Acoustics & Music Theory
why it sounds like that
Series

Voice-leading — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. How far the voices have to move. Three chord changes with their voices joined by the assignment that moves the least in total. The distance is computed over every way of pairing the notes, so a change that shares two of its three notes shows as two flat lines.

    The shortest move, which is what a chord change is

    Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones.

    part 1 · harmony
  2. The twenty-four triads, and the three moves between them. Every major and minor triad, placed on the single cycle that alternating two of the three transformations produces. Neighbours round the ring differ by one voice; the chords across the middle are the third transformation. Every edge's cost is the measured voice-leading distance, and none of them is more than two semitones.

    Chords as a space

    Three operations turn any triad into another by moving one voice. They generate all twenty-four major and minor triads in a single cycle, their costs are one, one and two semitones, and the map that results is a geometry rather than a list of rules.

    part 2 · harmony
  3. Which partials two notes have in common. The first 12 partials of a note and of the note an interval above it, on a logarithmic frequency axis, with every partial of the upper tone that lands on one of the lower marked. octave: 12 of 12, fifth: 6 of 12, fourth: 4 of 12, major third: 3 of 12, minor third: 2 of 12, minor second: 0 of 12. Sharing partials is what fusion is: two voices whose spectra mostly coincide stop being heard as two voices, which is a measurable property of the interval rather than a matter of taste.

    Two voices that stop being two

    The ban on parallel fifths is the most famous rule in Western music and it is usually taught as taste. It is not taste. At an octave the upper voice contributes no frequency the lower one did not already have, at a fifth it contributes half of them, and the number can be counted — which turns a prohibition into a measurement.

    part 3 · harmony
  4. Every delay, scored — a round and a tune that is not one. Mean Plomp-Levelt roughness of the simultaneities a tune makes against a copy of itself entering a whole number of bars later, for Frere Jacques and Twinkle, twinkle. The two do not overlap: the worst delay of the first is smoother than the best delay of the second. The model sees roughness and nothing else — no voice leading, no parallels, no distinction between a passing dissonance and a structural one.

    A melody that can accompany itself

    Whether a tune works as a round is decidable before anyone sings it. Score every simultaneity it makes against a delayed copy of itself, at every delay, with the roughness model already in use, and two nursery tunes separate completely — every delay of one is smoother than every delay of the other. What the scan does not do is pick out the entry the tradition uses, and that refusal is the most informative thing in it.

    part 4 · harmony
  5. What each rule costs, in semitones of extra motion. Each prohibition priced on I – ii – iii – IV as the difference between the cheapest realisation that obeys it and the cheapest realisation of all, averaged over every one of the 144 melodies the progression admits inside an octave. The dearest rule costs 1.27 semitones a melody and the cheapest costs nothing, so the bars run from 1.27 to zero; the notes beside them give the share of melodies that pay nothing, or that cannot obey the rule at all.

    What the rules cost

    The prohibitions of counterpoint are constraints on a minimisation already computed here, so each one has a price in semitones of extra motion. Priced over every melody a progression admits, most of them turn out to be free, the dearest is not the famous one, and two of them cost no motion at all — they cost tunes.

    part 5 · harmony
  6. Both criteria, measured from C major. The 23 chords other than C major placed by the smallest total voice motion and by how many notes are held in common. Agreement between the two criteria would be a staircase; the cells drawn in the discrepancy colour are the ones that break it, and over the whole set of 24 chords the two criteria order a pair of destinations oppositely in 72 of 6072 comparisons. A minimal assignment drops a common tone it could have held in 0 of the 552 pairs.

    Two criteria that are taught as one

    Move the least and keep the common tones arrive in one breath, as though they were one instruction. They are two different functions, and enumerating every pair of chords settles exactly where they agree: as a way of connecting two given chords they never once disagree, and as a way of saying which chord is nearer they disagree about one comparison in eighty.

    part 6 · harmony
  7. Every triad, by how far it is from G7. All 24 chords ranked by the smallest total motion that takes G7 onto each, every note of the target sounded and one voice doubled where the sizes differ. The spread is narrow — 2 semitones at the nearest and 6 at the furthest — and the ranking is flat in the middle: 12 chords tie at four semitones.

    The resolution the metric cannot find

    Rank every triad by how far its voices are from the dominant seventh and the tonic is not at the minimum. It sits in a twelve-way tie in the middle, the chord a deceptive cadence goes to is the furthest away of all twenty-four, and the two nearest are chords no cadence ever uses. What selects the two resolutions is not a distance — it is a direction, and it picks out exactly two triads from the twenty-four.

    part 7 · harmony
  8. Four voices, placed by the arithmetic. The rules in force are: no parallel octaves; no parallel fifths; no voice crossing; no gap over an octave above the tenor; the leading note is not doubled; the leading note resolves, outer voices; no augmented melodic interval. The four parts are drawn lowest to highest — bass, tenor, alto, soprano. I – vi – ii – V – I in C major, realised in four voices by the cheapest set of voicings obeying 7 rules, at 16 semitones of motion in all. Every gap between adjacent voices narrower than the fission boundary of 5.2 semitones is marked, and below that boundary two parts cannot be heard as two however hard a listener tries.

    A voice is a stream, and the ear decides which

    Seven earlier essays have assigned voices to notes. Whether a listener follows the assignment is a separate question with laboratory numbers attached, and the numbers are unkind to it: two parts closer than about five semitones cannot be heard as two at any speed, a third of the gaps in the cheapest four-part writing are inside that limit, and in a third of chord changes the ear's own rule for continuing a line does not recover the parts as written.

    part 8 · perception
  9. How much of the writing the prohibitions forbid. I – vi – ii – V – I in C major, written in 3, 4, 5, 6 parts, with every ensemble covering the same total compass. A triad has three pitch classes, so n parts double n − 3 of them and a duet cannot state one at all. Of every ordered pair of complete voicings of V and I, the share with no parallel fifth or octave between any pair of voices: 91.2% at 3, 75.0% at 4, 44.3% at 5, 17.9% at 6.

    Why the exercise is in four parts

    Eight earlier essays move four voices, and nothing here ever chose four. Cut one choir's compass into three parts instead and the two most famous prohibitions in music cost exactly nothing — the cheapest realisation already obeys them. Cut it into six and they cost more than half a semitone per voice per chord change, because a parallel octave needs a doubled note and six voices sharing three notes can hardly avoid one. Four is the smallest number of parts at which the rules have a price at all, and it is the largest at which the price is small.

    part 9 · harmony

All series