Acoustics & Music Theory
why it sounds like that
Concept

Inversion — where it appears

A rearrangement that puts a different note of a chord lowest, or the reflection of an interval or a set of pitches about a fixed point. The two senses are unrelated in origin and both are standard, so which is meant has to be said.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

Chords as stacked intervals. Each of 4 chords — major, minor, diminished, augmented — drawn as the semitone distances above its root, with the nearest simple frequency ratio beside each note where the 5-limit table has one. Where the chords are triads they differ only in the size of the two stacked thirds, and that difference is the whole of their character.

Three notes at once, and why these three

A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.

harmony · The triad Every interval, and what it becomes when it is turned over. Each interval within the octave paired with its inversion — the interval left when the lower note is raised by an octave. The two always add to twelve semitones, so exactly one interval can be its own inversion, and it is the one at six.

The interval that inverts to itself

Six is the only number that divides twelve into two equal halves, so the tritone is the only interval unchanged by being turned over. That symmetry is not a curiosity — it is why one tritone belongs to two dominant chords, and why they resolve to keys a tritone apart.

harmony · The triad The thirteen intervals at C4, ordered by roughness, against Fux's species, 1725. Every interval within an octave above 262 Hz, ordered by the Plomp–Levelt roughness of a string spectrum — from 0.0029 for the unison to 0.2635 for the minor second — beside whether Fux's species, 1725 counts it a consonance and how many of the thirteen may follow it under that rule set. The best single threshold on roughness misclassifies 2 of the thirteen, among them the major third and the minor third, and the gap it falls in is 4.2 per cent of the roughness either side of it.

A dissonance is what has to be resolved

The perfect fourth is a consonance between two upper voices and a dissonance against the bass, and the same three notes are involved either way. Score both arrangements with the roughness model and the one the rules call a dissonance comes out thirty per cent smoother. Whatever the rule is tracking, it is not the sound.

harmony · Consonance Every voicing of a second-inversion major triad, least rough first. All 27 arrangements of the same three pitch classes within 3 octaves from 131 Hz, scored for roughness. The best is spaced 19 then 9 semitones — wide below, close above — and the worst is the chord in close position at the bottom of the range, 7.6 times rougher with exactly the same notes in it.

The inversion that cannot end a phrase

Three positions of one triad, three measurements, and the practice contradicted at every turn. The second inversion is the smoothest of the three by roughness, the best explained of the three by a virtual-pitch model, and the one that three centuries of practice will not let a phrase end on. What forbids it is a rule about a two-voice skeleton.

intervals · The triad What one progression leaves open. Realisations of I–IV–V–I in four parts with no parallel fifths or parallel octaves, counted exactly by a dynamic programme over the voicings rather than sampled. The chord symbols admit 16,100,352,296; the Roman numerals 59,418,496; the figured bass 2,042,672. The three notations differ by four orders of magnitude, and every one of them was in daily professional use.

Three notations, one progression

A figured bass, a Roman numeral and a chord symbol are three professional notations for the same four chords, and the number of four-part realisations each of them admits can be counted exactly rather than argued about. With no parallel fifths or octaves the counts are sixteen billion, fifty-nine million and two million: a factor of eight thousand between the loosest and the tightest. What each one collapses is what its tradition thought a chord was, and the three do not agree.

harmony · Notation Which note a chord would rather have twice. Every complete four-part voicing of each chord inside the SATB ranges, grouped by which member sounds twice and scored for roughness — 480 voicings for a triad. The order for a major triad is root < fifth < third, which is the rule every part-writing treatise states. For a minor triad it is fifth < root < third, which is not. The numbers printed under each bar are the mean error, in cents, with which the four sounding notes fit a single harmonic series, and that measure separates the three far more sharply than roughness does.

The note that sounds twice

A triad has three notes and a four-part texture has four voices, so one note is doubled — and the voicing model used here leaves the choice free because the rules have an opinion about it. Asked properly, the arithmetic agrees with the treatises for the first time in nine essays: root, then fifth, then third. For a minor triad it does not agree, and for a symmetric chord it correctly has nothing to say.

intervals · The triad A major triad's combination tones, against its own notes. The three notes of a major triad on C4 in root position, voiced C4–E4–G4, as tall lines, and every combination tone its pairs make, as short ones: difference tones lowest, cubic products taller. In just intonation 2 cubic products land exactly on a note of the chord, and none comes within forty hertz of one. In equal temperament no cubic product lands on a note of the chord, and the nearest miss is 5.63 hertz.

A major triad's combination tones are its own notes

Play a just major triad of pure tones and two of the ear's cubic products land exactly on its root and its fifth. The reason is a condition rather than a coincidence — a chord's cubic products fall on its own notes when its middle note is the mean of the outer two in hertz — and it holds for the major triad in root position and in the six-four, and for no minor triad in any position or tuning. Equal temperament misses the landing by one number, 5.6 hertz on middle C, which is a beat that belongs to no pair of notes in the chord.

intervals · Combination tone An inversion lasts as long as its outer sixth. The six three-note voicings of a major and a minor triad, each over a bass of C3 struck at 80 decibels, with how long the strongest partial coincidence of each of its three intervals survives the strike. The interval that goes first is marked, and its time is how long the chord keeps the evidence of all its intervals at once. major, root position: major third 5:4 1.26 s, minor third 6:5 1.03 s, fifth 3:2 2.14 s; the chord 1.03 s. major, sixth chord: minor third 6:5 1.04 s, fourth 4:3 1.62 s, minor sixth 8:5 0.74 s; the chord 0.74 s. major, six-four: fourth 4:3 1.60 s, major third 5:4 1.27 s, major sixth 5:3 1.26 s; the chord 1.26 s. minor, root position: minor third 6:5 1.04 s, major third 5:4 1.27 s, fifth 3:2 2.14 s; the chord 1.04 s. minor, sixth chord: major third 5:4 1.26 s, fourth 4:3 1.63 s, major sixth 5:3 1.26 s; the chord 1.26 s. minor, six-four: fourth 4:3 1.60 s, minor third 6:5 1.03 s, minor sixth 8:5 0.74 s; the chord 0.74 s. The major six-four lasts longest and the major sixth chord shortest; every root position is held to its minor third's life.

An inversion lasts as long as its outer sixth

A struck interval keeps the partial coincidence that names it for a time set by its ratio, and a chord is three intervals at once. Voiced over one bass and struck on a piano, a triad keeps the evidence of all three only as long as its weakest one lasts, and for an inversion that is the sixth on the outside: a major sixth lasts as long as a major third, a minor sixth dies first. So the major six-four and the minor sixth chord are the most durable voicings of their triads and the major sixth chord and the minor six-four the least — and unlike a dyad, a triad's inversions keep their order by roughness through almost the whole decay.

timbre · Harmonic series The ninety per cent was a ceiling. The share of 188 scheme bars read right on both key and degree with a bass note worth 1.5, for three bass lines — every chord's root, a line moving to the nearest chord tone, every chord's fifth — and two ways of using the bass: rewarding the triad rooted on it, or any triad containing it. Wide bars are with no profile in the emission, narrow bars with the profile alone. root line, root rule: 89% and 86%; root line, member rule: 60% and 29%; smooth line, root rule: 68% and 46%; smooth line, member rule: 61% and 47%; sixfour line, root rule: 3% and 17%; sixfour line, member rule: 52% and 30%. The reading with no bass at all is 47%.

A bass line is not a list of roots

Every bass note the key-finder has been given was its chord's root, and under that line a bass cue reads 89 per cent of scheme bars on the right degree. Give the same chords an economical bass that moves to the nearest chord tone, as a keyboard reduction would, and nearly half of them are inverted. The cue that rewards the triad rooted on the bass then reads 68 per cent at best and worse as it is trusted more; the cue that rewards any triad containing the bass cannot be fooled and stops at 61. The same inverted line does one thing the roots never did: it puts the leading note of each new key at the bottom, and finds the rondo's modulations.

scales · Key-relations Opening a triad changes which interval goes first. The six voicings of a major and a minor triad over C3, each close and with its middle note raised an octave, struck at 80 decibels, with how long each keeps the coincidences of all three of its intervals and which interval goes first. major root position: close 1.03 s, held by its minor third 6:5; open 1.26 s, held by its major tenth 5:2. major sixth chord: close 0.74 s, held by its minor sixth 8:5; open 0.47 s, held by its minor tenth 12:5. major six-four: close 1.26 s, held by its major sixth 5:3; open 0.74 s, held by its eleventh 8:3. minor root position: close 1.04 s, held by its minor third 6:5; open 0.47 s, held by its minor tenth 12:5. minor sixth chord: close 1.26 s, held by its major third 5:4; open 1.26 s, held by its major sixth 5:3. minor six-four: close 0.74 s, held by its minor sixth 8:5; open 0.74 s, held by its minor sixth 8:5.

An open triad lasts as long as its tenth

Close, every triad's weakest link is a third or a sixth. Raise its middle note an octave and the link becomes a compound interval, and compound intervals do not last alike: a major tenth, 5:2, lives as long as a major sixth, while a minor tenth, 12:5, needs the twelfth partial and lives 0.47 seconds. So the spacing orchestration manuals recommend for a major chord in the bass is the longest-lived voicing a struck triad has, and the same spacing halves the life of a minor chord — and the six-four's advantage reverses.

intervals · Harmonic series Both qualities reach the same ceiling. The longest-lived spacing of a major triad and of a minor one, over four basses, taken over every arrangement of the three pitch classes within 2 octaves. They are the same number at every bass — 1.16 seconds over C2, 1.26 seconds over C3, 1.26 seconds over C4, 1.41 seconds over C5 — and at each bass 2 major and 2 minor spacings are tied at it. The faint line is the worst a minor spacing can do, which is 3.5 times shorter. So the asymmetry found earlier is a fact about the minor tenth rather than about the minor triad: a minor chord has a spacing that avoids it, and that spacing is its first inversion, where the minor third between two of its notes appears as a major sixth instead.

A minor triad can be spaced to last

Two spacings of each triad, drawn side by side, say that opening lengthens a major chord and halves a minor one. Drawn over every arrangement of the three pitch classes within two octaves of a fixed bass, the asymmetry disappears: a minor triad reaches 1.26 seconds over C3 and so does a major one, with two spacings of each tied at the top. The short-lived chords were never the minor ones. They were the ones with a minor tenth on the outside, and a minor triad has three spacings that avoid it.

intervals · Harmonic series No seventh chord can be spaced to last as long as a triad. Every inversion and spacing within 2 octaves over C3, struck at 80 dB, for two triads and five seventh chords: the bar is the longest any spacing keeps every pair's partial coincidence, and the tick is the bound set by the chord's worst pitch-class distance — the longest any presentation of that distance lasts. major triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; minor triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; dominant seventh: 0.66 s over 32 voicings, bound 0.64 set by the tone; major seventh: 0.42 s over 32 voicings, bound 0.38 set by the semitone; minor seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; half-diminished seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; diminished seventh: 0.86 s over 32 voicings, bound 0.87 set by the tritone. Every seventh chord contains a distance worse than any a triad contains, except the diminished seventh, whose distances are only minor thirds and tritones.

A seventh chord cannot be spaced to last like a triad

A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.

timbre · Harmonic series Holding the bass through one mid-bar change in six finds the barline five times in six. Passages of eight bars at two chords a bar, with the bass arpeggiating on 50 per cent of its beats, and each barline reading's share of passages it places correctly, against the share of mid-bar chord changes voiced over the bass already sounding. 0% held (convention strength 0): metre then bass 50%, bass alone 13%, metre alone 50%, chord changes alone 10%; 4% held (convention strength 0.1): metre then bass 62%, bass alone 34%, metre alone 50%, chord changes alone 10%; 6% held (convention strength 0.2): metre then bass 68%, bass alone 44%, metre alone 50%, chord changes alone 15%; 9% held (convention strength 0.3): metre then bass 75%, bass alone 57%, metre alone 50%, chord changes alone 18%; 15% held (convention strength 0.4): metre then bass 82%, bass alone 68%, metre alone 50%, chord changes alone 13%; 16% held (convention strength 0.5): metre then bass 85%, bass alone 74%, metre alone 50%, chord changes alone 14%; 20% held (convention strength 0.6): metre then bass 88%, bass alone 79%, metre alone 50%, chord changes alone 11%; 26% held (convention strength 0.7): metre then bass 92%, bass alone 86%, metre alone 50%, chord changes alone 13%; 27% held (convention strength 0.8): metre then bass 92%, bass alone 86%, metre alone 50%, chord changes alone 13%; 28% held (convention strength 0.9): metre then bass 95%, bass alone 92%, metre alone 50%, chord changes alone 10%; 33% held (convention strength 1): metre then bass 96%, bass alone 93%, metre alone 50%, chord changes alone 10%. The metre ties the barline with the half-bar and the chord changes are at chance, since the chords change at both; the bass's holds are the only evidence that separates them.

A bass that holds through a change marks the barline

At two chords a bar the chords change on the barline and on the half-bar alike, so a reading of where they change is at chance, and the metre ties the two. The bass has one more piece of evidence: a change inside the bar can be voiced over the note already sounding, and a change on the barline is voiced over its root. Hold the bass through one mid-bar change in six and the metre and bass together place the barline in 85 per cent of eight-bar passages; one in three, 97. The convention cannot be stronger than that, and the chord changes, asked first, only get in the way.

harmony · Progression

Named alongside it

The objects these essays reach for when they reach for this one.

RoughnessTriadVoicingDecayHarmonic seriesCounterpointPartialCadenceChord qualityConsonanceDominant seventhInference

All concepts