The tuning a string quartet cannot change

A string quartet is the most flexible tuning instrument in common use. No fret, key or hole fixes any stopped note, so every third can be made just, every leading note raised, every chord adjusted as it sounds, and somebody has to pay the comma is an argument about exactly the freedom this gives an ensemble. Four players and sixteen strings, and almost every pitch they produce is a decision made while playing.

Almost. The open strings are not decisions. Once the pegs have been turned, the violin’s G, D, A and E, the viola’s C, G, D and A and the cello’s C, G, D and A an octave lower sound where they were tuned and nowhere else, and every time a player lifts the left hand off a string that pitch comes back. An orchestra is given a note followed the A from the oboe into the room. What happens after the A is the subject here, because the remaining twelve strings are tuned from it by a method that fixes their pitches as firmly as a keyboard fixes its keys.

Four pure fifths from one given A #

A string player tunes a fifth by bowing two open strings together and turning the peg until the beat between them stops. The beat is between the lower string’s third partial and the upper string’s second, and when it stops the fifth is exactly 3:2 — which is the only fifth the method can find, because a pure fifth is the only setting with nothing to hear. A tuner counts beats to set a tempered interval to a rate; a string player listens for no beat at all.

So from the A, the D below is pure, the G below that is pure, the C below that is pure, and the violin’s E above the A is pure. The quartet’s open strings are four consecutive pure fifths — C, G, D, A, E — and twelve fifths and seven octaves is the account of why a chain like that cannot be squared with a keyboard. Each pure fifth is 1.96 cents wider than an equal-tempered one, and the widths accumulate outward from the A: the D is 1.96 cents flat of the keyboard’s D, the G 3.91 flat, the C 5.87 flat, and the E 1.96 sharp.

In hertz the cello’s C2 is at 65.19 against a piano’s 65.41, a fifth of a hertz low; the viola’s C3 and the violin’s G3 are each 0.44 hertz low; the violin’s E5 is 0.75 high. The chain is four fifths long, and its two ends, the cello’s C and the violin’s E, are a Pythagorean major third apart plus three octaves.

Every third two open strings can make is a comma out #

Four fifths up from C is E, and four pure fifths make an interval of 81:64 — 407.8 cents, where a just major third is 386.3. The difference is the syntonic comma, 21.5 cents, and it is the whole of what separates Pythagorean from just intonation.

The chain C–G–D–A–E contains exactly three thirds whose two notes are both open strings: the major third C–E, and the minor thirds A–C and E–G. All three are a syntonic comma from just — the major third wide and the two minor thirds narrow, each by 21.5 cents — and all three are worse than the equal-tempered thirds a piano plays, by 7.8 and 5.9 cents.

That is the result the flexibility hides. An ensemble that can make every stopped third just carries three Pythagorean thirds in its fixed pitches, and a keyboard, the instrument a quartet is supposed to be freer than, has better thirds on the same notes. Where to hide the comma describes four tuning systems as four decisions about where the comma goes. A quartet’s open strings take the Pythagorean decision without anybody deciding it, because the tuning method admits only pure fifths.

Against a keyboard, the discrepancy beats partial by partial #

A string 5.87 cents flat of a piano is not obviously out of tune. The cello’s C and the piano’s C differ by 0.22 hertz at their fundamentals, which is one beat every four and a half seconds and nothing a listener would call a beat. But every partial beats at its own rate, and the k-th partial of each note beats k times as fast.

At the fourth partial the cello’s C beats 0.88 times a second against a piano’s, at the eighth 1.8, and at the sixteenth 3.5. The fundamentals agree to within a fifth of a hertz and the upper spectrum does not agree at all. The viola’s C and the violin’s G, 0.44 hertz out, beat twice as fast at every partial.

Two things follow, and they pull in opposite directions. A cello playing its open C in unison or at the octave with a piano produces a slow shimmer in its upper partials that a careful listener hears as the two instruments not quite agreeing, and no amount of listening to the fundamentals will say why. And the same number, a fifth of a hertz, is small enough at the fundamental that a cellist comparing the two Cs by pitch alone would call them the same note: at 65 hertz it is under half the smallest difference a listener can detect.

The quartet agrees with itself #

The discrepancy is with the keyboard and not inside the quartet. Every open-string pitch class that two instruments share is tuned by the same fifths from the same A, so the viola’s C3 and the cello’s C2 are both 5.87 cents flat of a piano and exactly an octave apart; the violin’s G3 and the viola’s G3 are the same frequency; the cello’s G2 sits exactly an octave under both. Every octave and unison between two open strings in the ensemble is pure.

So a quartet playing alone never meets its own Pythagorean offset as an offset. Its open strings coincide with each other wherever they meet, and the only intervals between them that are not pure are the thirds — which is where the offset surfaces, a comma wide or narrow, and nowhere else. A quartet’s fixed pitches are consistent with each other and inconsistent with every keyboard, and a listener hears the second fact only when a piano joins or when a chord puts two open strings a third apart.

That also means a tempering, if a quartet made one, would have to be made identically by all four players. A cellist who narrowed the lower fifths to meet a piano while the violist tuned pure would put the two instruments’ C strings out of octave with each other, and trade a discrepancy with the keyboard for one inside the ensemble.

Which keys the fixed pitches clash in #

A player never has to take an open string. Every open-string pitch can also be stopped on the string below, and a player who wants a just third takes it stopped. But an open string rings more freely than a stopped one, is easier to play in tune in a hurry and is the only pitch on a string instrument that sounds identically every time, so the question of where taking the open string forces a Pythagorean third is a question about real playing, and it has a precise answer.

A triad forces one when two of its notes a third apart are both open-string pitch classes. Run through the seven diatonic triads of every major key and count.

The census has a shape nobody would guess from the word “open”. The keys with the most open strings are the keys with the most clashes: F, C and G contain all five open-string pitch classes and each has four triads that force a Pythagorean third. D and B♭ contain four and have two each. A contains three — A, D and E — and has none, because none of those three is a third from another. E contains two and has none. The remote keys have none at all, for the uninteresting reason that they barely use the open strings.

Only two keys have a conflicted tonic triad, C major and F major, and only two have a conflicted dominant: F, whose C major chord on the fifth degree contains both C–E and E–G, and B♭, whose F major dominant contains A–C.

G major, the key with every open string #

G major is a key string players reach for readily, and it contains every open string on all three instruments.

The tonic and dominant are clean. G–B–D takes open G and D and a stopped B, so its third is free; D–F♯–A takes open D and A and a stopped F♯. The clashes are all on the secondary triads: the subdominant C–E–G has two Pythagorean thirds if all three notes are open, the supertonic A–C–E has two, the submediant E–G–B has one and the leading-note triad has one.

So G major’s resonance and G major’s clash come from the same place. The key contains every open string, which is what makes it ring, and it contains all three open-string thirds, which is what makes four of its triads unplayable just on open strings. A quartet in G major keeps the tonic and dominant open and takes the subdominant’s E stopped, or its thirds are a comma out.

C major, where the tonic itself clashes #

C major has as many clashing triads as G, and the difference is which ones.

The C major triad on open strings is the cello’s C, the viola’s G and the violin’s E: a chord of three open strings, and the most natural voicing of the tonic a quartet has. Its major third is 407.8 cents, its minor third 294.1, and the E is 21.5 cents above the just E the chord wants — a tonic chord seven cents worse than a piano’s, on the one ensemble that could have played it just. The practical advice this arithmetic stands behind is familiar to string players: take the E stopped in a C major chord.

That is also the surprising half of the census. Keys that had characters found that eighteenth-century key character on keyboards was a measurable difference in the size of a key’s thirds. A quartet has no keyboard temperament, and it has key character anyway, from the other direction: the keys whose chords force its fixed pitches into Pythagorean thirds are C, F and G, and the keys that never do are A and E. What is fixed about a string ensemble’s tuning is not a temperament distributed round twelve keys but a five-note chain, and a key’s character is how much of the chain its harmony runs into.

Minor keys, where the tonic clashes in two more #

The same count run over the minor keys finds the clash in the same neighbourhood and more often at the tonic. In natural minor, A minor, E minor and D minor each have four triads with a third between two open strings; G minor and B minor have two; the rest have none.

The difference is where the clash falls. A minor’s tonic triad, A–C–E, has two open-string thirds, A–C and C–E, so the tonic chord of A minor taken on open strings is a Pythagorean minor third under a Pythagorean major third. E minor’s tonic, E–G–B, has one. D minor’s tonic, D–F–A, has none, since F is not an open string, and its clashes are all on the chords round it. With a raised seventh the picture moves only at the dominant: F minor’s dominant becomes the C major chord and acquires both of C major’s clashes, and B♭ minor’s becomes F major and acquires A–C.

So the four keys whose tonic chord on open strings is Pythagorean are C major, F major, A minor and E minor. C major and A minor share a key signature and so do their clashes; F major’s relative, D minor, is clean at its tonic, and E minor’s, G major, is clean at its tonic too. The conflict is not a property of a signature. It is a property of which pitch classes the tonic chord puts a third apart.

The arithmetic under the chain #

The pitches are four pure fifths and the octaves the strings sound in, measured against equal temperament at A = 440. The A is taken as exact, which is generous to the chain: who listens to whom when an orchestra tunes put the error of one match at about two cents, and every string inherits it. The thirds are differences of positions on the chain, and the census is a count over the diatonic triads of each major key with no weighting for how often a triad is used.

Nothing here assumes how players actually play. The census says where an open string forces a Pythagorean third, not where one is taken; the beats say how a string tuned by pure fifths differs from a keyboard, not whether a player leaves it that way.

What the chain on the page leaves out #

Players do not always tune pure fifths. A cellist who knows a piano is coming may narrow the lower fifths to meet it, and whether that narrowing can be set by ear on the cello’s lowest strings is a separate and less comfortable question. Everything above describes the tuning the method produces when it is applied as taught.

Strings do not stay put. A gut or synthetic string flattens as it warms under the bow, a steel one differently, and a standard is a point and a performance is a band priced the drift a room’s temperature gives a string. Over a movement the chain’s pure fifths become approximate fifths, and the drawing is of the moment after tuning.

An open string does more than sound. It rings in sympathy with any stopped note that shares its partials, and that resonance is audible to a player as a reward for playing a note in tune with it — which pulls stopped notes toward the Pythagorean positions of the open strings. None of that is in the census, which counts only notes taken open.

And the census counts triads, not music. A piece in C major that avoids the subdominant, or voices every tonic chord with the E on top stopped, never meets the clash. What the count measures is how much harmony in a key is exposed to it.

Whose ensemble #

The ensemble is the string quartet and, by extension, any bowed-string group tuned in fifths from a given A — orchestral strings included, whose open strings are the same five pitch classes. It is not a guitar, whose open strings are mostly fourths and whose frets fix everything else, and a guitar cannot be in tune is about that different problem. It is also not a viol consort, tuned mostly in fourths with a third in the middle, whose open strings would make a different chain with a just third already in it.

Still open: whether the chain can be tempered by ear #

The obvious repair is to narrow every fifth by the 1.96 cents that would meet a keyboard, and the method that tuned the fifths pure is also the method that would have to narrow them — by listening for a slow beat instead of for none. A narrowed fifth beats at its lower string’s third partial times the narrowing, so the same 1.96 cents is a faster beat on the violin’s A–E than on the cello’s C–G, by the ratio of their frequencies: six and two-thirds. Whether a cellist can hear a beat that slow in the time a bow lasts, how the error of each fifth adds up down the chain to the C, and whether what arrives at the bottom of the quartet is more or less wrong than the Pythagorean string it replaced, is arithmetic on the beat rates and the chain.