Instruments and their design

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.

Assumes: The fourth player is a spectrum, not a decision · The dissonance arrives and the dynamic does not

Seven rungs of this anchor have scored an orchestration. Three of them made the objective a functional over a passage — a written dynamic is an instruction to a running impression, roughness has an integration window of its own, and a chord is masked by the one before it — and then the seventh put four players on three notes, enumerated the thirty-six arrangements, and held the chord perfectly still while it did so.

That is the retreat every enumeration makes when the enumeration gets large, and it is worth naming because a doubling is exactly the operation it cannot afford to make. An orchestrator does not double a chord. They double a line. The pair carrying a note is, almost always, the pair that carried the note before it, because the parts are written down a phrase and not decided at each vertical.

The good news is that the object survives the move. An arrangement is a map from players to positions in the chord — who is on the bass, who is in the middle, who is on top — and that is precisely what carries forward when the chord changes. A player on the bass stays on the bass while the bass moves.

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. — Fig. 1 One arrangement of four players on three notes, held down a five-chord phrase, with its rank among the thirty-six at each chord and the arrangement each chord would have chosen on its own. It is the chord’s own first choice four times out of five.

What holding one arrangement actually costs

Two policies, and the whole rung is the gap between them. Per chord: solve each vertical as the seventh rung does and play the sequence of those answers. Held: choose one arrangement for the whole passage, which is what a score contains.

On a five-chord cadence with a flue pipe, a trumpet, a clarinet and a violin, the best held arrangement is a flue pipe on the bass, a trumpet in the middle and the clarinet doubling the violin on top. Playing that arrangement all the way through costs 16.2 per cent more roughness than re-solving at every chord.

Sixteen per cent is not obviously small, and it is the wrong number to look at on its own. The arrangements available at any one of these chords differ from each other by a factor of 7.8 on average — the worst way of placing four players on a triad is nearly eight times rougher than the best. So the whole penalty for never changing the scoring is 2.4 per cent of the range the choice spans, and the practice the arithmetic was supposed to be second-guessing turns out to have been buying almost the entire prize with its first decision.

Doubling a line rather than a chord is nearly free. For each of three passages, the whole range the arrangements of 4 players on three notes span at one chord — the pale bar, drawn as the factor above the smoothest — with the cost of holding one arrangement for the whole passage marked on it. The choice of arrangement is worth a factor of 7.5 to 9.9; holding one across the passage costs 16.2, 12.3, 0.0 per cent, which is 2.4, 1.9, 0.0 per cent of that range. On a rising sequence it costs nothing at all, because the same arrangement is the smoothest at every chord. What that prices is the practice: an orchestrator doubles a line, and the arithmetic that would ask them to re-score every chord is buying a few per cent of a decision the first choice has already made. — Fig. 2 The same comparison over three passages: the pale bar is the range one chord’s arrangements span, the dark one is what holding a single arrangement gives up. On a rising sequence it gives up nothing, because the same arrangement is the smoothest at every chord.

On a chromatic descent holding costs 12.3 per cent, which is 1.9 per cent of the range. On a rising sequence it costs nothing at all — the same arrangement is that passage’s best answer at every one of its five chords, so the two policies are the same policy.

Why it is cheap, which the seventh rung already knew

The reason is in the seventh rung’s own decomposition and was not read that way at the time.

Fig. 3 The earlier variance decomposition run on this quartet: which players sit on the lowest note accounts for eighty-five per cent of the spread across the thirty-six arrangements, which two players form the pair for twenty-four, and which note carries the pair for two.

Which players sit on the lowest note accounts for eighty-five per cent of the spread. Which note carries the doubling accounts for two.

That is a fact about one chord, and its consequence over a passage is the result above. A quantity that is eighty-five per cent determined by which instrument is on the bass is a quantity that barely notices the bass moving, because the identity of the bass instrument is exactly what a held arrangement holds. Roughness is dominated by the pairs of partials inside a critical band, a critical band is a fixed width in hertz, and so the low notes of a chord contribute most of it wherever the low note happens to be. Transposing the bass down a third changes which partials are in the band; it does not change whose partials they are.

The scoring practice and the arithmetic agree here for a reason neither of them states: the decision an orchestrator makes first — who takes the bass line — is the decision that carries nearly all of the objective, and it is a decision about a line by construction.

Which computation produced the number

Each chord’s thirty-six arrangements are the surjections of four players onto three notes, not the permutations: a note left empty is a different chord and not a different orchestration. Each is scored by summing the Plomp–Levelt roughness over every pair of partials belonging to different parts, with the two players sharing a note splitting that note’s power so that nothing is won by being louder, and with every arrangement bisected to the same total loudness in sones — because a scoring compared at a different level is being compared on a different set of audible partials.

The held answer is the arrangement whose sum over the passage is smallest, found by scoring all thirty-six at all five chords rather than by any heuristic; a hundred and eighty numbers, of which the drawing reads five.

The four players are a flue pipe, a trumpet, a clarinet and a violin, and that quartet is not the seventh rung’s. It replaces the oboe with a trumpet for a reason that decides the second half of this essay: the collection has a measured attack time for five of its six radiators and none for the oboe. The spectrum table and the attack table were built by different ladders for different arguments and they do not have the same members. A doubling whose timing cannot be priced is not a doubling this rung can put on a clock, and inferring the missing row from its family would put a number in a figure that no measurement supports.

The one chord that pays for all of it

The cost is not spread over the passage, and the drawing of it says so.

Four of the five chords take the held arrangement as their own first choice, at no cost whatever. The fifth — a close-position tonic with the third on top — ranks it seventh of thirty-six and costs 103 per cent, which is the whole of the passage’s penalty in one vertical.

What the fifth chord wants instead is worth reading, because it does not touch the bass at all. It keeps the flue pipe on the lowest note and swaps the other two positions: the clarinet and violin come down to the middle note and the trumpet goes to the top. And the fifth chord is the only compact one in the passage — its bass sits an octave under the note above, where the other four have fifteen to nineteen semitones there.

So the entire penalty of holding falls inside the fifteen per cent of the objective the bass does not explain, at the one chord whose geometry makes that fifteen per cent matter. That is the honest reading of the decomposition rather than a second mechanism: what carries a passage is the bass assignment, and the chords that punish a held scoring are the ones where the rest of the chord has closed up around it.

That is the shape a practising orchestrator would recognise and it is worth stating as a rule rather than as an observation. The cost of holding a doubling is not a tax; it is an occasional bill. A phrase whose chords are voiced alike pays nothing, and a phrase containing one chord voiced unlike the rest pays for that chord alone. Which means the arithmetic does not recommend re-scoring every vertical. It recommends finding the one chord in the phrase where the held scoring is wrong, which is what a rehearsal does.

A doubled note is a pair, and a pair has an owner

What makes the doubled note different from the other two is not that it is louder — the power is split so that it is not — but that it sounds like neither of its players.

Which pairs blend, and what blending is. Every pair of radiators playing one note at equal level, ranked by how much of the composite belongs to neither of them. The best is a clarinet with a violin at 0.57, and the worst is under a tenth of that. The quantity that sorts the list is not similarity: it is whether the level at which the composite changes owner falls inside the range a player can reach. A pair whose crossing is at 2 decibels blends, and one whose crossing is off the end of the dial does not, because there is no balance at which the quieter instrument is doing anything but colouring the louder. That is a definition of blend rather than a description of one, and it is computed from two spectra and nothing else. — Fig. 4 Every pair of the quartet on one note, ranked by how much of the composite belongs to neither of them. A clarinet with a violin is the best blend of the four instruments at 0.57, and it is the pair the held arrangement chose.

The held answer doubles the clarinet with the violin, and that pair is the best-blending pair of the four by the definition the seventh rung of the spectrum ladder gave: the level at which the composite changes owner sits inside the range a section could actually be asked for, so there is a balance at which the pair belongs to neither player. It was not chosen for that. It was chosen by a roughness sum that knows nothing about blend, and the two criteria happen to agree.

Which of two instruments owns the pair, at 262 hertz. Every pair of 4 radiators on one note at equal level, with the one the composite belongs to marked. Read across all 6 pairs this is a tournament, and the question a trio raises is whether it has a cycle: if the composite of A and B belonged to A, that of B and C to B and that of C and A to C, a trio would have no strongest member at all. There are 0 such cycles among the 4 triples here, and the win counts are 3, 2, 1, 0 — every count different, which is what a total order looks like. The order is trumpet over clarinet over violin over flue pipe. Nothing in the construction forbids a cycle; instrument spectra simply do not make one. — Fig. 5 Which of each pair owns the composite, read as a tournament. The four instruments fall into a total order with no cycles — trumpet over clarinet over violin over flue pipe — so a doubling always has a single strongest member.

The ownership relation across the quartet has no cycles, so a doubled note always has one member that owns it, and the arrangement that keeps the flue pipe on the bass has put the weakest owner where eighty-five per cent of the objective lives. That is not a coincidence either, and the reason is a count. Whichever instrument is on the bass supplies one member of nearly every rough pair in the scoring, so what matters about it is how many partials it has to offer. The flue pipe’s list is six partials long and stops; the trumpet’s is still at full strength at its eighth, because a bell radiates everything above its cutoff and its cutoff is low. Putting the shortest list at the bottom is putting the fewest candidates where the critical band is widest.

The doubling is a timing decision as well

Everything above is computed on a chord already sounding. Here is what the same thirty-six arrangements do to the moment the chord arrives.

A doubling decides which note of the chord arrives first. For each of the 36 ways of putting 4 players on a three-note chord, when each note's perceptual centre falls — the moment its envelope crosses six decibels below its own peak, which is the criterion Vos and Rasch measured — with the arrangements sorted by how far apart the three are. The three notes never arrive together: the spread runs from 14.1 to 19.0 milliseconds and the arrangement chooses which. A doubled note is heard from whichever of its two players speaks first, so the pair's note is pulled forward by the gap between them — up to 19.0 milliseconds, and 11.0 on average — and the doubled note leads the chord in 18 of the 36. Nothing in the roughness ranking can see any of this, because a roughness is computed on a chord already sounding. — Fig. 6 When each note’s perceptual centre falls, for all thirty-six arrangements, sorted by how far apart the three are. The notes never arrive together; the spread runs from 14.1 to 19.0 milliseconds and the doubling pulls its own note forward by up to 19.0.

A note’s perceptual centre lags its physical onset by an amount set by its attack, so three players starting together do not sound together. A doubled note is the one place in the chord where that arithmetic changes shape: the pair is two independent sources and a listener takes the earlier of the two, so the doubled note’s centre is the smaller of its members’ rather than either of them.

A doubling can therefore only pull its own note earlier, never later, and the pull is exactly the gap between the pair’s two members — up to 19.0 milliseconds in this quartet, eleven on average. The three notes of the chord are spread across 14.1 to 19.0 milliseconds depending on the arrangement, and which note leads is decided by where the fastest player sits.

The extremes are worth reading off, because they show what the spread is made of. The tightest arrangement puts the trumpet and violin together on the bass, the clarinet in the middle and the flue pipe on top, and its three centres fall at 9.7, 14.2 and 23.7 milliseconds — the trumpet is the fastest player in the quartet and the doubling has pulled the bass to its centre, 18.8 milliseconds ahead of where the violin alone would have put it. The loosest puts the flue pipe on the bass, the violin in the middle and the clarinet doubled by the trumpet on top, at 23.7, 28.5 and 9.5, so the top note arrives first and the middle one fourteen milliseconds later. A chord whose top note leads by that much is not a chord a listener assembles in the order it is written.

The two objectives are not aligned. The held arrangement puts the clarinet and violin on top, and a violin is the slowest of these four; the doubling pulls that note forward to the clarinet’s own centre, which still leaves it behind the flue pipe on the bass. Nothing in the roughness ranking can see any of this, and nothing in this figure can see the roughness. They are two functionals over the same thirty-six candidates and the collection has no way of trading them off — which is a shortfall this rung records rather than one it repairs, because a trade-off needs a listener and this one had none.

The mask that does not reach

The fifth rung of this anchor ended by naming a third temporal constant, neither of its own two: forward masking, at a couple of hundred milliseconds, sitting between the roughness window and the loudness release. The debt this rung pays predicted that it would reach the next chord through whichever instrument was loudest in the one before.

The mask cannot reach the next chord's doubling. How many partials the previous chord takes away from this one, against the gap between them, summed over all 36 arrangements of 4 players on three notes. The mask is at its whole strength while the two chords overlap and is gone by 20 milliseconds — far short of the 200 milliseconds forward masking is usually quoted at, and short of this passage's own 49-millisecond roughness window. The reason is not that the mask is weak: it is that a chord with four players on it masks itself, so every partial a predecessor could hide is already under the threshold its own neighbours set. Which arrangement is the smoothest never changes at any gap on this axis, so the third temporal constant named earlier does not enter the doubling decision at all. — Fig. 7 How many partials the previous chord hides from this one, against the gap between them, over all thirty-six arrangements. It is gone by twenty milliseconds — shorter than the roughness window, and a tenth of what forward masking is quoted at.

It does not. Summed over all thirty-six arrangements the previous chord takes thirty-four partials away while the two chords overlap, six at ten milliseconds, and none at all by twenty — against the two hundred milliseconds forward masking is usually quoted at, and against this passage’s own forty-nine-millisecond roughness window. Which arrangement is the smoothest never changes at any gap on the axis.

The reason is not that the mask is weak, and this is the part worth carrying. A chord with four players on it masks itself. One sound hides another and it hides upward, and four spectra on three notes put enough energy in every critical band that any partial a predecessor could hide is already under the threshold its own neighbours set. The forward mask has nothing left to remove. It is not a small effect being outrun by a fast tempo; it is an effect with no work to do at any tempo, on a texture this dense.

That is a correction to the expectation and not to the fifth rung, which measured the mask on a four-part chord with one player on each note and found it pushed the same way as the other windows. Adding a doubling makes the chord denser, and denser is exactly what removes the mask’s opportunity.

What the pictures cannot show

The roughness figures rank thirty-six candidates on one criterion, and no orchestrator has ever chosen a scoring on one criterion. Range, register, colour, the passage before it and the entrances of the players are all absent, and so is the reason most doublings exist, which is loudness rather than smoothness: a line is doubled because it must carry, and this arithmetic holds the total loudness fixed precisely so that it cannot see that.

The timing figure has a sharper limit. It computes where each note’s perceptual centre falls and says nothing about whether a listener could hear the difference. The spread across the thirty-six arrangements is under five milliseconds from tightest to loosest, and an ensemble’s own asynchrony is larger than that. So the ranking by onset spread is almost certainly below the noise of any real performance, while the pull — nineteen milliseconds — is not.

And the model of a doubling here is two independent sources adding in power at one pitch, which is the spectrum ladder’s own and is exactly right for two players and exactly wrong for two players in tune with each other to within a few cents, where the partials beat instead.

Whose music, and when

The practice this prices is European orchestral scoring from Berlioz onward, where doubling is treated as a property of a part — first flute colla parte with first violin for a phrase, not for a chord — and where the treatises discuss which doublings blend at length and take the choice of bass instrument as given by the scoring.

The arithmetic says that emphasis is upside down, and it says so twice over. The seventh rung found that which note carries the pair moves the objective by a few per cent and which instrument is on the bass moves it by nearly the whole range. This one finds that holding the pair fixed for a whole phrase — the thing the treatises describe as a strong constraint on the writing — costs two and a half per cent of that range.

Both readings point the same way. Doubling is written as a property of a line because it is nearly free to be one, and the decisions that are not nearly free are made once, at the top of the score, when the bass is assigned.

Where this ladder goes next

Eight rungs. Who plays what and how loud is one question; the ranking survives the dynamic and the chord does not; a dynamic mark changes what a note is; a subito piano is a rate rather than a level; the other quantity succession has arrives whole; the mask arrives with it and pushes the same way; more players than notes, which is the smallest decision in the problem; and now the doubling put back into time, where holding it costs two and a half per cent and the mask costs nothing.

What is owed after this is the entrance. Every figure on this anchor, including both of the temporal ones, assumes all the players are playing at every chord — the ensemble is fixed and only the assignment moves. The commonest thing a score actually does with a fourth player is not to double a note with them but to bring them in and take them out, and an entrance is a different object from an assignment: it changes the total loudness, it changes which partials are masked, and it arrives with an onset of its own that no chord before it prepared. What that would settle is whether the place an instrument enters is worth as much as the note it enters on, which is the same comparison this rung made between the pair and the bass, run on a decision every score makes many times a page. The machinery is here — a scoring is already a list of players with levels and the running impression already carries one chord into the next, so an entrance is a scoring whose player list grows between chords. It needs arithmetic and a passage, and no listener, and the first thing to compute is whether the loudness ladder’s two-second release makes an entrance on a loud chord audible at all, because a part that adds nothing to the impression is a part the score is paying for and the listener is not receiving.