Every essay — page 21
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Perception and the listener
Loudness is not amplitude and a beat is not in the signal. What the ear adds, hides, merges and supplies — measured, with the numbers every other field has been assuming.
Where a wrong head gives itself away
Every claim so far maps a delay to a direction through one fixed geometry, and the listener acquires that map while the geometry grows under them by seventy per cent. So the map can be wrong — and the essay before this one said the error would be largest on the median plane, where the delay curve is steepest. It is exactly zero there. The steepness is in the error and in the threshold and cancels between them, which leaves a listener whose internal head is 1.3 millimetres out with one place to catch it: hard to the side, where nobody localises well.
A subito piano is four seconds longer in the bass
Every loudness figure with time in it converts level to loudness at one kilohertz, and the equal-loudness contours say that no other frequency works that way. Joining the two sorts the published numbers into those that were about the treble and those that were not. Three move a great deal — a twenty-decibel crescendo is worth 27 phons on a bass note and 20 on a high one, and the seven seconds a subito piano takes becomes eleven and a third. Three do not move at all, and the reason they do not is the same reason in every case.
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.
The error that moves straight ahead
The essay before this one found that a listener whose internal head is the wrong size makes no error at all on the median plane, and has to look hard to the side to catch it. Every head drawn here has its ears at equal radii, which makes the delay curve odd and every error a factor — and a factor cannot move a zero. Real heads are not symmetric. A constant offset of twenty microseconds displaces a listener's straight ahead by two and a quarter degrees, and it displaces every other direction by the same number of just-noticeable steps, exactly.
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.
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.
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.
A quiet bass is further off-centre than a loud violin
An ear whose threshold is a decibel worse than the other is a little later to report every sound, and the published brainstem latency law says how much later: a gap that halves every 43 decibels and grows toward the bass. Taken at full size, a one-decibel asymmetry would put straight ahead twelve degrees round at a moderate level — seven times what the ear's own two thresholds say a decibel is worth, so most of that latency cannot be reaching the comparator that reads direction. What survives any scaling is the shape. The displacement is not one number but a curve in level and in pitch, and it deforms an orchestra rather than moving it.
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.
A bowed note is placed as if it were quieter
An ear a decibel or two worse than the other is later to report every sound, by an amount that grows as the sound gets quieter, and a listener takes a note's direction from its onset. A plucked or struck note reaches its dynamic before the onset is over and is placed almost where a steady tone would be. A bowed note is still near the threshold of hearing while its onset is being read, so it is placed as though it were 27 decibels quieter than it is. The prediction was that slow attacks would soften the displacement; they sharpen it, and an asymmetric listener's orchestra is deformed by its articulation as much as by its register.
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.
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.
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.
A page has twelve phons, and its entries always said so
An ending built up one part at a time was found to move its loudness by under two decibels, and five essays have leaned on that number since. It came from a way of grouping partials into critical bands that sorts an eight-part chord's sixty-four partials into two bands. Counted along the ear instead, the same ending gains twelve phons at every step, which is what the collection's own price for a single entry always implied.
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.
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.
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.
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.
The page can write a diminuendo and not a climax
Counted on the excitation pattern rather than in grouped bands, the four ending textures the closure essays compare come out the right way round at last: a thinning ending is a ten-phon diminuendo written into the score, and an ending that adds parts arrives louder than what came before rather than quieter. But the gain is tiny. Three more parts on a six-part final chord buy under a phon and arrive a twentieth of a phon above the listener's running impression; only a full chord straight after a thinning stands out, and only at a quick tempo. A page can take loudness away. It cannot make a climax.
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.
A hall leaves a thinning ending most of its diminuendo
A written diminuendo made by dropping parts arrives about four and a half phons under the listener's impression in a dry room. A hall's tail sounds under every bar and holds the level up as parts leave, and the obvious worry is that it fills the gaps a thinning leaves, as it fills a rest. It barely does. A concert hall leaves the thinning 86 to 100 per cent of its contrast and a stone church about three quarters, where the same rooms leave a two-second general pause three quarters and a half. A thinning steps down a decibel or two a bar, and a hall's tail falls nine decibels in the time a listener takes to register the arrival; only a cathedral is slow enough to blunt the last step, and it takes half.
A blown note parts by its register, not its family
A wind note's partials reach their steady levels at different rates, and one ranking of five instruments had the saxophone's parting soonest and the horn's latest. Run at every note rather than one, that ranking dissolves: it was read at pitches from 126 to 428 hertz, and pitch was doing most of the sorting. Moving up one instrument shifts the point of parting three times as far as changing instrument at one pitch, and read at half of steady level, all five come together within a fifth of each other, between about 290 and 440 hertz.
A hall forgets almost everything but the last part to leave
A hall's bass outlasts its treble, so a thinning ending ought to care which parts it drops first: top down should leave the tail's darkness to fill the gap, and bass up should remove what the tail holds longest. Searched over all 720 orders, it barely does. Give the tail a spectrum and the drop moves by under half a phon, in the direction that helps the order expected to suffer. What sets the ending is two notes: the one left sounding, and the one that went just before it. Everything earlier has gone.