter a five-chord passage, choosing each part's moment the way one part's moment is chosen puts three of them into the same chord and lands in the bottom fifth of all 15,625 schedules. Placing them one at a time does no better. The schedule under which the least audible entrance is heard best is unique, and it brings the low and middle parts in while the texture is thin and holds the three highest back for the last three chords — because a high part keeps its room over a full texture and a middle part does not.

form · Orchestration The thirds' difference-tone line, note by note, against the dynamic. Each note of the line the difference tone f₂ − f₁ draws under a scale in just thirds, as its level above the higher of the threshold of hearing and the primaries' masking, against the level of the primaries. The product sits 50 dB below the primaries at 60 dB and grows twice as fast as they do. C2 above the limit from 73.5 dB; A1 above the limit from 76 dB; C2 above the limit from 73.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; C3 above the limit from 66 dB.

A combination-tone bass needs a forte

A scale in just thirds draws a diatonic bass line through its difference tones, and in sixths the cubic product draws one. Given the two published level laws, with their constants swept, the thirds' bass is not heard at all below primaries of about 66 dB and is heard whole only from 71 to 81. The cubic products are a different kind of object: the primaries mask them decibel for decibel as they rise, so no dynamic changes whether they are heard. Most of the thirds' inner line never is, and the sixths' bass needs a forte and a gentle law.

intervals · Combination tone Every product of every pair of partials is a harmonic of one absent note. A 4 : 5 interval on 261.6 and 327.0 hertz, just, each note carrying 6 partials at one over n, with the fundamental at 80 decibels. Every pair of partials makes its own difference tone, and there are 19 distinct frequencies among them. All of them are exact multiples of 65.41 hertz — the note neither instrument is playing — because partial j of a p·f₀ note and partial k of a q·f₀ note differ by (kq − jp)·f₀ whatever j and k are. The heavy line is what each product has to clear — the threshold of hearing at its own frequency, or the masked threshold the two primaries cast there, whichever is higher. 11 of the 19 get through.

Three harmonics of the bass arrive before the bass

Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.

intervals · Combination tone With 4 of six required at the end, the best schedule hands parts over. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 4 of the six sounding at the last chord. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: oboe on E4 enters at 0.3 dB, and clarinet leaves; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and oboe leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: flue pipe on C6 enters at 4.0 dB. The least audible entrance is -0.72 dB against -2.65 for the best schedule in which nobody leaves; the walk has 2 exits and 6 entrances.

An exit is worth nothing until the tutti is given up

Six parts entering a five-chord passage have a best schedule when each enters once and stays, and letting parts leave and come back was supposed to improve it. Searched over every set of sounding parts at every chord, it improves it by exactly nothing, with or without the chord before still masking — as long as all six must be playing at the end. Let one part be missing from the final chord and the weakest entrance gains 1.4 decibels; let two be missing and it gains 1.9, by a relay in which the parts with least room come in, are heard for one chord, and give way.

form · Orchestration The arch belongs to hearing, and the spacing only moves it. The share of a close major triad's twenty-four components that stand above what the rest of the chord masks, at 70 dB, with the root from C1 to C7, for three spectra given the same amplitude law and different frequencies: the harmonic series, a founder's bell, and a stiff string with B = 0.01. harmonic series: 0.04 at C1, peaking at 0.79 on E3, 0.42 at C7; a founder's bell: 0.04 at C1, peaking at 0.75 on C4, 0.38 at C7; a stiff string: 0.04 at C1, peaking at 0.71 on E3, 0.46 at C7. Only one of the three is a harmonic series, and all three rise out of the bass, peak in the middle of the compass and fall in the treble.

The arch belongs to hearing, not to the series

A chord delivers most of its partials in the middle of the compass and loses them in the bass and the treble, and every spectrum that showed that arch was built on whole multiples of a fundamental. Give the same amplitudes to a bell's eight modes and to a stiff string's stretched partials and the arch is still there, peaking within a major third of where the harmonic series peaks. What the spacing changes is the detail: a bell crowds its tierce and quint into a quarter of a critical band in the bass and loses them, and a stiff string's stretch buys the bass back.

perception · Masking Counted over what arrives, the balanced bass is not the roughest register. The mean roughness of the I – vi – IV – V – I arrivals with each chord played as loud as the written register's, relative to the written register, counted over every partial and over the partials that stand above what the rest of the chord masks. Every partial: 70 −2 octaves, 7.48 −1 octave, 1.00 as written, 0.20 +1 octave. Delivered partials only: 6e-9 −2 octaves, 2.83 −1 octave, 1.00 as written, 0.19 +1 octave. Over every partial the lowest register is 343 times rougher than the highest; over what arrives it is the smoothest of the four, and the roughest is −1 octave, 2.8 times the written register.

A bass chord low enough to balance has already hidden its tenor

Played as loud as the written register, a progression two octaves down is 343 times rougher than the same progression an octave up — if every partial on the page is counted. Count only the partials that stand above what the rest of the chord masks and that register is the smoothest of the four, with nothing left that beats. The balance is not what does it: the extra thirteen decibels move no voice by more than two partials. The register had already buried the tenor at the written dynamic.

perception · Tonal-expectation A bell's tierce does not arrive late; it leaves early. The tierce's margin over what the bell's other modes put across it, in decibels, against time since the strike, for four bells, with the hum's sixty-decibel time set to 40 seconds and each mode's decay time that divided by its frequency ratio to the hum raised to 1. G1: 0 s -3.5 dB, 2 s -5.3 dB, 3 s -7.4 dB, 5 s -9.8 dB, 6 s -12.4 dB, 8 s -15.2 dB; C2: 0 s -2.2 dB, 2 s -3.7 dB, 3 s -5.6 dB, 5 s -7.7 dB, 6 s -10.1 dB, 8 s -12.6 dB; G2: 0 s 0.1 dB, 2 s -1.0 dB, 3 s -2.4 dB, 5 s -3.9 dB, 6 s -5.8 dB, 8 s -7.9 dB; C3: 0 s 1.9 dB, 2 s 1.1 dB, 3 s 0.2 dB, 5 s -0.9 dB, 6 s -2.2 dB, 8 s -3.8 dB. Above the line the tierce is heard. The earlier prediction was that the quint's faster decay would lift the tierce clear of its masking a second into the note; every curve here falls instead.

A bell's tierce does not arrive late, it leaves early

The essay before this one found a low bell's tierce masked at the strike and predicted it would emerge a second into the note, because the quint above it decays faster than the prime and tierce below. It does not emerge, and the reason is that the quint was never masking it: removing the quint moves the tierce's threshold by a fifth of a decibel and removing the prime moves it by three to five and a half. The tierce is masked from below, by two modes that outlive it under any law that falls with frequency — so a bell that has a tierce at the strike loses it within a few seconds, and a bell that has none never gains one. The minor third a carillon's bass is known for is a property of its attack.

perception · Masking Re-struck under a chord that has been decaying, a buried tenor gets its partials back. How many of the tenor's eight partials stand above the masking of the other three voices, for the first chord of I–vi–IV–V–I balanced for loudness in four registers, on a struck instrument whose held voices decay while the tenor is re-struck a step away at a stated time after the chord; beside it, the same tenor held from the start. Two octaves down: 0.25 s 1 moving, 0 held; 0.5 s 3 moving, 0 held; 1 s 6 moving, 0 held; 2 s 8 moving, 0 held; 4 s 8 moving, 0 held. An octave down: 0.25 s 6 moving, 1 held; 0.5 s 8 moving, 1 held; 1 s 8 moving, 1 held; 2 s 8 moving, 1 held; 4 s 8 moving, 0 held. As written: 0.25 s 8 moving, 4 held; 0.5 s 8 moving, 5 held; 1 s 8 moving, 5 held; 2 s 8 moving, 2 held; 4 s 8 moving, 1 held. An octave up: 0.25 s 8 moving, 3 held; 0.5 s 8 moving, 4 held; 1 s 8 moving, 3 held; 2 s 8 moving, 1 held; 4 s 8 moving, 0 held. Two octaves down, where the held tenor keeps nothing at any time, the moving one has six partials back by a second and all eight by two.

A moving tenor is heard because it is struck again

A bass-heavy chord balanced for loudness hides its tenor under the bass's upper skirt, and arrangers keep inner voices moving in the low register as though motion were the cure. Counted as masking, motion alone is worth nothing: the note the tenor leaves stops masking the new one within ten milliseconds, and on a sustained instrument the moving tenor keeps exactly what the held one keeps. On a struck instrument everything changes, and for a reason that is not the motion. The held voices have been decaying since they were struck and the tenor's new note is struck now, so an octave down a tenor re-struck half a second into the chord has all eight of its partials back where the held tenor has one.

perception · Tonal-expectation Quietly, or quickly, the scale's Pythagorean third is heard with the bass of the just third a comma away. The third D4–F4 of a just C major scale, a Pythagorean 32 : 27, fitted as a whole object with every component that has run 4 of its cycles and every product only if it is audible at the level played: what the fit names at the end of a note of each length, at each dynamic. 40 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 50 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 60 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 66 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 70 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 74 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 76 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 80 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 85 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 90 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz. 58.1 Hz is the bass of D–F with D a syntonic comma lower, a just 6 : 5; the Pythagorean third's own difference tone is 54.5 Hz and is audible from 76 dB, and once it has run its cycles the fit breaks.

The dynamic chooses which D the scale has

A just C major scale has one third that is not 5 : 4 or 6 : 5 — D–F, a Pythagorean 32 : 27 — and a pitch mechanism fitting the notes' partials hears it as the just minor third a comma away, with that third's bass, B♭1, beneath it. Only the Pythagorean third's own difference tone can contradict that, and it is audible only from 76 dB. So below a forte the ghost line under a scale in just thirds is C, B♭, C, F, G, F, G, C for the whole of every note, and the syntonic comma is resolved by the listener rather than the players; above it, and only in notes longer than 73 milliseconds, the B♭ is taken away.

intervals · Combination tone Made to stay 2 chords, the parts change over in groups rather than one by one. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord, and every part that enters staying at least 2 chords before it may leave. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: nobody enters; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and clarinet leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: oboe on E4 enters at -1.1 dB, flue pipe on C6 enters at 4.7 dB, and brass and voice leave. The least audible entrance is -1.07 dB against -2.65 for the best schedule in which nobody leaves; the walk has 3 exits and 6 entrances.

A part that must stay changes over in shifts

The best way to keep every entrance audible, when the passage need not end with everybody in, was a relay: a part enters, is heard for one chord, and gives way. Make every part stay at least two chords once it is in and the relay does not collapse into overlaps, as predicted. It changes over in shifts: two parts enter together, one chord passes in which nobody enters, and both leave on the chord their successors arrive. Half of what exits were worth survives a two-chord stay and nearly two fifths a three-chord one — and with the chord before still masking, the stay costs less, not more, because it was the one-chord handover that memory had already made leaky.

form · Orchestration A moving tenor on a sustained instrument can hand over only the partials the chord leaves it. A four-voice arrival balanced for loudness, played on a sustained instrument, with the tenor moving a second after the chord: how many of the tenor's partials stand above the masking of the other three voices, on average over four arrivals, against how far the chord is transposed, for a string, an organ and a clarinet spectrum. The held tenor keeps exactly the same partials. string: 24 semitones down 0.0, 21 semitones down 1.0, 18 semitones down 1.0, 15 semitones down 1.0, 12 semitones down 1.0, 9 semitones down 2.5, 6 semitones down 1.8, 3 semitones down 2.3, as written 3.5, 3 semitones up 4.3, 6 semitones up 2.5, 9 semitones up 3.8, 12 semitones up 3.8; organ: 24 semitones down 0.0, 21 semitones down 1.0, 18 semitones down 1.0, 15 semitones down 1.8, 12 semitones down 2.3, 9 semitones down 2.3, 6 semitones down 1.0, 3 semitones down 2.8, as written 3.0, 3 semitones up 3.3, 6 semitones up 1.3, 9 semitones up 3.0, 12 semitones up 3.0; clarinet: 24 semitones down 0.5, 21 semitones down 2.5, 18 semitones down 1.0, 15 semitones down 2.5, 12 semitones down 3.5, 9 semitones down 3.0, 6 semitones down 3.3, 3 semitones down 3.3, as written 3.8, 3 semitones up 3.3, 6 semitones up 3.5, 9 semitones up 3.5, 12 semitones up 3.5. Whatever the onset does, it can only do it to these.

An onset can only hand over what the chord leaves

On an organ, a tenor that moves while the chord holds keeps exactly the partials a held tenor keeps, so the masking count cannot tell them apart. What can is the onset: a component that starts thirty milliseconds after the sound around it is heard out as something new. A moving tenor starts hundreds of milliseconds after the chord and gets that for every partial it has left — and what it has left, an octave down, is one partial on a string and three on an organ, which still name the tenor's own pitch. Two octaves down there is nothing to hand over. And a tenor that moves at the chord change, with the other voices, gets nothing, which is why the motion that carries an inner voice is the motion between the harmony's changes.

perception · Tonal-expectation Scored as one arrival, a pair enters as a loud part carrying one that cannot be heard. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord. I, open: clarinet on G4 enters at -10.9 dB, violin on G5 enters at 19.6 dB; vi, close below: oboe on E4 enters at -1.5 dB, voice on C5 enters at 14.5 dB, and clarinet and violin leave; IV, close above: nobody enters, and oboe and voice leave; V, bracketing: brass on E3 enters at -0.0 dB, flue pipe on C6 enters at 25.0 dB; I, hollow: nobody enters. Scored with the parts entering on a chord taken together, at most 2 at a time, as one arrival against what was already sounding: the weakest arrival is 10.93 dB, and the most buried single entrant, measured as itself, is -10.94 dB.

A pair entering as one hides the part that cannot be heard

Scored as two entrances, a pair of parts coming in together is charged for masking itself; scored as one arrival — a new colour against what was already sounding — it is not, and the best schedules change completely. Allowed two at a time, the weakest arrival rises from 0.6 dB above the mask to 10.9, and every pair the search chooses is a loud part carrying a quiet one: a violin 19.6 dB clear beside a clarinet 10.9 under. Allowed any number, the six parts enter at once. The staggered build-up of every earlier schedule exists only because each part was meant to be heard as itself.

form · Orchestration Every decibel taken off the passenger comes off the arrival. The best walk with parts entering together scored as one arrival, at most two at a time, against a floor under each member: no entrant, heard as itself, more than the floor below its mask. Solid lines are the weakest arrival, dashed the most buried member, for two parts required at the end and for all six. 2 at the end: floor 0: arrival 0.59, most buried 0.59; floor 1: arrival 0.59, most buried -0.86; floor 2: arrival 0.59, most buried -1.57; floor 3: arrival 0.59, most buried -2.87; floor 4: arrival 0.59, most buried -3.47; floor 5: arrival 0.59, most buried -3.47; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.16; floor 10: arrival 10.93, most buried -9.90; floor 12: arrival 10.93, most buried -10.94; every part heard as itself, the earlier walks' score, 0.59. 6 at the end: floor 0: no walk; floor 1: no walk; floor 2: no walk; floor 3: arrival -2.65, most buried -2.65; floor 4: arrival 0.50, most buried -3.27; floor 5: arrival 0.59, most buried -4.24; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.13; floor 10: arrival 4.50, most buried -6.13; floor 12: arrival 4.50, most buried -6.13; every part heard as itself, the earlier walks' score, -2.65.

A floor under each member is paid for in steps

Score parts entering together as one arrival and the best pairs are a loud part carrying one that cannot be heard. Put a floor under each member — no part more than so many decibels under its own mask — and the prediction was a schedule between the two, neither staggered nor buried. There is none. The arrival's margin climbs in three steps as the floor is loosened, each step bought with a passenger buried about as deep as the step is high, and below a floor of five decibels nothing beats bringing the parts in one at a time, because the clarinet can clear its mask only alone.

form · Orchestration In the bass a bowed step keeps the entrance's displacement; in the treble every note is placed afresh. A listener whose ears differ by 2 dB, with 14 per cent of the brainstem latency difference reaching the delay comparator: how far the second note of a bowed legato pair at 60 dB is displaced, in detectable steps of 15 µs, against the interval it moves by, in three registers. A note with no onset of its own keeps the first note's position, drawn at the entrance's displacement; the dotted lines are each register's entrance from silence and the dashed its steady tone. G2, a double bass (entrance 6.16, steady 4.00): -12 6.19, -7 5.02, -4 4.35, -2 4.06, -1 no onset, +1 no onset, +2 no onset, +4 4.08, +7 4.26, +12 4.53. A4, a violin (entrance 3.73, steady 2.15): -12 3.72, -7 3.56, -4 2.82, -2 2.35, -1 2.19, +1 2.15, +2 2.22, +4 2.39, +7 2.67, +12 3.07. A6, a violin's top (entrance 2.13, steady 1.21): -12 2.13, -7 2.13, -4 1.77, -2 1.40, -1 1.26, +1 1.22, +2 1.29, +4 1.43, +7 1.66, +12 1.93. At the dial's up 2 semitones: G2 no onset, keeps 6.16; A4 2.22; A6 1.29.

A bowed bass line is placed by its entrance; a violin line by every note

A listener whose ears differ slightly places a note by its onset, and a bowed entrance is placed as though it were far quieter than it is. A legato line's later notes do not start from silence. Whether one of them has an onset at all depends on how much of the note before is already in its own auditory filter — and in the bass, where the filters are wide and their lower skirts shallow, a step of a tone or less has none, so the note keeps the entrance's displacement. In the treble nearly every note has its own onset and is placed near its steady position, except where the line leaps down.

perception · Localisation Read in its partials' filters, every step of a bowed bass line is an onset. The second note of a bowed legato pair on the double bass's G2 at 60 dB, for a listener whose ears differ by 2 dB: its displacement in detectable steps against the interval, placed by the filter at its fundamental alone (dashed, with pale dots where that filter sees no onset and the note keeps the entrance's position) and by the first of its eight partials' filters to see an onset (solid). By partials: -12 6.19 in partial 1, -7 5.02 in partial 1, -4 3.68 in partial 2, -2 3.12 in partial 3, -1 2.76 in partial 4, +1 2.61 in partial 5, +2 3.10 in partial 3, +4 3.56 in partial 2, +7 3.15 in partial 3, +12 3.51 in partial 2. The entrance, placed the same way, is 4.49 steps by its partial 4, against 6.16 by its fundamental. At the dial's +2: 3.10 steps, first in partial 3.

A bowed bass line has an onset at every step, in its upper partials

A legato note's onset was read in the auditory filter at its fundamental, and in the bass a step of a tone or less left that filter nearly as full as before: no onset, and the note kept its entrance's displacement. A bowed note is not a fundamental. Read in every filter its partials reach, every step of a bowed bass line has an onset, first seen in the third to fifth partial, where a tone moves a partial right out of its filter. The line is re-placed at every note after all — and nearer the centre, because a partial's filter is at a frequency where the ears' latency difference is smaller. The bowed bass entrance itself is displaced a quarter less than its fundamental said.

perception · Localisation A bell's tierce is best placed at the first instant of the strike, and only for an instant. The tierce's margin above its masked threshold, in decibels, through the first 400 milliseconds after the strike, for bells on G1, C2, E2, G2, C3, with every mode starting at full amplitude and built up in the listener's own auditory filters. G1: -1.3 at 0.5 ms, -3.9 at 400 ms, never above threshold; C2: 0.3 at 0.5 ms, -2.5 at 400 ms, above threshold from 0.5 to 0.7 ms; E2: 1.6 at 0.5 ms, -1.2 at 400 ms, above threshold from 0.5 to 4.3 ms; G2: 2.7 at 0.5 ms, -0.2 at 400 ms, above threshold from 0.5 to 104.2 ms; C3: 4.7 at 0.5 ms, 1.7 at 400 ms, above threshold from 0.5 to 400.0 ms. Every curve falls from its first moment: there is no peak inside the window.

A bell's strike gives its tierce a millisecond

A low bell's tierce is masked from below by its prime and hum, so it is present at the strike or never — and the strike was the one moment the count treated as a steady state. Give the modes a rise as well as a fall and the prediction was that the tierce's margin would peak somewhere inside the first two hundred milliseconds. It peaks at the very first instant and falls from there, because the shadow a masker casts upward grows faster than the masker. On a C2 bell the tierce is above its threshold for less than a millisecond, and under the most generous rise anybody could propose for twenty-five: ten to thirty decibels too brief to be heard. What the onset visits for an instant, a softer stroke keeps for the whole note.

perception · Masking How far under its mask a passenger is, and how much it still changes. Every pairing of two parts entering together on any chord of the passage in which the quieter part — the passenger — sits under its mask on average: 71 of them. Across, the passenger's mean margin, the measure every earlier walk used; up, how far its partials together change the arrival above the threshold of detection. Every one of the 71 is above zero: the lowest is 0.11 dB. 35 of the 41 buried more than 3 dB by the mean still change the arrival by more than 3 dB. The few near the zero line are the clarinet on the bracketing dominant, where every one of its partials is just under.

A passenger under its mask is still heard in the arrival

The best way to bring parts in together was a loud part carrying a quiet one, and the quiet one was scored as buried — ten decibels under its mask, on average over its partials. An average is the wrong question. A masked threshold is the level at which a sound becomes a detectable change, and nearly every buried passenger has partials well clear of it. Scored as a change to the arrival rather than as a line of its own, every one of seventy-one passengers is heard, and the carriers' walk passes almost untouched.

form · Orchestration A bell is never minor alone. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels: whether the masked-threshold count delivers the tierce, the minor third above the prime, and the deciem, the major third above the nominal. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MBBBMMMMMMMMMM····; E2: ·BBBBBBBBMMMMMMM···; G2: ·BBBBBBBBBBMMMMMM··; C3: BBBBBBBBBBBBBMMMMM·; E3: BBBBBBBBBBBBBBBMMM·; G3: BBBBBBBBBBBBBBBBMMM; C4: BBBBBBBBBBBBBBBBBMM (B both thirds, M the major only, m the minor only, a dot neither; levels in steps of five from 20). Of 190 cells, 82 deliver both thirds, 59 the major third only, 49 neither, and 0 the minor third alone. A dot in a cell marks a stroke at which fewer than two of the three modes that imply the strike note survive.

A bell is never minor alone

A tuned bell carries two thirds: the tierce, a minor third above the prime, which is why a bell is called minor, and the deciem, a major third above the nominal. Run the masking count across every register and every stroke and the minor one is never delivered without the major one. Struck softly a bell delivers both; struck harder, only the major; and below C2 the tierce is never delivered at all. The deciem survives because it is masked by one mode barely louder than itself, the tierce by two that outweigh it.

perception · Masking A bell whose deciem is the tierce's octave is minor at every stroke. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels, with the deciem tuned from 2.5 times the prime to 2.4 — a minor third above the nominal and exactly an octave above the tierce: whether the masked-threshold count delivers a third at all. C1: ···mm··············; E1: ···mmmmmm··········; G1: ··mmmmmmmmm········; C2: ·mmmmmmmmmmmmm·····; E2: ·mmmmmmmmmmmmmm····; G2: ·mmmmmmmmmmmmmmm···; C3: mmmmmmmmmmmmmmmmm··; E3: mmmmmmmmmmmmmmmmmm·; G3: mmmmmmmmmmmmmmmmmm·; C4: mmmmmmmmmmmmmmmmmmm (m a minor third, a dot none; levels in steps of five from 20). 131 of 190 cells read minor and none major. Of the 59 cells in which the bell as cast delivered only its major third, 49 now deliver a minor one.

A bell can be made minor by the third nobody tunes

A low bell struck hard delivers its major third, the deciem, and loses its minor one, the tierce, to masking. A founder could flatten the deciem from a major to a minor third above the nominal, which puts it exactly an octave above the tierce. The masking count says that move costs the deciem only one or two decibels of stroke at the top of its window, so the flattened deciem still outlasts the tierce in every register. A bell tuned that way reads minor in every cell of register and stroke that delivers a third at all. In-between tunings read as neutral thirds, so the move only works if it goes all the way.

perception · Masking A bell with a major tierce is major wherever it delivers a third. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the deciem as cast at 2.5 times the prime and the tierce at 1.25, 386 cents above the prime: which third the masked-threshold count delivers. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MMMMMMMMMMMMMM····; E2: ·MMMMMMMMMMMMMMM···; G2: ·MMMMMMMMMMMMMMMM··; C3: MMMMMMMMMMMMMMMMMM·; E3: MMMMMMMMMMMMMMMMMM·; G3: MMMMMMMMMMMMMMMMMMM; C4: MMMMMMMMMMMMMMMMMMM (M major only, b a minor tierce with the major deciem, n a neutral tierce, a dot none; levels in steps of five from 20). 141 cells major, 0 with thirds that disagree, 49 with none. Dots mark the 10 cells in which this bell delivers a third and the minor bell with its deciem flattened to the tierce's octave delivers none.

The major-third bell has no mirror problem

A minor bell struck hard loses its tierce to masking and delivers its major deciem instead. Raise the tierce to a major third and the two thirds agree — but the raised tierce sits closer to the quint, and it was fair to fear that the quint would bury it as the prime buried the minor one. It does not. The prime is what masks a tierce, and moving away from it gains four to twelve decibels of stroke while the quint stays nine decibels too weak to matter. Tuned to agree, the major bell delivers its colour in 141 of 190 cells; the minor bell, made consistent, in 131.

perception · Masking With the tierce at 1.25 times the prime, the lower modes point only at the strike note's own name. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the tierce at 1.25 times the prime, 386 cents: the series the delivered lower modes — hum, prime, tierce and quint — fit best, within thirty cents. C1: ···················; E1: ·······SSS·········; G1: ·····SSSSSSS·······; C2: ··SSSSSSSSSSSS·····; E2: ·SSSSSSSSSSSSSSS···; G2: ·SSSSSSSSSSSSSSS···; C3: SSSSSSSSSSSSSSSSS··; E3: SSSSSSSSSSSSSSSSSS·; G3: SSSSSSSSSSSSSSSSSS·; C4: SSSSSSSSSSSSSSSSSSS (S the strike note's name, 3 the note a major third below it, a dot fewer than two lower modes delivered; levels in steps of five from 20). 124 cells point at the strike's name, 0 a major third below, 66 at nothing.

The raised tierce points the bell at its own note

A bell's four lowest modes — hum, prime, tierce and quint — can imply a pitch of their own. With a major tierce they are the second, fourth, fifth and sixth harmonics of the prime two octaves down: the strike note's own name, with one empty slot. It was the minor tierce that pointed elsewhere all along. Its lower modes are the fifth, tenth, twelfth and fifteenth harmonics of the note a major third below the strike, and in 82 of 190 struck bells that is the series the delivered modes fit. Raising the tierce removes a competitor rather than adding one.

perception · Masking A drum that is always felt is worth no more than one that is always heard, until the crowd drowns the hearing. Ten metres known, entered at a random step, recognised within the 16-step present, the crowd covering half the time at a stated level above the band: the points marking the downbeat adds, for each way of marking it heard through the crowd, and for a low drum that is felt as well and so always arrives. a louder melody note: +0 28.9, +5 20.4, +10 14.7, +15 11.6, +20 11.3, +25 11.3, +30 11.3; a bass note an octave below: +0 28.9, +5 22.1, +10 19.8, +15 18.0, +20 13.7, +25 11.4, +30 11.3; a low drum: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 17.3, +25 12.9, +30 11.4; a high bell: +0 28.9, +5 22.1, +10 19.8, +15 18.2, +20 14.3, +25 11.5, +30 11.3; a low drum, felt as well: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 19.8, +25 19.8, +30 19.8. A quiet room's mark is worth 23.4.

A felt drum reaches only where the heard one already does

A low drum near a stage is felt in the body as well as heard, and talk cannot mask a vibration. Given that second channel, a drum-marked downbeat always arrives — and is then worth 19.8 points to a late dancer, exactly what an always-heard drum was worth, because the crowd has already taken the soft strokes that made a mark worth more. The felt channel pays only under crowds twenty decibels over the band. And it is driven by the drum's own level, which in a reverberant room is high enough to be felt only where the drum is already loud enough to be heard.

rhythm · Additive metre Two parts entering together mask each other unequally: the carrier loses a partial or two, the passenger up to eleven. Every pairing of two parts entering together on a chord of the passage in which the quieter part sits under its mask on average, 71 of them, sorted. Up: how many of the carrier's partials that are clear of the mask with the carrier alone over the chord are pushed under it by the passenger. Down: how many of the passenger's are pushed under by the carrier. In all, carriers lose 19 partials and passengers 135. 54 carriers lose none, 15 lose one and 2 lose two; passengers lose 0 in 25, 1 in 25, 2 in 6, and three or more in 15, the most being 11.

A carrier gives up room, not partials

A passenger masks its carrier as the carrier masks it, so the arrival might be a new spectrum belonging to neither part. Across the seventy-one pairings of the passage it almost never is. Carriers lose nineteen partials in all to their passengers and passengers lose a hundred and thirty-five to their carriers; a voice beside a violin takes thirteen decibels of the violin's room and not one of its partials. The nine pairings that leave neither part a majority are mostly on one chord, and there the chord had already done it.

form · Orchestration A third part takes from the carrier what it would have taken alone; the room the passenger spent adds almost nothing. Every pairing of carrier and buried passenger, with each other part of the roster entering beside them as a third: 284 arrivals. For each third part, the carrier's partials — clear of the mask over the chord alone — that are under it with all three sounding, split into those the passenger alone already took, those the third part would take on its own, and those lost only to the two together. clarinet: 46 arrivals, 19 partials (14 to the passenger, 3 to the third alone, 2 only together), the third alone costing 1.5 dB of room; oboe: 50 arrivals, 20 partials (9 to the passenger, 9 to the third alone, 2 only together), the third alone costing 1.3 dB of room; brass: 46 arrivals, 23 partials (8 to the passenger, 14 to the third alone, 1 only together), the third alone costing 1.0 dB of room; voice: 47 arrivals, 40 partials (14 to the passenger, 20 to the third alone, 6 only together), the third alone costing 7.3 dB of room; violin: 47 arrivals, 127 partials (17 to the passenger, 108 to the third alone, 2 only together), the third alone costing 8.8 dB of room; flue: 48 arrivals, 209 partials (14 to the passenger, 187 to the third alone, 8 only together), the third alone costing 14.1 dB of room. In all 438 partials: 76 to the passengers, 341 to the thirds on their own, 21 only to the combination.

A third part takes what it would have taken alone

A passenger costs its carrier room rather than partials, and the obvious worry is that the room compounds: that a third part, entering beside a carrier already ten decibels nearer its limit, finds it fragile. Across 284 three-part arrivals it does not. Of the 438 carrier partials lost with three parts sounding, 21 are lost only to the combination; the rest are taken by one of the two parts alone. Masking adds in power, so a threshold the passenger has raised rises less when a third part adds to it. What decides the loss is which part enters third, and two parts — the flue and the violin — take five sixths of it.

form · Orchestration In a ring of minor bells, five of the eight hidden pitches fall off the scale the ring rings. A ring of eight bells with minor tierces tuned to a major scale, the tenor's strike note C4: for each bell and each stroke with the hum at 40 to 90 decibels, what the delivered lower modes — hum, prime, tierce and quint — imply. C4 (hidden A♭, off it): 333333333SS; D4 (hidden B♭, off it): 3333333333S; E4 (hidden C, on the scale): 33333333333; F4 (hidden D♭, off it): 33333333333; G4 (hidden E♭, off it): 33333333333; A4 (hidden F, on the scale): 33333333333; B4 (hidden G, on the scale): 33333333333; C5 (hidden A♭, off it): 33333333333 (3 the note a major third below the strike, S the strike's own name, a dot too few lower modes delivered). 33 strokes imply a note on the ring's scale and 52 a note off it.

A ring of minor bells rings a second scale a major third below

A minor-tierce bell's four lowest modes, read as a residue, imply a note a major third below the bell's strike note. Put eight such bells in a ring tuned to a major scale and the implied notes are that scale moved down a major third — a ring in C carries the scale of A♭ major underneath it. Only three of the eight land on notes the ring rings, from the mediant, the submediant and the leading note; the tonic, the octave, the supertonic, the subdominant and the dominant all imply notes the ring never sounds, each a semitone from two that it does. Above a tenor on C3 the lower modes deliver the implied note at nearly every stroke. A reading that weights the whole spectrum still hears none of it.

perception · Masking Felt on every landmark, the stream recovers most of a quiet room's recognition even when no accent is felt. Ten dances known, entered at a random step, the crowd covering half the time at the level on the axis above the band: the share of entries in which the dance is recognised within the present, for four ways of carrying the landmarks — a drum on the downbeat and the melody's softer strokes between, heard; the same with the downbeat felt as well; a kick drum on every landmark eight decibels softer than the downbeat, heard; and the same felt, with a felt accent recognised 0 per cent of the time. A quiet room gives 80.9. drum downbeat, melody between: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 34.9, 25 dB 30.4, 30 dB 28.9; the downbeat felt: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 37.3, 25 dB 37.3, 30 dB 37.3; a kick on every landmark, heard: 0 dB 80.9, 5 dB 80.9, 10 dB 68.1, 15 dB 50.2, 20 dB 35.7, 25 dB 30.4, 30 dB 28.9; a kick on every landmark, felt: 0 dB 80.9, 5 dB 80.9, 10 dB 80.9, 15 dB 80.9, 20 dB 75.8, 25 dB 71.6, 30 dB 70.1.

The felt channel carries the inner beats, not the downbeat

A low drum felt in the body gets the downbeat through any crowd, and is then worth less than it was in a quiet room, because the crowd has taken the soft strokes between. Put a kick drum on every landmark of the dance and let all of it be felt, and in a crowd thirty decibels over the band a late dancer recognises the dance in seventy per cent of entries against thirty-seven for the felt downbeat — with no felt accent at all. The accent, felt or not, is worth about ten points on top. The worry that a stream felt alike has no downbeat turns out to be the small half of the arithmetic.

rhythm · Additive metre Marked down, the violin gives its carriers back their partials while it is still heard; the flue does not. Every arrival in which a carrier and its buried passenger enter with a third part beside them (284 arrivals), the third part marked 12 dB below the others: for each third part, the carrier partials clear beside the passenger that the third still pushes under the mask (filled bar; the outline is its cost at full level), and the share of its arrivals in which the third itself still changes the sound audibly (printed). flue: 114 of 195, heard in 100%; violin: 8 of 110, heard in 66%; voice: 1 of 26, heard in 28%; brass: 0 of 15, heard in 0%; oboe: 0 of 11, heard in 2%; clarinet: 1 of 5, heard in 2%. In all 124 of 362.

A quieter third part gives back only as deep as it took

A high part entering third costs its carrier partials, and the masking rule suggests that marking it down should return them all at once. It does not. Each partial comes back when the third part has been lowered by about as far as it had pushed that partial under, so the return is as gradual as the depths are spread. The violin's are shallow: twelve decibels down it gives back nine-tenths and is still heard in two arrivals of three. The flue's run to thirty decibels, and there is no level at which it is both heard and harmless.

form · Orchestration Moved down an octave a third part gives back most of what it takes and is still heard; moved up it takes more. The 284 arrivals in which a carrier and its passenger enter with a third part beside them, the third moved 1 octave down. Carrier partials it pushes under the mask, against where it stood: flue 50 (was 195), heard in 100%; violin 3 (was 110), heard in 98%; voice 23 (was 26), heard in 100%; brass 7 (was 15), heard in 98%; oboe 5 (was 11), heard in 100%; clarinet 7 (was 5), heard in 100%. Altogether 95, against 362.

An octave down a third part is heard and harmless, with nothing to spare

A flue pipe cannot be marked down, but it can be moved, and the expected answer was upward: an octave higher clears the carrier's main region and spreads over partials that are already faint. The count says the opposite. Up an octave a third part takes half as many partials again. Down an octave it takes a quarter of what it took, gives a voice back its whole upper spectrum, and is still heard in nearly every arrival. No level reaches that. The price is that the part has spent its spare level, and three decibels soft takes it out of a third of them.

form · Orchestration

Named alongside it

The objects these essays reach for when they reach for this one.

RegisterOrchestrationCritical bandwidthDynamicsSpreading functionTexturePartialRoughnessVoicingLoudnessSpectrumDoubling

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