A dissonance is what has to be resolved

Eight rungs of this ladder have computed roughness and then found out what bounds it. It depends on register, on the spectrum, on how long the note lasts and on where the notes are put; it is only half of what listeners mean by consonance; and it cannot pick a scale. Each of those is a limit on how far a number reaches. None of them questions whether the number is measuring the right thing at all.

This rung does, and it uses the interval that has embarrassed the theory for a thousand years.

The interval that will not sit still #

The perfect fourth is a consonance when it occurs between two upper voices, and a dissonance when it occurs against the lowest sounding note. That is not a subtlety or a special case; it is stated in that form in every counterpoint manual descended from Fux’s Gradus ad Parnassum of 1725, and it is the rule students find hardest to believe.

It is worth stating the awkwardness precisely. The same two notes, at the same two frequencies, in the same room, are a consonance or a dissonance depending on whether a third note is above them or below them. Nothing about the pair has changed. Something about its position has.

Whatever else that is, it is not a property the roughness model can have an opinion about — but the model can be asked, and the answer is worth having in numbers rather than in principle.

Which computation produced the number #

Three notes, given as MIDI numbers so the frequencies are frequencies rather than an idealisation, and the roughness model this ladder built at its second rung summed over every pair of partials of every pair of notes.

E3 G3 C4 — the fourth between the upper voices, a free consonance — scores 0.480.

G3 C4 E4 — the same fourth against the bass, a dissonance requiring resolution — scores 0.335.

The chord the rules forbid to move freely is thirty per cent smoother than the chord they leave alone. Per sounding pair the gap is the same, 0.112 against 0.160, so it is not an artefact of counting pairs. And the fourth’s own contribution — the beating between the partials of G3 and those of C4 — is bit-identical in the two, because it is the same two frequencies. Every difference between the two numbers comes from the third note, which is the note the rule does not mention.

The essay was slated on the expectation that the two would score the same, which would have been embarrassing enough for a rule presented as acoustic. They do not score the same. They score in the wrong order, and the wrong order is the stronger result: a threshold on roughness would not merely fail to separate these two cases, it would separate them backwards.

There is a further twist in the same direction, and the manuals supply it themselves. Not every second-inversion triad is treated as a dissonance. The cadential six-four — the one that arrives on a strong beat before a dominant — must resolve; the passing six-four and the pedal six-four, which arrive on weak beats and leave the way they came, are allowed to pass without preparation. The three are acoustically indistinguishable, being the same chord in the same position, and the rules separate them by where they fall in the bar and by what surrounds them. A single sonority is therefore sorted into two classes by its context, which is already an admission that the class is not a property of the sonority.

The rule has no register and the model has nothing else #

The failure is not confined to one pair of chords. Ask the same question of the whole consonance list at once and the shape of the mismatch becomes visible.

At middle C the best single threshold on roughness misclassifies two of the thirteen intervals, and the two are the major third and the minor third. In the bass it misclassifies five.

That is the third rung of this ladder arriving where it does the most damage. Roughness depends on where the notes are; a consonance list does not. Counting from the smoothest end, the fourth is eleventh of thirteen at C2 — third roughest of everything — sixth at C4 and fourth at C5, and its classification in the manuals is one word, unqualified, at every pitch a voice can sing.

So the rule and the model do not even have the same arguments. A model whose answer is a function of frequency is being asked to reproduce a classification that is a function of interval alone, and it was never going to, for the same reason the previous rung’s scale search failed: a transposition-varying measure cannot produce a transposition-invariant answer.

The list the model does fit is eight centuries out of date #

Here is where the computation stops being a refutation and becomes something more interesting.

Run the same threshold test against the consonance list of Musica enchiriadis, the ninth-century treatise in which the consonances — the symphoniae — are the octave, the fifth and the fourth, and thirds and sixths are not consonances at all. At middle C the best threshold gets one of thirteen wrong. At C5 and C6 it gets none wrong, through gaps of thirty-three and fifty-nine per cent.

The medieval list of consonances is, at the registers plainchant is sung in and above, very nearly a roughness threshold with the fourth added by hand. Fux’s list is not a roughness threshold at any register tried.

Two octaves up the fit is not approximate. Every interval the treatise names is smoother than every interval it does not, and the boundary is a gap no listener would need a fine judgement to place. What closed the one error is the register dependence of the previous section running the other way: the fourth is eleventh of thirteen counting from the smoothest at C2 and fourth of thirteen at C6, so up here it sits with the other perfect intervals, which is where the ninth century put it.

The obvious reading of that is the wrong one. Nothing about ears changed between 900 and 1725, and neither list is more correct. What changed is which sonorities a musical practice was willing to leave alone, and the intervals that were added to the list — the thirds and the sixths — are precisely the ones the acoustic measure cannot place, because they sit in the middle of the ordering and move about with register. A list assembled where the acoustics are ambiguous is a list assembled by something other than acoustics.

That is a fact about the history and it is consistent with the site’s own evidence that consonance judgements are about half preference: the half that varies between listeners is the half that varied between centuries.

Three periods, one interval, and the roughness runs the wrong way #

The fourth’s status has changed three times in the written record, and each change can be given its actual pitches and scored.

Per sounding pair, the ninth-century organum fourth scores 0.201, the eighteenth-century six-four scores 0.112 and the 1959 quartal voicing scores 0.101 — the last being smoother than the same four pitch classes in close position, at 0.163.

So across the three periods the per-pair roughness of the textures the fourth is used in varies by a factor of two, and it varies in the opposite direction from the interval’s respectability. The century that treated it as a paradigm consonance is the century that put it lowest, where the model says it is roughest; the century that treated it as a dissonance put it higher up, where the model says it is smoother.

A ranking that runs backwards against the history is not a weak explanation of the history. It is not an explanation of it.

Two intervals the theory splits and the sound does not #

There is a reason the fourth in particular is where the theory strains, and it is arithmetic rather than acoustics.

Every interval and its inversion swap places when the lower note is moved up an octave — a fourth becomes a fifth, a major third a minor sixth — and the theory treats one member of each pair as a consonance and the other as conditional. The sound does not: an interval and its inversion share most of their partial coincidences, so the roughness model puts them close together and the rule puts them on opposite sides of a line.

Inversion is the site’s own subject one field over, and the fourth is what the fifth becomes when its lower note is moved up an octave. The fifth is the least rough interval after the octave at every register this ladder has tested.

The tempting next sentence is that the two are therefore acoustically much the same, and that the theory is splitting an identical pair. The machinery refuses it.

So there is an acoustic difference between the fourth and the fifth and it is not small. It is the wrong shape to be the rule, and the shape is what decides the argument. The acoustic gap is unconditional and the theory’s gap is conditional. The fourth is rougher than the fifth wherever it occurs — above the bass, between upper voices, in 900 and in 1725 — while the theory calls it a dissonance in one position and a consonance in the other, in one of those centuries and not in the other. A constant cannot account for a variable.

What the theory is reading instead is the criterion the model has no access to: which note is at the bottom. A fifth’s lower note is the note the pair is heard as being of; a fourth’s is not. That is a statement about how a sonority is read — about roots, and about what the bass is doing — and this site has computed elsewhere that a chord’s function depends on more than the notes in it.

If the fourth is not enough, the tritone settles it, because there the sound is not merely similar but identical.

Written out, the pair is a diminished fifth in the first chord — B up to F spans five letter names — and an augmented fourth in the second, where B is spelled C♭ and C up to F spans four. Equal temperament gives both spellings one size, six semitones, and the roughness model gives both one number, because the input is the same number.

The tritone’s two resolutions drawn as motion are the same two voices moving outward by a semitone each, into the tonic and its third, or inward into the third and the root of the chord a tritone away. Every voice moves the minimum and the destination is decided by which of the two chords is wanted, not by the interval — which is why one tritone serves two keys.

The two continuations are opposite and the sound is one sound. Whatever decides which obligation applies is reading something that is not in the air, and the site’s harmony field has already identified what — the progression the chord is in, which is a claim about a sequence rather than about a sound.

What a dissonance is, then #

The right-hand column of the hero is the essay’s answer, and it is the only definition that survives all of the above.

A dissonance is a sonority whose continuation is constrained. Not one that beats, not one that scores above a threshold, not one that a listener dislikes — one that the style will not let stay. Under Fux’s rules a consonance may be followed by any of six or seven of the thirteen intervals and a dissonance by two or three; the fourth has three, and the major third has seven. Under Musica enchiriadis the fourth has four and the major third has one. Under a modal jazz voicing after 1959 every interval has thirteen, because the constraint has moved out of the sonority and into the mode.

The fourth goes four, three, thirteen. The major third goes one, seven, thirteen. Their roughness is the same number in all three centuries and the two counts cross between them.

That definition is deflationary and it is not empty. It is checkable — a successor count is derived from a stated rule and can be wrong — it makes the rule a claim about a repertoire rather than about hearing, and it explains why the fourth is the hard case: the fourth is the one interval on which the practices that share an acoustics do not share a rule.

It is also worth recording that this is not an attack on the roughness model, and that its authors would not have recognised the claim being tested here as theirs. Plomp and Levelt published a curve fitted to judgements of pairs of pure tones, and were explicit that what they had measured was a sensory phenomenon rather than a theory of harmony. The extension from that curve to a consonance list, and from a consonance list to a rule of continuation, is two further steps, each taken by other people. The first step this ladder has already computed and found conditional. The second is the one that fails outright.

Whose music, and when #

Every rule in this essay has been given a book and a date, and that is the discipline the subject requires rather than a courtesy.

The rule about the fourth is Fux, Gradus ad Parnassum, Vienna 1725, a pedagogical abstraction of sixteenth-century vocal practice — not a description of what Palestrina did, and criticised on that point ever since. The list of symphoniae is Musica enchiriadis, anonymous, late ninth century, describing organum at the fourth and fifth. The quartal voicing is Bill Evans’s, on Kind of Blue, 1959, and its currency is modal jazz and what came after it.

Outside those three, the rule does not apply and the question often cannot be posed. Much of the world’s music has no bass in the sense that makes the fourth’s position meaningful, and the maqam and raga traditions the site has drawn elsewhere organise pitch without a chord to be inverted at all. A rule stated without its jurisdiction is the failure mode of this entire subject, and “the fourth is a dissonance” without a century attached is that failure exactly.

Where the model stops, and what the picture cannot show #

The model stops here, and this rung is where the ladder says so. Roughness is a good account of why two nearby partials grate. It is not an account of a consonance list, a resolution or a rule, and the eight rungs of computation before this one are what earn the right to say so with numbers rather than as an opinion.

The threshold test is a generous test that still fails. It is allowed to choose its own cut, on the whole thirteen at once, at the register that flatters it most. A real classifier of consonance would have to work at every register with one cut, and no such cut exists.

The rule sets are encodings, and the encoding is a claim. Each is derived in code from that source’s stated consonance list plus its stated resolution policy, rather than typed out interval by interval — but the choice of what to encode is a reading of a book, and a different reading gives different successor counts. What survives any reading is the direction: the fourth loses continuations between 900 and 1725 while its roughness stays put.

The picture cannot show the voice leading. A sonority is drawn here as a set of simultaneous pitches, and every rule the essay is about is a rule about what happens next. The site’s own motion metric is the object that has the missing dimension, and no static ranking of chords has it.

And the picture cannot show the metre. A dissonance on a strong beat and the same dissonance on a weak one are treated oppositely by the same book, and the previous rung put a number on how long one has to last to be heard at all. Position in a bar is a third variable the roughness model has no term for, after position in the register and position in the chord.

Where the ladder ends #

One rung remains, and it removes the last free variable the model has. Every roughness number printed anywhere on this site — every number in this essay included — was computed at a level nobody stated, and level turns out to be the one parameter that changes the answer by four orders of magnitude.