A tied note is the oldest thing in its own dissonance

A suspension is the most carefully regulated dissonance in the teaching of harmony. A note is sounded as a consonance, held while the harmony changes under it, and so becomes a dissonance without having been attacked as one; then it steps down, usually by a semitone or a tone, into a note of the new chord. The three moments have names: preparation, suspension, resolution. Every harmony textbook draws them as three vertical sonorities on a stave, and the word resolution carries an acoustic promise. Something is rough and then it is not.

A dissonance is what has to be resolved has already shaken half of that promise. Scored by roughness, the perfect fourth against a bass, which the rules call a dissonance, comes out smoother than the fourth between upper voices, which they call a consonance. That essay asked about an interval. It did not ask about the event the rule was written for, and an event has something an interval lacks. The note that is dissonant has been sounding for longer than the notes it is dissonant with.

On a choir that difference is nothing. On a piano it is most of the answer, and it runs against the word.

Three moments, each with its own clock #

The model needs one thing a chord symbol throws away, and that is when each note began. Every sounding note keeps its own moment of attack. Each partial of it decays exponentially at a rate proportional to its number, the first approximation in the note that gets duller as it dies, so that the eighth partial goes eight times as fast as the first. The fundamental’s T60, the time to fall sixty decibels, is 12 seconds at middle C and halves every eighteen semitones up the compass, which is a stated round figure for a middle-register piano rather than a measurement of any one instrument. An instrument that sustains is the same calculation with the T60 set to infinity. The roughness is Plomp and Levelt’s, summed over every pair of partials of every pair of notes on the string spectrum of eight, with each pair weighted by the product of its two amplitudes, and read over the fifth of a second after the moment in question.

Nothing else is added. There is no model of expectation, no key and no theory of tension. The question is narrow: what does the most-cited object in the study of dissonance do to the quantity that the word dissonance most plainly refers to in the ear, once the notes have ages?

The example in the figure above is deliberately ordinary. C5 belongs to a C major chord; the bass moves to G3 and the inner voices to D4 and G4, and the C stays; a beat later it steps to B4. Each chord lasts a second, which at a chorale’s crotchet of sixty is one beat. That is the textbook 4–3 over a dominant, in four parts, in the spacing a student would write first.

On an instrument that sustains, the step does almost nothing #

With nothing decaying, the suspended chord’s roughness is 0.327 and the resolved chord’s is 0.325. That is less than one per cent, and it is what the word resolution is supposed to name.

It could be a bad choice of spacing, so the census takes all of them. Hold the bass, put the suspended note anywhere from middle C to G5, put the two other voices anywhere above the bass within a twelfth of it, and refuse a spacing in which another voice already sits on the note of resolution. A 4–3 has eight spacings that survive that, and the other three common suspensions have between six and twelve.

The 4–3, which is the suspension a student meets first and the one whose name is on the cadence, is close to a coin toss. Two spacings fall, four rise and two stay within two per cent, with a median of 0.97. The two that fall are the two with D5 directly above the C5, where the suspended note is a major second from a neighbouring voice, and the roughness the step removes is that second’s rather than the fourth’s. The four that rise are the four with the suspension at C4, resolving to B3, where the resolved chord has acquired a third low in the register, and a third is rougher in the bass. Those spacings are rougher after they have resolved, by up to a third, and no rule distinguishes them.

The others are kinder, in a pattern that makes sense once it is printed. The 7–6 has seven spacings falling and two rising, median 1.16; the 6–5 has four falling and none rising, median 1.11. The 9–8 has eleven of twelve falling and a median of 1.42, and it is the only one of the four that resolves onto an octave of its own bass: the D sits a tone from one of the bass’s partials, and on stepping down it lands on that partial, where its own partials coincide with the bass’s instead of rubbing against them. So the one suspension that reliably resolves in the ear is the one that resolves onto something already sounding.

This much is an extension of a known embarrassment. A rule written for voices does not track roughness on voices. The rule was stated by Fux in 1725 as an abstraction of sixteenth-century vocal polyphony, and consonance is half learned, so nothing here says the rule is wrong. What it says is that whatever a singer’s 4–3 is resolving, it is not, in general, a roughness.

On a piano the tied note is not where the roughness is #

Now strike the same notes. The C5 was struck at the start of the preparation, so at the moment the harmony changes it is a second old. With a T60 of 7.6 seconds at that pitch, its fundamental has fallen 7.9 decibels and its second partial twice that. The bass and inner voices arrive at full strength beneath it.

On the struck instrument the note the theory blames supplies 29 per cent of the chord’s roughness. Sustained, the same pairs supply 59. The three pairs among the fresh notes, which form an open G sonority that no book would call dissonant, supply the other 71 per cent, because they are three fresh notes at full amplitude with their upper partials intact, and their upper partials interact with each other far more than anything does with a C that has already lost its top.

That needs a moment’s care, since it is easy to overstate. The fresh G chord has a fair amount of roughness on any instrument; a fifth and a fourth in that register are not smooth by this measure, as the sustained bars show. Over the fifth of a second after the attack those pairs lose about a fifth of their sustained value, while the pairs with the tied note lose about three quarters. The dissonance has not moved anywhere. The note carrying it has simply aged, and the rule that requires the note to be tied is the rule that ensures it has.

The size of that discount depends on how long the chord before it lasted, and the census can price it directly by comparing each tied chord with the same notes struck together.

At a second a chord a tie removes between a third and a half of the suspension’s roughness: 48 per cent on the 4–3, 41 on the 7–6, 39 on the 6–5 and 34 on the 9–8. At half a second it removes about a quarter to a third. On a slow chorale at two seconds a chord, the 4–3 has 34 per cent of what the same notes would have if they were struck together.

And that comparison is the connection worth having, because the same notes struck together also have a name. A dissonance attacked with its chord and then stepping down, with no preparation at all, is an appoggiatura. What separates the two in the textbooks is syntax, one prepared and one not. On a keyboard at a second a chord, they turn out to differ by a factor of between 1.5 and 1.9 in the roughness they deliver at the moment of dissonance. C. P. E. Bach’s Versuch of 1753, writing specifically about keyboard playing, asks that an appoggiatura be played louder than the note it resolves to. That instruction is a fact about the style, and nothing in the calculation shows it was motivated by this; but it was written for instruments on which the alternative, the tied dissonance, was already fading by the time it mattered.

The fall belongs to the clock #

Since a struck instrument’s roughness falls all the time, a suspension’s resolution on a piano is measured against a falling floor. So the fair comparison is the control that no textbook example runs: the same suspension at the same moment, with no step taken and the C held on.

The step makes the sound rougher than waiting would have. At a second a chord, the 4–3’s resolved chord is 3.45 times as rough as the chord left hanging, the 6–5’s 2.53, the 7–6’s 2.31 and the 9–8’s 1.28. For the example above, the suspended chord reads 0.149, the resolved chord 0.054 and the unresolved one at the same moment 0.024. Resolving does lower the sound below where the suspension began, and it does so by a factor of 2.8. Leaving the note alone lowers it by a factor of 6.2.

The mechanism is not obscure once the values are printed. By the time of the resolution the bass and inner voices are a second old and have lost most of their upper partials, and the tied note is two seconds old and has lost more. Then a fresh B4 arrives, at full amplitude with all eight partials, and is the loudest thing sounding. Its partials interact with whatever remains of the G chord, and every one of those products has a fresh amplitude in it. The resolution of a struck suspension is an attack.

That is the same arithmetic as a moving tenor is heard because it is struck again, which found that on a struck instrument the value of moving a part lies in its new attack and not in its motion. A resolution is exactly that, in the top voice: it is heard because it is new. What a listener receives at that moment on a piano is an arrival, a change of pitch at a moment when almost everything else has gone quiet, and it would be a mistake to read that as a release of roughness when the roughness has been falling on its own for two seconds.

The further the instrument decays, the larger this gap, which is what the last figure draws.

So the two claims that open this essay meet at a single variable. As the decay slows towards nothing the step’s fall goes to 0.97 for the 4–3, and resolution is a coin toss. As the decay quickens, the fall grows to five or thirty times, and almost all of it is time. Nowhere on this axis does the step of a 4–3 do what the word says, which would be to lower the roughness below the alternative. The 9–8 does, at the slowest decays only: at T60s of 24 and 48 seconds it falls further stepping than waiting, because the octave’s coincidence is worth more than the small amount of decay.

Which of these numbers depend on which assumptions #

Three choices inside the calculation could have produced these results by themselves, so each was changed.

The weighting. Roughness weighted by the product of two amplitudes penalises an old note twice over, and the alternative in the literature weights by the smaller of the two. With the smaller, at a second a chord, the fall from suspension to resolution is 3.01 on the 4–3 and 4.02 on the 9–8, and the tie’s discount is about the same, at 51 and 71 per cent of the struck chord. The step’s excess over waiting is what shrinks: 1.90 on the 4–3, 1.45 on the 6–5, 1.20 on the 7–6 and 0.92 on the 9–8. So the claim that the resolved chord is rougher than the hanging one is robust for three suspensions and is a tie on the fourth, and the size of the number is the weighting’s. The same chord is harsher when it is louder has already shown how much the quadratic carries.

The decay law. A piano string actually decays twice, with a fast prompt sound over a slow aftersound, as a string that decays twice is counted early worked out for beats. A model with a single rate understates the early fall, which is the fall the tie exploits, so it should understate the discount at short chords. It should not reverse the sign of anything here, since the sign comes from one note being older than another; a two-stage decay was not run.

The window. Everything is read over the fifth of a second after each change. Read over fifty milliseconds or four hundred instead, the medians move by less than a tenth of their value and no ordering changes. That range starts below the seventy milliseconds a dissonance has to last to be heard as one, so nothing here depends on the fifth of a second.

What a roughness total cannot say about a suspension #

A roughness total for a chord says nothing about what the note is doing. A listener following a melody hears the C as the C it was a moment ago, and hears it become the wrong note under a changed harmony. That is a claim about streams and expectation, about a voice held across a change, and the shortest move has the dimension for it that this figure does not. The tension a musician reports in a suspension may have nothing to do with roughness, and on the evidence above it would be better for the theory if it did not.

Nor can it hear a syntax. Everything here is one chord pair in one key at one tempo with no context, and a suspension in a real cadence comes with a phrase’s worth of expectation about what the chord is about to do. Consonance is half learned is the reminder that the half this model cannot reach is not small.

The model is a piano with its dampers up and nobody’s hand on it. A pianist voices a chord, bringing out one note and leaving the others under it. A pianist playing a suspension may well re-strike it, or spread the chord as in the chord that is not played at once, and those are exactly the measures this calculation suggests a player would want. Nobody has been asked. There is no listening test here and no recording: the numbers state what a decaying instrument does to a written suspension if nothing is done about it, and what is done about it is not known from this.

And a room would soften the whole result. All of the above is direct sound. Reverberation carries each chord into the next, which should lift the hanging chord above its dry value by more than the resolved one and close part of the gap the step opens. That has not been computed, so the dry struck case should be read as the extreme of the effect rather than its typical size.

Whose suspensions, and on what #

The rule is vocal. Fux’s fourth species of 1725 codified the way sixteenth-century polyphony treats the suspension, and the instruments of that repertoire, voices and viols and organ, are the ones at the left of the last figure, where the step is a coin toss for the 4–3 and a modest fall for the others and nothing ages. There the suspension’s whole interest lies in the line and the syntax.

The numbers on the right of that figure are a claim about a different repertoire: keyboard music, plucked strings, anything whose notes die. A lute’s or a harpsichord’s T60 is far shorter than a modern piano’s, so all of this is stronger on them, and the Baroque keyboard’s ornament tradition is the obvious place to look for what players did about it. What did they do? That is a question about history, and this essay has no data to answer it with. Everything said here about a 4–3 is about four-part writing in a major key, with four specific dissonances over one bass note each, and says nothing about a suspension in a dense orchestral texture, where the dissonance arrives and the dynamic does not sets a different question.

Still open: what a tied note could do instead #

The obvious repairs are all things a player already has. Re-strike the tied note softly, which restores some of the roughness at the cost of the very property the tie was for; or trill it, keeping it alive with a stream of fresh attacks, which is what the Baroque ornament does and which would make the suspended note the newest thing in the chord instead of the oldest. Each is a small change to the same arithmetic: the note’s attack times, at a stated level. The computation would say how much of the struck chord’s roughness a trill on the suspended note gives back, at what rate of trill the dissonance stops fading, and whether a player can keep a suspension as rough as a sustained one’s at a slow tempo, or whether that is something only a voice can do.