Acoustics & Music Theory
why it sounds like that
Pitch and tuning

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.
14 min read 5 figures The comma that will not closeThe same shape, movedThe design was forced

Assumes: Counting beats moves the price of a chord, not the tuning · One tuning has no comma to place

Two searches have now asked which tuning of a guitar’s six strings brings a short piece in standard tuning closest to just intonation. The first counted cents and found a tuning that makes the E major and A major chords exactly just and leaves the G major chord 13.29 cents out. The second counted beats and got the same six offsets back, and noticed something in the bookkeeping: at every corner of the cost surface the search can stop at, the E and A chords’ mean error plus the G chord’s is exactly 13.29 cents. Whatever the objective, the search is dividing one fixed quantity between two groups of chords.

A quantity that survives every change of objective is a property of the problem, not of the objective, and it should be possible to point at it. It is possible, and what it turns out to be is not a comma, not a temperament and not a chord. It is a difference between two ways of putting a hand on the fingerboard.

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.
Fig. 1 Each of the fifteen pairs of strings, and the difference between their two offsets that would make the interval on that pair exactly just — once as asked by the E major and barred A major chords, once as asked by the open G major chord. The two disagree on thirteen of the fifteen pairs. The ring on each row is where the piece’s cheapest tuning puts the pair, and it sits on the first group’s demand every time.

What a tuning actually fixes

A tuning fixes six numbers, and nothing a chord does can change them. But no chord hears a string’s offset on its own. What a chord hears is the difference between two strings’ offsets, because every interval it sounds lies between two strings — and so what a tuning fixes, from the point of view of the music, is fifteen differences, one for each pair of strings, of which only five are free.

A chord shape puts a particular interval on each pair of strings it sounds. The open E major chord puts a major third plus two octaves between the bottom string and the G string, because the G string is stopped at its first fret and sounds G♯3 over E2. To make that interval exactly just — 5:1 — the G string’s offset must sit 13.69 cents below the bottom string’s. The open G major chord puts an octave on the same two strings, G2 at the bottom string’s third fret against G3 on the open G string, and an octave is just when the two offsets are equal. There is no tuning that does both. Thirteen point six nine cents on that pair of strings belongs to one chord or the other, and somebody has to pay in exactly the sense a keyboard temperament does.

The figure above is that comparison for every pair. The E chord and the barred A chord, which are the same fingering five frets apart, never disagree with each other: on every pair both of them sound, they ask for the same difference. The open G chord disagrees with them on thirteen of the fifteen pairs, by amounts from 1.96 to 29.33 cents depending on which intervals collide. The two pairs they agree on are the ones where both shapes happen to place an octave or a double octave, which just intonation and the frets agree about.

That is the fixed quantity. The piece’s cheapest tuning satisfies every demand of the E-and-A group exactly — the rings sit on the round dots — and every demand of the G chord that disagrees with them is left unmet. The search was never choosing a compromise; it was choosing which of two sets of fifteen numbers to obey.

No way of counting gets past it

If the conflict is between fixed demands, then the objective can only decide how the unmet part is shared out, never whether there is one. The prediction is testable, because some objectives share error out and some concentrate it.

Every way of counting gives the same chord away. The piece in standard tuning searched four times, each time minimising a different measure of how far its intervals are from just: the mean departure in cents, the mean beat rate, the mean squared departure, and the mean beat rate less one beat every 2 seconds, so that a beat too slow to be heard while the note lasts costs nothing. For each optimum, how far each chord then sits from just, in cents. cents: E major, open 0.00, A major barred at 5 0.00, G major, open 13.29; beats: E major, open 0.00, A major barred at 5 0.00, G major, open 13.29; squared cents: E major, open 2.05, A major barred at 5 2.05, G major, open 11.25; audible beats: E major, open 1.18, A major barred at 5 1.18, G major, open 12.84. Squaring shares the error and the free zone relaxes the other chords by a cent, and neither brings G major, open within ten cents of just.
Fig. 2 The piece searched under four objectives: the mean departure in cents, the mean beat rate, the mean squared departure, and the mean beat rate with beats slower than one every two seconds counted as free. Under each optimum, the bars are how far each chord sits from just, in cents. Every objective leaves the G chord more than ten cents out.

Counted in cents, the G chord is at 13.29 and the other two are exactly just. Counted in beats, the result is identical, for the reason the previous essay gave: a beat rate is a cent times a positive weight, and reweighting does not move a corner.

Squared cents are the objective that ought to behave differently, because a square penalises one large error far more than several small ones and so the minimum of a sum of squares sits at a weighted average rather than at a corner. It does behave differently. The E and A chords come off exact just intonation and sit at 2.05 cents each, and the G chord comes down to 11.25. That is the whole of what a least-squares objective can buy: two cents of error handed to forty-four beats of music in exchange for two cents taken off eight. The G chord is still more than eleven cents from just, because squares divide the conflict in proportion to how much music sits on each side of it and one side holds five and a half times as much.

The fourth objective is the one designed to rescue a chord if anything could. A beat that does not complete a cycle while a note is sounding is not a beat anyone hears, so a departure small enough to beat slower than one cycle in two seconds is given away free. That builds a dead zone into every interval, several cents wide on a low one and under a cent on a high one, inside which any tuning costs nothing. The optimum uses it — and uses it on the wrong chords. The E and A chords drift 1.18 cents off just into their free zones, and the G chord goes to 12.84.

A shorter note does not help the chord that needs help

The dead zone’s width is a parameter, and the obvious objection to one value of it is that a strummed guitar note is loud for well under two seconds. So it can be swept.

A free zone for slow beats relaxes the chords that were already in tune. The piece in standard tuning searched with a cost that ignores any beat slower than one per window, for windows of 0.25, 0.5, 1, 2, 4, 8 seconds: a stand-in for how long a strummed note is loud enough for a beat to complete. How far each chord sits from just at each optimum: 0.25 s — E major, open 8.34, A major barred at 5 8.34, G major, open 11.37; 0.5 s — E major, open 4.03, A major barred at 5 4.03, G major, open 11.69; 1 s — E major, open 2.24, A major barred at 5 2.24, G major, open 12.54; 2 s — E major, open 1.18, A major barred at 5 1.18, G major, open 12.84; 4 s — E major, open 0.59, A major barred at 5 0.59, G major, open 13.06; 8 s — E major, open 0.20, A major barred at 5 0.20, G major, open 13.16. A short note opens a zone several cents wide, and the search spends it on the two chords that were exactly just, letting them drift, while G major, open stays more than ten cents out.
Fig. 3 The piece searched with a free zone for beats slower than one cycle per window, for windows from a quarter of a second to eight seconds. The lines are how far each chord sits from just at each optimum. As the window shortens, the E and A chords drift further from just and the G chord barely moves, staying between 11.4 and 13.2 cents throughout.

At an eight-second window the zone is a fraction of a cent wide and the optimum is almost exactly the cents optimum: the E and A chords at 0.20, the G chord at 13.16. At a two-second window, 1.18 and 12.84. At one second, 2.24 and 12.54. At a quarter of a second — a pluck heard for a single beat of a brisk tempo — the E and A chords sit 8.34 cents from just and the G chord 11.37.

The shape is the conflict again, seen through a different lens. Widening the free zone does not reduce the thirteen point six nine cents on the bottom and G strings; it only lets the side that holds the tuning give some of it back for nothing. Because the side holding the tuning is the E-and-A group, the freedom lands on the chords that did not need it. By the time the free zone is wide enough to matter to the G chord, it is wide enough that nearly everything is free and the search has almost nothing to decide.

Moving one note does not dissolve it either

If the conflict is between two fingerings, the natural repair is to change the one that disagrees. A guitarist has several ways to play G major, and each changes which intervals sit on which pairs of strings.

The chord comes back when it changes shape, not when it changes weight. The piece in standard tuning with its G major, open chord fingered 5 ways and the whole search re-run for each. For each voicing: the piece's cost on the frets, its cost at its own optimum, and how far G major, open then sits from just. open, as written (3 2 0 0 0 3): 5.97 cents on the frets, 2.046 at the optimum, the chord itself 13.29; open, top string on 3 (3 2 0 0 3 3): 5.60 cents on the frets, 1.565 at the optimum, the chord itself 10.17; open, G string muted (3 2 0 x 0 3): 6.13 cents on the frets, 1.565 at the optimum, the chord itself 10.17; open, A string muted (3 x 0 0 0 3): 5.71 cents on the frets, 1.866 at the optimum, the chord itself 12.12; barred at 3, the E shape (3 5 5 4 3 3): 5.60 cents on the frets, 0.001 at the optimum, the chord itself 0.00. Muting a string or moving one note inside the open shape leaves the conflict in place. Barred as the E shape at the third fret — the shape the other two chords already are — the chord asks every pair of strings for what they ask, and one tuning makes the whole piece exactly just.
Fig. 4 The piece re-searched with its G chord fingered five ways. The upper bar is the whole piece’s error on the frets; the lower bar is its error at the tuning that suits it best; the right-hand label is how far the G chord itself then sits from just. Four open voicings leave the piece between 1.6 and 2.1 cents out. Barred as the E shape at the third fret, the G chord and the whole piece are exactly just.

The open shape with the top two strings both stopped at the third fret, which many players use for a fuller top, costs the piece 1.565 cents at its optimum and leaves the G chord 10.17 out. Muting the G string, which removes the octave the earlier essay singled out, costs exactly the same 1.565 and leaves the chord at the same 10.17 — because the octave was one of thirteen disagreements, and removing it removes one row of the figure at the top of the page, not the conflict. Muting the A string instead, which takes out the lower of the two Bs in the fastest-beating sixths, costs 1.866 and leaves the chord at 12.12.

Barred at the third fret in the E shape, the G chord asks every pair of strings for exactly what the E chord asks, because it is the E chord moved up three frets. The search returns the same six offsets, and the piece costs 0.001 cents — zero, to the grid the search works on. Every chord in it is exactly just.

This is the result one tuning has no comma to place found about open tunings, turned into a general rule. That essay observed that a chord barred straight across an open tuning is exactly as pure as the open chord it copies, and treated it as a fact about barre chords. It is a fact about shapes: a tuning makes a shape just or not, and everywhere a shape is moved without changing its fingering, it keeps whatever the tuning gave it.

Every open shape is its own family

That rule makes a strong prediction about ordinary guitar playing, where a player’s chords are mostly open shapes that are not transpositions of one another.

Every open chord shape asks the strings for something different. Ten chord shapes in standard tuning — eight open shapes and two barre chords — compared two at a time. Each cell is the largest disagreement, in cents, between what the two shapes ask of any pair of strings both of them sound, and the small number is how many pairs they share. All 28 pairs of the eight open shapes disagree, by between 11.7 and 29.3 cents. The only exact agreements are the barre chords with the shapes they are moved from: F barred at the first fret with E, and B minor barred at the second with A minor. A player who uses all eight open shapes equally is served best by G3 at +1.96 cents with every other string on its fret, at 7.096 cents from just against 7.193 on the frets — the lowest of all 42568 corners of that surface.
Fig. 5 Ten chord shapes in standard tuning compared two at a time: each cell is the largest disagreement between what the two shapes ask of any pair of strings both of them sound, with the number of shared pairs beneath it. All twenty-eight pairs of the eight open shapes disagree, by 11.7 to 29.3 cents. The only exact agreements are F barred at the first fret with E, and B minor barred at the second with A minor.

Every pair of the eight commonest open shapes — E, E minor, A, A minor, D, D minor, C and G — disagrees about at least one pair of strings, and the smallest worst disagreement is 11.7 cents, between C and E minor. The largest is 29.33, which is a just major third and a just minor third placed on the same two strings by two shapes a fret apart: E major and E minor differ only in the G string’s fret, so they ask that string for a major third above the bottom string in one case and a minor third in the other, and the just major third sits 13.69 cents below its fretted version while the just minor third sits 15.64 cents above it. The only cells that read zero are the barre chords against the open shapes they are moved from.

So a tuning can make at most one family of shapes exactly just, and every open chord shape is a family of one. The consequence for a player of all eight is severe and a little funny. Weighted equally, the eight open shapes have 42,568 distinct corners on their cost surface, and the cheapest of them all is equal temperament with the G string raised 1.96 cents — the size of the gap between a pure and a tempered fifth — at 7.096 cents from just against 7.193 on the frets. The best tuning for a player of open chords is, to within a tenth of a cent, the one the frets already give — which is why a guitar cannot be in tune, measured from the other side.

That is the other face of the finding the earlier searches made. A piece built on one shape moved about the neck can be tuned to perfect intonation, and a piece built on two shapes can be made perfect for whichever holds more of the clock. A piece built on eight shapes has eight sets of fifteen demands, all disagreeing, and the minimum over their corners is the democratic answer — equal temperament — not because anybody designed the frets to be optimal but because an equal division of the error is what many mutually contradictory demands average to. It is the same reason a temperament cannot do what a single just tuning does for a single key, arrived at by a guitarist’s hand rather than a keyboard’s keys.

Which computation produced the numbers

The piece, its durations and the ten chord shapes are stated rather than transcribed. The shapes are the standard open fingerings any guitar method teaches, and the piece is the one the two earlier essays searched.

A shape’s demand on a pair of strings is the offset difference that makes the interval between them just. The interval is named by the frets, so a stopped note’s span is counted in semitones, reduced to a pitch class, matched to the five-limit ratio for that class, and the demand is that ratio’s size in cents less the equal-tempered span. Two shapes agree on a pair if their demands are equal to a millionth of a cent. The search under each objective is the coordinate descent the earlier essays used, with the cents optimum offered as a start. The minimum over the open shapes is taken from the full list of corners of the cents cost surface — every point where five of its forty-eight just-interval planes meet, pinned at the bottom string — so it is the global minimum rather than the best of several descents.

The squared-cents objective minimises the mean of squared departures and reports its result in plain cents for comparison. The audible-beat objective subtracts one beat per window from every interval’s beat rate and floors the result at zero.

What the arithmetic leaves out

Partial strength. As in the earlier essays, every interval’s error counts regardless of how strong the partials that carry it are. That changes how a conflict is divided and cannot change whether there is one, since the demands are fixed by the intervals, not by the spectrum.

Intonation by the fretting hand. A guitarist bends strings a few cents by pressure and lateral push, and a player of a G chord can pull its B up or its G down. That is a way to satisfy two families at once, and it is the same precision problem a cellist setting fifths by ear faces — but only on stopped notes, since an open string cannot be bent, and the open G string is exactly the note the G shape needs moved.

Five-limit targets. Each interval is scored against its five-limit just ratio. A minor seventh has two plausible just forms, and a tritone several; none of the shapes here contain one, but a piece with seventh chords would need the choice made.

What no search can say about a hand

Whether a player would choose the barre. The barred G chord makes the piece exactly just, and it is a harder chord to play, with a different sound: every note is stopped, so nothing rings on past the hand, and the B and D that the open shape leaves sounding on open strings are pressed down inside the barre. A guitarist choosing the open shape is choosing that sound, and the thirteen cents come with it.

Whether anyone hears the conflict. Equal temperament puts a player of open chords 7.19 cents from just on average and nobody plays open chords as if they were out of tune. A comma under the threshold is the nearest this collection comes to saying how large a departure has to be before a listener objects, and the answer there depended on what else was sounding.

Whether tunings were ever chosen this way. The open tunings that make a chord are played mostly as one shape moved about, and on the arithmetic here they are exactly the tunings where a search has something to find. Whether that is why their players like them, or a coincidence of the same whole numbers, is a question about players.

Still open: a piece that changes shape to stay in tune

The findings so far pull in one direction. A tuning can serve one family of shapes perfectly, a family is a fingering moved about the neck, and a player who uses many fingerings is served best by the frets. What has not been asked is the choice a player actually makes: given a tuning, which fingering of each chord to use.

Every chord on a guitar has several fingerings, and each belongs to a family. A piece in a non-standard tuning set just for one family could be re-fingered chord by chord to stay inside that family wherever the hand can reach, and that is a search over a finite set rather than over six continuous numbers — for each chord, the fingerings within a stated stretch of the fretboard, scored by the tuning’s error and by the distance the hand has to move. The prediction the arithmetic here makes is that the cheapest route through a piece stays in one shape family far longer than a player choosing by comfort would, and the cost of leaving it is thirteen cents or more every time. What would come out is a number that belongs to guitar pedagogy rather than to tuning: how many frets of hand movement a cent of intonation is worth.

Part 9 of 9

One essay in the series on open strings. The essays either side of this one:

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 Part 8 — 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.

Where this goes next

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. 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. 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.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Just intonationOpen stringOptimisationTranspositionTuning by earVoicing