One guitar tuning serves every key, and it is the D shape's

A guitar tuned for a set is tuned for the D shape tuned a guitar once for an evening rather than once per song. For the I–vi–IV–V ground in C, G, D, A and E, at the valuation a typical player puts on intonation — five frets of hand movement for a cent — the set’s best tuning was exact for the D shape, the one fingering every key’s route used for some chord, with that shape’s two silent strings placed for the set. Not retuning between songs cost a fifth of a cent.

It ended by turning the question over. A player chooses the set as well as the tuning, and five keys are not twelve. Given one tuning, which songs does it serve best? And how many keys can a set span before its best tuning stops being any shape’s — before the guitar has to be tuned for nothing in particular, as a guitar tuned for a still hand was? The answer would also say something about an old practice: retuning between sets rather than between songs, one tuning per half of the evening. Is that about the strings stretching, or about the keys changing?

A set never stops sharing a shape. Grown one key at a time round the circle of fifths, the set’s tuning is exact for the C shape up to three keys, and exact for the D shape from four keys to all twelve, with the same offsets to within a fifth of a cent from five keys on. The tuning chosen for five keys serves the keys outside the set better than it serves C and G, which are in it.

The same search, one key at a time #

The search is the fifteenth essay’s. Every song in the set is routed at the same exchange rate under the current tuning — each chord given the fingering that, with the moves between chords, costs least in cents from just plus frets moved over the rate — and the six strings are then retuned for the union of all the songs’ chords, each weighted by its share of its song and each song by its share of the evening. The two steps repeat until no route changes, from eight starting tunings: the five placed shape tunings, the frets and two seeded random ones. Each set’s tuning is compared with retuning for every song, which is what the still-hand essay’s single-song search gives, and with equal-tempered frets.

The sets are grown round the circle of fifths from C: C, then G, D, A, E, B, F♯, D♭, A♭, E♭, B♭ and F, the order in which a guitarist’s keys get progressively less open. The exchange rate is held at five frets a cent, the figure tuned for one shape, a cent is worth five frets measured for a typical player.

The C shape, then the D shape, and never nothing #

With one key, C, the set is one song, and its tuning is the C shape’s with its silent low string placed — the tuning a silent string tunes the minor chord found best for this ground on its own. G added changes nothing: the ground in G is also played mostly in the C shape at a moderate valuation, and the two songs want the same tuning. D added still leaves the C shape’s tuning best, but now at a cost: 0.16 over retuning for each song, because the ground in D would rather have something else.

At four keys, with A added, the answer changes. The set’s tuning becomes exact for the D shape, and from there to all twelve keys it stays exact for the D shape. The prediction that a set spanning many keys would end up tuned for nothing — as the still hand’s route did, with every string within two cents of its fret — does not come true at any span. The set never stops sharing a shape, because every key’s route contains the D shape.

The routes show why. The D shape has no open string in it, so it can be played at any fret, and at a moderate valuation the routes use it wherever it keeps the hand near the chords around it. Under the five-key tuning, every key’s route plays at least three of its eight chords in the D shape, and A major five. No other shape is in every route. The C shape is in the routes of the keys where open strings are useful and disappears from the others; the E-minor and A-minor shapes serve the vi chord in some keys and not in others. A tuning must make some shape exact to be worth anything over the frets — tune a guitar for the shape of its key’s open chord found that a player who keeps every chord in the tuned shape plays exactly in tune while the hand travels — and the only shape every key can use is the one without an open string. The minor chords cannot share it: a tuned guitar has no minor chord in tune, because every minor shape disagrees with every major shape about some pair of strings, so the vi chord in each route is left to whichever minor shape is nearest.

The tuning settles #

The offsets make the change of shape concrete. The C shape’s tuning raises the A string 13.7 cents, the G string 15.6 and the B string 13.7 — the just third and fifth of the C shape’s major chord, measured against the low E and the D string. Neither tuning is tuning by ear in the usual sense: a guitar tuned by harmonics hides a comma, because standard tuning’s four fourths and a third, tuned pure, close a comma short of two octaves, and both shape tunings decide where that comma goes. The D shape’s tuning leaves those strings within four cents of their frets and lowers the high E about twelve cents, a just major third below where the frets put it, which is the D shape’s own third.

At four keys the D tuning’s silent strings, the low E and A that the D shape does not play, sit exactly on their frets. From five keys on they move two cents up, placed for the chords of the added keys that do sound them. After that nothing moves. The tunings for five, eight and twelve keys agree on every string to within a fifth of a cent. Past five keys, adding keys to a set does not change what the guitar should be tuned to.

Which songs the tuning serves #

The converse question can now be answered directly: take the five-key tuning and play the ground in every key under it.

The keys it serves worst are C and G, which were in the set. C gives up half a cent against its own best tuning and G a third. The keys it serves best are B, D♭, A♭ and B♭, none of which was in the set, which give up two thousandths of a cent — nothing a player could hear or measure. The rest, D, A, E, F♯, E♭ and F, give up about a tenth.

The reason is which shape each key’s own tuning is for. C and G are the open keys: their best routes at this valuation use the C shape’s open chords, and their own best tuning is the C shape’s, so a D-shape tuning is a compromise for them. The keys a guitarist plays in barre shapes — B, D♭, A♭, B♭ — have routes built almost entirely of movable shapes, and their own best tuning already is the D shape’s. For them the set’s tuning is their tuning.

So the answer to “which songs does a given tuning serve best” is not “the songs it was chosen for”. A tuning serves best the songs whose own best tuning is for the same shape, and the D-shape tuning is every barred key’s own. A set chosen from the open keys is served worst by the tuning the set itself chose, because it was the open keys that pulled against it.

That gives a player a rule that does not need the search. An evening in the open keys alone — C and G — wants the C shape’s tuning, and loses nothing against retuning for each song. Add D and the C shape’s tuning still wins, at a sixth of a cent. Add A, or any key further round the circle, and the D shape’s tuning wins and keeps winning. So the choice is between two tunings, not twelve, and it is made by one question: does the evening include a key in which the guitarist barres more than they play open? If it does, the D-shape tuning; if the evening is all open-position songs in C and G, the C-shape tuning. Nothing in between is ever best at this valuation, and the frets are never best at all.

Not retuning costs less the more keys there are #

The cost of tuning once for the whole set, rather than for each song, is largest at four keys — 0.32 of a cent — where the set has just left the C shape’s keys and still contains two of them. After that it falls, to 0.22 at five keys, 0.15 at eight and 0.12 at twelve. Each key added round the circle past E is a barred key whose own tuning is the D shape’s, so it adds a song the set’s tuning serves perfectly, and the set’s average loss goes down. Meanwhile the gain over the frets stays between 1.9 and 2.4 cents at every span.

This answers the question about the old practice. An evening split into two halves by key, with the guitar retuned at the interval, is not justified by the keys. A set of all twelve keys loses an eighth of a cent against retuning for every song, and three different five-key halves of an evening want the same tuning to a fifth of a cent.

The three halves are the C-to-E set of the fifteenth essay, the next five keys round the circle, and the five flat keys. All three want the D-shape tuning with offsets identical to a fifth of a cent. The flat-side sets lose four hundredths of a cent against retuning for every song, because their keys are barred keys; the sharp-side set loses a fifth, because it contains C and G. If a guitarist retunes between halves of an evening, on this arithmetic it is not because the second half is in different keys. It is because the strings have moved.

The decision that matters is the frets #

The numbers put the two decisions a guitarist could make about tuning on one scale. Tuning once for an evening rather than once per song costs between nothing and a third of a cent, depending on the span, and an eighth of a cent for all twelve keys. Tuning to the set’s D-shape tuning rather than leaving the strings on the frets gains between 1.9 and 2.4 cents, at every span. The second decision is worth ten to twenty times the first. At twelve keys it is seventeen times: 1.94 cents gained over the frets against 0.12 lost to not retuning.

So for a player choosing how to tune for an evening, the question is not how many tunings the evening needs. It needs one. The question is whether that one is the frets or the D shape’s, and on this arithmetic the D shape’s is worth about two units of the combined cost — two cents of intonation, or the equivalent in hand movement — in every key. That is not large on its own — the chord a tuning gives up is a fingering found single chords abandoned by thirteen cents where a tuning cannot serve them — but it is spread over every chord of every song, and it is paid for by one adjustment made before the evening starts: the high E string lowered about twelve cents and the three middle strings raised two to four.

The cost measure could be argued with. Error counted in beats rather than cents weights a mistuned third more heavily than a mistuned octave, and counting beats moves the price of a chord, not the tuning found that for single pieces the best tuning came back unchanged when it was counted that way. That recount has not been run on the sets here. The span result rests on which shape every route contains, which is a fact about fingerings rather than about how error is counted, so it would be expected to survive it.

The model the search rests on #

The tuning search, the fingering routes and the costs are the earlier essays’: five chord shapes from the CAGED family and the E-minor and A-minor shapes, each playable wherever its fingering fits, a chord’s error measured as the mean departure from just intonation over its sounding pairs of strings, and hand movement measured in frets. The ground is I–vi–IV–V twice round, eight chords of equal length; a song in each key is that ground transposed. The exchange rate is five frets a cent throughout.

What the search leaves out #

One ground. Every song is the same four chords. Real sets mix grounds — the fifteenth essay’s mixed set added I–vi–ii–V — and songs with chords outside the ground, a borrowed iv or a secondary dominant, whose best shapes may not be the D shape. A set containing many such songs could lose the shared shape the pure ground always keeps.

One valuation. At fifty frets a cent the fifteenth essay found every set’s tuning to be the C shape’s, and at one or two frets a cent a mixed set tuned for nothing. The span result is at five frets a cent, the typical player’s valuation, and it is not claimed at others.

Stable strings. The model’s strings stay where they are tuned. The whole of the argument about halves of an evening rests on the one thing it does not model, which is that real strings drift with temperature, humidity and playing, and a tuning set at the start of an evening is not the tuning at its end.

What the picture cannot show #

Whether players do this. The D-shape tuning lowers the high E string by about twelve cents, which a guitarist tuning by ear to the frets would never do. Whether any players, finding their D-shape chords sweeter with the high E lowered, arrive at something like it by habit is not a question the arithmetic can answer.

The listener. The costs are cents from just and frets moved, an intonation measure and an effort measure. Neither says how a listener would hear the difference between a set’s tuning and the frets, which is two cents on average and up to about a quarter of a semitone on the high E.

The open strings’ own sound. An open string rings longer and brighter than a stopped one, and a player who chooses the open keys often chooses them for that. The model prices intonation and hand movement and nothing else, so it cannot say what a set in C and G gives up in resonance by being played in barre shapes under a D-shape tuning — only that, measured in cents and frets, the tuning serves those keys least well of all twelve.

Still open: songs outside the ground #

The span result depends on every song sharing a ground whose routes all contain the D shape. The natural test adds songs that break it — a minor-key song whose i and iv are best in the E-minor and A-minor shapes, a blues whose I7 wants a shape of its own, a tuning is right for some chords and wrong for the rest and the chord it gives up is a fingering. The computation adds them to a twelve-key set one at a time and asks how many a set can hold before its tuning leaves the D shape, which would say whether the shape every key shares survives a real evening’s repertoire or only the ground’s.