Tuning and Temperament Explorer

The arithmetic that will not close

A keyboard makes a promise it cannot keep. Climb by fifths from C, through G, D, A and the rest, and after twelve steps the keys return you to C. Pure intervals say you missed.

A pure fifth is two notes whose frequencies stand as 3 to 2, and an octave is 2 to 1. Twelve fifths multiply out to 531441/4096, a little under 129.75. Seven octaves come to exactly 128. The leftover, 531441/524288, is the Pythagorean comma. Musicians measure such things in cents, a unit Alexander Ellis set out in the nineteenth century: 1200 to the octave, 100 to each equal semitone, found as 1200 times the base-two logarithm of the frequency ratio. On that scale a pure fifth is 701.955 cents, twelve of them make 8423.46, and seven octaves make 8400. The comma is 23.46 cents.

No cleverness with whole numbers removes it, because no power of three is ever a power of two. An instrument with twelve fixed pitches in each octave therefore has to lose 23.46 cents somewhere among its twelve fifths. The history of keyboard tuning is a long argument about where.

There is a second comma, and it concerns thirds. A major triad is a root with a major third and a perfect fifth above it, and the sweetest major third is 5 to 4, or 386.31 cents. Four pure fifths up from C, dropped two octaves, give an E at 81/64, which is 407.82 cents. The difference between the two, 81/80, is the syntonic comma of 21.51 cents. Pure fifths and pure thirds cannot both be had from one chain of notes.

Pythagorean tuning and its wolf

Pythagorean tuning is the plainest answer. It uses nothing but the 3:2 fifth and the octave. Eleven pure fifths in a chain, for example from E flat up to G sharp, supply all twelve notes. The twelfth fifth is never tuned; it is simply what remains between the two ends, and it absorbs the whole comma.

That leftover interval, G sharp to E flat in the usual layout, measures about 678.5 cents instead of 701.96. It is called a wolf, because badly mistuned intervals reminded listeners of a howl. The wolf can be moved by starting the chain on a different note, so a chain built around C puts it between F sharp and D flat, but there is always exactly one. The system is thought to have been widespread, and in music that seldom changes key the wolf rarely sounds. Its real weakness lies elsewhere. Every usable major third is the wide 81/64, a syntonic comma sharp of pure, and a triad built on it is restless.

Pure thirds, and a worse wolf

Once musicians wanted their thirds to sound as settled as their fifths, the wide third became the problem to solve. Quarter-comma meantone solves it directly. Each fifth is narrowed by a quarter of the syntonic comma, so that four of them, less two octaves, land on an exact 5:4. The fifth becomes the fourth root of 5, which is 696.58 cents, a little over five cents flat. A harpsichord tuning manual by Pietro Aron in 1523 contains a possible description of the scheme, and Gioseffo Zarlino set it out in clear mathematical terms in 1571. It was in common use from the early sixteenth century until the early eighteenth, and on church organs well into the nineteenth.

The price is paid at the far side of the circle. Eleven fifths that are each too narrow undershoot seven octaves by a wide margin, and the twelfth interval must make up all of it. In quarter-comma meantone it is 737.64 cents, more than 35 cents wider than a pure fifth and a far uglier wolf than the Pythagorean one. It was generally placed between G sharp and E flat.

Eight of the twelve major thirds are pure. The other four are not thirds at all but diminished fourths, since three pure thirds fall 41.06 cents short of an octave and something has to take up the slack. Music stayed in the keys that avoided them. Some instruments grew split keys, with separate pitches for D sharp and E flat, to push the trouble further away.

Well temperaments and the title Bach chose

A well temperament gives up pure thirds in the common keys in exchange for no wolf anywhere. The term is modeled on the German word wohltemperiert, and one of its first appearances is in the subtitle of Andreas Werckmeister’s Orgelprobe of 1681. It usually means an irregular tuning: the fifths are of different sizes, the narrowest lying among the natural notes, and no key has a very impure interval. Each key has its own pattern of intervals and so its own character.

The temperament now called Werckmeister III shows the method. The fifths C to G, G to D, D to A and B to F sharp are each narrowed by a quarter of a comma. The other eight are pure. Four quarters make one whole comma, so the circle closes exactly and all twelve notes can serve as a tonic. Thomas Young, writing to the Royal Society in 1799, put the aim of such schemes in one phrase: to make the harmony most perfect in those keys which are the most frequently used.

Bach dated the first book of Das Wohltemperirte Clavier 1722: a prelude and a fugue in each of the 24 major and minor keys. A second book followed twenty years later. The title suggests a twelve-note tuning in which every key sounds acceptable. It does not say which one.

For much of the twentieth century it was presumed, possibly mistakenly, that he meant equal temperament. Since then scholars have argued for various unequal schemes, some close to Werckmeister’s, some read out of the loops drawn across the top of the 1722 title page. Nobody knows. A tool can let you play a progression in each candidate and hear how the keys differ; it cannot tell you what was in Bach’s ear.

Equal temperament, the comma spread thin

Equal temperament makes the simplest division of all. Each of the twelve fifths gives up one twelfth of the Pythagorean comma, about 1.955 cents, and becomes exactly 700 cents. Every semitone is the twelfth root of 2. The calculation was worked out long before the practice settled: Zhu Zaiyu published a precise numerical specification in 1584, and Simon Stevin’s manuscript on the same idea dates from about 1605.

The compromise is easy to state. Fifths are very nearly pure. Major thirds are 400 cents, which is 13.69 cents wider than 5:4, in every key alike. Nothing but the octave is in tune, nothing is a wolf, and no key sounds different from any other. What well temperaments offered as color, equal temperament removes on purpose.

When it became the standard depends on what is being counted. It has been called the predominant system of Western music since the eighteenth century, yet meantone organs survived well into the nineteenth. Tuning it accurately by ear took longer still: methods of near modern accuracy date from the first decades of the nineteenth century, and counting beats became common only in its second half. By 1885 Ellis could call it the system at present used throughout Europe.

The pitch all this hangs from was fixed separately. The American Standards Association recommended 440 Hz for the A above middle C in 1936, the same year the radio station WWV added a 440 Hz tone to its broadcast at the request of several music organizations. The International Organization for Standardization took the figure up in 1955 and formalised it in 1975.

What beats tell a tuner

Two tones of nearly the same frequency do not sound like two tones. They sound like one tone that swells and fades, and the number of swells each second is the difference between the two frequencies. A string at 440 Hz against one at 442 Hz beats twice a second.

Wider intervals beat too, through their overtones. A string sounds its fundamental and also partials at whole multiples of it. In a fifth, the third partial of the lower note sits beside the second partial of the upper note. If the fifth is pure the two coincide. If it is tempered they differ slightly and beat, at three times the lower frequency minus twice the upper. For a major third the pair is the fifth partial of the lower note and the fourth of the upper.

Take the triad C4, E4, G4 with A at 440 Hz. In equal temperament the notes are 261.626, 329.628 and 391.995 Hz. Five times the first is 1308.13 and four times the second is 1318.51, so the third beats 10.38 Hz, a fast shimmer. The fifth gives 784.88 against 783.99, under one beat a second. In quarter-comma meantone, with A still at 440, the notes become 263.181, 328.977 and 393.548 Hz. The third now gives 1315.91 against 1315.91 and is still. The fifth gives 789.54 against 787.10, about 2.45 beats a second. The roughness has moved from the third to the fifth.

This is how a tuner works without a meter: set one note from a fork, then count. It is also where the arithmetic stops being the whole truth. The sums assume partials at exact whole multiples and tones that hold steady. Real piano strings are stiff, their partials run slightly sharp, and tuners stretch the octaves to suit. A table of cents describes an ideal instrument, not the one in the room, and it says nothing about singers and string players, who adjust every chord as they go.

Other ways to cut the octave

Twelve is a habit, not a law. Divide the octave into 31 equal steps and the fifth is 696.77 cents, almost the meantone fifth, while the major third is under a cent from pure. Christiaan Huygens wrote about the division in 1691, and in 1951 an organ with 31 notes to the octave was installed in a museum in Haarlem. Fifty-three steps come closer still. Jing Fang observed more than two thousand years ago that 53 pure fifths very nearly equal 31 octaves, and William Holder pointed out in 1694 that the same division holds the pure major third to within 1.4 cents. No keyboard for ten fingers has room for it.

Every one of these systems pays the same debt. The comma does not go away. It can be dropped in one place and avoided, shared among a few fifths so that keys take on characters, or ground so fine that it is everywhere and nowhere. Hearing the choices side by side is the quickest way to understand why the argument lasted four hundred years.