Three notations, one progression

Three notations for harmony are in professional use and have been for a long time. A jazz musician reads C – F – G – C. A theory student reads I – IV – V – I. A continuo player in 1720 read a bass line with numbers under it and no chord names at all.

They are usually described as three levels of detail, with the figured bass at the specific end and the chord symbol at the general one. That description is wrong twice over, and the way to see it is to stop describing and start counting.

The thing to count #

A figured bass writes out one voice. Four bass notes on a staff, in pitches, and above them nothing but numbers — and where the chord is a plain triad in root position there are no numbers either, because the absence is the instruction. What a Roman numeral would write for the same four chords is four symbols and no notes at all; what a chord symbol would write is four letters. Three notations, three different things committed to paper, and only one of them writes down a pitch.

A notation is a map from what is written to what may be played. If two performances satisfy the same notation, the notation has thrown away the difference between them; the number of performances it admits is the exact size of what it threw away.

For four-part harmony inside ordinary vocal ranges, that number is computable. This site already has the machinery: every arrangement of a chord’s pitch classes across four voices in range, and the rules that say which consecutive pairs are allowed.

Four hundred and eighty, one hundred and fifty-two, seventy-six, one. Those are the sizes of the equivalence classes each notation maps a chord onto, and the ratios between them — a little over six, then two — say exactly how much each successive specification buys.

Note the shape of it. Naming the inversion cuts the field by a factor of three; naming the bass octave as well cuts it only by two more. The inversion is the expensive piece of information and the register is the cheap one, which is a fact about the ranges rather than about the notations and is the reason chord symbols get away with omitting the second.

Over a progression, where the rules bite #

One chord is the easy case. What a notation is really for is a sequence, and a sequence brings the part-writing rules with it, because the rules relate consecutive chords rather than describing single ones.

Sixteen billion is a large number for four chords, and it is worth saying where it comes from: 480 × 480 × 334 × 480 arrangements is thirty-seven billion before any rule is applied, and forbidding parallel fifths and octaves removes fifty-six per cent of them.

That figure is itself a result. The rules of part-writing have a price, and this collection has measured it in voice-leading distance; here it is measured as a fraction of the space removed. Adding the leading-note and augmented-interval rules on top takes the count down to 4.7 billion, so the four rules together forbid seven realisations in eight — and the remaining eighth is still four billion things a chord symbol permits.

There is a second reading of the same count, and it is the one a continuo player would give. Two million realisations of four chords is not a measure of vagueness; it is a measure of how much of the music the notation is deliberately handing to the performer. A figured bass was written for an improvising keyboardist, and it specifies the part of the texture nobody else can supply — the bass, which is a real line played by a real instrument — and leaves the rest, which is the part the keyboardist is there to invent. Read that way the notation is not underspecified at all. It is precisely specified about a division of labour.

The same reading applies to the chord symbol, whose sixteen billion is not carelessness either: a chart is written for a rhythm section that will voice the chords differently every chorus, and a notation that fixed the voicings would be describing a performance that is not going to happen twice.

Most of the headline spread is a bass player #

The chord symbol’s count is taken with no constraint on the bass at all, which the note below admits is generous, since a chart implies a bass player. Putting one in is a one-line change: fix the bass pitch class to the root.

The count falls from 16.1 billion to 59.4 million — which is, to the last digit, the Roman numeral’s figure. That is not a coincidence and it is not a coincidence worth hiding: fixing the bass pitch class is exactly what a root-position Roman numeral does, so a chart played by a rhythm section with a bass player is, on this measure, precisely as specific as a numeral.

So the factor of two hundred and seventy-one between the loosest and the middle notation is the bass player and nothing else. What is left of the four orders of magnitude is a factor of twenty-nine, between naming the bass note and naming its octave — which is the same small factor the single-chord figure reports, and is the honest size of the gap between a numeral and a figured bass.

That reorders the essay’s own conclusion. The three notations are not spread over four orders of magnitude by anything they say about harmony; they are spread by one absent instrument and one octave.

And the ratios are not range-independent #

The computation note below says the counts depend on the vocal ranges but the ratios between notations barely do, because each constraint is applied to the same enumeration. Widening and narrowing all four ranges together:

The first ratio moves by a factor of two and the second is not defined at all below the standard ranges, because narrowing every voice by two semitones leaves the figured bass with no legal realisation whatever. Fixing four bass pitches and forbidding parallels over four chords is unsatisfiable in ranges only slightly tighter than the ones the tradition uses.

That last is the more interesting number. It says the standard four ranges are not comfortably inside the feasible region; they are close to its edge, and the exercise the whole apparatus was built for stops having answers a couple of semitones away. Whether that is a fact about the ranges having been chosen to make the exercise possible, or about the rules having been written for the ranges, is not something a count can decide — but the two are evidently fitted to each other much more tightly than a notation-independent ratio would suggest.

Where the ordering breaks #

So far the three look like a chain: chord symbol ⊂ Roman numeral ⊂ figured bass, loosest to tightest. That is the description this essay set out to test, and here is where it fails.

A figured bass does not name the root. The figures are intervals above the bass. A bass C with a 6 under it means a sixth and a third above the bass — C, E, A — which is an A minor chord in first inversion, and nowhere on the page does the letter A appear. The player works it out, or does not need to.

A Roman numeral does not name any pitch at all. It names a scale degree in a key, and the key is stated separately. Transpose the piece and the numerals are unchanged; that is the whole point of them and it is exactly the property a staff has for melodies, applied to harmony.

It is also the only one of the three that can name a chord outside any key at all — a quartal voicing has no Roman numeral and a perfectly good chord symbol.

A chord symbol names the pitch classes and no key. C – F – G – C is the same four symbols in C major and in the key of F where they would be V – I – II – V. The symbol is below the numeral in one dimension and above it in another.

So the three notations are not a chain. Each fixes a different quotient:

The empty cells are the interesting ones. The chord symbol and the figured bass both lack a key; the numeral and the figured bass both have the inversion; only one of the three says what a chord is for.

Which of the three is right about what a chord is #

Read that table as three answers to one question and they become three theories.

A figured bass says a chord is a bass note with intervals over it. That is a contrapuntal answer: the bass is a real voice with a real line, and the harmony is what happens above it. The notation was invented in an idiom where the bass part existed and was played, and its shape follows from that.

A Roman numeral says a chord is a degree of a scale acting on a key. That is a functional answer and it is the youngest of the three, arriving with nineteenth-century theory. It is the only one that survives transposition and the only one in which a progression is a path with a direction, because a path needs somewhere to be going and only a key supplies one.

A chord symbol says a chord is a set of pitch classes with a root. That is a vertical answer and it is the answer a keyboard or a guitar wants: play these notes, in whatever arrangement the hands prefer. It is also the answer the space of chords this site drew is built on, where a chord is a point and a progression is a move between points, with no key anywhere in the geometry.

Ranked by roughness, the voicings a chord symbol admits are not interchangeable: the smoothest arrangement of a C major triad and the roughest differ by a factor a listener hears immediately, and where the third goes accounts for most of it. The notation is silent about all of that. There is nothing wrong with the silence — the player is the one who can hear the room — but a notation that leaves the choice open is asserting that the choice is not part of the composition.

The three theories are not compatible, and they were not meant to be. Each is right about the music it was written for.

And there is a dimension none of the three notations carries well, which is time. A chord symbol sits over a bar and takes its duration from the bar; a Roman numeral takes its duration from whatever is written above it; a figured bass’s numbers sit under a note whose length is written out, so only the third carries the rhythm inside the notation itself. That matters more than it sounds, because which notes are the chord is a decision the metre makes, and a notation that does not carry the metre has left the segmentation to be redone by the reader.

The root, which two of the three leave to the ear #

There is one more asymmetry and it is the sharpest, because it is where a notation depends on a perceptual fact.

A figured bass gives a bass note and a set of intervals and never says which note is the root. In root position that is trivial; in a first inversion it is not, and in a diminished seventh it is genuinely ambiguous — four notes, four possible roots, and nothing in the sound to decide.

This collection has already found where the root comes from when the page does not supply it: the ear supplies it, by something very like the harmonic-template fit that produces a residue pitch, and the answer it gives is stable for a major triad and unstable for a diminished seventh.

So the figured bass is not being vague. It is declining to write down a quantity that the listener computes anyway and that the sound sometimes does not determine — which is a defensible position and is the opposite of the usual complaint about it.

The moves a numeral is about are distances, and the numeral does not name them: V–I and IV–I are the same distance in voice-leading terms, which is a finding a Roman numeral cannot state and does not encode. Which notes actually go where is chosen by the player from among the realisations the notation admits — two million of them, for the four chords above.

What a Roman numeral does encode, and neither of the others can, is how well each triad fits the key it is in. That column of numbers is the numeral system’s whole content: a chord symbol carries none of it and has to be handed a key from outside before any of it can be recovered.

Seen as a graph of common-tone relations, a chord symbol is a vertex and a Roman numeral is a vertex plus a choice of origin — the same picture with one node marked home. Two notations, one object, and the difference between them is a single label.

Which computation produced the numbers #

The realisations are counted, not sampled. For each chord, every assignment of its pitch classes to four voices within the standard ranges is enumerated; consecutive layers are then joined by a dynamic programme that admits a pair only if it violates none of the named rules, and the total is the number of paths through the resulting graph. For I–IV–V–I that is a count over a space of thirty-seven billion, and it takes a few milliseconds because the graph has 1,774 nodes.

The ranges are the standard four: bass 40 to 60 in semitones above C0, tenor 48 to 67, alto 53 to 74, soprano 60 to 81. Every count in this essay depends on them, and the ratios between notations were recorded here as barely depending on them. The section above sweeps the ranges and finds otherwise: the symbol-to-numeral ratio moves by a factor of two, and the numeral-to-figured ratio is undefined two semitones in, because the figured bass admits nothing at all there.

The sweep needed a small change to the machinery, which is worth recording. notationCollapse takes the ranges as a parameter and countRealisations, which does the progression, does not — it names SATB_RANGES inside its own body. So the claim the note made could not have been checked with the function that produced the numbers it was about.

The rules are the same four the voice-leading ladder uses, implemented once and shared: no parallel fifths, no parallel octaves, the leading note resolves, and no augmented melodic interval. “Every rule” above means all four; “parallels” means the first two.

The chord symbol’s count is taken with no bass constraint at all, which is generous to it — in practice a jazz chart implies a bass player. The Roman numeral’s count fixes the bass pitch class from the inversion. The figured bass’s count fixes the bass pitch, which is what writing it on a staff does.

What the picture cannot show #

It counts arrangements and not music. Most of the two million realisations a figured bass admits are bad, and a continuo player produces one of the good ones without enumerating anything. The count measures what a notation permits, which is a property of the notation; what a competent reader will do with it is a property of the reader and is not in any of these numbers.

And the headline is a comparison between notations under different assumptions about the ensemble. Sixteen billion is a chord symbol read by a pianist with nobody else in the room; fifty-nine million is the same symbol read by a pianist with a bass player. Neither number is wrong and they are not measuring the same situation, so the four orders of magnitude in the dek should be read as three claims of one order each and one claim about who is playing.

It has four voices. A continuo realisation is not in four parts — it is whatever the keyboard’s hands are doing, often five or six notes with doublings, and the count would be much larger. Four is chosen because it is what the rules were written for and what this site’s part-writing machinery already models.

It ignores everything a real figured bass says. Suspensions, accidentals under the bass, the horizontal line meaning hold, and the tasto solo instruction are all part of the system and none of them is modelled here. The count is for plain triads with no figures beyond the inversion.

The dynamic programme counts paths and not distinct sounds. Two realisations that differ only by an octave doubling somewhere inaudible are two paths, and the count treats them as two. Any coarser equivalence would be a judgement about hearing rather than about notation, and this essay is about notation.

And the table above is a caricature of three living traditions. Chord symbols carry extensions, alterations and slash basses; Roman numerals carry inversions, applied dominants and mode mixture; figured bass carries a shorthand that a good player reads as gesture rather than as arithmetic. The claim being made is about what each system’s core fixes, and every one of the three has been extended in the direction of the others.

The ladder from here #

This closes the notation ladder at six rungs, one in each of six fields, and what it has found in all six is the same shape: a notation fixes some coordinates of a musical object and leaves the rest to be supplied, and which ones it fixes is a claim about what the object is. The staff fixes the scale degree and leaves the pitch; the signature fixes the bar and leaves the grouping; the dynamic mark fixes an order and leaves the spectrum; the tablature fixes the action and leaves the sound; and here, three notations fix three different things about one chord and disagree about which of them the chord is.

What the ladder does not have is a rung about the clef, which the first rung named and left owing, and one about the notations invented for music this system cannot hold — the graphic scores, the tablatures for instruments that do not exist yet, and the several attempts at a chromatic staff. Both are about the same question from outside: what a notation does when the thing it was built for stops being the thing being written.