A bass line is not a list of roots

Two cues meet in a corner swept the ordered key-finder’s two added cues together and found the best readings in a corner of the plane: a probe-tone profile strong enough to find the tonic, and a bass note strong enough to find the degree. The bass had done most of the work on function. With it weighted, 89 per cent of the 188 bars of the six form schemes were read in the right key and on the right degree, against 47 with no bass at all.

Every one of those bass notes was the root of its chord. That is the one bass line under which a cue that rewards the triad rooted on the bass cannot be misled, and it is not a line anybody plays. A bass part moves — it takes the chord tone nearest the note it just played, it walks, it sits on the third of a chord to make a line rather than leaping to every root — and on every bar where it does, the note at the bottom of a C major chord is E or G rather than C. A cue that rewards the triad rooted on E will reward E minor.

Three lines under the same chords #

The chords are the same 188 bars as before, and only the note at the bottom changes. Three lines are drawn.

The root line is the one every earlier figure used: each bar’s bass is its chord’s root. The smooth line is the economical one: the first bar’s bass is its root, and every bar after it takes whichever of its own chord’s three tones is nearest the previous bass note, preferring the root on a tie. That is a crude description of a bass part and a fair one of a keyboard reduction’s left hand, and it has no knowledge of the model. The six-four line puts every chord’s fifth in the bass. Nobody writes it; it is the control, the line on which a root rule is wrong on every bar.

The smooth line inverts 44 per cent of its chords — 21 per cent in first inversion and 23 in second. That is not a feature of these schemes so much as of what a nearest-tone rule does to any progression. A triad’s three tones leave no gap wider than five semitones around the octave, so every triad has a tone within two semitones of any bass note whatever, and for a chord a fourth or fifth away that tone is usually not its root: after C in the bass, a G major chord offers B a semitone away and D a tone away, and G itself five semitones off.

None of this is a claim about how the ear hears a root. The root an ear supplies found that a major triad fits one harmonic series with its fundamental below the bass and a minor triad fits two, so even the root is an inference a listener makes rather than a label a chord carries. The model is handed the root as a fact, which is the kindest assumption available to it, and it is still misled by where the bass sits.

The ninety per cent was a ceiling #

On the root line the root rule rises from 47 per cent with no bass to 89 at a weight of one and a half and stays there. On the smooth line it rises to 65 at a weight of a half, stays between 64 and 68 up to one and a half, and then falls to 62 at two.

Trusting the root rule more makes it worse on a real line. Each inverted bar votes, with the bass’s full weight, for the triad rooted on the note at the bottom, and that triad is the wrong one. Weighted lightly, those votes are outvoted by the triad overlap and the transitions; weighted heavily, they win. The line of fifths shows the same mechanism at its limit: every bar votes for a chord a fifth above the one sounding, and by a weight of one and a half the model reads 3 per cent of bars on the right degree — a sixteenth of what it read with no bass at all.

So the 89 per cent the corner was built on is a ceiling rather than a floor, and a ceiling set by the one bass line the cue was designed around. On an economical line the same cue gives about two thirds of bars, 68 per cent, which sits exactly halfway between the 47 of no bass at all and the 89 of a line of roots: an economical line throws away half of what the bass was worth.

The rule that cannot be fooled says less #

The obvious repair is to reward any triad that contains the bass note rather than the one rooted on it. That rule cannot be misled by an inversion, because a chord’s bass note is in the chord whichever note it is.

It is also much less informative. A C in the bass is a tone of three triads in C major — the tonic, the subdominant and the submediant — so the member rule narrows seven degrees to three and leaves the triad overlap to choose among them. On the root line, which the member rule treats exactly like any other line, it reads 60 per cent. On the smooth line it reads 67 at its best, at a weight of a quarter, and settles at 61. On the line of fifths it reads 52. The member rule’s answer hardly depends on the line at all, which is its virtue, and it is never good, which is its price.

The two rules therefore bracket the problem rather than solve it. The root rule is excellent on roots, fair on an economical line and useless on a line of fifths; the member rule is mediocre on all three. On the line a bass player might actually write, the two are a point apart at their best, 68 to 67, and once the bass is trusted at a weight of one or more the rule meant as the repair settles seven points below the rule it was meant to repair. What would do better is a rule that knows which tone of the chord the bass is — and that is exactly the information an inversion is, and exactly what a bar of pitch classes does not contain.

The profile makes both worse #

The plane’s corner needed the profile for the tonic, so the realised line has to be asked about with the profile in.

On the root line the root rule recovers almost all of its advantage under the profile: 87 per cent at a weight of two. On the smooth line it does something worse than failing to recover. At light bass weights it drops to 22 per cent — below the 46 the profile reads with no bass at all — and it climbs back only to 50 at a weight of two. The member rule settles at 47.

The reason is the one a tonic bought with the function found: the profile scores a key and not a degree, so under the profile the triad overlap no longer distinguishes degrees and the bass is the only cue left doing so. On roots the bass is a good cue and that is enough. On a real line the bass is a misleading cue on nearly half the bars and nothing else is left to outvote it.

The corner survives, twenty-two points lower #

On the realised line with the root rule the degree share’s best cell is 68 per cent, at the left edge of the plane where there is no profile and so no tonic. Moving right costs degree share in every row with a bass in it, but slowly at first: at a bass of one and a half the row runs 68, 64, 65, 65, 65, 57 and 46. The profile can be raised past the step where the minor tonic appears, between 0.5 and 0.6, for three points of degree share. It is the last two columns that cost, where the profile has taken the emission away from the triad overlap that a misleading bass needs to be outvoted by.

So the corner survives. The product of the degree share and the minor share reaches 65 per cent at a profile share of 0.6 with the bass worth one and a half, which is the cell that read 87 on roots. The shape is the one the line of roots drew — a step for the tonic, a rise for the degree down the column — at three quarters of the height.

The member rule’s corner is lower still: its best cell on the same line is 54 per cent, at the same profile share with the bass worth one or more. A root rule that is fooled on nearly half its bars reads every question better than a member rule that cannot be fooled at all, 65 to 54, because a misleading cue that is right on 56 per cent of bars still carries more than a truthful cue that never narrows below three chords.

The leading note in the bass times the modulation #

One number rises on the realised line rather than falling, and it is the one the bass was never introduced to supply. Nowhere on the plane does the line of roots read more than 91 per cent of bars in the right key. The smooth line reads 94 under the member rule with no profile at all, and 95 under the root rule with the profile at a half or 0.6.

Nearly all of the gain is one scheme, and it is the scheme two cues meet in a corner found no setting of either cue could help. The rondo is forty bars in five sections of eight — refrain, an episode in G major, refrain, an episode in A minor, refrain — and on the line of roots, with the bass alone, it is read in C major from the first bar to the last, which is exactly 60 per cent. On the smooth line with the profile at a half it reads 93 per cent. Its three wrong bars are the two before the first episode, which the model moves into G early, and the first bar of the refrain after it, which it leaves in G one bar late. Both episodes are otherwise found whole.

The bars that found them are the inverted dominants. In the first episode the dominant seventh D–F♯–A–C comes round three times, and the nearest-tone line puts F♯ under it twice. In the second, E–G♯–B–D comes twice, and the line puts G♯ under both. On the line of roots those bars have D and E at the bottom, and D and E are notes of C major, so nothing in the bass says the key has changed: the roots of the first episode’s chords are G, D and C, all three of them home notes. The leading note is the one bass note in either episode that the home key does not contain. Under the root rule it rewards a triad rooted on F♯ or G♯, which no reading in C major can offer; under the member rule it rewards any triad containing it, which C major cannot offer either. Both rules read the key change from one note.

That is the evidence how much evidence a modulation needs asked for, arriving from a direction that essay did not look in. It is also the note that names a key in the key-finder with no tonic, where a handful of raised sevenths was enough to tell A minor from its relative major in the upper voices; here the same note does it from the bottom. And it is one reason, stated in information rather than in voice leading, why the first-inversion dominant with the leading note rising to the tonic is among the commonest bass moves in tonal writing: one dominant among seven is about why that chord exists in only one rotation of the scale, and a bass that sits on its third is pointing at the rotation.

The line did not do this on purpose. The nearest-tone rule put F♯ under D–F♯–A–C because it was a semitone from the G before it, and a bass player walking into a dominant does the same for the same reason. An economical bass line tells a model less about the chords and more about the keys, which is the opposite of what the bass cue was introduced to do and exactly what a model of a modulating piece was missing.

The arithmetic under the three lines #

The model is the one a key-finder that keeps the order built and the two cues are the ones the plane swept, with nothing changed but the bass notes and the rule for using them. The root rule adds the bass’s weight times the log of one when the candidate triad’s root is the bass note and of a fifth otherwise; the member rule does the same when the candidate triad contains the bass note at all. The keys are twelve majors and twelve harmonic minors, a key change costs 2.2 in the same log units, and every share is the most probable path through the whole passage rather than a bar-by-bar guess. The smooth line is realised once from the chords, by a nearest-tone rule that knows nothing of the model and cannot adapt to it, and the blues’ sevenths are never put in the bass, so every line is root position, first inversion or second inversion.

The schemes are skeletons and the realisation is a rule, so the percentages are shares of 188 bars of constructed music under one constructed line. What survives that is the ordering: with no profile and the bass trusted, the root rule beats the member rule on roots and on the economical line and loses to it only on a line of fifths; the member rule barely notices the line; trusting a root rule more on a real line reads worse; and a line that inverts its dominants finds a modulation that a line of roots cannot.

What a line of pitch classes cannot say #

The inversion is information the bars do not carry. A bar here is a set of pitch classes and a bass note. Whether the E at the bottom of a C–E–G is the third of C major in first inversion or the root of an E minor chord whose B is missing is decided by what else is sounding and where, which is a chord is a register’s subject and not this model’s. A rule that knew the inversion would be a rule given the voicing, and the voicing is the one thing the key-finder has never been handed.

Inversions are not acoustically alike, and the model treats them as if they were. An inversion lasts as long as its outer sixth found that a struck major six-four keeps the evidence of all its intervals for 1.26 seconds at C3 and a major six-three for 0.74. A listener deciding what an inverted chord is has that evidence, fading at different rates for different inversions; the model has a pitch class at the bottom and nothing about how long the chord above it stays legible.

The six-four is not only a control. Second inversions are rare and constrained in real bass lines — the inversion that cannot end a phrase is about why — so a line of fifths misrepresents real music badly. The economical line’s 23 per cent of second inversions is also more than most repertoires would show, since a nearest-tone rule does not know that a six-four needs preparing.

Real bass lines do more than choose among chord tones. They pass through notes outside the chord, anticipate the next chord and repeat a pedal under several. Every one of those puts a note at the bottom that is in no triad of the bar, and both rules would reward nothing on those bars — or, where the passing note is chromatic, would read it as the leading note of a key the music never visits.

And the margin is unmeasured. The shares are counts of right answers, and a reading can be right with less certainty on a realised line than on roots. That is likely and it is not drawn.

Whose bass lines #

The practice is keyboard and ensemble bass writing in tonal music, where a bass part is a line with its own logic and inverts a large share of its chords, and the claim is about what an automatic analysis of such music can take from the bass note alone. Taking it as the root is right in the one texture that puts roots there — chorale-style cadences, the bass of a harmonic reduction written for analysis — and wrong wherever the bass is a line. The corner the two cues made is a corner for skeletons, and the modulations a real line finds are a finding about real lines.

It is not the only bass line in this collection that is not a list of roots. The bass line under a passage in thirds is one no instrument plays at all: the difference tones under a scale harmonised in just thirds spell C, A, C, F, G, F, G, C, a diatonic line the ear supplies from two upper voices. That line and this one make the same point from opposite ends — the note heard at the bottom is a fact about the sounding texture, and neither a chord symbol nor a root can be relied on to say what it is.

Still open: the bass as a register rather than a pitch class #

The rule that would recover the ceiling needs to know which chord tone the bass is, and that is a fact about register: a bar voiced in real octaves says whether the lowest note is the third of the chord above it or the root of a different one, by what lies between the bass and the upper voices. A cue that rewards a triad according to where the bass sits among the chord’s own sounding notes would be one more term in the emission, given a voicing of each scheme in registered octaves. What it would settle is whether the degree share on a realised line returns to 89 once the model is given the register the line was realised in — and whether it can do so without giving back the 93 per cent the inverted dominants found in the rondo, since a rule that learns to discount an inverted bass may learn to discount the leading note with it.