Two names for one key

There is one key between G and A on every keyboard ever built. The page has two names for it, and a musician who writes the wrong one is told they have made a mistake.

That is a strange thing for a notation to insist on. Notation is a compression, and an expensive one — its vertical axis has five values missing and has to patch them with accidentals — so a distinction it maintains at the cost of extra symbols is a distinction it is paying for. Either it is paying for nothing, or the two names are not names for the same thing.

They are not names for the same thing, and this essay is about how far apart they are.

Where the two names come from #

A written pitch is a letter and an alteration, and both of those are positions on one line: the chain of fifths. C is at zero, G at one, D at two; F is at minus one. A sharp adds seven steps and a flat subtracts seven, because seven fifths is what it takes to get from a natural to its own sharp.

A chain of fourteen fifths in meantone reaches past the twelve a keyboard has, and the two extra links are exactly the notes the enharmonic pairs disagree about — the chain does not stop, the keyboard does.

G♯ is eight steps up the chain from C. A♭ is four steps down. The distance between them is twelve steps, which is twelve fifths, and twelve fifths is exactly the quantity every essay in the comma ladder is about.

Every spelling a staff position can carry, at up to two accidentals, is a census of the names available: the page has more names than the instrument has keys, and the surplus is the whole subject.

So the question “how different are G♯ and A♭” has a completely determinate answer as soon as the size of a fifth is fixed. It is twelve fifths minus seven octaves, and there is nothing else in the calculation.

The answer, and it changes sign #

Twelve pure fifths overshoot seven octaves by the Pythagorean comma, 23.46 cents. So in Pythagorean tuning, where every fifth is pure, G♯ is 23.46 cents above A♭.

Narrow each fifth and the overshoot shrinks. Narrow them enough and it reverses.

In quarter-comma meantone — the tuning of most of the sixteenth and seventeenth centuries, and the one in which the major third is pure — A♭ is forty-one cents above G♯. That is a third of a semitone, it is not subtle, and it is in the opposite direction from Pythagorean.

The interval has a name, the lesser diesis, and it is what is left when three pure major thirds fail to make an octave: 128/125, or 41.06 cents. That the same number arrives from twelve narrowed fifths and from three pure thirds is not a coincidence, because a quarter-comma fifth is defined as a quarter of the way to making the third pure, and both quantities are the same statement about the same lattice.

The point where the two names mean one thing #

Between plus 23.46 and minus 41.06 the curve crosses zero, and the crossing is worth locating exactly because of what sits there.

The gap is 23.46 minus twelve times the narrowing, and the narrowing is a fraction of a syntonic comma of 21.506 cents. So it vanishes at 23.46 / (12 × 21.506) = 0.0909037 of a comma.

One eleventh is 0.0909091. They are not the same number.

Narrowing each fifth by exactly one eleventh of a syntonic comma leaves G♯ 0.0014 cents below A♭ rather than exactly on it, which is a part in ten million of a semitone and is not a distinction anybody can act on. But it is worth writing down, because “one-eleventh-comma meantone is equal temperament” is repeated everywhere and it is an approximation rather than an identity. Equal temperament is the point where twelve fifths close, and the meantone family passes through it at an irrational fraction of a comma that happens to be very near a simple one.

A chain that closes has no wolf and no enharmonic distinction, and it cannot have one without the other. Equal temperament is not a tuning that happens to spell G♯ and A♭ alike; it is defined by the closure that makes them alike, and every property of it follows.

It is worth saying what that means for the page, because it is the reverse of the usual story. The notation is not an approximation that equal temperament tidied up. The notation is exact, in the sense that it names a unique point on a chain of fifths and always has; what equal temperament did was make two of its names refer to one sound, which is a loss of information in the instrument rather than a gain in precision. A reader who takes the page at face value is reading a system with infinitely many pitches, and the twelve-key instrument is the approximation.

What the instruments did about it #

If G♯ and A♭ are forty-one cents apart and a keyboard has one key, somebody has to decide which of the two the key gives. In meantone the usual chain runs E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯, which supplies G♯ and E♭ and refuses A♭ and D♯.

On a keyboard the five black keys are the ones the two spellings compete for, and a meantone instrument tuned for sharps plays flats badly and the reverse — which is why split keys were built and why they were built for exactly those five.

Which of the two a chain supplies is the decision the second rung of the comma ladder is about, seen from the other side: hiding the comma somewhere means choosing which twelve of the infinitely many written pitches an instrument will have, and every choice leaves some spellings unplayable.

The response was not to give up the distinction. It was to build instruments with more keys. Split black keys, with the front half sounding G♯ and the back half A♭, were made from the sixteenth century onward; so were keyboards with fourteen, nineteen and thirty-one notes to the octave. The nineteen-note instrument is exactly the meantone chain extended until it nearly closes, and nineteen is one of the divisions this site has already found arriving from a completely different direction.

Each temperament has its own keyboard, and the arithmetic names it #

“Extended until it nearly closes” is the whole design rule and it produces a different number for every temperament, which is worth computing because it explains why the surviving instruments have the key counts they do.

A chain of n fifths closes when n times the fifth is a whole number of octaves. Sweeping n for each temperament and keeping the length whose residual is smallest:

The two key counts history actually built are the top two rows. Vicentino’s thirty-one-note archicembalo is not an arbitrary act of enthusiasm; thirty-one is where a quarter-comma chain runs out of new pitches, and building a thirty-second key would add one six cents from a key already there. Nineteen is the same statement for third-comma, which is the other meantone in serious use, and a nineteen-note instrument in third-comma closes to within one cent.

This also says what the equal divisions are. The fifth of the equal division that closes at each of those lengths — 694.74 at nineteen, 696.77 at thirty-one, 701.89 at fifty-three — is within two cents of the temperament’s own fifth in every row. An equal division is a meantone whose chain was made to close exactly, and the three divisions this site found by scoring approximations to the simple ratios are the three that close the chains anybody was already tuning. Two independent criteria, one list, which is the check.

It also explains a gap in the historical record. Sixth-comma temperament was widely used in the eighteenth century and no fifty-five-note instrument was built, and the table says why: sixth-comma’s chain is the worst-closing of the five, ten cents out after fifty-five fifths. A temperament whose chain does not close tidily has no natural instrument, and the response to sixth-comma was a circulating temperament on twelve keys rather than more keys.

Whether anybody can hear it #

Forty-one cents is a third of a semitone and nobody argues about it. The Pythagorean case is smaller, and this site has a measurement of where a comma stops being audible that the question can be put to.

Drawn as a spiral the twelve fifths overshoot and never close, which is what makes the twelfth link a leftover and the enharmonic pair a disagreement rather than an identity.

Twenty-three cents is roughly an eighth of a semitone and is comfortably above the difference limen for a sustained tone at any musically useful frequency, which this collection puts at three to six cents in the middle register. So the Pythagorean distinction is audible as a pitch difference, in isolation, by anybody. What is much harder is hearing it as the distinction — telling that a note just heard was the sharp rather than the flat, with no reference to compare it against — and that is a question about memory rather than about resolution.

The meantone distinction needs no such care. Forty-one cents played against a sustained chord is not a shading; it is a wrong note, and the wolf it produces when the wrong spelling is used is the same object the comma ladder is about heard on a single key.

The thing the page is telling a player, and the two answers to it #

Equal temperament removed the acoustic distinction and left the notational one standing, and what happened next is that performers gave it a meaning.

The rule taught to string players and singers is that sharps are high and flats are low — that a G♯ leading up to A should be played sharper than an equal-tempered G♯, and an A♭ falling to G should be played flatter. Read as a claim about tuning, this says G♯ is above A♭, which is the Pythagorean answer and the opposite of the meantone one.

And against the twelve equal steps the just ratios sit where they sit; a G♯ and an A♭ are the same key on the ruler and two different points on the chain, which is the entire difference between a notation and a tuning.

So there are two consistent positions and they disagree in sign, and this collection has already reached each of them from somewhere else.

The vertical criterion — make the simultaneities pure — produces meantone, because meantone is what happens when the major third is prioritised, and in meantone the flat is the higher of the pair. The horizontal criterion — make the melodic tendency strong — produces the Pythagorean answer, because a chain of pure fifths sharpens the leading note, and that is exactly the pull the melody ladder measured: a syntonic comma of disagreement between the leading note taken as a pure third above the dominant and the same note taken as five pure fifths up.

That essay reached a syntonic comma of 21.51 cents by asking what the seventh degree should be. This one reaches a Pythagorean comma of 23.46 by asking whether G♯ is A♭. They are two different commas and they are the same argument, arrived at from the two ends of the same lattice.

The spelling on the page is therefore not an instruction about pitch at all, in either reading. It is a statement about function — this note is the leading note of A, or it is the flattened sixth of C — and the pitch a player takes from it depends on whether they are tuning to a chord or to a line.

Written out, a C–E–G♯ and a C–E–A♭ are two chords on the page and one chord on a modern keyboard — an augmented triad either way — and in meantone they are two different sounds, one of them usable.

Which computation produced the numbers #

The enharmonic gap is 12 × fifth − 8400 cents, where the fifth is the meantone fifth for the stated fraction. That is the whole of it: no table, no lookup, and the straight line in the figures is the arithmetic being linear rather than a fit to anything.

The zero-crossing is 23.46 / (12 × 21.506), computed from the two comma sizes this file already carries rather than entered. The gap at exactly one eleventh is then evaluated and comes out at −0.0014 cents.

The chain-of-fifths position of a written pitch is its letter’s position plus seven per sharp. The claim that two spellings of one pitch class are always twelve or twenty-four fifths apart is an enumeration over every written pitch inside the stated alteration limit, and it is also provable: a chain position n sounds pitch class 7n modulo twelve, seven is invertible modulo twelve, so 7m ≡ 7n forces m ≡ n and the two positions differ by a multiple of twelve. This was previously recorded here as “seven or twelve”, and in the caveat below as fourteen. Seven fifths is the distance from a natural to its own sharp, which is a distance of one semitone and not zero; it is where the number came from and it is not a distance between two spellings of one sound.

The closing lengths are a sweep of chain length from five to sixty for each temperament, keeping the length whose residual against a whole number of octaves is smallest in absolute value. Nothing selects for the historically attested numbers: nineteen and thirty-one come out of third-comma and quarter-comma respectively without being looked for, and fifty-five comes out of sixth-comma with a residual four times worse than any of the others, which is the row that has no instrument.

The matching equal division is read off the same calculation — the number of octaves the chain spanned, over the chain length, times 1200 — rather than taken from the division-scoring essay, so the agreement between the two is a comparison of independent results and not a restatement.

The lesser diesis is quoted as 128/125 and checked against the quarter-comma figure: three pure major thirds are 3 × 386.31 = 1158.94 cents, an octave is 1200, and the difference is 41.06, which is the same number the twelve-fifth calculation gives to two decimal places.

What the picture cannot show #

It cannot show how anybody actually played. Every number here is a property of an idealised tuning system, and the historical evidence for what was done on unfretted instruments and by singers is thin, contradictory and mostly indirect. The meantone figures describe keyboards, which are the instruments that had to decide.

It cannot show the third dimension. Septimal music — anything using the seventh partial — needs a spelling axis this notation does not have, and the usual expedients are a downward arrow or a modified flat. The page is a two-dimensional lattice reading and the lattice has more than two dimensions.

And the closing table treats a temperament as one fifth repeated, which is what a meantone is and is not what most historical keyboards were. A circulating temperament gives different fifths to different parts of the chain, so it has no single fifth to raise to a power and no natural chain length at all — which is the point made about sixth-comma above, arriving from a different direction. The table says what an instrument-builder committed to a regular temperament should do, and the eighteenth century’s answer was to stop being committed to one.

It cannot say what a modern player should do, and the two answers above are not reconcilable by measurement. They are two different criteria, and a performer moving between a chord and a line changes criterion mid-note. This site can say what each criterion implies and cannot say which is right.

It says nothing about fretted instruments, where the frets run straight across and every string gets the same division, so a guitar cannot supply two spellings at all and never could. The instruments that carried this distinction are the ones with a separate mechanism per note.

And it is nearly silent about the double accidentals. F𝄪 and G are the same key and are twelve fifths apart, exactly like G♯ and A♭, so in quarter-comma meantone they differ by the same 41 cents. The larger case needs three spellings of one pitch class: F𝄪, G and A𝄫 span twenty-four fifths, which is two dieses, 82 cents — most of a semitone. The notation admits those spellings and no instrument has ever been built that distinguishes the outer pair.

The ladder from here #

This rung took the page’s one visible redundancy and found a tuning system in it. The next moves to the other axis: the bar line, which is not a redundancy but a claim — an assertion about where the accents are, made by the notation, which the sound may or may not support and which this site has a model that can be asked.

The debt this rung leaves is that the enharmonic gap is computed and never heard. Nothing here plays a sixteenth-century keyboard, and the difference between the two accounts of a leading note is a difference in performance practice that a synthesised organ tone cannot settle.