Counting beats moves the price of a chord, not the tuning

A tuning is right for some chords and wrong for the rest searched all six strings of a guitar for the tuning that brings a short piece closest to just intonation, and found something that is not a compromise. The piece in standard tuning — an E major chord, an A major chord barred at the fifth fret, and a G major chord in its open shape — is best served by a tuning that makes the E and the A chords exactly just and leaves the G chord 13.29 cents out, worse than equal temperament leaves it.

That search counted every cent alike, and the essay said plainly that this was the assumption with the most in it. A cent of error on an octave and a cent of error on a major third are not the same event to a listener. What a listener hears in a mistuned interval is a beat, and the arithmetic a tuner works in says the beat rate is the departure in cents times the frequency at which the two notes’ partials coincide, divided by 1731. An octave’s partials first meet at the lower note’s second harmonic, a fifth’s at its third, a major third’s at its fifth. So a cent on a third beats two and a half times as fast as a cent on an octave with the same bass, and weighting each interval by its beat rate is a change of one line in the cost.

The essay predicted what the change would do: the narrow octave the optimum puts inside the G chord would be heard as a slow wobble, the weighting would refuse the trade, and the abandoned chord would come back. None of that happens. The tuning does not move by a hundredth of a cent. What moves is somewhere else, and it is the more useful thing to know.

Six offsets that do not care what is counted #

On every piece the beat-counted optimum is the cents-counted optimum.

The open G blues costs 3.10 beats a second on the frets and nothing at all on its own tuning, because every interval it plays is exactly just and a just interval does not beat. The D A D G A D air costs 0.87 on the frets and 0.265 at its optimum, which is the same tuning the cents search found; its residue is the handful of fretted notes inside shapes that also ring open strings, and no peg reaches a fret. The song in standard tuning costs 3.40 beats a second on the frets and 1.056 at its optimum, and the optimum is E, A, D, G, B, E at 0, +1.96, 0, −13.68, +1.96 and 0 cents — the tuning the earlier essay printed, to the second decimal place on every string.

The reason is structural, and it is the same reason the earlier essay gave for why the optimum never shares its error out. The cents cost is a weighted sum of absolute departures, each of which is a linear function of the six offsets. A beat rate is that same absolute departure multiplied by a positive number — the coincident partial’s frequency over 1731 — and that number depends on the sounding pitch only through a factor of 2 raised to the offset, which over fifteen cents is a change of under one per cent. So counting beats replaces one set of weights with another, and a weighted sum of absolute values has its minimum at a corner of its surface, where several departures are exactly zero. Reweighting a corner does not move it. It can only change which corner is lowest, and a change of that kind happens all at once, at a threshold, not by a drift.

At the piece’s own proportions — the E chord for thirty-two beats, the A chord for twelve, the G chord for eight — the threshold is not crossed, and the corner that was cheapest in cents is cheapest in beats. The G chord is still sacrificed, and it now has a price in the unit a player hears: 6.86 beats a second, averaged over its fifteen intervals.

The octave was never the problem #

The earlier essay named the narrow octave as the audible cost of the trade, because it was the interval a guitarist would recognise — a G on the bottom string at the third fret against the open G string thirteen and a half cents flat. Taken apart interval by interval, the chord says the octave is the mildest thing wrong with it.

The low octave, G2 against G3, is 13.68 cents narrow and beats at 1.54 a second, which is exactly the wobble the earlier essay described and slower than its estimate of two and a half. The upper octave, G3 against the G on the top string, beats at 3.09. Neither is where the chord is loud with beating. The fastest pairs are the two minor sixths from B up to G, at 17.79 beats a second, and the major third G3–B3 at 16.61 — because flattening the G string by 13.68 cents while the B string is sharpened by 1.96 widens that third to 29.33 cents, and a third’s partials meet at the fifth harmonic of G3, near a thousand hertz.

So a cent-counting search and a beat-counting one disagree completely about what the G chord’s problem is, and they agree completely about the tuning. That is not a coincidence to be explained away; it is the corner property again. The G chord’s intervals are not individually negotiable. Every one of them is a difference between two strings’ offsets, and the same offsets are set to make the E and A chords exact, so the whole chord is dragged along by a decision made elsewhere.

There is a further consequence for the reading of the eighteen-beat pairs. A beat at seventeen or eighteen cycles a second is past the rate at which a beat is heard as a beat at all and inside the range where it is heard as roughness instead — the sound goes sour rather than wobbling. What a listener would report about the optimum’s G chord is not “the octave wavers” but “the chord is harsh”, and the harshness is in its thirds and sixths.

A staircase where there was a cliff #

What the change of units does move is the answer to the question the earlier essay asked last: how much of the piece would the G chord have to hold before the tuning gave it back?

Counted in cents, the answer was a cliff. The G chord is abandoned at 13.29 cents until it sounds for exactly as long as the E and A chords together — forty-four beats — and past that point it is exactly just and the other two are abandoned instead. There is nothing in between: at forty-four itself every tuning on a whole flat face of the surface costs the same.

Counted in beats, the answer is a staircase with three treads. The G chord is rescued partially at 30.4 beats, left 4.96 cents from just while the E and A chords give up 8.34 each; further at 47.4 beats, to 3.91 cents; and completely only at 73.1 beats, by which point it holds more of the piece than both other chords together by a wide margin. The first step comes fourteen beats earlier than the cents cliff did, and the last comes twenty-nine beats later.

The thresholds are computed exactly rather than read off the sweep. At each corner the piece’s cost is the rest of the piece’s weighted departure plus the G chord’s own times its beats, all over the total, so two corners cost the same where one straight line crosses another, and each crossing is a ratio of two numbers the corners already have. The dots on the figure are independent searches at eleven weights, and the figure is drawn only if every one of them lands on the envelope the crossings predict.

Why the cents version had only one step #

The difference between a cliff and a staircase is a fact about conservation, and it can be seen by drawing each corner’s cost as a line against the G chord’s weight.

In cents, every corner’s line passes through a single point at forty-four beats, where each costs 6.648 cents. That is not luck. At each of the four corners the E and A chords’ mean departure plus the G chord’s mean departure is exactly 13.29: 0 and 13.29, 8.34 and 4.95, 9.38 and 3.91, 13.29 and 0. The conflict between the two groups of chords is a fixed quantity, and every corner is a different division of it — the same bookkeeping where to hide the comma describes for a keyboard, done on six strings instead of twelve keys. With the two groups weighted equally, every division costs the same, so the envelope has one crossing and the rescue is one step.

In beats, a cent is worth a different number of beats depending on which intervals carry it. Moving error off the G chord’s thirds, which coincide near a kilohertz, and onto the E chord’s intervals, several of which coincide lower, converts the same cents into fewer beats — and at forty-four beats the four corners cost 3.432, 2.651, 2.667 and 3.134 beats a second rather than one common number. The lines have fanned, the lowest of them changes hands three times, and each change is a tread of the staircase.

The general statement is worth having because it applies to any tuning problem with this shape. A weighting that is proportional to cents moves thresholds and never moves an optimum between them. A weighting that converts cents at different rates on different intervals also splits a threshold into several, because it breaks the conservation that made every division of a comma cost the same.

A search that has become a proof #

The earlier essay ended its method with a sentence it could not improve on: the search was the best of seventeen descents and forty random restarts never beat it, which is evidence rather than an argument. The argument turns out to be cheap.

Every just interval a chord shape puts between two strings is satisfied exactly on a flat surface in the space of offsets — the set where one string’s offset minus the other’s equals a constant. The cents cost is a weighted sum of distances to those surfaces, so it is flat between them and its minimum lies at a point where enough of them meet to pin all five free strings. Those points can be listed.

The song in standard tuning has twenty-eight distinct surfaces, which meet five at a time in 2,050 distinct corners. The cheapest costs 2.0454 cents; the descent reported 2.0462, which is the same point on the descent’s finest grid. For the D A D G A D air the list is 2,038 corners and the minimum is 0.6028, again the descent’s answer. For the open G blues all fifteen surfaces pass through one point: the named tuning is the only corner there is. Evaluating every corner of every piece takes under a tenth of a second, which is less than one descent.

So the earlier result now stands as a theorem about these three pieces rather than a strong suspicion. The same list gives the beat-counted minimum, and it is at the same corner on all three — with one honest qualification. A beat rate’s weight moves by under one per cent as a string’s pitch moves, so the beat-counted surface is not exactly flat between corners, and the list proves the cheapest corner rather than the cheapest point. A departure from a corner shrinks the departure it moves away from by the same amount it grows the one it moves toward, and a one-per-cent change in their weights cannot make that trade profitable against the tens of per cent that separate the corners here — but that is an argument about these numbers, not a certificate.

Which computation produced the numbers #

The pieces, the shapes and their durations are those of the earlier essay, stated as progressions rather than transcriptions and supporting a comparison between tunings rather than a claim about any repertoire.

Each interval’s beat rate is the magnitude of p times the lower note’s frequency times (2 raised to the departure over 1200, minus 1), where p is the numerator of the just ratio in lowest terms, with octaves folded in — so a tenth, 5:2, meets at the lower note’s fifth harmonic and a twelfth, 3:1, at its third. The lowest string’s frequency is read from its note name at A = 440. A chord’s cost is the mean over its sounding pairs; a piece’s is that mean weighted by beats. Nothing about the strength of a partial enters, which is the largest simplification and has its own section below.

The search is the coordinate descent the earlier essay used, and counting beats makes it markedly worse. Of eighteen starts, the only one that arrived was the one handed the cents optimum as its starting point; the other seventeen stopped at corners of the beat-counted surface that are not its bottom, and the worst stopped at a tuning 9.03 cents from just. That is why the beat-counted results above are quoted from the corner list and the staircase thresholds from the corner lines, and why the descent is used only to find which corners the sweep reaches. A beat-counted search run alone, from its own starts, would have reported a tuning worse than equal temperament as the answer.

The partial that is not there #

The weighting assumes every coincident partial is as strong as every other. The fastest beat in the G chord is between B2 and G4, meeting at B2’s sixteenth harmonic. A plucked guitar string’s sixteenth harmonic is a small fraction of its fundamental, and a beat has a depth that is set by the weaker of the two partials involved — a beat between a strong partial and a weak one barely modulates the sum at all. Weighting by rate alone makes the high coincidences count for most exactly where they are least audible. A weighting by rate and depth would pull the price of the G chord down, and because depth also varies by interval it would move the treads of the staircase again.

A beat faster than about fifteen a second is not counted, it is felt as roughness. Every partial beats at its own rate, and past the rate at which a listener can follow the fluctuation the percept changes kind. The cost here treats an eighteen-beat pair as six times worse than a three-beat one, and a listener would call the first harsh and the second wavering, which is not a ratio.

A strummed note decays. The beat rates are those of steady tones. A guitar’s upper partials die in a fraction of a second, so the high-coincidence beats are present only at the attack and the low ones outlast them; the ranking of pairs by rate is a ranking at the moment of the strum and not over the note.

What a count of beats cannot tell a player #

Whether the tuning is settable. The optimum is unchanged, so this essay inherits the earlier one’s difficulty without adding to it: a G string set 13.68 cents flat is set by nulling the beat of the just third it makes against a fretted E-chord note, and whether a player can hear that beat against a decaying fretted note is a measurement nobody here has made.

Whether a partial rescue is worth anything. The staircase’s middle treads leave every chord between four and nine cents out — which is the neighbourhood of equal temperament’s own thirds, and possibly less pleasant than two perfect chords and one harsh one. Whether a listener prefers the error concentrated or spread is a preference, and minimising a mean has no opinion about preference.

And whether the chord is really what is being abandoned. The G chord is sacrificed under both counts at the piece’s own proportions, by a corner that is now proved to be the cheapest one there is. A search over offsets can only divide a conflict between the chords; it cannot say what the conflict is made of.

Still open: whether it is the chord or the shape that is given up #

Under cents and under beats alike, the four corners are four divisions of one fixed quantity: 13.29 cents that belong either to the E and A chords or to the G chord. That quantity does not depend on how error is counted, which is why no weighting has removed it, and it does not look like a comma — it is smaller than the syntonic comma’s 21.51 and it is not spread round a chain.

It has a more concrete candidate. The E major chord and the A major chord barred at the fifth fret are one fingering: the barre chord is the open E shape moved up five frets. The G major chord is a different fingering altogether, putting its root on the bottom string and its octave on an open string. If the conflict is between two fingerings rather than between two chords — each asking the same pairs of strings for different intervals — then it should survive every possible way of counting error, it should vanish when the G chord is played in the E shape at the third fret, and a barred chord is exactly as pure as the open chord it copies would be not a curiosity about open tunings but the whole of the answer. That is a direct computation on the pairs of strings, and nothing in it needs a listener.