The chain of fifths in quarter-comma meantone
Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.
It makes a noise. 185 of its 195 placements carry sound, built from the same numbers as the drawing, offering these buttons: A good fifth, A ground song under the C shape's tuning, the one the evening starts in, A ground song under the E-minor shape's tuning, where a set that cannot retune between its runs leaves it, A ground song under the ground's tuning, the one the evening starts in, A major barred at 5 at the piece's optimum: just, to the search's own grid, A major on twelve keys, A major, tuned for the A shape, every chord in it, A minor at the nut, the first of a pair a tone apart that can share a fret and 209 more.
Called by 32 essays
the blast radius of changing it
The wolf at the end of the chain
Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.
A tuner counts beats, and that is the whole method
Laying a temperament sounds like the most subjective job in music, and it is arithmetic. Every interval in the bearing octave has a target rate in beats per second, the rate follows from the ratio and the register, and a tuner who hits the numbers has produced equal temperament without ever thinking about a cent.
A fraction of a comma
The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.
The same distance, under two names
Four hundred cents is a major third or a diminished fourth, and on a keyboard nothing in the sound distinguishes them. An earlier essay was about the boundary between two categories; this is about two categories at one acoustic value, and the surprise is where the ambiguity comes from. In quarter-comma meantone a major third is 386 cents and a diminished fourth is 427 — two names, two pitches, forty-one cents apart. Equal temperament collapsed them, and what a listener now supplies from context used to be in the sound.
Who listens to whom when an orchestra tunes
One match is good to two cents and an orchestra is sixty of them, arranged in an order that nobody chose deliberately. Passed along a line, the errors accumulate and the last player is fifteen cents from the first; given to everybody at once, nobody is more than two. The convention every orchestra uses — principals to the oboe, sections to their principals — is two matches deep, costs 2.8 cents whether there are four players or a hundred, and sits within a cent of the best arrangement that exists.
The tuning a string quartet cannot change
A string quartet can put every stopped note wherever it likes, and it has five pitches it cannot move once the pegs are turned: the open strings C, G, D, A and E, tuned in pure fifths from the A it was given. That makes its only fixed tuning a five-key Pythagorean keyboard — the cello's C nearly six cents below a piano's, and every third two open strings can make a syntonic comma from just. Counted key by key, the clash is worst in G, C and F, the keys that use every open string, and absent from A and E.
The cello cannot hear its own tempering
Narrowing a quartet's fifths to meet a piano is one number, 1.96 cents a fifth, and it is a different beat on every string: once every two thirds of a second on the violin's A–E and once every four and a half seconds on the cello's C–G. Set by ear for two seconds a fifth, the violin's E lands within two thirds of a cent and the cello's C within 5.7 — which is as large as the Pythagorean error the tempering was meant to remove. The string whose tuning is most wrong is the string whose tuning is least certain, and a cellist tuning down the chain cannot tell pure from tempered.
An open string pulls the quartet flat
Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.
A guitar tuned by harmonics hides a comma
A quartet's stopped notes go wherever its players put them, so its open strings can sit on a pure chain. A guitar's stopped notes are fixed by frets that are already an equal temperament, and its six open strings — four fourths and a third — close two octaves exactly a syntonic comma short when tuned pure, since (4/3)⁴·(5/4) is 4 × 80/81. Tuning by harmonics decides where the comma goes, not whether: a pure third puts it in the octaves of an open E major chord, 17.6 cents narrow; a B from the low E's twelfth puts it in the G–B third, 21.5 cents wide.
One tuning has no comma to place
Standard guitar tuning is four fourths and a third, and tuned pure it closes two octaves exactly a syntonic comma short — so tuning by ear decides where the comma goes rather than whether. An open tuning replaces the interval list. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four, so its chain closes and there is nothing to place: its six strings sit within four cents of the frets. The open-chord tunings close too, and pay for it with one string fifteen cents flat — which every chord barred straight across inherits exactly, and which is why every such chord is precisely as pure as the open one.
The quartet settles at two pitches, not four
The quartet's open strings have been treated as five fixed pitches on one chain, and they are not: the violin, the viola and the cello each tuned a G string by ear and the three are expected to sit two and a half cents apart. Giving each player their own strings, with their own scatter, and pulling each toward only their own, changes the ensemble's settled pitch by a hundredth of a cent. What it does change is systematic rather than random: a violin has an E string and no C, the lower instruments have a C and no E, so the quartet splits by section by a tenth of a cent in every key.
A tuning is right for some chords and wrong for the rest
An earlier essay asked which tuning minimises a piece's total departure from just, and guessed the answer would be a few cents off a named one. Searched over all six strings, two of three pieces get back the tuning they were already in — because their chains close and there is nothing to improve. The third lands on a tuning nobody names, three and nine tenths of a cent better than anything a player would have tried, and it is not a compromise: two of its three chords are exactly just and the third is abandoned by thirteen cents.
Counting beats moves the price of a chord, not the tuning
The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.
The chord a tuning gives up is a fingering
A guitar piece's best tuning makes two of its chords exactly just and abandons the third by thirteen cents, and no way of counting error — cents, beats, squared cents, or only the beats slow enough to hear — brings the third chord back. Fingered as a barre chord in the shape the other two already use, it comes back completely, and the whole piece is exactly just. The conflict was never between chords. It was between hands.
Tuned for one shape, a cent is worth five frets
A guitar can be tuned so that one chord shape is exactly just wherever it is moved on the neck, and every other shape is then between twelve and fifteen cents out — worse than on the frets. So a player of such a tuning chooses, chord by chord, between intonation and the distance the hand travels. Searched exactly over every fingering of a I–IV–V ground, the route that keeps every chord in the one shape is the cheapest as soon as a cent of the piece's mean error is worth 4.8 frets of movement. On equal-tempered frets the same route needs a cent to be worth fifty-seven.
Tune a guitar for the shape of its key's open chord
A guitar can be tuned for any of five moveable chord shapes, and a player who then keeps every chord in that shape plays exactly in tune while the hand travels. Which shape should the tuning serve? Searched over seven keys and all five shapes, in every key whose tonic is an open chord the answer is that chord's own shape — C for C, D for D, G for G, A for A, and a tie between the E and A shapes in E. The reason is not the key and not the progression. It is that an open chord needs no barre: its hand sits one or two frets above the nut, and that saving is the whole of the difference.
A tuned guitar has no minor chord in tune
A guitar tuned to make one major chord shape exactly just can play every major chord in tune by moving that shape. It cannot play a minor chord in tune anywhere: every minor shape disagrees with every major shape about some pair of strings by eleven cents or more, and on any of the five tunings the least-bad minor chord — the A minor shape on a guitar tuned for the E shape — is 8.60 cents out, worse than on the frets. Put one minor chord into a I–vi–IV–V ground and the most-in-tune tuning in every key becomes the E shape's, because it is the one that hosts that chord least badly.
A silent string tunes the minor chord
A continuous search over a guitar's six strings, run on a piece with one minor chord in it, never settles between two chord shapes: at any share of the piece, the best tuning is exact for one chord and makes the other pay. But the tunings the earlier essays called exact for a shape were exact only on the strings that shape sounds. The C shape never sounds its low E string, and tuned to the C chord's own E it hosts an E minor chord at 6.26 cents — not the 8.60 that was said to be the best any tuned guitar could do. The pop ground's most in-tune tuning is then the C shape's, in every key, at 1.57 cents.
A guitar tuned for a still hand is tuned for nothing
Tune a guitar for a whole route through a ground rather than for one chord shape — routing, retuning and routing again until neither changes — and the answer depends on how much the player is willing to move. Valuing intonation highly, the best tuning is one major shape with its silent string placed, as before. In the middle it can be exact for a minor shape instead, because the route that keeps the hand near the nut plays its minor chords open. And for a player who will not move at all, the best tuning is exact for no chord whatever: every string within two cents of its fret, beating equal temperament by a few tenths of a cent. It is the compromise the earlier search went looking for, and it lives where nobody looked.
A guitar tuned for a set is tuned for the D shape
Tune a guitar once for an evening of songs in five keys rather than once per song, and the tuning the set wants is not the compromise the single-song search predicted. At the valuation a typical player puts on a cent, it is exact for the D shape — the one fingering every key's route uses for some chord — with that shape's two silent strings placed for the set rather than for any song. Not retuning between songs costs a fifth of a cent. Only a set that mixes two grounds, played by a player who will not move, wants a tuning exact for nothing.
One guitar tuning serves every key, and it is the D shape's
A guitar tuned once for a set of pop songs in five keys is tuned exact for the D shape. The open question was the converse: which songs a given tuning serves, and how many keys a set can span before its best tuning stops being any shape's. Grown one key at a time round the circle of fifths, the set never stops sharing a shape: up to three keys it is the C shape's tuning, and from four keys to all twelve it is the D shape's, with the same offsets to a fifth of a cent. The five-key tuning serves the seven keys outside the set better than the two open keys inside it, and not retuning between songs costs less the more keys the evening spans.
Two minor songs retune a guitar set for their dominant
A guitar tuned once for an evening of pop songs in every key is tuned exact for the D shape, because every key's route through the ground passes through it. Add songs that are not that ground and the tuning holds or breaks by a rule nobody would guess. Rock songs on I, ♭VII and IV never move it off the D shape, however many are added: they retune only the two strings the D shape leaves silent. One minor-key song leaves it alone too. Two move it to the C shape's tuning, and the chord they move it for is not one of their minor chords but their one major chord — the dominant, fingered in the C shape. Twelve move it again, to the E-minor shape's.
A guitar set keeps its tuning until the third minor song
Two minor-key songs were found to move a pop set's guitar tuning from the D shape to the C shape. Priced as a vote — what the move costs the twelve ground songs, against what it saves each minor song — two songs do not pay for it and three do. The earlier search had never tried keeping the tuning the set already had. And the key a minor song is in matters only at the nut: ten keys vote alike, and G minor, whose dominant is the open D chord, hardly votes at all.
A guitar set tunes for its minor chords only when nearly half its songs are minor
Add a third candidate to the vote — the tuning that makes the minor songs' own tonic chord exact — and a pop set's guitar passes through three tunings as minor songs are added: the ground's to the second, the dominant's from the third, and the tonic's only from the tenth, when ten songs in twenty-two are minor. Nothing in between is ever chosen: the two minor-song tunings differ on two strings, and every compromise between them costs the ground more than either end. Two keys break the pattern at the nut, and one earlier figure turns out to be wrong by a song.
One retuning is worth what the pop songs pay for the minor tuning
A guitarist allowed to retune once in an evening plays the pop songs in their own tuning and the minor songs in theirs. What that saves rises with every minor song until the tenth, and then stops: from there on it is exactly what the twelve pop songs would have paid to be played in the minor songs' tuning, and adding minor songs cannot raise it. The C shape's tuning, which a one-tuning guitar spends most of the range in, is never used. And the whole saving belongs to the running order. Put the same ten minor songs in two runs instead of one and a single retuning is worth three-fifths as much; spread them through the evening and it is worth a fifth.
The minor half's own tuning belongs to one key
A guitarist who retunes once between the pop songs and the minor songs has been choosing among three tunings found for whole evenings. Searched for the minor songs alone, the best tuning is none of them — but only in A, B and D minor, where it gives the tonic and subdominant one shape at the nut and both are in tune. In A it saves six-tenths a song, and the refund a retuning buys stops having a ceiling. In the other nine keys, and in any minor half that mixes keys, the E-minor shape's tuning was already the answer.
A capo is a transposition the tuning cannot see
Tempered frets make a chord shape's error the same wherever it sits, so a song with a capo at the fifth fret is, to the tuning, the song five semitones lower on a shorter neck. That lets every minor key borrow B minor's open-string fingerings, and eight of the twelve take the raised-B tuning's best price with a capo no higher than the seventh fret. A minor half in mixed keys, which gave a retuning nothing, now nearly doubles its refund — but the capo pays the single-tuned evening as well, and a minor half all in A gets less from its retuning than before.
A capo left in place serves one key
Moved for every song, a capo lets nearly every minor key borrow B minor's open-string fingerings. Left at one fret for a whole run of minor songs, as most guitarists leave it, it does something much weaker: it renames the keys. Over a run in every key that is a reshuffle on a shorter neck and can never beat the nut; over the four keys guitarists use most it gains nothing either, because two of them are already the tuning's best. A fixed capo pays only a run in very few keys, spaced the way the tuning's cheapest keys are.
A set ordered by key relation moves the capo most
Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.
A set steps by fifths where the capo moves
A set of twelve minor songs in E, A, D and B can be ordered for the capo or for key relation, and the two orders pull apart. Searched over every one of its 41,304 orders, the trade between them is exact and simple: a set can put a fifth between neighbouring songs at each capo change for nothing, because a fifth always crosses frets and a capo change is where crossing is paid for. Beyond that each extra fifth costs about seven points of the capo's saving. The tone steps the earlier essay suspected of carrying the compromise are where the capo stays still.
A set free to change key becomes a set in two
A set list orders its songs for the music and a capo wants them ordered by fret, and the trade between the two looked as if it might disappear if a singer would move a song a tone. It does not. Searched exactly, moving songs adds five to twenty points of the capo's saving at any count of fifths and stops improving at six moved; at its limit the best set rocks between B minor and E minor, because a count of fifths rewards a pendulum. A set that really walks the chain of fifths has three to spend, not eleven.
A split key buys one of everything, and two buy C and E
An organ builder with money for two extra keys could have put them anywhere, and the arithmetic of triads cannot say where. On a meantone keyboard every added key buys exactly one major triad, one minor triad, one major key and one minor key, wherever it goes. So the count is no guide, and the choice was about which keys. The two accidentals most often divided, G♯/A♭ and D♯/E♭, are one on each side of the chain, and they are exactly what completes the minor keys on C and E. What they buy by the roughness sum is seven per cent.