One tuning has no comma to place

The essay that found a guitar’s comma found that a guitar’s six standard strings — four fourths and a major third — close two octaves exactly a syntonic comma short when every interval is tuned pure. The frets are an equal temperament, the strings are not, and the difference is 21.5 cents that has to be left somewhere — the mirror of the situation the cello cannot hear its own tempering describes, where the reference is a piano and the instrument has no frets at all. Tuning by harmonics puts it in one place and tuning at the fifth fret in another; no method removes it.

Its closing paragraph named the case it had not computed. Open tunings replace the fourths-and-a-third with fifths, octaves, tones and fourths, chosen so that the six open strings make a chord rather than a chain. Whether that moves the comma into the notes a player seldom frets, or into the chords a player reaches for most, is the same arithmetic on a different interval list.

It does neither, and the reason is a small whole number.

Some chains close and some do not #

Take each tuning’s adjacent intervals and set every one to the pure ratio nearest the interval it spans — a fourth to 4:3, a fifth to 3:2, a major third to 5:4, a whole tone to 9:8. Multiply the chain and compare it with the two octaves the top string’s fret position asks for.

Standard tuning’s chain is $(4/3)^4 \times (5/4)$, which is $320/81$ against the 4 that two octaves ask for — short by exactly 81/80, the syntonic comma, 21.5 cents. That is the fourth essay’s result.

D A D G A D’s chain is $(3/2) \times (4/3)^2 \times (9/8) \times (4/3)$, and it is exactly 4. Not nearly four; four. So there is no comma, nothing to place, and no decision to make: every string can be set pure against its neighbour and the chain arrives exactly where the frets say the top string should be.

The arithmetic is worth doing once because it is short. Multiply the first three: $\tfrac32 \times \tfrac43 \times \tfrac43 = \tfrac83$. The whole tone takes it to $\tfrac83 \times \tfrac98 = 3$. The last fourth takes 3 to 4. The tuning closes because a fifth, a fourth and a fourth make a minor seventh of 16/9, the tone turns that into a perfect twelfth, and a fourth above a twelfth is two octaves.

What closing buys and what it costs #

Closing is not the same as sitting on the frets, and the difference is the useful part.

D A D G A D’s strings sit at 0, +2.0, 0, −2.0, +2.0 and 0 cents from the frets — a spread of 3.9 cents, which is the pure whole tone against the tempered one and is the largest departure anywhere in the tuning. Standard tuning’s strings run to −21.5 with a spread of 21.5. A player tuning D A D G A D by ear is within two cents of equal temperament on every string, and one tuning standard by ear is not.

The open-chord tunings — open G and open D — also close, and they pay differently. Their chains contain a major third, which is where the syntonic comma lives, and a major third tuned pure is 13.7 cents flat of the tempered one. Open G’s B string sits 15.6 cents flat and every other string is within two cents; open D’s F♯ sits 13.7 flat.

So the three ways of closing are not one thing. D A D G A D avoids the third entirely and has almost nothing to distribute. Open G and open D contain a third, put the whole of its flatness into one string, and close anyway because the rest of the chain compensates. Standard tuning contains a third and does not close.

Why a barred chord is exactly as pure as the open one #

The consequence for playing is sharper than the tuning table suggests, and it is a one-line argument.

In an open tuning the characteristic gesture is a bar straight across at one fret. A bar at the fifth fret raises every string by exactly five frets, which is exactly five hundred cents on every string, because the frets are an equal temperament and a fret is a fret whatever string it crosses. Adding the same number to every pitch leaves every interval unchanged. So the chord at the fifth fret has precisely the intervals of the open chord — not approximately, exactly — whatever the tuning did to the individual strings.

Open G’s G major is pure by construction. Its C major barred at the fifth fret and its D major barred at the seventh are pure by the same construction, to within nothing at all. Standard tuning’s E major, G major and barred A major are 29, 33 and 29 cents from just at their worst, because each of them mixes strings at different frets and so inherits a different combination of the tuning’s displacements.

That is the answer to the question the fourth essay asked, and it is not one of the two answers it offered. An open tuning does not move the comma anywhere. It removes it from the chords the tuning is for, and it does so by making those chords translations of one another rather than shapes.

Where the cost reappears #

The cost is real and it arrives the moment a player leaves the translated shapes.

D A D G A D’s open chord is pure and its G major shape — a bar at the fifth on the lower three strings with the upper three open — is within 2.0 cents. Its A minor shape reaches 19.6, because it fingers strings unevenly and the tuning’s whole-tone offsets stop cancelling.

So the purity is a property of the gesture rather than of the tuning. A tuning that closes gives a player a set of chords that are exactly right and a much larger set that are no better than standard tuning’s, and the boundary between the two sets is whether the shape is a translation.

That is a reasonable description of what open tunings are for. The repertoire that uses them leans heavily on bars, drones and open strings, and on chords that keep several strings open while moving one or two — which is also what makes an open string an attractor for every stopped note that shares its pitch class. The first of those is exactly pure, the second is exactly pure, and the third is the case where the arithmetic goes back to being ordinary.

Drop D closes nothing, and that is the useful control #

One tuning in the table is neither standard nor an open chord, and it is the one that shows the finding is about arithmetic rather than about open tunings as a category.

Drop D is standard tuning with the lowest string dropped a tone. Its intervals are a fifth, three fourths and a third — the same third standard tuning has — and multiplying them leaves the chain 17.6 cents short of two octaves. It closes no better than standard tuning, and its strings spread over 19.6 cents.

So dropping a string does not fix anything. A player who drops the low E to D has not changed the problem; they have moved the whole chain down by a tone relative to the frets and left the same third in the same place. The comma is 17.6 cents instead of 21.5 because the fifth that replaced a fourth sits differently against its fret, which is a difference of four cents and is not a difference in kind.

That is the control the finding needs. If every tuning with a chord in it closed and every tuning without one did not, the result would be about chords. Drop D has no chord and does not close; D A D G A D has a suspended chord and closes exactly; open G has a triad and closes. What decides it is whether the interval list multiplies to a power of two, and that is a fact about small whole numbers rather than about harmony.

What this says about the quartet #

The first three essays here are about a string quartet, whose problem is the opposite one: its open strings sit on a pure chain and its stopped notes are free, so the chain is the only fixed tuning it has and the stopped notes adjust round it.

A guitar inverts that. Its stopped notes are fixed by the frets and its open strings are free, so the open strings have to be reconciled with a temperament they were not tuned to. Both instruments have five or six pitches they cannot move and a much larger set they can, and they differ in which set is which.

The comparison makes the guitar’s situation the harder one to describe. A quartet’s Pythagorean chain is wrong against a piano and consistent with itself; a guitar’s pure chain is wrong against its own frets, which are a foot away from the strings. An open tuning is the one case in either instrument where the two systems agree exactly, and the tunings that manage it do so by containing no major third or by containing exactly one.

Why nobody says a tuning closes #

There is a small puzzle in the fact that this is not something guitarists say, given how much is written about open tunings and how flatly the arithmetic comes out.

Part of it is that the quantity is invisible in the usual way of describing a tuning. A tuning is named by its notes — D A D G A D — and the closure is a property of the intervals between them, which the name does not show. Two tunings whose names differ by one letter can differ by twenty cents of closure, and no amount of looking at the letters says so.

The larger part is that it does not feel like a property of the tuning. What a player notices is that the instrument goes into tune easily, that the open chord rings, and that the bars up the neck ring too. Every one of those is the closure showing up as an absence of trouble, and an absence of trouble is not something anybody reports. The tunings that do not close announce themselves and the ones that do are simply described as sounding good.

That is worth saying because it is the same shape as the fourth essay’s own finding. Standard tuning’s comma is famous — every guitarist knows the G–B third is a problem and that the B string is the one that will not settle — and the reason it is famous is that it is a decision a player has to keep making. A tuning with nothing to decide leaves no trace in the practice, and the practice is where the vocabulary comes from.

Which computation produced the numbers #

Each tuning is a list of six pitches as semitones above the lowest string. The equal-tempered position of each is a hundred cents a semitone; the pure position is the running product of the ratio nearest each adjacent interval — 9:8 for a tone, 6:5 for a minor third, 5:4 for a major third, 4:3 for a fourth, 3:2 for a fifth, 8:5 for a minor sixth, 5:3 for a major sixth and 2:1 for an octave. The offset is the difference, and the closure is the offset of the top string.

A fretted note sits at its string’s pure pitch plus a hundred cents a fret, so every string carries its own offset up the neck. A chord’s intervals are the differences between the notes it sounds, and each is compared with the just interval its span names, reduced by whole octaves.

The chord shapes are the ones a player of each tuning uses, which is a judgement rather than a computation and is stated as one: the open chord, a bar at the fifth and a bar at the seventh for the open tunings, and the common open-position and barred shapes for standard.

The pure ratio “nearest” an interval is a choice at two places and neither matters here. A tritone has no small-number ratio and none of these tunings contains one; a minor seventh could be 16:9 or 9:5, and none of them contains one adjacent either.

Where the model stops #

A guitar’s frets are not an exact equal temperament. They are placed by the rule of eighteen or by its modern equivalent, the intonation is compensated at the bridge for each string’s stiffness, and a fretted note is sharpened by the finger pressing the string down. A stiff string’s partials are stretched as well, so a string’s own octaves are wider than 2:1. Every one of those is a few cents and they do not all point the same way.

The frets are also not where a maker put them for this. A guitar’s neck is cut for standard tuning and every open tuning inherits it, so a tuning that closes does so against a temperament chosen for a tuning that does not — which is why the closure is a property of the interval list and owes nothing to the instrument. An open string pulls the quartet flat is the case where the instrument has no frets at all and the pull goes the other way.

And nobody tunes to the ratios exactly. A player tuning by harmonics is nulling a beat, which is accurate to a fraction of a cent on a long note and to several cents on a short one; The second of these essays priced that for a cellist and found the lowest string the least certain. The offsets above are what a perfect ear would produce.

The open chord is a chord and the fretted ones need not be. A bar across an open tuning gives the open chord transposed, and a shape that fingers strings unevenly gives something else — so the purity result is about a gesture and the sample of gestures here is six. The cello cannot hear its own tempering is the essay that shows how much of a tuning’s accuracy is the ear’s rather than the arithmetic’s.

The shapes are a small sample. Five tunings and a dozen chords is enough to show that barred chords inherit exactly and shapes do not, and it is not a survey of what anybody plays.

What the picture cannot show #

It cannot show a capo, which is the same translation argument made with a bar of metal and which therefore also preserves every interval exactly — including, in standard tuning, its errors.

Nor the open chord’s own beating. A pure chord and a chord 30 cents from pure differ in more than a number: the second beats, at a rate the arithmetic here gives elsewhere, and the beating is what a player hears rather than the cents. A pair tuned apart on purpose is the essay that prices what a deliberate mistuning sounds like.

And it cannot show what a player does about a bad chord. A guitarist who finds a chord sour bends a string, or re-fingers it, or tunes the guitar slightly wrong on purpose so that the chords of the piece in hand come out better — which is a per-piece temperament arrived at by ear and is what a quartet’s consensus does continuously rather than once, and is the same manoeuvre a keyboard tuner makes deliberately.

Still open: whether a tuning could be chosen for a piece #

Every tuning here is a fixed list and the frets are a fixed temperament, and the arithmetic showed that the two agree exactly in one gesture and not otherwise. That suggests a question nobody on the account here has asked: given a piece, which tuning minimises its total departure from just?

It is a small search and the pieces are now all present. A piece is a list of chords, each a set of strings and frets; a tuning is a list of six offsets; the departure of a chord is the worst or the mean of its intervals from just; and the objective is a sum over the piece weighted by how long each chord sounds. 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.

What makes it worth running rather than merely stating is that the answer is probably not any named tuning. A piece in D that uses three chords heavily would have an optimum a few cents off D A D G A D, and the interesting number is how much it buys: if the best tuning for a piece is two cents better than the nearest named one, the named tunings are already optimal and the search says so. If it is fifteen cents better, then guitarists tuning by ear for a particular piece — which they do, and describe as making it sound right rather than as tempering — have been doing something the arithmetic can name.