Pitch and tuning

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.

Assumes: One tuning has no comma to place · A guitar tuned by harmonics hides a comma

One tuning has no comma to place worked through five guitar tunings and found that some of their chains close and some do not. It ended with a question it could state and had not run:

Given a piece, which tuning minimises its total departure from just? It is a small search and the pieces are now all present. The free variables are the six offsets themselves — which need not be the pure chain at all, since a player tuning by ear can put a string anywhere.

The pieces were not present. What was present was a handful of chords a player of each tuning reaches for, with no order and no durations, and the objective is a weighted sum — so the weights had to be written before anything could be searched. With them written, the search costs a few hundred milliseconds and the answer is in three parts, of which the third is the one worth having.

Fig. 1 Three short pieces, each scored by how far its intervals sit from just on average, and each tuned three ways: with every string on its fret, with every adjacent interval of its tuning set pure, and with all six strings free and searched. Two of the three get their own tuning back.

The coordinates the question has to be asked in

A tuning is six numbers, and which six matters.

The obvious six are the intervals — a fifth here, two fourths, a tone. That is how a tuning is named, how it is taught and how it is set by ear, and it is the wrong coordinate for a search, because the intervals are not independent: choosing five of them fixes the sixth against the octave the frets ask for. The coordinate a player actually has is the peg, and a peg moves a string rather than an interval. So the six variables here are six offsets in cents from where the frets would put each string, and any six numbers are a legal tuning.

That immediately gives one degree of freedom away. Every quantity in the objective is a difference between two strings’ offsets, so moving all six together changes nothing at all: the optimum is a line rather than a point. A player hears that line as tuning the whole guitar up or down, which is a decision about pitch standard and not about temperament, and the search pins the lowest string to its fret to pick one member of it.

The cost of a chord is the mean, over every pair of notes it sounds, of how far that pair sits from the just interval its span names. The cost of a piece is that averaged over its chords, each weighted by how long it sounds. Nothing in it is a psychoacoustic claim — a cent of a fifth counts the same as a cent of a third, which is not how they are heard, and the section on what the picture cannot show is about exactly that.

Where the chain closes, the search has nothing to add

Two of the three pieces come back with the tuning they started in, and the two are the two whose chains close.

The open G piece costs 5.74 cents with every string on its fret — which is the ordinary price of equal temperament, most of it in the thirds, and is what a guitar cannot be in tune is about and exactly nothing when every adjacent interval is set pure. The search, given six free strings and seventeen starting points, finds nothing better than nothing. The D A D G A D piece costs 1.79 on the frets and 0.60 on its pure chain, and the search again returns 0.60.

That is the finding about barred chords arriving from the other direction, and it is worth being clear about why it is a different statement. That finding was that a chord barred straight across a tuning whose chain closes is exactly as pure as the open chord it copies — a fact about a shape. This says that no tuning of any kind, related to the named one or not, does better on a piece made of such shapes. The first is a property of the gesture; the second is a property of the whole six-dimensional space, and only a search can say it.

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. — Fig. 2 The earlier figure: five tunings with every adjacent interval set pure, drawn as how far each string then sits from its fret. The two rows whose chains close exactly are the two whose pieces the search leaves alone.

The D A D G A D piece keeps 0.60 cents rather than reaching zero, and the residual is not the tuning’s fault. Two of its chords are shapes fretted at the fifth and seventh frets on three strings while the other three ring open, so they mix a pure interval with a tempered one, and no setting of the pegs removes a fret. That is the general shape of everything below: a tuning can only fix what the open strings decide, and a fretted note carries equal temperament with it wherever it goes.

Where it does not close, the answer has no name

The piece in standard tuning behaves differently in every respect.

On the frets it costs 5.97 cents. On the pure chain — which is what a player tuning by harmonics arrives at — it costs 13.80, more than twice as much, because standard tuning’s four fourths and a third fall a syntonic comma short of two octaves and tuning them all pure puts the whole 21.5 cents somewhere. That is the situation every keyboard temperament was invented for, and somebody has to pay the comma is the general statement of it. The search finds 2.05, which is 3.92 cents better than the better of the two and a third of what the pure chain costs.

So for this piece the two tunings a player would actually consider are both a long way from the best one, and the guess that earlier essay offered — that the optimum would be a few cents off a named tuning — is wrong in a specific way. It is not a few cents off; it is a different object.

Fig. 3 The six strings of the standard-tuning piece under three tunings: on the frets, on the pure chain, and at the piece’s own optimum. The pure chain spreads its comma across five strings. The optimum leaves four of them on their frets and puts the entire thing on one.

The optimum is E, A, D, G, B, E at 0, +1.96, 0, −13.68, +1.96 and 0 cents. Four of the six are within two cents of equal temperament and the fifth string of the six carries everything.

Both of those numbers are recognisable. +1.956 cents is exactly the amount by which a pure fifth exceeds a tempered one, so the A and B strings are set as pure fifths above the strings the piece actually uses them against. And 13.686 cents is exactly the amount by which a tempered major third exceeds a pure one — 400 cents against 386.314 — so the G string is flattened by precisely enough to make the tempered third that lands on it just. The search was given a grid and no theory, and it arrived at two intervals out of the ratio table.

Two chords exactly right, and one abandoned

What the optimum does to the piece, chord by chord, is the finding this essay is for.

Fig. 4 Each distinct chord of the piece, with how far its intervals sit from just under the three tunings and how many beats it holds. The optimum does not shade the error across the three; it removes it from two of them completely and leaves all of it on the third.

The E major chord sounds thirty-two beats and is exactly just at the optimum — every one of its ten sounding intervals, the fifth, the octave, the tenth, the twelfth and the double octave, at its whole-number ratio to the resolution of the search. The A major chord barred at the fifth fret sounds twelve beats and is exactly just too, by the same six offsets. The G major chord sounds eight beats and is 13.29 cents from just, which is worse than it was on the frets and worse than it was on the pure chain.

The mechanism is concrete and can be checked by counting frets. In the open G shape the low G is the third fret of the bottom string, which the optimum leaves where the frets put it, and the G an octave above it is the open fourth string, which the optimum has taken down by 13.68 cents. The chord contains an octave that is thirteen and a half cents narrow, between two notes of the same name, and that is an interval a guitarist tuning by ear would notice immediately and reject. The optimum does not care, because it is not listening to the G chord; it is spending the G chord to buy the other two.

That is the opposite of what a temperament does. Every historical temperament drawn here — and where to hide the comma is the question they all answer — is a distribution of a comma, an attempt to make no key intolerable by making every key slightly wrong. The best tuning for a particular piece is the reverse: make most of it perfect and one part of it intolerable.

Why it never compromises, and the reason is in the shape of the sum

That looks like a quirk of one piece and it is a property of the objective, which is worth stating because it says what any such search will always return.

The cost is a sum of absolute values of quantities that are linear in the offsets: each interval’s departure from just is one string’s offset minus another’s plus a constant, and the cost takes its size without regard to sign. A weighted sum of absolute values of linear functions is minimised where a weighted median of them sits, not where a weighted mean does — and a median lands on one of the values rather than between them. That is why an optimal tuning always makes some intervals exactly just: the minimum is at a corner of the surface, and a corner is a place where several of the absolute values are zero.

The consequence is that the optimum cannot slide. Give a chord more of the piece and the answer does not drift toward it; it stays exactly where it was until that chord’s weight passes a threshold, and then it jumps.

Fig. 5 The same search with the abandoned chord given more and more of the piece, from nothing to ninety-six beats, with the rest unchanged. The two lines do not approach each other. They hold, cross in a single step, and hold again at the reflected values.

The threshold is exactly where it should be. The E and A chords hold thirty-two and twelve beats, forty-four between them. At forty-three beats the G chord is still abandoned at 13.29 cents; at forty-five it is exactly just and the other two are abandoned at 13.29; and at exactly forty-four the search returns a third answer, a tuning two cents from the frets that splits the difference at 8.36 and 4.94. That is the tie, and the tie is a whole flat face of the surface rather than a point — every tuning on it costs the same, and the one reported is the one the descent happened to reach.

So a piece does not have a best tuning that is a little better for its common chords. It has a best tuning that is perfect for whichever chords hold more than half the clock, and a player who repeats the bridge one more time can change the answer completely.

Which computation produced the numbers

Each piece is a list of fretted shapes with beats, stated rather than transcribed: the commonest harmonic ground of the repertoire its tuning belongs to, in the proportions that ground gives. They support a comparison between tunings on one piece and they support no claim about any composer’s music, and none is made here.

A shape is a fret per string. A sounding note sits at its string’s equal-tempered pitch plus a hundred cents a fret plus that string’s offset, and the span of a pair — which decides which just interval it is scored against — is taken from the frets rather than from the sounding pitches. That detail is load-bearing: forty cents of offset on each of two strings can move a major third far enough that rounding it to the nearest semitone names a fourth, and the chord would then be scored against an interval nobody is playing.

The search is coordinate descent on a grid, coarse to fine: each string in turn is moved to its best value with the other five held, the passes repeat until nothing moves, and then the window narrows to the last step either side and the whole thing runs again, four times over. The finest grid is four thousandths of a cent, which is why the chords called just above are called just to that resolution rather than to machine precision.

Fig. 6 Seventeen starting tunings for each piece, run to convergence and sorted by where each stopped. Between forty-seven and fifty-nine per cent of them arrive; the rest stop at corners the descent cannot leave, some of them worse than doing nothing.

Under half the starts arrive, and that is the honest number rather than an aside. The objective is convex, which is usually the end of the argument, and convex is not enough here: a piecewise-linear surface has corners at which no single-string move improves anything while a joint move still would, and that is precisely the shape of a tuning problem where two strings a just third apart are both wrong by the same amount. A pass that moves two strings together frees some of those corners and not all. Run from one start, the search would have reported 3.11 or 3.39 cents as the answer on standard tuning, either of which would have looked like a result.

Where the model stops

A cent is not a cent. Every interval in the objective counts equally per cent of error, and a cent of a major third is a very different event from a cent of an octave: the octave beats against a low partial and the third against a high one, and every partial beats at its own rate prices that difference directly. Weighting the intervals by their beat rates would very likely move the answer, and it would move it toward protecting the octaves — which is the one thing the optimum above sacrifices.

A chord is not its notes held forever. The cost of a chord here has no attack, no decay and no order. A strummed guitar chord is a rolled arpeggio whose lowest note has been decaying for eighty milliseconds by the time its top note starts, and the pairs that beat audibly are not the same pairs in the first half second as in the last — which is what a string that decays twice is counted early found happening to a tuner counting a beat.

The pieces are grounds, not scores. Three chords in a fixed proportion is a skeleton of a piece, and a real piece has passing chords, single-note lines and open strings ringing into chords that do not contain them. Adding a rarely-played chord to a piece changes nothing at all on this objective until its weight crosses a threshold, so the shape of the answer is robust to the detail — but which side of a threshold a real piece sits on is a question a real piece would have to be counted to settle.

And the search is not a proof. Nothing here establishes that 2.05 cents is the global minimum. It is the best of seventeen descents from seventeen places, forty of forty-five further random starts never beat it, and that is evidence rather than an argument.

What the picture cannot show

Whether anybody could set it. The optimum asks for a string 13.68 cents flat of its fret, and the essay that priced a cellist’s ear measured how precisely a player setting an interval by ear can land: a fifth nulled over two seconds is good to a fraction of a cent at the top of a violin and to several cents at the bottom of a cello. A guitarist has no beat to null for an offset like this one, because it is not an interval anybody names — it is a third made just, which does have a beat, and whether a player can hear that beat against a fretted note is a separate measurement nobody here has made.

Nor whether a listener would prefer it. The piece at its optimum has two perfect chords and one containing a narrow octave. The same piece in equal temperament has three chords equally and mildly wrong. Which a listener would call better in tune is an experiment, and the arithmetic has no opinion: it was asked to minimise a mean and it did. A comma under the threshold is the nearest this collection comes to the answer, and what it says is that the size at which a mistuning stops being audible depends on what else is sounding.

Whose guitars, and when

Guitarists retune for pieces constantly, and describe it as making the piece sound right rather than as tempering anything. That practice is what this essay is about, and the arithmetic says two things to it that are not the same.

For a player in an open tuning the practice buys nothing measurable, because the tuning already is the optimum for the chords such tunings are played with. For a player in standard tuning it buys almost four cents on this piece, most of it on one string, and it buys it by giving up a chord — which is the choice where the chain was never closed describes traditions making at a larger scale.

The open tunings are much the older practice in the traditions that use them, and the arithmetic does not say they were chosen for this — a tuning whose open strings make a chord was chosen because its open strings make a chord. That it also happens to be the optimum for its own repertoire is a consequence of the same small whole numbers that make twelve fifths and seven octaves nearly agree, and is the kind of coincidence the essay on the chains themselves found in their arithmetic.

Still open: whether the beats change the verdict

The single assumption above with the most in it is that every cent counts equally, and it is the one this collection is best equipped to drop.

A departure from just is heard as a beat, and the rate of that beat is the departure in cents times the frequency of the partial where the two notes coincide, divided by seventeen hundred and thirty-one, which is the arithmetic a tuner works in. That coincidence sits at the second partial of an octave, the third of a fifth and the fifth of a major third — so a cent of error on a third beats roughly two and a half times as fast as a cent on a fifth in the same register, and a cent on a low octave beats slowest of all. Weighting each interval by its beat rate rather than counting its cents is one line in the cost, needs nothing the arithmetic here does not have, and the prediction is specific: the narrow octave the optimum accepts in the G chord beats at about two and a half a second on the low strings, which is a wobble rather than a shimmer, and a weighting that heard it would refuse the trade.

If that turns out to be right, the optimum for the standard-tuning piece would move off its corner and the abandoned chord would come back — and a search that returned perfect chords and one ruined one would turn out to be an artefact of counting cents instead of counting beats.