A minor triad can be spaced to last

The essay that opened the triads took the six voicings of a triad, raised the middle note of each an octave, and found that the same operation lengthens a major chord and halves a minor one. Its explanation was the outer interval: opened, a major root position acquires a major tenth, which lasts 1.26 seconds, and a minor root position acquires a minor tenth, which lasts 0.47. Its closing paragraph asked whether that is a fact about the minor triad or about one spacing of it, and said what would settle the question — every arrangement of the three notes within two octaves, over the same bass.

Two spacings cannot answer it, and four can. Over a fixed bass, each of the two upper notes can be placed in either of two octaves, and the arrangements that leave three distinct pitches are four per position, twelve per quality, twenty-four in all.

The answer is the interval #

Over C3 the longest-lived major spacing lasts 1.26 seconds and so does the longest-lived minor one. They are not close; they are the same number, because they are held by intervals that need the same partial. Four spacings are tied at the top and two of them are major and two minor.

The best major arrangement is C–G–E, the root position opened, whose outer interval is a major tenth. The best minor is C–E–A — the minor triad in first inversion, close — whose three intervals above the bass are a major third, a fourth and a major sixth. Not one of them is a minor third or a minor tenth.

So the previous essay’s asymmetry was real and was attributed to the wrong object. A minor triad is not a short-lived chord. A minor tenth is a short-lived interval, and the operation that essay performed — raise the middle note — happens to put a minor tenth on the outside of a minor root position and a major tenth on the outside of a major one. Change the operation and the asymmetry goes.

That is worth stating as a correction rather than as an extension, because the earlier essay’s own summary reads as a claim about chord quality: “the same spacing that lengthens a major chord cuts a minor one to less than half.” That sentence is true of that spacing and false of the chord.

The tie is a symmetry, and it is exact #

Four spacings tied at 1.26 seconds is the kind of result that invites a check, because a tie between two families of twelve numbers is either an accident of rounding or a structure. It is a structure, and once seen it says the previous essay’s question could have been answered without the census.

A pair of notes three pitch classes apart can be presented, within two octaves, as a minor third, a major sixth, a minor tenth or a major thirteenth — three, nine, fifteen and twenty-one semitones. A pair nine pitch classes apart can be presented as exactly the same four intervals, because nine semitones and three semitones are the same pair of notes with the octave assigned differently. The menu available to a distance $d$ and the menu available to a distance $12-d$ are one menu, so nothing that depends only on the menu can tell $d$ from $12-d$.

A chord’s lifetime depends only on which intervals its pairs are presented as. So the whole census is invariant under reflecting a chord’s pitch classes about any axis — and a minor triad is exactly the reflection of a major one. Mirror $\{0, 4, 7\}$ to $\{0, -4, -7\}$, reduce modulo twelve and normalise, and the answer is $\{0, 3, 7\}$.

With the register effect taken out — every pair scored at the bass rather than at its own lower note — the two lists agree at all twelve entries, to machine precision. The residue in the census proper is one hundredth of a second in two places: the major list reads 1.09 and 1.03 where the minor reads 1.04 and 1.04, because a pair whose lower note sits above the bass has its partials where the threshold of hearing is lower and lasts fractionally longer. That is register, and register is the axis the previous two essays already isolated.

So a difference in lifetime between a major and a minor triad is impossible in this model, and the twelfth essay’s finding was a difference between two spacings that the reflection does not map onto each other. Raising the middle note of a major root position and raising the middle note of a minor root position are not mirror operations, which is why they gave mirror-breaking answers.

The chords the symmetry does not rescue #

The reflection argument applies to any chord whose mirror image is itself transposed — which is true of the major and minor pair and is not true of everything.

A diminished triad has distances $\{3, 3, 6\}$ and is its own reflection, so it has no partner to tie with; its best spacing lasts 0.87 seconds, held by the tritone, which needs a seventh partial and is the shortest-lived interval any common triad is obliged to contain.

An augmented triad has distances $\{4, 4, 4\}$ and is the case worth stopping on. Its best spacing lasts 0.74 seconds, held by a minor sixth — and its three pairs’ own menus would each allow 1.26. The reason it cannot reach that is its symmetry: because transposing it by a major third gives the same set of pitch classes, it has only four distinct spacings within two octaves where every other chord here has twelve. Two notes a major third apart can be presented as a major third or a major tenth, both long-lived, but three notes each a major third apart cannot all be so presented at once — whichever arrangement is chosen, one of the three gaps comes out as a minor sixth or a minor thirteenth.

Symmetry costs it the arrangements that would have saved it. That is a different kind of limitation from the diminished triad’s, which is simply obliged to contain a tritone, and it is the one case here where a chord’s ceiling is set by what its spacings cannot do rather than by what its intervals cannot do.

What the census reduces to #

The whole of it is one lookup table and one rule. A chord keeps the evidence of all three of its intervals for exactly as long as the shortest-lived of them, and how long an interval lasts is decided by the higher of the two partial numbers its defining coincidence needs.

The table sorts by partial number and not by width, which is why it is not the ordering anybody would guess. An octave needs partial two and lasts 3.16 seconds. A fifth and a twelfth both need partial three and last 2.14. A fourth needs partial four and lasts 1.60. Then everything that needs a fifth partial — major third, major sixth, major tenth — arrives together at 1.26. The minor third needs a sixth and lasts 1.04; the minor sixth and the eleventh need an eighth and last 0.74; the minor tenth needs a twelfth and lasts 0.47; the minor thirteenth needs a sixteenth and lasts 0.36.

A triad cannot reach the top of that table, and the menu argument says why without any enumeration. Its three pairs sit at pitch-class distances of three, four and five, and the menus for three and for four both top out at 1.26 seconds — a major sixth and a major third or tenth respectively. Only the pair at five semitones can do better, reaching a fifth or a twelfth at 2.14. A chord lasts as long as its worst pair, so no arrangement of a triad can beat 1.26, and both qualities have arrangements that reach it.

That puts a ceiling on the whole object. However a triad is spaced, however it is inverted, and whatever its quality, it cannot keep all three of its coincidences for longer than a major third can keep one.

What inversion actually does to a minor chord #

The route by which a minor triad reaches the ceiling is worth isolating, because it is a familiar musical object arriving from an unfamiliar direction.

A minor triad contains a minor third between two of its pitch classes. That is what makes it minor and there is no spacing in which those two notes stop being a minor third apart as pitch classes. What a spacing can do is decide how the pair appears as an interval. Put the lower of the two on top and the pair appears as a major sixth — 5:3, needing a fifth partial, 1.26 seconds — instead of as a minor third, 6:5, needing a sixth partial, 1.04 seconds.

Inversion removes the minor third from the chord’s interval list without removing it from the chord. In C–E–A the notes A and C are a minor third apart, and no pair of sounding pitches in that voicing is separated by three semitones. The interval C–A that carries the relationship is a major sixth, and a major sixth is one of the three longest-lived intervals a triad can contain.

This is not the usual reason given for the first inversion of a minor triad being treated as a settled, sonorous object rather than as an unstable one. The usual reasons are about the bass line and about which note an ear supplies as the root. This is a different reason and it is about the decay: of the minor triad’s twelve spacings within two octaves, the two that last longest are both first inversions, and they last as long as anything a major triad has.

It cuts the other way too. The chord with the shortest life in the whole census is not a minor chord: it is the major sixth chord spaced C–E♭–A♭ with the A♭ raised, at 0.36 seconds, held by a minor thirteenth. Three of the four worst spacings are major.

The ceiling holds across the compass #

Over C2 the ceiling is 1.16 seconds, over C3 and C4 it is 1.26, and over C5 it is 1.41. The best spacing is the same one at every bass and so is the tie: two major arrangements and two minor, at every bass drawn.

The rise with register is the same effect the previous two essays found and it has the same cause: a higher note’s partials sit where the threshold of hearing is lower, so the fifth partial that carries a major third survives longer above a higher bass. It moves every lifetime in the table together and no interval past another, which is why the ordering never changes and why one table drawn over one bass is the whole of the census at every bass.

What does not survive across the compass is the claim that any of this is audible. The lifetimes here are the times at which a coincidence’s weaker partner falls under the threshold of hearing, which is a statement about the signal and not about what a listener attends to. It is the same convention the two essays below this one use, and the numbers are comparable to theirs and to nothing else.

What this does to the advice the previous essay gave #

the twelfth essay ended on a practical-sounding conclusion: that the spacing orchestration manuals recommend — spread the low notes, put the third high — “is the best a major chord can do and the worst a minor chord can do.” Half of that stands and half of it needs restating.

It is still true that the manuals’ spacing is the best a major triad has and among the worst a minor triad has. What is no longer true is the implied advice, which reads as a minor chord is at a disadvantage and there is nothing to be done. The census says there is something to be done and names it: put the minor triad in first inversion. C–E–A over a C bass lasts 1.26 seconds, exactly as long as the major root position spread the manuals’ way, and it is not an exotic voicing — it is the commonest inversion of a minor triad in four-part writing.

The rule the manuals could have given, if lifetime were what they were after, is one sentence: avoid putting a minor third or a minor tenth on the outside of the chord. Both qualities then reach the same ceiling, and the choice of spacing stops being a choice about quality at all.

Whether any of that is advice worth taking is a separate question this essay cannot settle, because the manuals are about roughness and blend and a lifetime is neither. The two measures agree for the major chord and disagree for the minor one — the manuals’ spread minor chord is the smoothest at the strike and the shortest-lived afterwards — and a chord is two measurements through a strike rather than one.

Which computation produced the numbers #

Each note of a chord is the eight-partial struck-string model used throughout at the stated level, with partial $n$ falling sixty decibels in the fundamental’s decay time divided by $n$. An interval’s coincidence is the lowest pair of partials $p$ of the lower note and $q$ of the upper that agree within the stated tolerance, and its life is the moment the weaker of that pair falls under the threshold of hearing at its own frequency. A chord’s life is the shortest of its three intervals’ lives, which the census establishes rather than assumes: every spacing’s computed life agrees with the minimum of its three dyads computed independently.

The spacings are every assignment of the two upper pitch classes to octaves that keeps the chord within two octaves of the bass and leaves three distinct pitches. Two arrangements that produce the same set of sounding pitches are one spacing, which is why each position has four rather than nine, and the bass is held fixed throughout so that the comparison is between voicings of one chord over one note rather than between transpositions.

Two octaves is a choice and it is the one the previous essay’s question implied. Widening it to three adds arrangements whose outer interval is a double octave plus a third, whose coincidences need still higher partials, and which are therefore shorter-lived: the ceiling does not move and the floor falls. The interesting range is the one a player would use.

Where the model stops #

A coincidence is not a chord’s only evidence. Nothing here says a listener identifies a triad by watching a partial pair survive. The lifetime is the survival of the strongest physical cue the interval has, and it is a bound on one kind of evidence rather than a model of recognition — a chord whose coincidences have gone is still three notes with a pitch each.

The tolerance is a convention. A coincidence counts while the two partials agree within twenty cents, which is wide enough to include the mistuning equal temperament leaves and narrow enough to exclude an interval that is merely nearby. Every life in the table moves if that number does, and the ordering does not, because the ordering is set by partial numbers.

And the model has no inharmonicity. A stiff string’s partials are stretched, more so higher up, so the twelfth partial a minor tenth needs is not where a harmonic series puts it. The intervals that depend on high partials are exactly the ones inharmonicity disturbs most, so the bottom of the table is the least trustworthy part of it and the bottom of the table is where the previous essay’s asymmetry lived.

What the picture cannot show #

It cannot show the chord getting rougher or smoother, which is the other axis the previous two essays measured and which does not sort the same way. A spacing can last a long time and be rough throughout, and a rough arrival is rough because of its spacing rather than because of its quality. Lifetime and roughness are two numbers and this essay computes one.

Nor doubling. Every spacing here has three notes. A four-part chord doubles one of them, and which note sounds twice adds an interval to the list — usually an octave or a unison, which are the two longest-lived entries in the table and so cannot lower the minimum. The prediction is that doubling never shortens a chord, and it is a prediction rather than a result.

It cannot show what a player hears while choosing. A spacing is chosen at the keyboard in a moment, against the previous chord and the next one, and the quantity being optimised is not a lifetime. The census says what the arithmetic favours; it says nothing about why the arrangements it favours are also the ones that sound settled, and the agreement may be a coincidence of two different mechanisms.

And it cannot show a chord that is not struck. Everything here depends on partials decaying at different rates, which is what a driven note does not do. A triad held by three bows has no lifetime at all in this sense: its coincidences last as long as the notes do, and the whole of this table is about instruments that are hit.

Still open: whether the ceiling is a ceiling for seventh chords too #

The structural argument above — that a triad always contains a third or a sixth, so 1.26 seconds is its ceiling — is an argument about three pitch classes a fifth and a third apart. It does not obviously extend, and the direction it fails in is not obvious either.

A dominant seventh has four notes and six intervals, so it has more chances to contain something short-lived and its minimum should be lower. But it also contains a minor seventh, 9:5, whose coincidence needs a ninth partial and lasts 0.64 seconds — which would make every spacing of a dominant seventh shorter-lived than every triad, and would say something quite strong about why seventh chords are unstable objects that resolve. Whether that survives being computed, and whether some spacing of a seventh chord dodges its own worst interval the way the minor triad’s first inversion dodges the minor third, is the same census run over four notes instead of three. It is the same arithmetic, it is larger by one loop, and the answer is either a striking result about dissonance or a demonstration that the lifetime measure stops meaning anything past three notes.