Where to put the third

Every book on orchestration says the same thing about spacing, and says it as a matter of experience: keep the low intervals wide, keep the close intervals high, and do not write a major third down at the bottom of the bass staff. Some of them add that the harmonic series is a good model for spacing. None of them computes anything.

It is computable, and it is the smallest exhaustive search on this site. Three pitch classes and three octaves to place them in give twenty-seven arrangements, the roughness model scores each one, and the whole ranking fits on one page. What comes out is not a tendency or a correlation; it is a complete ordering of every choice available, which is the kind of answer the previous three rungs of this ladder each failed to produce for their own questions.

The answer, and what it is a copy of #

At a root of C3, the least rough voicing is 0, 19, 28 semitones: C3, G4, E6.

Partials one, three and five of a harmonic series on C3 are at 0, 19.02 and 27.86 semitones.

The best voicing is the harmonic series itself, to within a seventh of a semitone. The search was given no information about the harmonic series — it was given a roughness model, three pitch classes, and every arrangement — and it returned the spacing of the overtones of the bass note.

The reason is not mysterious once stated. A chord spaced like the harmonic series of its own root has every note landing on a partial that is already there, so the coincidences the previous rungs of this ladder are about are maximal and there is almost nothing left to beat. Any other arrangement puts a note between two partials of the bass instead of on one.

The advice inverts up the compass. Wide-below-close-above is right in the bass and becomes something much more extreme in the treble, because the critical band is a fixed fraction of the frequency and the same interval in semitones is a different number of bands at each register.

It is worth being careful about what “the search was given no information about the harmonic series” means, because the model is built out of partials and the partials are a harmonic series. The information the search did not have is any preference for particular chord spacings: it was handed three pitch classes, twenty-seven placements, and a function that adds up beating between partial pairs. What it discovered is that arranging the notes to coincide with the bass note’s own partials minimises that sum — which is a consequence rather than an input, and it is a consequence with a specific shape that could have come out otherwise.

It could have come out otherwise in an obvious way: the widest possible arrangement also minimises beating, by putting every note out of every other’s critical band, and at two of the three registers tried it wins. The harmonic-series voicing is not simply “as spread out as possible” — it is 0, 19, 28 rather than 0, 24, 31 or 0, 28, 31 — and its being competitive at all is the finding.

The other end, which is the more useful half #

Close root position at the bottom of the range is twenty-seventh of twenty-seven — the worst arrangement available — and it is worst at C2, at C3 and at C4.

That is a stronger result than the winner, because the winner shuffles a little with register and the loser does not move at all. A major third and a minor third stacked immediately above a low C are always the roughest thing that can be done with those three pitch classes, and the reason is the one the third rung of this ladder computed: down there, a third fits inside a critical band, and two tones inside one band are the definition of rough.

So the practical rule has a firm computed basis in its prohibition and a softer one in its recommendation. Do not write the third low is an instruction the model supports emphatically. Write it like a harmonic series is an instruction the model supports at some registers and shrugs at in others.

How stable the answer is #

Three searches at three registers, and it is worth reading the ranking rather than only its top row.

At C3 the harmonic-series voicing wins outright. At C4 it is second by a third of a per cent. At C2 it is third by four per cent. In every case the top five voicings are all wide at the bottom and close at the top, and in every case the bottom five are all clustered somewhere low.

So the ranking is stable in its shape and unstable in its first place, which is a different situation from the scale search of the previous rung, where the shape itself inverted between registers. The difference is that a voicing is a set of frequencies rather than a set of pitch classes — it is already the kind of object roughness has an opinion about, so moving it changes the answer by an amount rather than reversing it.

And the same is true across timbres, which the caveats say and do not draw #

Running all twenty-seven arrangements at three registers for each of the six spectra this site carries — eighteen searches — separates the two halves of the rule completely.

Close root position at the bottom is the worst of the twenty-seven in sixteen of the eighteen, and in the two exceptions the winner of the wooden spoon is another low close cluster. Pure tones, a string, a clarinet, a reed, a bell, an organ; C2, C3, C4. The prohibition does not care what the instrument is.

The winner cares a great deal. It is the harmonic-series voicing at C3 for every timbre, and elsewhere it is whichever wide arrangement the spectrum happens to favour — [4, 24, 31] for pure tones at C2, [0, 28, 31] for a string at C4, [24, 28, 31] for a clarinet at C4. The harmonic-series voicing’s rank runs from first to ninth of twenty-seven across the eighteen searches.

So the asymmetry the essay found between register and register holds across timbre as well, and more sharply. One half of the orchestration rule is a result and the other half is a tendency, and no amount of varying the inputs changes which is which.

The strength of the prohibition does not track brightness #

The caveat below predicts that the prohibition strengthens with brighter timbres and weakens with darker ones. Measuring it as the ratio between the worst arrangement and the best, at C3:

There is no relationship. The darkest and the brightest of the five give 8.1 and 6.9, and the two extremes of the ratio — 6.5 and 8.3 — are adjacent in brightness. Across every complex spectrum the prohibition is worth between six and a half and eight and a half times, which is a spread of under thirty per cent on a quantity that spans a factor of eight.

That is a better result than the caveat expected, and it is worth having for the same reason the register sweep is. The manuals state the prohibition as an absolute and this is why they can: it is not a rule that has to be softened for a dark instrument or hardened for a bright one, because within the range of spectra an orchestra contains the penalty is the same size.

Pure tones are the exception and they are the degenerate case rather than the extreme one — the ratio there is over a thousand, because a widely spaced chord of sine tones has essentially no roughness at all to divide by. Nothing in an orchestra is a sine tone.

Four notes, where the rule gets sharper #

A dominant seventh has four notes and eighty-one arrangements, which makes the coincidence harder to arrive at by accident.

The least rough is 0, 19, 28, 34 semitones. Partials one, three, five and seven of the harmonic series are at 0, 19.02, 27.86 and 33.69.

The match is within a third of a semitone across four notes, and it holds at C2, C3 and C4 — the four-note case is more stable than the three-note one, not less. That is worth stating as a checked result rather than an impression: over eighty-one arrangements the harmonic-series voicing is first at all three registers, where the three-note version is first at one of them and ninth at worst across the timbres. Adding a note quadruples the search space and makes the answer more determinate, because a fourth note gives the coincidences one more place to fail and the series is the only arrangement where none of them does.

The seventh partial is the notoriously flat one, 31 cents below a tempered minor seventh, and the search still puts the chord’s seventh there because that is where the beating stops.

Why the traditional voicing is not the winner #

Nothing in this contradicts what four-part writing actually does, but it does need reconciling with it, because a chorale texture is nothing like a harmonic series.

Four-part vocal writing keeps the upper three voices within an octave of each other and the bass anywhere below. That arrangement is nowhere near the top of this ranking, and the reason is that four-part writing is not optimising roughness. It is optimising voice independence and singability — every part has to be a line a person can hold — and a voicing spread over four octaves has no lines in it at all.

The two objectives pull in opposite directions and the model only knows one of them. Maximal coincidence of partials is exactly the condition under which two voices stop being heard as two, which is the thing counterpoint most wants to avoid. A perfectly smooth chord is a chord that sounds like one note.

That is the sharpest reconciliation available: the ranking is a ranking of fusion, and fusion is desirable in an orchestral tutti and undesirable in a fugue.

Whose music, and when #

The spacing rule is a claim about a practice, and the practice is European orchestral writing of roughly the last two hundred years, as codified in the scoring manuals from Berlioz onward and taught since.

Those manuals are consistent about the prohibition and vaguer about the recommendation, which is exactly the pattern the computation produces. Rimsky-Korsakov’s guidance to keep the bass wide and the upper parts close is the harmonic-series spacing described qualitatively; the warning against low close thirds is universal and is stated as an absolute.

And the rule is genre-specific in a way the model is not. Close low thirds are avoided in orchestral scoring and are a characteristic sonority in other traditions — in barbershop close harmony they are used deliberately at moderate pitch, and in some rock and metal writing a low close third is the point rather than the problem. The model predicts that all of them are rough, which is true and is not an objection: roughness is a description and not a preference, and how much of consonance is preference is a question the site has already put a number to.

The rule generalises past chords #

The result is not really about triads. It is about a fact the whole site keeps meeting from different directions: a critical band is a fixed width in hertz down low and a fixed width in semitones high up, so the same musical interval is a different fraction of a band depending on where it is.

Stated that way, the spacing rule is the same statement as several others already on the site.

A third is muddy two octaves below middle C and clean two above — the same fact about one interval. A bass semitone’s roughness is carried by high partials rather than by its fundamentals — the same fact seen from the rate. A drum’s low modes are unusable as harmony for a related reason: things close together down low do not resolve.

And the rule generalises past music entirely. Any system that mixes signals in a channel with frequency-dependent resolution has the same constraint, which is why perceptual audio coders allocate their bits by critical band and why the masking curves the perception field draws are drawn on the same axis. A voicing decision and a bit-allocation decision are the same decision about the same organ.

The ranking changes, and it changes because half the partials that were doing the colliding are gone. Where to put the third is therefore not one answer but one answer per timbre — the width of the ear’s band decides the mechanism and the spectrum decides which pairs are in a position to use it.

Where the model stops #

The spectrum is the string spectrum in the figures, and the section above runs the other five. The prohibition on low close thirds was predicted here to strengthen with brighter timbres and weaken with darker ones; it does neither, holding at between 6.5 and 8.3 times across every complex spectrum with no relation to the centroid. What the spectrum does change is the winner, whose rank runs from first to ninth. The six spectra here are stated amplitude sets rather than measured instruments, so what the sweep establishes is insensitivity to a broad family and not to any particular orchestration.

Every note is at the same level. Real voicing decisions are made together with dynamic decisions, and a quiet third under a loud fifth is a different object from three equal notes. The model’s amplitudes are the partial amplitudes of one timbre, and it has no per-note level.

The pitch classes are fixed and doubling is not considered. Real four-part writing doubles the root, and a doubled root at the octave is a fifth voice this search does not enumerate. Doubling is one of the strongest tools a scorer has and the model would score it as free — an octave adds almost no roughness — which is very nearly right and not exactly.

And the search is over octave placements only. Every arrangement here keeps the same three or four pitch classes. A scorer choosing between an open fifth and a full triad, or between a third in the bass and a root, is choosing between different chords, which is a different and larger search.

What the picture cannot show #

It cannot show the melody. A voicing is chosen to put a particular note on top, because the top voice is heard as the tune. The search treats all three notes symmetrically and would happily recommend a voicing whose highest note is the one the composer wanted buried.

It cannot show the instruments. Three notes on one piano and three notes on three players are different problems: the second involves onset asynchrony, vibrato and separate room positions, all of which reduce fusion and therefore change what smooth means. Partials that behave differently stop being one note, and an orchestra is a machine for making partials behave differently.

It cannot show the room. A hall adds a reverberant copy of every note at every other note’s position, so a widely spaced chord in a cathedral is a densely spaced one by the time it has been round the building twice — and the smoothest arrangement on paper is not the smoothest arrangement in the seat.

And it cannot show what happens next. Voicing is a decision about a chord, and a progression is a sequence of them in which each voice’s motion is the constraint. The smoothest arrangement of one chord is often not reachable from the smoothest arrangement of the previous one, and no static ranking can say so.

Where this ladder ends #

Eight rungs. The first four established that consonance is computable, register-dependent and only half a fact about the sound; these four asked the model to decide things, and it decided two of the four. It cannot pick a scale, because a scale is transposition-invariant and roughness is not. It cannot say a dissonance is rough without being told how long the note is. It can say which spectrum makes which intervals smooth, exactly and in both directions. And it can rank the arrangements of a chord decisively enough to reproduce a rule from a different discipline — which is the strongest thing a model built from nineteenth-century listening data has done on this site.