Every essay — page 27
Pitch and tuning Intervals and chords Scales and modes Harmony and voice leading Rhythm and metre Timbre and acoustics Perception and the listener Instruments and their design Form and structure Series Objects Sounds Search
Form and structure
The shape a piece has in time, computed rather than labelled. A distance matrix finds the sections before anybody names them, a phrase turns out to be a number of seconds rather than of bars, and an ending is five separate signals that can each arrive without the others.
A form is sharp at the bottom and vague at the top
A movement is divided into sections, each into phrases, each into bars, and every level is a ratio of two estimates. Timed, the hierarchy is almost uniformly blunt — 2.1 distinguishable proportions at the top and 5.2 at the bottom, because a Weber fraction is a step function of duration and six of a piece's seven levels fall in one step of it. Counted, the same hierarchy runs from 2.2 to 12.2 and sharpens monotonically downward. At no level of either does timing separate 3 : 2 from the golden section.
A golden section is a coin toss with six coins
An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.
The ceiling everybody names is the loose one
Ask why phrases are the length they are and the answer given is the breath. It is arithmetic — usable lung volume over the air a note costs per second — and it comes out between fifteen and twenty-three seconds at a comfortable dynamic and between seven and twelve at a loud one. The ceiling the present moment imposes, the two-to-eight seconds inside which a stretch is heard as one thing rather than as a series, is two to three times tighter at almost every note and dynamic. A singer in an adagio is not running out of breath at the phrase end. They are running out of present.
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.
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.
Checkpoints sharpen the middle of a form, not its top
A listener who counts bars loses the count somewhere in a long section and is thrown back on timing the whole of it. A listener who also counts phrases can mend the lapse at the last phrase. If the two counts fail independently, the level a minute long goes from four distinguishable proportions to nineteen; the whole eight-minute piece goes only from two to four, because thirty phrases are long enough to lose a count as well. And if a fifth of lapses take both counts at once, three quarters of the gain is gone.
Only a lapse of attention keeps a form's top vague
A listener who keeps a count at every level of a form — bars, phrases, periods, sections — each mending the one below when it lapses, does not have a vague top at all. The whole eight-minute piece becomes the sharpest level, at seventy-six distinguishable proportions against two. Failures shared between neighbouring counts barely dent it, because a shared failure has to climb every level to reach the top. What puts the vagueness back is a lapse that takes every count at once, and one lapse in a hundred of that kind is enough.
A repeat sign loses a listener that a refrain does not
A lapse of attention loses every count a listener is keeping, and when it ends the listener has only the music to say where they are. A listener who knows the form comes back to a rondo's refrain and is almost never misplaced, because the episodes between refrains make only one of them reachable. They come back near the join of a section repeated immediately and are misplaced nearly half the time, and a form built of such repeats loses between a third and a half of the precision a count at every level had restored. One of the most familiar signs in notation does more damage than the most repetitive form.
No rest balances a louder chord held past a ninth of a second
A rest before a final chord leaves the passage quieter than the tutti and a louder mark on the chord leaves it louder, so the two together can leave it exactly where it was, while the chord stands out by both at once. They can — but only for a stamp. A chord a step louder and longer than 115 milliseconds lifts even a listener's fully fallen impression past the tutti, so no rest of any length balances it; two steps louder, the limit is 67 milliseconds, and three, 45. The balanced ending exists, and it is decided by the release, not by the rest.
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.
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.
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.
In a hall the balanced ending is a louder chord
In a dry room a rest before a final stamp a step louder can leave a passage finishing exactly at its tutti's weight, but only for a stamp shorter than a ninth of a second. The prediction was that a hall would make it impossible: its tail fills the rest and lengthens the stamp. That is right for a small room and wrong for a hall. Read when the chord's tail has died, a live hall pulls every ending below its tutti, and the ending that finishes at its own weight there is a held chord marked louder than the passage — five phons louder at a second of reverberation, a whole step at about 1.3 seconds, two at 2.3.
New music says only that it is new
A listener who knows a piece comes back from a lapse of attention and is placed at once by whatever they hear, and a through-composed piece, which never repeats anything, is the easiest of all forms to come back to. A listener hearing it for the first time is not so lucky. What they hear on returning is either something they have heard before, which names itself, or something they have not, which says only that it is new. For a rondo that costs almost nothing. For the through-composed piece it costs its whole lead: at lapses of half a minute or more, a first-time listener is placed no better in it than in a rondo — and being told the piece's plan in advance gives back almost none of the difference.
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.
A section that says what it is for places a first hearing
A first-time listener who loses count in a through-composed piece comes back into music they have never heard, and new music says only that it is new. Suppose instead each section announces what it is doing — opening, continuing, intensifying, closing — and the listener recognises it. How much of their place that gives back depends hardly at all on how many functions there are and almost entirely on how soon the same function comes back. Four functions cycled restore 85 per cent of what a first hearing costs at forty-second lapses; the same four in the familiar arc, two sections each, restore 42; two functions alternating restore 85 per cent at twenty seconds and 14 at a minute.
Two functions far apart outlast four that blur
A first-time listener who loses their place in a through-composed piece gets most of it back if each section announces what it is doing and the same function does not come back too soon: four functions cycled restore 85 per cent of what a first hearing costs. That rests on hearing each function exactly. Let a listener mistake a function for its neighbour in the arc — a continuation for an intensification — one time in ten, and the cycle keeps less than half of its gain. Two functions at the opposite ends of the arc, opening and closing alternated, lose nothing to such confusion, and from a little over one confusion in five no arrangement does better.
A listener who learns the plan keeps only the far pair's immunity
A first-time listener who recognises what each section of a through-composed piece is doing gets most of their place back after a lapse — if they know the plan of functions. A listener who has to infer the plan from the sections heard so far loses most where the plan said most: four functions cycled give up more than half of what their labels were worth, nearly all of it to lapses in the first two minutes, before the second function has settled which plan the piece has. Opening and closing alternated lose a little and stay immune to confusing neighbouring functions. The far pair's immunity was never knowledge of the plan; it was the arc.
An announced plan pays the cycle and punishes the blocks
A slow introduction, a title or a programme note can tell a first-time listener the plan of a piece before the first lapse — and be wrong. Trusted as far as it deserves, an announcement gives back half of what being told the plan is worth to four cycled functions only once it is right nine times in ten. But how far to trust it depends on the piece: a cycle rewards believing even a coin-toss announcement completely, while functions in blocks of two are made worse by an announcement right half the time, and a confident wrong one can cost a listener forty points of the first minute.
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.
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.
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.
A singer needs a quarter of the breaths the page offers
A breath in a sung phrase looks like evidence of a boundary, and also like a physical necessity, and it cannot be both at once. Solved exactly over three tunes, a trained singer is forced to breathe inside a phrase on 8 settings of 147, all of them slow and loud; at a comfortable level the air asks for about a quarter of the phrase ends a tune offers. So nearly every breath is a choice, which is what makes it a cue. Where the air does force one, it lands on a listener's boundary no more often than chance.
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.