Acoustics & Music Theory
why it sounds like that
Concept

Open string — where it appears

A string sounded without a finger stopping it, so that its pitch is set by the instrument's tuning rather than by the player. The violin family tunes its open strings in pure fifths, which carries a Pythagorean tuning into every ensemble those instruments play in.

Named by 25 essays across 2 fields — each of them below, with the objects they name alongside it.

The worst interval in each open chord, before and after the best possible compensation. For each shape, the largest departure of any interval from the just interval its name implies, in cents. Before compensation the worst is 15.6 cents; after a search over per-string saddle compensation it is 15.6. The search has six numbers to fix eight shapes with, and on this measure it finds nothing worth changing — because what is being measured is almost entirely equal temperament's own thirds, and a saddle moves a length rather than a temperament.

A guitar cannot be in tune

Three errors land on the same instrument — equal temperament's own thirds, the sharpening that comes from pressing a string down to a fret, and the fact that the only correction available is one length per string. Searching over every setting a luthier could choose leaves a worst-case error of about fifteen cents, and most of what is left is the temperament, which no saddle can reach.

tuning · The comma C4, in every place it can be played. A guitar neck with the 4 places C4 can be stopped, drawn at the fret spacing a 64.8-centimetre scale actually has. The stave writes one note and the tablature writes one of these; each notation says exactly what the other leaves out. The speaking lengths run from 61.2 down to 27.2 centimetres, so a hand plucking 12 centimetres from the bridge meets between 20 and 44 per cent of the string.

What a tablature keeps

Middle C can be stopped in four places on a guitar. The speaking lengths run from 61 to 27 centimetres, so a hand plucking twelve centimetres from the bridge meets between a fifth and nearly a half of the string, and the comb of missing partials is different at every one: the second partial is thirteen decibels stronger in the best position than in the worst. A stave writes one note for all four. A tablature writes four different things and cannot say which note any of them is.

instruments · Notation A string quartet's open strings are five keys of a Pythagorean keyboard. The five pitch classes a string quartet's open strings sound — C, G, D, A and E — laid out as the chain of fifths they are tuned along, outward from the A the ensemble is given, with each fifth pure. The bars give each string's departure from the same note on an equal-tempered keyboard: C −5.87 cents, G −3.91 cents, D −1.96 cents, A 0.00 cents, E +1.96 cents. Above, the strings each instrument owns: the violin G, D, A, E; the viola C, G, D, A; the cello the same four an octave lower. The cello's C2 is 0.221 hertz below the keyboard's, and the widest span of the chain, from the cello's C to the violin's E, is a Pythagorean third and two octaves, 21.5 cents wider than a just one.

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.

tuning · Open strings A tempered fifth is a beat a second on the violin and one in four and a half seconds on the cello. How fast each open fifth of a string quartet beats when it is narrowed by 1.955 cents, the narrowing that meets an equal-tempered keyboard. The beat is the lower string's third partial against the upper string's second, so it is proportional to the lower string's frequency. Cello C2–G2: 0.22 a second, one beat every 4.5 seconds; cello G2–D3: 0.33 a second, one beat every 3.0 seconds; cello D3–A3: 0.50 a second, one beat every 2.0 seconds; viola C3–G3: 0.44 a second, one beat every 2.3 seconds; viola G3–D4: 0.66 a second, one beat every 1.5 seconds; violin G3–D4: 0.66 a second, one beat every 1.5 seconds; violin D4–A4: 1.00 a second, one beat every 1.0 seconds; violin A4–E5: 1.49 a second, one beat every 0.7 seconds. The slowest, the cello's C2–G2, is 6.7 times slower than the violin's A4–E5.

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.

tuning · Open strings An unaccompanied quartet settles where its open strings put it. The average pitch of a quartet correcting toward itself over 480 corrections, in cents from the note it was given, averaged over 24 runs. With no pull from the open strings the ensemble random-walks, and the shaded band is how far: 3.7 cents root-mean-square by the end. With each open string pulling the notes that share its pitch class at a weight of 0.05, the ensemble settles at −0.97 cents in A major, against −1.01 from the open strings' weighted mean; −2.46 cents in C major, against −2.42 from the open strings' weighted mean; −2.53 cents in E♭ major, against −2.54 from the open strings' weighted mean.

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.

tuning · Open strings Four pure fourths and a pure third leave a comma on the top strings. Each of a guitar's six open strings, in cents from the note its own frets make, when the strings are tuned against each other three ways. fretted unisons: E2 +0.00, A2 +0.00, D3 +0.00, G3 +0.00, B3 +0.00, E4 +0.00. harmonics, pure third on G–B: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 −19.55, E4 −21.51. harmonics, B from the low E's twelfth: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 +1.96, E4 +0.00. Tuned with pure fourths and a pure third the high E closes two octaves 21.51 cents flat, a syntonic comma; taking the B from the low E's twelfth lands the high E exactly and moves the error into the third between the G and B strings.

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.

tuning · Open strings One of these tunings has no comma to place. Five guitar tunings, each with every adjacent interval set to the pure ratio nearest the interval it spans, drawn as how far each string then sits from where the frets put it. standard, spanning 5–5–5–4–5 semitones, closes 21.5 cents short and spreads its strings over 21.5 cents; drop D, spanning 7–5–5–4–5 semitones, closes 17.6 cents short and spreads its strings over 19.6 cents; D A D G A D, spanning 7–5–5–2–5 semitones, closes exactly and spreads its strings over 3.9 cents; open G, spanning 5–7–5–4–3 semitones, closes exactly and spreads its strings over 15.6 cents; open D, spanning 7–5–4–3–5 semitones, closes exactly and spreads its strings over 15.6 cents. Standard tuning's four fourths and a third fall a syntonic comma short of two octaves, which is what was found earlier. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four — two octaves — so nothing has to be put anywhere.

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.

tuning · Open strings Three G strings, and they are not one pitch. Each instrument's four open strings, at the Pythagorean position its own chain of fifths puts them, with the uncertainty its own tuning leaves drawn as a band. The A is given and carries no error; every other string is reached from it one fifth at a time, and a fifth set by ear is set by nulling a beat whose rate falls with frequency — so the error accumulates down the chain and is worst at the bottom. violin: G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00, E5 2.0 ± 0.44; viola: C3 -5.9 ± 2.83, G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00; cello: C2 -5.9 ± 2.83, G2 -3.9 ± 1.77, D3 -2.0 ± 0.98, A3 0.0 ± 0.00. The three G strings share a pitch class and are expected to sit 2.5 cents apart; the two C strings 4.0.

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.

tuning · Open strings Which tuning a piece wants, and when the answer is one that has a name. Three short pieces, each a list of fretted chords with durations, scored by how far their intervals sit from just on average — and each played three ways: with every string on its fret, with every adjacent interval of the tuning set pure, and with all six strings free and searched. open G blues costs 5.74 cents on the frets, 0.00 on the pure chain and 0.00 at its own optimum; D A D G A D air costs 1.79 cents on the frets, 0.60 on the pure chain and 0.60 at its own optimum; standard song costs 5.97 cents on the frets, 13.80 on the pure chain and 2.05 at its own optimum. The two pieces whose tuning closes gain nothing from the search: the named tuning already is the optimum, to a thousandth of a cent. The piece in standard tuning, whose chain falls a syntonic comma short, gains 3.92 cents on a tuning that has no name.

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.

tuning · Open strings Counted in beats, the abandoned chord comes back sooner and by steps. The piece in standard tuning with its G major, open chord given more and more of the piece, searched twice: once minimising the mean departure from just in cents, once minimising the mean beat rate. The vertical axis is how far that chord sits from just at each optimum. Counted in cents it stays at 13.29 cents until it holds exactly 44 beats — the same as the rest of the piece — and then drops to zero in one step. Counted in beats it drops at 30.4 beats to 4.96, then at 47.4 beats to 3.91, then at 73.1 beats to 0.00 cents. At the piece's own 8 beats the two counts agree exactly: the corner does not move, and what the beats change is where the steps are.

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.

tuning · Open strings Two shapes asking fifteen pairs of strings for different things. Every pair of the six strings, and the difference in cents between the two strings' offsets that would make the interval each shape puts on that pair exactly just. The piece's chords fall into two families that never disagree with themselves: E major, open and A major barred at 5 (44 beats), and G major, open (8 beats). They disagree on 13 of the 15 pairs: E2–A2 is asked for 1.96 and -13.69; E2–D3 is asked for 0.00 and 1.96; E2–G3 is asked for -13.69 and 0.00; E2–B3 is asked for 1.96 and -13.69; A2–D3 is asked for -1.96 and 15.64; A2–G3 is asked for -15.64 and 13.69; A2–E4 is asked for -1.96 and 13.69; D3–G3 is asked for -13.69 and -1.96; D3–B3 is asked for 1.96 and -15.64; D3–E4 is asked for 0.00 and -1.96; G3–B3 is asked for 15.64 and -13.69; G3–E4 is asked for 13.69 and 0.00; B3–E4 is asked for -1.96 and 13.69. The ring on each row is where the piece's cheapest tuning actually puts the pair. It sits on the first family's demand every time, which is what abandoning the other chord means: no weighting of the error can put a ring on two different places.

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.

tuning · Open strings Tuned for the E shape, a cent of intonation is worth about five frets of hand movement. Every way through a ground on E major — I, IV, I, V, IV, I, for 16, 8, 8, 4, 4 and 8 beats — that is the cheapest route at some exchange rate between the piece's mean departure from just and the frets the hand moves, choosing each chord's fingering from its five moveable shapes; once with the guitar tuned to make the E shape exactly just and once on equal-tempered frets. Tuned for the E shape: 12.09 cents for 0 frets; 4.30 cents for 2 frets (worth it above 0.3 frets a cent); 2.09 cents for 10 frets (worth it above 3.6 frets a cent); 0.00 cents for 20 frets (worth it above 4.8 frets a cent). On the frets: 6.85 cents for 0 frets; 6.15 cents for 2 frets (worth it above 2.8 frets a cent); 5.95 cents for 4 frets (worth it above 10.2 frets a cent); 5.78 cents for 10 frets (worth it above 34.5 frets a cent); 5.60 cents for 20 frets (worth it above 57.5 frets a cent). Tuned for the shape, the route that keeps every chord in it is the cheapest once a cent is worth 4.8 frets; on the frets it needs 57.5.

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.

tuning · Open strings In every key with an open chord, the shape to tune for is that chord's. For a I–IV–I–V–IV–I ground in each of seven major keys, how far the hand moves on the route that keeps every chord exactly just, with the guitar tuned for each of the five moveable shapes, in frets. C: C 20, A 24, G 26, E 28, D 28; D: C 24, A 24, G 20, E 28, D 16; E: C 24, A 20, G 28, E 20, D 24; G: C 24, A 28, G 16, E 24, D 26; A: C 28, A 16, G 24, E 24, D 20; F: C 28, A 28, G 28, E 24, D 24; B♭: C 28, A 24, G 24, E 30, D 28. The keys whose tonic is an open chord — C, D, E, G and A — are served best by the shape of that chord in 5 of 5 cases; F and B♭ have no open tonic, and every shape leaves the hand further from the nut.

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.

tuning · Open strings On a guitar tuned for any major shape, no minor chord is within eight and a half cents of just. Each moveable minor shape — the open E minor, A minor and D minor chords — with its mean departure from just when the guitar is tuned to make one of the five major shapes exactly just, and on equal-tempered frets. E minor shape: 10.30 tuned for C, 10.95 tuned for A, 10.04 tuned for G, 9.78 tuned for E, 10.92 tuned for D; 5.74 on the frets. A minor shape: 10.17 tuned for C, 11.73 tuned for A, 14.47 tuned for G, 8.60 tuned for E, 12.20 tuned for D; 6.65 on the frets. D minor shape: 16.94 tuned for C, 9.78 tuned for A, 10.75 tuned for G, 11.41 tuned for E, 14.66 tuned for D; 8.15 on the frets. The least-bad is the A minor shape on a guitar tuned for the E shape, at 8.60 cents, and every one is worse than the frets.

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.

tuning · Open strings With its silent string placed for the minor chord, the C shape's tuning is the most in tune in every key. The I–vi–IV–V ground twice round in five major keys: the mean departure from just of the most-in-tune route under each major shape's tuning, with the strings that shape does not sound left where the one-shape tuning put them and placed for the minor chord. C: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. D: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. E: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. G: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. A: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. Under the C shape's tuning every major chord is exact and the minor chord is an E minor shape 6.26 cents out, for a quarter of the piece.

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.

tuning · Open strings Tuned for the whole route, the I–vi–ii–V ground wants no shape at all until the hand is allowed to move. The best tuning for the I–vi–ii–V ground twice round, found by routing and retuning in turn from seventeen starts, in five keys and at eight exchange rates between a cent of mean error and a fret of hand movement: what chord shape it makes exact, its cents from just and the frets its route moves. C major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 30 frets); 50, C (3.13 cents, 30 frets). D major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (6.52 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 33 frets); 50, C (3.13 cents, 33 frets). E major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, E (4.60 cents, 15 frets); 12, A minor (4.11 cents, 20 frets); 20, A minor (3.78 cents, 26 frets); 50, C (3.13 cents, 49 frets). G major: 1, nothing (7.71 cents, 0 frets); 2, nothing (7.71 cents, 0 frets); 3, nothing (7.71 cents, 0 frets); 5, E minor (4.71 cents, 9 frets); 8, E minor (4.27 cents, 12 frets); 12, D (3.45 cents, 21 frets); 20, C (3.13 cents, 27 frets); 50, C (3.13 cents, 27 frets). A major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (5.90 cents, 3 frets); 8, C (4.11 cents, 15 frets); 12, C (4.11 cents, 15 frets); 20, C (4.11 cents, 15 frets); 50, C (3.13 cents, 35 frets). At the bottom of the range the best tuning is exact for no chord and the hand does not move; at the top it is one major shape with its silent string placed.

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.

tuning · Open strings The set's tuning is the D shape's on the strings it sounds, and the set's own on the two it does not. The six strings' offsets from equal temperament, low E pinned, of the single tuning found for the I–vi–IV–V ground in five keys at two, five and fifty frets a cent, beside the D shape's placed tuning, dashed. 2: low E +0.0, A +2.0, D +2.0, G +3.9, B +2.0, high E −11.7, exact for D; 5: low E +0.0, A +2.0, D +2.0, G +4.0, B +2.0, high E −11.7, exact for D; 50: low E +0.0, A +13.7, D +0.0, G +15.6, B +13.7, high E +0.0, exact for C. The D shape's placed tuning: low E +0.0, A +2.0, D +8.5, G +10.5, B +8.5, high E −5.2. On the D, G, B and high E strings, measured from the D string, the five-frets tuning and the placed D tuning agree; they differ only on the two strings the D shape leaves silent.

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.

tuning · Open strings However many keys the set spans, its tuning is exact for a shape — the D shape from four keys on. The I–vi–IV–V ground in a set of songs whose keys are added one at a time round the circle of fifths from C — C, G, D, A, E, B, F♯, D♭, A♭, E♭, B♭, F — at five frets a cent: the set's mean cost with each song given its own tuning, with one tuning for the set, and on equal-tempered frets, with the shape the set's tuning is exact for above each point. 1 key: own 5.17, set 5.17 (C), frets 7.60; 2 keys: own 5.27, set 5.27 (C), frets 7.67; 3 keys: own 5.27, set 5.43 (C), frets 7.54; 4 keys: own 5.01, set 5.33 (D), frets 7.39; 5 keys: own 5.14, set 5.36 (D), frets 7.43; 6 keys: own 5.23, set 5.41 (D), frets 7.46; 7 keys: own 5.29, set 5.47 (D), frets 7.48; 8 keys: own 5.34, set 5.49 (D), frets 7.49; 9 keys: own 5.38, set 5.51 (D), frets 7.51; 10 keys: own 5.41, set 5.54 (D), frets 7.52; 11 keys: own 5.43, set 5.55 (D), frets 7.53; 12 keys: own 5.45, set 5.57 (D), frets 7.51. The set never stops sharing a shape: up to three keys it is the C shape's, and from four to all twelve the D shape's.

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.

tuning · Open strings A set in twelve keys keeps its shape through any number of rock songs and loses it to two minor ones. The I–vi–IV–V ground in all twelve keys, with songs outside it added one key at a time — minor-key songs on A, E, B, F♯ and on round the circle, or rock songs on G, D, A, E and on — at five frets a cent: the set's mean cost under its one best tuning, with the shape that tuning is exact for above each point. minor-key songs, i–iv–V–i: 0 added 5.57 (D), 1 added 5.77 (D), 2 added 5.97 (C), 3 added 6.01 (C), 4 added 6.04 (C), 6 added 6.08 (C), 8 added 6.12 (C), 12 added 6.03 (E minor). rock songs, I–♭VII–IV–I: 0 added 5.57 (D), 1 added 5.49 (D), 2 added 5.42 (D), 3 added 5.36 (D), 4 added 5.22 (D), 6 added 5.14 (D), 8 added 5.09 (D), 12 added 4.96 (D).

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.

tuning · Open strings What each minor-key song gains from the C shape's tuning. For a minor-key song, i–iv–V–i, in each of the twelve keys, the saving in cents plus frets over five from playing it under the C shape's tuning rather than the twelve-key ground's own, the D shape's: C minor 1.69, C♯ minor 1.69, D minor 1.69, E♭ minor 1.69, E minor 1.69, F minor 1.69, F♯ minor 1.69, G minor 0.33, A♭ minor 1.69, A minor 1.69, B♭ minor 1.69, B minor 1.61. Switching the tuning costs the twelve ground songs 3.90 between them, so the switch pays when the added songs save more than that: the lines mark a half and a third of it, what each of two or three songs would have to save. Ten keys save 1.69 each — less than half, more than a third. G minor saves 0.33, because under the ground's tuning its dominant is the open D chord, already exact.

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.

tuning · Open strings The minor songs' tonic takes the tuning only at the ninth minor song. The twelve-key ground with minor-key songs added on A, E, B, F♯ and on round the circle: the set's mean cost, in cents plus frets over five, under three fixed tunings — the ground's own (the D shape's), the C shape's (the minor songs' dominant exact) and the E-minor shape's (their tonic exact). 0: 5.567, 5.892, 6.499; 1: 5.767, 5.937, 6.436; 2: 5.938, 5.975, 6.327; 3: 6.081, 6.008, 6.285; 4: 6.211, 6.037, 6.247; 5: 6.326, 6.062, 6.214; 6: 6.428, 6.085, 6.185; 7: 6.520, 6.105, 6.159; 8: 6.602, 6.123, 6.135; 9: 6.638, 6.102, 6.076; 10: 6.707, 6.118, 6.058; 11: 6.711, 6.134, 6.042; 12: 6.772, 6.148, 6.028. The ground's tuning is cheapest to two songs, the C shape's from three to eight, and the E-minor shape's from nine — when minor songs are nine of twenty-one.

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.

tuning · Open strings One retuning is worth what the pop songs pay for the minor tuning, and no more. The twelve-key ground with minor-key songs on A added, and what the evening saves, in cents plus frets over five summed over its songs, if the guitarist may retune once between two songs, against tuning once for the whole set. With the minor songs together after the ground: 1 songs 2.48, 2 songs 4.95, 3 songs 6.25, 5 songs 7.81, 9 songs 10.94, 10 songs 11.18, 14 songs 11.18, 20 songs 11.18. From ten songs on it is 11.18 exactly — what the ground's twelve songs pay for the E-minor shape's tuning, which the one-tuning guitar has by then moved to — and it never grows again. With the same songs spread one at a time through the ground, one retuning saves 1.33 at 3, 1.07 at 5, 2.30 at 10, 1.30 at 20. The dashed line is what any number of retunings could save, every song in its own cheapest tuning: 12.04 from ten songs on.

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.

tuning · Open strings Only three minor keys want a tuning of their own, and each is a key with open strings in its chords. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five: under the E-minor shape's tuning, the best of the one-tuning vote's three for every minor key, and under the tuning a search finds for that song alone. C minor 5.69 and 5.69; G minor 5.69 and 5.69; D minor 5.69 and 5.35; A minor 5.69 and 5.09; E minor 4.91 and 4.91; B minor 5.69 and 4.53; F♯ minor 5.69 and 5.69; C♯ minor 5.69 and 5.69; A♭ minor 5.69 and 5.69; E♭ minor 5.69 and 5.69; B♭ minor 5.69 and 5.69; F minor 4.89 and 4.89. Nine keys keep the E-minor shape's tuning. D, A and B minor each find a tuning of their own and save 0.34, 0.60, 1.16; A minor's makes its tonic and subdominant one shape, the open A-minor chord and the same shape at the fifth fret, both exact.

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.

tuning · Open strings With a capo up to fret 7, 8 of the twelve minor keys are played as B minor in one tuning. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five, under the tuning searched for the song on A (the B string raised 13.7 cents): at the nut, and at the capo position up to fret 7 that suits it best, with the E-minor shape's tuning at its best capo for comparison. C minor 5.72 → 4.53 (capo 1); G minor 5.72 → 5.72; D minor 5.72 → 4.53 (capo 3); A minor 5.09 → 5.09; E minor 5.72 → 4.53 (capo 5); B minor 4.53 → 4.53; F♯ minor 5.72 → 4.53 (capo 7); C♯ minor 5.72 → 4.53 (capo 2); A♭ minor 5.72 → 5.72; E♭ minor 5.72 → 4.53 (capo 4); B♭ minor 5.72 → 5.09 (capo 1); F minor 5.72 → 4.53 (capo 6). 8 keys reach 4.53, B minor's price, by putting the capo where B minor's open-string chords become theirs. E minor costs 4.53 at capo 5, against 4.91 in the E-minor shape's tuning.

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.

tuning · Open strings With the capo left at fret 5, E minor becomes B minor and the other keys pay for it. Twelve minor-key songs, i–iv–V–i, one in each key round the circle, in the raised-B tuning, in cents plus frets over five: at the nut, with the capo left at fret 5 for every song, and at each song's own best capo up to fret 7. C minor 5.72, 8.62 at fret 5, 4.53 at its own; G minor 5.72, 5.72 at fret 5, 5.72 at its own; D minor 5.72, 5.09 at fret 5, 4.53 at its own; A minor 5.09, 6.13 at fret 5, 5.09 at its own; E minor 5.72, 4.53 at fret 5, 4.53 at its own; B minor 4.53, 8.62 at fret 5, 4.53 at its own; F♯ minor 5.72, 5.72 at fret 5, 4.53 at its own; C♯ minor 5.72, 8.62 at fret 5, 4.53 at its own; A♭ minor 5.72, 5.72 at fret 5, 5.72 at its own; E♭ minor 5.72, 5.72 at fret 5, 4.53 at its own; B♭ minor 5.72, 7.82 at fret 5, 5.09 at its own; F minor 5.72, 5.72 at fret 5, 4.53 at its own. The twelve cost 66.82 at the nut, 78.04 with the capo at fret 5 and 57.87 with it moved for every song. At fret 5 the song on E takes B minor's fingerings and the song on D A minor's; every other key is moved to a key the tuning does not favour, on a neck 5 frets shorter.

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.

tuning · Open strings

Named alongside it

The objects these essays reach for when they reach for this one.

IntonationEqual temperamentJust intonationOptimisationVoicingTuning by earChain of fifthsSyntonic commaKey colourBeatingResonanceTransposition

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