Critical bandwidth — where it appears
Named by 53 essays across 8 fields — each of them below, with the objects they name alongside it.
Beats are arithmetic that anybody can hear
Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.
Roughness can be computed, and the answer looks like a scale
Feed two spectra into a model of how nearby frequencies interfere, sweep one past the other, and the curve that comes out has dips exactly where the consonant intervals are. Nobody put them there.
A third is rougher in the bass
Consonance is usually presented as a property an interval has. It is not. The same major third is muddy two octaves below middle C and clean two octaves above it, the ratio never changed, and the frequency where it stops being muddy can be solved for.
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.
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.
Roughness cannot choose a scale
Score all four hundred and sixty-two seven-note selections from the twelve for roughness and the answer at middle C is a mode of the diatonic set, first out of four hundred and sixty-two. Ask the same question an octave lower and the same selection ranks a hundred and forty-first. The model is not wrong; it is answering a question about frequencies, and a scale is not one.
A dissonance has to last
Roughness is a fluctuation, and a fluctuation needs cycles. A minor second at the bottom of a cello fluctuates thirty-three times a second, so a semiquaver holds four of them and a demisemiquaver holds two — which turns the counterpoint rule that a dissonance may pass if it is short into a number, and puts that number at about seventy milliseconds through most of the range, once the pairs beating too slowly to be roughness at all are taken out of the average.
Where to put the third
Take three pitch classes, four octaves to put them in, and score all twenty-seven arrangements. The smoothest is root, fifth an octave up, third two octaves up — which is partials one, three and five of the harmonic series — and the roughest, at every register tried, is the chord in close root position at the bottom of the range. Every orchestration manual states that rule and none of them derives it.
Which harmonics carry the pitch
A missing fundamental is inferred from a pattern, and a pattern has to be legible before it can be matched. Counting how many harmonics of a note land in separate auditory filters prices the inference — and the answer at the bottom of a bass guitar's range is none of them.
Two pipes for a note neither makes
The resultant stop sounds a sixteen-foot pipe and a ten-and-two-thirds-foot pipe and asks the listener for a thirty-two-foot note. It is the missing fundamental built on purpose from the fewest partials that can imply anything — and counting how many fundamentals two partials actually imply is the arithmetic behind three centuries of builders disagreeing about whether it works.
A pitch with nothing to match
Filter a click train into a band where no partial is separable from its neighbours and it still has a pitch at its repetition rate. Displace each click by a fraction of a millisecond, leaving the average rate and the long-term spectrum exactly where they were, and the pitch goes. The mechanism is reading the timing — which bounds the account endorsed here from the start.
The same chord is harsher when it is louder
Every roughness number so far was computed at a level nobody stated. Roughness is the product of two partial amplitudes, so it is quadratic in pressure, while loudness is compressive — which makes a minor third at middle C thirty-two thousand times rougher at fortissimo than at pianissimo and only twenty-five times louder. A chord has no single consonance to report.
A chord is a register
The same three pitch classes are five times rougher at the bottom of a piano than in the middle, and the arrangement that minimises the roughness over a low bass turns out to be the bass's own fifth and sixth partials. The orchestration rule about low thirds is not a convention. It falls out of the width of a critical band, exactly.
The series has three tops
How far up the harmonic series can an ear go? The question has three answers and they are an order of magnitude apart. Consecutive partials stop being separately resolvable somewhere around the eighth, and where exactly depends on the fundamental. They stop being a semitone apart at the seventeenth, at every fundamental, because the ratio does not know what it is measured in. And they stop being distinguishable in pitch at all between the thirty-fourth and the hundred and fortieth. Every claim about where the series runs out is a claim about which of the three was meant.
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.
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.
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.
Every partial beats at its own rate
Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.
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.
Sixteen sweeps against sixteen
Every intonation figure about the voice treats a singer as a frequency. A singer is a frequency being swept a hundred cents wide six times a second, and two sections singing an interval are two hundred and fifty-six pairs of sweeps. The beat rate between the partials the interval brings together stops being a number and becomes a function of time — and the pair spends four fifths of its time above the rate at which beating is beating at all.
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.
A roughness with a rate of its own
Every roughness figure so far computes one number for a steady spectrum. Evaluate the same sum at every instant of a vibrato and there are three numbers instead — a mean, a depth and a rate — and the mean is not the roughness of the mean frequency. On an octave it is nineteen times it, because an octave sits in a deep narrow minimum and a vibrato smears it out of one.
The mean survives the window
A roughness that moves has a mean, a depth and a rate — all three of which a listener could only have through a temporal window. Applying the window already to hand settles which of the three survives, and the answer refutes the guess: a running average cannot change an average, so the octave's factor of nineteen stands and the movement is what goes.
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.
The pitch that does not wobble
Three earlier essays have treated a vibrato as a modulation of roughness. The reason singers use one is what it does to the note, and there is an extractor here that turns a set of partials into a pitch and has never been asked what it does with partials that will not hold still. The period survives, at a cost that rises with the extent — and the practice stops within a hair of where the cost becomes total.
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.
The dissonance arrives and the dynamic does not
A scoring decides two things at once and both of them have to be integrated by a listener before they exist. The loudness smoother's release is two seconds and the roughness window is thirty-seven milliseconds, and that ratio of fifty decides which of the two survives at the pace music is actually played. Nothing anybody performs is fast enough to blur a dissonance, and a great deal of it is fast enough to average a dynamic.
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.
A page has two decibels
The account of closure called its dynamic component a corpus debt: it had no model of dynamics in a form. Loudness supplies one, and applying it answers the debt by refusing it. Adding parts to a final chord one at a time, each at the same level, moves the loudness by under two decibels and not monotonically — while a player has sixty. A texture that thins is not a diminuendo.
Eleven partials is one too many
Six earlier essays census exactly ten partials and no figure has ever passed another number. At eleven, the harmonicity census stops finding a perfect harmonic series' own fundamental, takes the octave above it, calls every odd partial inharmonic, and the competition cuts an ideal string in two. It is not the arbitration — the cost of a second stream was swept over a factor of fifty and every verdict came back identical — it is a cap that exists for a good reason and turns out to be the same number as the count.
A part entering is not a change of level
Five earlier essays measure a sonority that is already sounding. The one thing an orchestrator actually controls is the entry, and priced at every semitone it comes to 0.89 phons — under the difference limen — with only 29 of 61 available entries clearing it and none below A♭4. What an entry is instead is an object arriving, and the reason is that its own rise time is faster than the listener's.
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.
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.
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.
Three beats at most, and only in the middle of the keyboard
A mistuned octave on a real piano makes eight beats at once, and a listener attending to one of them is doing something that has a threshold. Two thresholds, in fact — a rate and a place — and once both are applied the eight become four at A3, one at A1 and one at A5. Every interval a tuner sets goes to zero at both ends of the compass and peaks at eight countable beats in the octave the bearing is laid in.
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.
The rate that does not rise with the partial
Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.
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.
Eighty-one chords, one number
A dominant seventh has eighty-one arrangements inside three octaves and their roughness spans a factor of five and a half. The tonal-expectation model gives every one of them the same 3.51 bits, because its states are scale degrees and there is no register anywhere in them. Conditioning the surprise on the voicing costs no corpus — and the arithmetic says the conditioning belongs beside the probability rather than inside it, for three reasons that can each be computed.
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.
Every member of a beat family is the same depth
A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.
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.
The release is on the wrong side
Whether the loudness model's two-second release makes an entrance inaudible has the answer no, for a reason the question did not anticipate. The smoother is asymmetric — ninety-nine milliseconds going up and two seconds coming down — so a rise is tracked twenty times faster than a fall, and an entrance is received promptly by every one of a listener's three readings. The colour of it arrives first, at twenty-five milliseconds against ninety, and the reading that moves with the ensemble is the one nobody would have picked.
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.
One fluctuation or two
Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.
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.
The smoothness is in the skips
Traditional scales come out smoother than random scales of their size because every pair of their degrees is counted once. Count only the pairs a melody on the scale actually sounds next to each other, and every seven-note scale here is rougher than ninety per cent of random ones, under every spectrum, and under a pure tone as well. The smoothness is carried by the pairs a melody reaches by skipping — its thirds and above all its fifths, which are smoother than all but two random scales in two thousand. Counted by their own ascents and descents, the two ragas that share one set of notes stand in different places at last: Deshkar's skipped Re buys a smoother ascent and costs it the fifths.
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.
A rough arrival is rough because of its spacing
The pair the expectation essays report for every chord — how surprising it was, how rough its voicing is — has no level in it. Putting level back in answers the question it left open, and not the way it was framed. At one written dynamic the arrivals keep their order from 40 to 90 dB at three registers of four, and the bass stays 8.6 times rougher than the treble. Made equally loud, the bass has to be played 12.8 dB harder, and it is 343 times rougher: level does not explain the register's roughness away, it multiplies it.
A scale is committed to how long its instrument rings
A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.
A louder final chord is a deeper silence and a brighter sound
A final chord marked a step louder than the passage was supposed to stand above a listener's running impression for as long as it sounded, since the impression can climb no higher than the chord. It climbs exactly that high, and the stand closes in half a second as it always did. What a louder mark actually buys is depth — about seven phons, the same depth a second of silence buys — and a spectrum whose balance point sits most of a whole tone higher, which, unlike the stand, lasts for the whole chord.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
RoughnessPartialOrchestrationMaskingBeatingRegisterSpectrumLoudnessVoicingDynamicsResolvabilitySensory dissonance