The stave is not a ruler

Every figure on this site draws pitch against an axis that had to be built. Cents up the side, or semitones, or hertz on a log scale, or positions on a circle — and in each case something had to decide what the spacing meant before anything could be drawn.

There is one pitch axis that did not have to be built, because the reader arrives holding it. It has five lines, it is five hundred years old, and it appears in fourteen essays here as scenery: a phrase drawn so that the thing under discussion is recognisable, and then argued about somewhere else.

This essay is about that object, and the first thing to say about it is that its vertical axis does not measure pitch. It counts letters.

Seven places and twelve pitches #

An octave holds twelve semitones. A staff holds seven positions for them — four lines and three spaces, or three lines and four spaces, depending where the count begins — and then repeats.

Seven of the twelve is the choice this whole collection keeps arriving at, and the page made it too. Seven into twelve does not go, so five of the twelve have no place of their own. They are written as an alteration of a neighbour, and the symbol that does the altering is the accidental.

It is worth being precise about what an accidental therefore is. It is not an extra refinement available on a complete axis, in the way that a decimal point refines a whole number. It is the repair for an axis with five of its values missing. A sharp does not mean “slightly higher than written”; it means “the pitch this position would have if the position it needs did not fail to exist”.

The consequence is immediate and it is the thing this rung is about. If the axis counts letters, then a distance measured on it counts letters too, and a count of letters is not a distance in pitch.

Every generic distance on the staff against every number of semitones it can stand for makes the ambiguity a count: with no accidental a third is three semitones or four; with one it is two, three, four or five. The page records the generic interval exactly and the specific one only with help.

Six of the eight generic distances inside an octave are ambiguous before anything is written in front of a note. A third on the page is three semitones or four; which one it is depends on where in the scale it sits, and the page does not say.

Why the axis is like that, and it is not laziness #

The obvious reading of the last figure is that the notation is badly designed — that a system with twelve positions and no accidentals would be simpler, and that this one has survived for historical reasons.

That reading is wrong, and the reason it is wrong is a property this site has already measured. The seven-note diatonic set has the property that every generic interval comes in exactly two specific sizes, and it is one of very few seven-note sets that do.

And that is not laziness. Every generic interval of the diatonic set comes in exactly two specific sizes, so a seven-position axis can name an interval unambiguously as a distance on the page while leaving its size to the key signature — a property a seven-note set that is not a mode of the diatonic does not have, and which is why an axis like this works for this repertoire and would not work for another.

How few is “very few” #

“One of very few” is the kind of phrase that should be a count, and the count is small enough to do exhaustively. There are 792 seven-note subsets of the twelve, which fall into 66 classes under transposition, and every one of them can be asked the same question: for each generic distance from a second to a seventh, how many different specific sizes does it take?

Two of the sixty-six. And the other one is not a rival. It is the seven consecutive semitones — a chromatic cluster — whose two sizes for each generic distance are five semitones apart rather than one: its “second” is either one semitone or six, its “third” either two or seven. Every generic distance on it is ambiguous by a tritone, which is not a near-miss that a key signature repairs but two unrelated intervals sharing a name.

So the diatonic set is the only seven-note set in the system whose two specific sizes for each generic size are adjacent, and adjacency is the whole of what makes the axis usable. A page that says “third” and means three semitones or four is telling a reader something; a page that says “third” and means two or seven is not.

The same census gives the other half of the argument a number. Count the total number of distinct generic-and-specific pairs a set produces — the size of the table a sight-reader has to carry — and the diatonic set needs twelve, which is the minimum. The average over the sixty-six classes is 20.6 and the worst are 28. A reader of a page written on the average seven-note set would have to hold nearly twice as many rules, and a reader of the worst would hold more than twice.

That is the strongest form of the claim this section is making, and it is worth stating as strongly as the arithmetic allows: the five-line staff is not a convenient axis for the diatonic scale among several that would have worked. It is an axis for the one seven-note set out of sixty-six that could carry an axis at all.

That is the trade the notation made, and once it is stated the design stops looking accidental. A seven-position axis works because the object it is an axis for has seven elements, and it is nearly a pitch axis because the seven are laid out as evenly as seven things can be laid out among twelve.

What the trade buys is that a shape on the page is a shape in the scale. A figure moved up two staff positions has moved up two scale degrees, whatever the key, whatever the mode, and without changing a single accidental.

This is not a small convenience. It is the reason a sequence can be written down as a sequence — the same three shapes climbing the page, each one a different chord — and the reason a musician transposing at sight moves their reading up two lines rather than recomputing eleven intervals.

The rotations the page cannot tell apart #

The same trade shows up in one more place, and it is the cleanest demonstration that the axis is the diatonic set rather than pitch.

Written in one key signature, all seven modes are the same seven positions. What distinguishes them is which note the music treats as home, and that is not a graphical fact at all — it is something a listener works out over several seconds from where the phrases land.

So the page encodes the set exactly and the rotation not at all. A reader looking at a Dorian melody and an Ionian one with the same signature sees an identical alphabet on both, which is correct — they use the same alphabet — and has to get the mode from somewhere else. That is a real loss and it is the same loss a pitch-class set formalism has, arrived at independently and five hundred years earlier.

What a real pitch axis looks like #

It is worth putting the page beside an axis that is a ruler, to see how differently the two behave.

A real pitch axis, with the ratios where they actually fall against twelve equal steps, is spaced evenly in cents and has twelve positions — which is what the stave is not, in both respects.

On this axis, the operation that is free on a stave is expensive: transposing a diatonic melody means recomputing every interval, because the pattern of tones and semitones does not translate. On the stave, the operation that is free on the ruler is expensive: reading an interval off requires knowing the key.

Neither is more correct. They are the duals of one another, and each is efficient at exactly what the other is bad at. A notation is a decision about which operations should be free.

The five that had to go somewhere #

There is a second thing the seven-position axis produces, and it looks like a defect until it is examined.

Because a written pitch is a letter plus an alteration, one sounding pitch has several spellings — and the spellings are not interchangeable in any system except the one everybody now uses.

The distance between two spellings of one pitch class is always seven fifths, and seven fifths is the distance a comma is measured over. That is not a coincidence and it is the next rung of this ladder: the page kept a coordinate that the twelve-key instrument threw away, and in the tunings that were actually played the two spellings are two different pitches.

For the moment it is enough to see that the redundancy is structured. A notation with a random extra symbol would produce spellings scattered anywhere; this one produces them at a fixed distance along a chain that everything else in tuning is built on.

It is also worth separating from a different ambiguity that runs the other way. This ladder is about one sound having two names; the case where one interval size has two names — four hundred cents heard as a major third or as a diminished fourth — is about the listener rather than the page, and the two are easy to confuse because both involve the word enharmonic. The distinction is which side of the encoding is being held fixed.

The key signature, which is where the specific size comes from #

If a staff position gives the generic interval and not the specific one, something has to supply the difference, and the thing that supplies it is written once at the left of the line.

A key signature is one number written as up to seven symbols, and the circle it is a position on is the same circle key distance is measured round — so a modulation to a neighbouring key is one symbol’s difference on the page and one step on the map, which is the notation and the theory agreeing for once. What it does is fix, for the whole system, which seven of the twelve the staff positions refer to; after that, position determines pitch exactly and no accidental is needed until the music leaves the set.

Written out, the same eight positions with two sharps at the front are D major, and the accidentals sit at the front rather than on the notes — which is the compression the system buys and the reason the page is readable at speed.

The efficiency of that is easy to underrate. A page of tonal music in the eighteenth century carries almost no accidentals, because the music stays in the set the signature names and the axis is exactly right for it. The notation is efficient in proportion to how diatonic the music is, which is a strong statement about what its designers thought music was.

Where the coordinate stops paying #

The corollary is the interesting half. A passage that does not live in one diatonic set gets no benefit from an axis built out of one, and pays the accidental cost on every note that leaves.

Count it. A twelve-note chromatic run written in C needs five accidentals, one for each pitch with no position of its own — five symbols for twelve notes, which is not terrible. A passage that modulates every bar needs a fresh accidental every time it enters a set the signature does not name, and the sharps and flats that result are describing the distance from the signature rather than anything about the sound.

And where the coordinate stops paying is the chromatic scale: twelve notes need five accidentals, the positions are no longer evenly spaced in anything, and a passage that uses all twelve equally is a passage the axis is actively working against.

That is why chromatic music of the late nineteenth century looks the way it does on the page, and why several people tried to replace the axis. A staff with twelve positions and no accidentals is easy to draw and has been proposed repeatedly. What it costs is everything in the section above: on a twelve-position staff a triad is three different shapes rather than one, the seven modes are seven different pictures rather than one, and diatonic transposition stops being a rigid translation.

The trade is exact and it can be counted. On the seven-position axis, the seven diatonic triads are one page shape and five pitches need an accidental. On a twelve-position axis, they are three page shapes — major, minor and diminished are three different spacings — and no pitch needs an accidental at all. Whichever axis is chosen, something that was one thing becomes three, or something that was three becomes one.

Which computation produced the numbers #

The census is exhaustive. For every pair of letters and every pair of alterations within the stated limit, the letter distance and the semitone distance are both computed and tabulated; the figure above is the whole table, not a selection from it. Intervals larger than an octave are excluded, which is why the row for a seventh runs out at twelve.

The chain-of-fifths position of a written pitch is its letter’s position — F, C, G, D, A, E, B are −1 to 5 — plus seven for each sharp and minus seven for each flat. That is a definition rather than a measurement, and everything in the spellings figure follows from it arithmetically.

The claim that the diatonic set has two specific sizes for every generic one is computed by genericSizes, which enumerates the set’s own rotations, and the comparison set is checked to be a genuine non-rotation rather than assumed to be one. The census is the same function applied to all 792 subsets, reduced to 66 transposition classes by taking each set’s lexicographically least rotation; nothing about the diatonic set is used to build the list, so its appearance at the top of it is a result rather than a construction. This site made that mistake once already: the first comparison ever handed to that figure was Phrygian, which is a rotation of the diatonic set and therefore has the property too, and the figure said so.

What the picture cannot show #

It cannot show the horizontal axis, which is the other half of the notation and has its own version of the same problem. A note’s horizontal position is a nominal duration in an implied metre, not a time; what a page can and cannot say about when a note happens is a rung of its own.

It cannot show the clef. Every figure here is on a treble stave, and a clef is a decision about which absolute pitches the seven positions refer to. That decision is a second coordinate choice, made for the same reason as the first — to keep a given voice inside the lines — and it is why there are so many of them.

It assumes the twelve are equal. Everything above counts semitones as though they were interchangeable units, which they are only in equal temperament; on the instruments the notation was designed for they were not, and a fretted instrument cannot make them so even now. The next rung is about exactly that gap.

It says nothing about legibility. The claim here is about what the axis encodes, and a notation is also a thing that has to be read at speed by somebody whose hands are busy. Several of the decisions that look inefficient as an encoding are efficient as an interface, and this site has no way to measure that at all.

And it treats the staff as a fixed object, which it is not. Four-line staves, staves with more than five lines, and staves whose lines mean different things were all in use for centuries, and the five-line form settled only when keyboard music made a fixed range worth standardising on.

The ladder from here #

This rung took the page’s vertical axis and found the diatonic set in it. The next takes the redundancy that produced — two spellings for one key — and shows it is not redundancy at all: in every tuning anybody played before the nineteenth century, G♯ and A♭ are different pitches, and the size of the difference is computable from twelve fifths and seven octaves and nothing else.

The debt this rung leaves is the clef, and the reason to name it is that it points at a question the melody ladder also owes: a clef exists to keep a voice inside the lines, so the number of clefs in use is evidence about how wide a voice’s working range is, and that width is the thing this collection cannot yet derive.