Two voices that stop being two

Of all the rules taught in first-year harmony, the prohibition on consecutive fifths and octaves is the one that has attracted the most contempt. It is arbitrary-sounding, it forbids something that is not unpleasant to hear, and generations of students have pointed out that plenty of music breaks it.

The rule is real, it has a mechanism, and the mechanism is countable.

For an interval p:q, every q-th partial of the upper note coincides exactly with a partial of the lower. The octave is 2:1, so every partial coincides. The fifth is 3:2, so every second one does. The fourth is 4:3, so every third. The count falls away as the ratio gets more complicated, and it reaches zero long before the intervals stop being usable.

What sharing partials does #

A voice, perceptually, is a stream — a thing the ear follows through time as a single source. Whether two simultaneous tones are heard as one stream or two is a question with a large experimental literature behind it, and one of the strongest determinants is how much spectrum they have in common.

Two tones an octave apart have everything in common. The upper note introduces no frequency the lower did not already contain; it strengthens the lower note’s even partials and adds nothing. What the ear receives is indistinguishable from one note with a different balance of partials — which is to say, one note with a different timbre.

That is the phenomenon the rule is about. Two voices moving in parallel octaves are not two voices. They are one voice with a brighter tone, and a piece written in four independent parts has quietly become a piece in three.

The fifth is a weaker version of the same thing. Half the upper voice’s partials coincide, the pair fuses substantially, and organ builders have exploited this for centuries: a mixture stop sounds fifths above the fundamental and is heard as brightness rather than as harmony.

Fusion is not consonance #

The two are constantly confused and they are different axes, which the same partial coincidences happen to drive in the same direction.

Consonance is the absence of roughness: no pair of partials close enough to grate. It is what the Plomp–Levelt model computes, and it is a statement about whether the mixture is comfortable.

Fusion is the collapse of two sources into one percept. It is a statement about how many things are heard, not about whether they are pleasant.

Roughness and fusion are not the same quantity and the roughness curve is where they come apart: its wells sit at the fourth, the fifth and the octave, which are also the intervals that fuse — but a major third fuses weakly and is perfectly consonant, and a unison fuses completely and is not an interval at all. Fusion is about how many voices a listener counts; consonance is about how rough the result is. The two orderings agree at the top and diverge below it.

They come apart in both directions. A major seventh is dissonant and does not fuse: two clearly separate voices, grating. A pair of flutes a twelfth apart is consonant and fuses so completely that orchestrators use it as a compound timbre. And the most useful case for counterpoint is the third — consonant, and fusing weakly enough that two voices a third apart stay two.

That is why the third’s arrival as a consonance in the fifteenth century mattered so much for texture and not only for harmony. It supplied an interval that was smooth enough to rest on and separate enough to write two lines in, which is exactly the combination the octave and the fifth do not offer.

Why parallel and not simply present #

The rule forbids consecutive fifths and octaves, not fifths and octaves. A single fifth is unremarkable and appears in nearly every chord; two in a row between the same pair of voices is the thing that is banned. That distinction is the second half of the mechanism and it is about motion rather than about the interval.

Two tones that begin and end together, at a fusing interval, with the same changes at the same instants, share every cue the ear uses for grouping: common onset, common offset, and common frequency modulation. Move both by the same amount and the coincidences hold throughout, so nothing ever arrives to separate them.

Two chord changes with their voices assigned to move least are the same object seen from the other side: a metric that counts total motion has no way to notice that two voices moved together, and the whole of the prohibition is about direction rather than distance.

Contrary motion breaks it. If one voice rises while the other falls, the interval between them changes, the coincidence pattern changes with it, and the two are continually re-separated. This is why counterpoint textbooks prize contrary motion far out of proportion to any acoustic property of the individual chords — it is the cheapest available way of keeping voices apart perceptually, and it sits oddly beside the preference for small total motion, which pulls the other way.

What the rule is protecting #

Once the mechanism is stated the rule stops looking arbitrary and starts looking like a constraint with an aim.

The aim is polyphonic texture: several independently audible lines, each followable, sounding at once. That is a specific and historically located ambition, it is difficult, and the fusion described above is its main enemy. A rule that forbids the two intervals which fuse most strongly, in the one kind of motion that sustains the fusion, is a well-aimed rule.

It also explains the exceptions, which are extensive and not embarrassing.

Doubling for weight is normal. Orchestral writing doubles at the octave constantly, and nobody counts it as a violation, because doubling is exactly the intent: one line, louder and brighter. The rule was never about the sound being bad.

Parallel organum was the point. Ninth-century European practice — the earliest written polyphony — moves entirely in parallel fourths and fifths. A tradition aiming at a single reinforced line rather than at independent voices has no reason to avoid the intervals that reinforce, and the chain of fifths those intervals come from was the whole of its harmonic material.

Debussy and after. Parallel fifths return in the late nineteenth century as a deliberate colour, in composers who were no longer aiming at independent voice leading. The rule was not overturned; the goal it served was set aside.

Written out, one line of a texture is three notes on a stave, and notation records which pitches sound without recording how many things a listener will hear — which is the gap this whole rule sits in. On a keyboard a fifth and the same fifth moved up a tone are two intervals a player thinks of as one gesture, and the hidden ones the rule extends to are the pairs that arrive at a fifth or an octave by contrary motion rather than by parallel: the fusion happens on arrival either way.

The counting, and how far it can be pushed #

The partial counts in the first figure are exact for perfectly harmonic spectra in exact just ratios. Both assumptions weaken the result in real music, in opposite directions.

Tempering. On a keyboard the fifth is 1.955 cents narrow, so the “coincident” partials are not coincident — they beat, and the beating is itself a separation cue. The rates are worth having, because they are not one number and they say where the cue exists:

Equal temperament leaves the octave exactly pure, so it supplies no cue there at all — the prohibition on parallel octaves is as necessary on a piano as it is in a choir, and only the fifth is weakened. That is an asymmetry the count cannot see and it runs the same way the textbooks do, since parallel octaves are the graver fault.

And the cue is a treble one. A beat at 0.22 hertz is a cycle every four and a half seconds, which is longer than most notes; at C6 the same coincidence beats three and a half times a second, which is unmistakable within a crotchet. Tempering separates two upper voices moving in fifths and does nothing for two low ones, which is the opposite of where a bass-heavy texture most needs help, and it is a reason the effect is real and small rather than real and useful.

A tempered parallel fifth therefore fuses slightly less than a pure one, in the treble, on a sustaining instrument.

Inharmonicity and vibrato. A real string’s partials are not quite at whole-number multiples, so the coincidences are approximate even in principle. Vibrato is the stronger effect and it runs the other way: two voices with independent vibrato have partials that wander independently, which is a powerful separation cue and is one reason two singers in octaves are easier to tell apart than two flutes.

Timbre. The count assumes both voices have the same partials. Two instruments with different spectra fuse less, and orchestration exploits this continuously. A fifth between a flute and a bassoon is a different perceptual object from a fifth between two flutes, and no rule stated in terms of pitch alone can capture that.

So the number in the figure is an upper bound on the fusion, achieved by identical harmonic timbres in exact tuning with no vibrato — which describes an organ and very little else. That the rule was codified in an era of vocal polyphony and organ accompaniment is not a coincidence.

The hidden ones, and why the rule extends to them #

Textbooks forbid more than the obvious case, and the extensions are a good test of whether the mechanism is the right one — a rule justified by fusion should extend exactly as far as fusion does.

Hidden or direct fifths — two voices arriving at a fifth by similar motion, without having been at a fifth before — are forbidden between the outer parts in strict practice. On the fusion account this is defensible but weaker: the voices are moving together, which supplies the common-modulation cue, and they land on a fusing interval. Only one of the two conditions is met throughout, and the textbooks correspondingly treat it as a lesser fault and allow it in inner parts.

Parallel unisons are forbidden most strictly of all, and here the mechanism is at its cleanest: two voices at a unison are not merely fusing, they are the same signal, and the part-writing has demonstrably lost a voice.

Parallel fourths are forbidden between the bass and an upper part and permitted between two upper parts. That looks inconsistent and is not. The fourth shares four partials of twelve, so it fuses moderately; what changes with the bass is that a fourth above the bass is heard as an inverted chord requiring resolution, which is a harmonic-function reason rather than a fusion one. Two different mechanisms, one rule, and the fusion account correctly declines to explain the whole of it.

The pattern across the three cases is consistent: the prohibition is strongest where fusion is strongest, weakens where fusion weakens, and where it does not track fusion at all — the fourth against the bass — there is a separate and openly harmonic reason. A rule that fitted its stated mechanism perfectly would be suspicious; one that fits it for three cases out of four and names a different mechanism for the fourth is behaving like a real description of practice.

Whose counterpoint, and when #

The prohibition is a rule of European contrapuntal practice from roughly the fifteenth century to the nineteenth, codified most influentially by Johann Joseph Fux in Gradus ad Parnassum (1725), which taught it to Haydn, Mozart and Beethoven and is still the basis of how it is taught.

Fux’s own justification is not the one given here. He argues from the variety of intervals and from the practice of Palestrina, which is an appeal to a repertoire rather than to a mechanism, and that is the honest description of what the rule was in its own time: a distillation of what good composers were observed to do. The spectral account is a nineteenth- and twentieth-century reconstruction of why the observation held.

Outside that jurisdiction the rule simply does not apply, and it is worth being blunt about the range of things it does not cover. It says nothing about music that is not polyphonic in this sense — most of the world’s music, and most Western popular music, where parallel fifths are a standard guitar texture. It says nothing about rhythm, where the analogous question of when two streams are heard as one has its own answers. It says nothing about music aiming at fusion deliberately, from organum to power chords. And it is not a claim about beauty in any case: the sound of parallel fifths is perfectly pleasant, which is precisely why the rule needs a mechanism to explain it rather than an appeal to taste.

The organ, which does this on purpose #

The clearest demonstration that fusion is real is an instrument built entirely around it.

An organ stop is a rank of pipes, and a large instrument has many ranks that can be drawn together. The ranks are labelled by length — 8′ sounds the written pitch, 4′ an octave above, 2⅔′ a twelfth above, 2′ two octaves above — and drawing several at once produces one sound rather than several. A player pulling out a 2⅔′ stop is adding a rank of pipes sounding a fifth above every note, and what arrives is not parallel fifths. It is a brighter tone.

A mixture stop takes this furthest: several ranks at once, sounding octaves and fifths above the fundamental, breaking back to lower pitches as the keyboard rises so that the top of the compass does not run out of pipes. It is, in strict counterpoint terms, an atrocity — parallel fifths in every voice throughout — and it is heard as timbre because that is what the intervals concerned do.

The design predates the rule it appears to break by centuries, and organ builders and counterpoint theorists were often the same people or in the same buildings. Nobody found this contradictory, which is the strongest evidence available that the prohibition was always understood as being about independent voices rather than about a sound.

The same principle is at work anywhere a fifth is used for reinforcement rather than harmony — the guitar power chord, the parallel fifths in a synthesiser patch, the octave doubling of a bass line. In every case the intent is one thickened line, and in every case the listener obliges.

Where the argument stops #

The strongest claim available here is that fusion is a real, measurable perceptual effect, that it is strongest at the octave and fifth, and that a tradition wanting independent lines would want to avoid exactly those in parallel motion. That is a good deal more than “it sounds bad” and a good deal less than a derivation.

What is missing is a quantitative link between the partial count and the amount of fusion. The count is an integer; fusion is graded, depends on level, register, timbre, duration and attention, and no published model takes a pair of spectra and returns a number for how likely they are to be heard as one source. Until such a model exists the account is a mechanism with the right shape and no calibration.

There is also a competing explanation that cannot be dismissed. Parallel perfect intervals reduce the harmonic information in a texture — two voices at a fixed perfect interval carry between them barely more than one voice does — and a rule against them might be protecting variety rather than separability. The two accounts predict the same prohibition and differ on the reason, and no experiment in the literature separates them cleanly.

What the picture cannot show #

A count of coincident partials is a static quantity, and everything about voice independence is temporal. What actually separates two lines is that they do different things at different moments — enter at different times, hold while the other moves, take different rhythms — and none of that is visible in a snapshot of two spectra.

There is also a cue the count cannot reach, and it is the stronger one. Fusion is decided by shared behaviour rather than by shared partials: two voices that enter thirty milliseconds apart stay two however many partials they hold in common, and a perfectly harmonic partial given a head start leaves the note it belongs to. The rule against parallel fifths is about a static coincidence, and the thing that actually keeps voices apart in performance is the timing of their onsets.

The picture also cannot show the listener. Whether two streams are heard as two depends on what is being attended to, and a trained listener can pull apart a texture that a naive one hears as a block. The rule was written by people for whom the separation was easy, about listeners for whom it may not have been.

The ladder from here goes back to motion: having established what keeps voices apart, the question is what makes a particular succession of chords feel like an arrival, which is not the size of the move and turns out to be which notes move rather than how far.