The note that sounds twice

The eighth rung of this ladder relaxed the assumption that a chord’s notes start at the same instant, and ended by naming the ninth: a triad has three notes and a texture that plays it has four voices, so one note sounds twice, and which one is a decision the four-part machinery leaves free and every treatise has opinions about.

The machinery says so itself. chordVoicings — the function every part-writing figure on this site is built on — has a line in its own docstring: which note gets doubled is left free, since that is one of the things the rules below have an opinion about. Roughness, virtual pitch and the masking figures all had something to say and none of them had been asked.

Asked, they say what the treatises say, which has not happened before in this ladder.

Four hundred and eighty voicings #

The census is exhaustive rather than sampled. Take the three pitch classes of a triad; find every pitch of each inside each of the four SATB ranges; take every combination in which all three pitch classes sound. That is 480 complete four-part voicings of one triad, and every one of them doubles exactly one member.

168 of them double the root. 168 double the fifth. 144 double the third — fewer, because the ranges are what they are and the third has fewer places to go.

Each voicing is then scored two ways, and the two ways are the ladder’s own.

Roughness is the summed Plomp–Levelt contribution over every pair of the four notes on a string spectrum, which is the measure the register rung and the census rung are built on.

Root strength is the virtual-pitch measure from the root an ear supplies: the four sounding frequencies are fitted to a single harmonic series, and the number reported is the worst partial’s error in cents. Small means the four notes look like members of one series, which is what makes an ear supply a fundamental beneath them.

The textbook rule, from the arithmetic #

For a major triad:

Root, then fifth, then third, on both measures. That is the rule as every harmony treatise states it — double the root by preference, the fifth where the root will not serve, and avoid doubling the third — and it is the first time in nine rungs that this ladder’s arithmetic has come out where the practice is.

The previous eight went the other way with some regularity. The second inversion is the smoothest of the three positions by roughness and the best explained by virtual pitch, and three centuries of practice will not let a phrase end on it. Eight of the eleven three-note stacks are smoother than the major triad. The masking asymmetry that ought to explain the rolled chord is two decibels where it would be audible. This ladder has a track record of refuting received rules, and this rung breaks it.

The two measures disagree about how much it matters #

They agree on the order and not on the size, and the difference is the more useful half.

Roughness separates the three by nine per cent. 0.4625 against 0.5039 is a difference nobody would notice as a difference in a chord, and it is a mean over a hundred and forty-odd voicings whose individual spread is far larger than the gap between the groups.

That last observation is a warning about the comparison as well as about the size, because the three groups do not have the same spacings in them — the third has fewer places to sit inside the SATB ranges, so its 144 voicings are distributed differently across the register from the root’s 168. Binning by the total span from bass to soprano and comparing within each band:

Controlled for spacing the effect is larger, not smaller — thirteen to eighteen per cent in the two bands that hold most of the voicings, against nine pooled. The pooled figure was diluting the difference rather than exaggerating it, because the groups’ span distributions partly cancel, and the honest number is half again as big as the one quoted.

The ordering survives the control too. The root is best or within a thousandth of best in every band, and the third is worst in every band that holds enough voicings to mean anything. So the roughness verdict is not an artefact of which voicings happen to be reachable; it is a property of the doubling at a fixed spacing.

What that does to the conclusion below is to soften it rather than overturn it. Eighteen per cent is still small beside the factor of six and a half that spacing itself commands, so a player choosing between arrangements should still be choosing the spread first. But it is no longer true that the doubling is nearly free on this axis, and the two measures are separated by a factor of two rather than the factor of thirty the pooled numbers implied.

The same control on the minor triad is worth a line because it does not tidy anything up. Within bands the fifth is smoothest in two of three and the root in the other, the third is worst in two of three, and the spread runs from 4.5 to 13.1 per cent — so the minor triad’s disagreement between the two measures is not a pooling artefact either. It is the same disagreement seen at every spacing, which is what a real one looks like.

The fit to one series separates them by a factor of 2.7. 14.9 cents against 39.5 is not a marginal preference; it is the difference between four notes that look like one harmonic series and four notes that do not.

So the rule is a virtual-pitch rule and not a roughness rule. Doubling the third does not make a chord harsher in any amount worth mentioning; it makes the chord’s root harder to find. Which is exactly what the treatises say when they explain themselves at all: a doubled third makes the chord “thick” or “muddy” or “ambiguous”, and none of those words is about beating.

Doubling is not a choice a four-part texture makes; it is a condition it is in. Four voices and three pitch classes leave exactly one note sounding twice, which is why this essay’s question — which note — has an answer at all, and why it stops having one below four parts and multiplies above them.

The minor triad does not obey #

The same census on a minor triad gives a different order.

The fifth is the smoothest doubling and the root is the best fit, and the third is last on roughness and second on fit. The two measures no longer agree with each other, and neither of them reproduces the treatise rule, which is stated for major and minor chords alike.

The reason is the ambiguity the fifth rung of this ladder measured. A major triad’s notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass; a minor triad’s fit two different series with two different answers a major sixth apart, and the model cannot choose between them. All three of its doublings fit badly — 43 to 50 cents against the major’s 15 to 40 — so the ordering among them is a comparison of three poor fits, and it is not stable.

Which means the treatise rule, insofar as the arithmetic supports it, is a rule about major triads that was generalised. The generalisation may be right for other reasons; it is not right for this one.

Where the chord has no members, the method says so #

The best evidence that the measure is measuring something is what it does on the two symmetric chords.

An augmented triad divides the octave into three equal parts, so its three notes are interchangeable and the words root, third and fifth are a spelling convention rather than a fact about the sound. A diminished triad is nearly as symmetric.

The census reports no preference for either. For the augmented triad the three groups’ roughnesses are 0.510, 0.517 and 0.518 — a spread of one and a half per cent — and their fits are 50, 49 and 50 cents. For the diminished triad, 0.572, 0.573 and 0.590, with fits of 44, 46 and 40.

That is the right answer and it was not put in by hand. A method that reported a doubling preference for an augmented triad would be reporting a property of the spelling, and this one does not.

At five parts the question this essay asks becomes two questions at once, and they interact: the two doublings cannot be chosen independently because the same voice-leading rules price the pair. That is why the four-part case is the one the tradition teaches — it is the largest texture in which the doubling decision is a single decision.

The exception the treatises make, which does not survive #

There is one place the practice is more specific than the rule, and the census can be pointed at it.

The prohibition on doubling the third is usually relaxed in first inversion — with the third in the bass, doubling the bass is tolerated and in some styles preferred, because the alternative doublings are awkward to reach.

The exception is the worst cell. Third in the bass with the third doubled fits a single series at 49 cents, against 11 for the best cell in the grid. Whatever justifies that relaxation, it is not that the resulting sound is well found by either of this ladder’s measures — and the honest reading is that the relaxation is about voice leading and hand position rather than about the chord, which is what makes it an exception rather than a rule.

The best cell in the grid is worth naming because nobody teaches it: fifth in the bass with the root doubled, at 11 cents. That is a second-inversion chord, which is the position a phrase may not end on, and the ladder’s fourth rung already found it the best-explained of the three positions. This rung finds its best doubling too, and the practice still forbids it.

What a doubling actually is #

There is a reading of the whole rung that makes it smaller and truer, and it belongs at the end rather than at the start.

Two voices on one note are not one note twice. They are one note about three decibels louder if the two voices are independent, or six if they are perfectly in phase, and between the two lies the whole difference between a choir and a duet. So a doubling is a balance decision taken with a very coarse dial: it says “this member of the chord shall be louder than the others” and it says it in the only unit a four-part texture has.

Which puts the result in a different light. The treatise rule says: make the root the loudest member. The virtual-pitch measure says: a chord whose root is emphasised is a chord whose root is easy to find. Those are the same statement, and the reason the census can find it is that raising a member’s level and doubling it are the same operation at the resolution a texture offers.

And it explains the minor triad’s failure. If the root cannot be found reliably anyway — two candidate series, a major sixth apart, and no way to choose — then making the nominal root louder does not settle anything, and the arithmetic has nothing to prefer.

Which computation produced the numbers #

Three functions and no new parameters.

chordVoicings enumerates the complete four-part voicings inside SATB_RANGES, which are this site’s own compass figures for the four voices and are used unchanged by every part-writing figure here.

chordRoughness sums Plomp and Levelt’s roughness over every pair of notes, on the eight-partial string spectrum this site uses throughout. The absolute values mean nothing outside this figure; the ratios are the content.

residueCandidates is the virtual-pitch matcher: it searches assignments of the sounding frequencies to harmonic numbers up to sixteen, fits a fundamental by least squares, and reports the worst partial’s error in cents. It is the same function the missing-fundamental ladder is built on and it is applied here to four fundamentals rather than to a set of partials, which is a use it was not written for and is a legitimate one — a chord’s notes are candidate members of a series, and that is the whole content of the virtual-pitch account of a root.

One number is a tolerance and it matters. The matcher is given 60 cents of slack, and about half the voicings find no fit inside it at all; those are excluded from the mean rather than counted as infinitely bad. Tightening the tolerance raises every group’s mean and does not change the order; loosening it lets in fits so poor that the mean stops meaning anything. The share that fits is reported alongside and is between 51 and 56 per cent in every group, so no group is being flattered by having its failures dropped more often than another’s.

Whose music, and when #

The rule is eighteenth- and nineteenth-century European part-writing pedagogy, codified from chorale style and taught since. It appears in Fux’s subject matter, in every nineteenth-century harmony treatise, and in every twentieth-century textbook that teaches four-part writing.

What the arithmetic adds is not authority but a mechanism, and a mechanism that is narrower than the rule. It supports the rule for major triads, does not support it for minor ones, and refuses to support the first-inversion exception at all. A pedagogy that stated it for major triads and gave a different reason for the minor case would be closer to what the numbers say, and no pedagogy does.

Outside that repertoire the question does not arise in the same form. A texture with more voices than notes is not a universal condition — a three-voice texture makes the prohibitions vacuous and a six-voice one makes them dominant — and a tradition that plays a triad on one instrument has no doubling decision to make.

What the picture cannot show #

The tolerance excludes half the voicings. Fifty-one to fifty-six per cent of each group finds a harmonic fit within sixty cents and the rest are dropped. The comparison is between the fitted subsets, and the assumption that the dropped ones are equivalently bad across groups is checked only by the share, not by their content.

Roughness is computed for a simultaneity and none of these chords is one — the previous rung’s whole point — so every roughness figure here is an upper bound multiplied by an overlap fraction a performer chooses.

No spacing constraint is applied. Part-writing forbids gaps larger than an octave between the upper voices and this census does not, so it contains voicings no treatise would accept. Excluding them would change the means and it would change them for all three groups.

And nothing here explains why a doubled third is “muddy”. The fit to one series is a model of how easily a root is found, and the connection between that and a listener’s report of muddiness is an assumption this collection makes throughout and has never tested.

Where this ladder closes #

Nine rungs, and this one closes the anchor.

The model is three simultaneous notes and what a listener makes of them. Name a variable it has and the ladder has a rung for it: which three notes (rung 1, the four stackings; rung 3, the census of every chord of every size; rung 7, the stacks that are not thirds), which interval is special (rung 2, the tritone that inverts to itself), which note is in the bass (rung 4, the three positions and the one that cannot end a phrase), what root a listener supplies (rung 5), which register (rung 6), whether the notes start together (rung 8), and which note sounds twice (this one).

What is not on that list belongs elsewhere and is a different model rather than a further rung. How a triad connects to the next one is voice-leading, which closed at nine. Whether its third is major or minor to an ear is categorical-hearing. Whether it is in tune is the-comma. How loud it is belongs to loudness.

One thing is genuinely missing and it is missing because it is not about the triad: balance. Every figure in this ladder gives every note of the chord the same amplitude, and a doubling is the coarse version of a balance decision — two voices on one note is a level increase of three decibels or six depending on their coherence. That is the parameter this rung’s own subject is a special case of, and it belongs to the loudness ladder rather than to this one.