Acoustics & Music Theory
why it sounds like that
Theme

The same shape, moved

A chord, a scale and a rhythm are all patterns that keep their identity when shifted. What survives the shift is what the theory is about.
The major scale as a cycle. The twelve semitones drawn as a cycle, with the notes of the scale filled in. The gaps between filled positions are the step pattern, and reading them round the circle is what makes the scale's asymmetry obvious. Scales and modes

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.

 The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time. Scales and modes

The same seven, started later

A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to.

 The circle of fifths. The twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering. Scales and modes

Keys are neighbours, and the map is computed

Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

 Stopping the chain of fifths at every length. The chain of fifths taken two notes at a time, three, four and so on, with each collection folded into a single octave. The smallest gap in the collection is measured at every stage: it is two semitones until the sixth note arrives, and then it is one. Five is as far as the chain goes without producing a semitone. Scales and modes

Five notes, and no semitones

The five black keys sound acceptable in any combination, in any order, over almost any bass note. That is not mysticism and not luck — five is the longest a chain of fifths can run before it produces a semitone, and the number is derived rather than chosen.

 The triads each scale builds on its own degrees. Every chord that can be stacked in thirds on each degree of each scale, with its quality worked out from the intervals the scale actually supplies. The quality of the chord on each degree is a consequence of each scale's own step pattern rather than of a convention. Scales and modes

What a raised seventh is for

The harmonic minor is usually taught as a scale with an odd gap in it. It is better understood as a repair to a single chord — raising the seventh degree turns the dominant triad from minor to major, and everything else about the scale is the bill for that.

 Chords as stacked intervals. Each of 4 chords — major, minor, diminished, augmented — drawn as the semitone distances above its root, with the nearest simple frequency ratio beside each note where the 5-limit table has one. Where the chords are triads they differ only in the size of the two stacked thirds, and that difference is the whole of their character. Harmony and voice leading

Three notes at once, and why these three

A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.

 How far the voices have to move. Three chord changes with their voices joined by the assignment that moves the least in total. The distance is computed over every way of pairing the notes, so a change that shares two of its three notes shows as two flat lines. Harmony and voice leading

The shortest move, which is what a chord change is

Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones.

 The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense. Harmony and voice leading

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

 Every interval, and what it becomes when it is turned over. Each interval within the octave paired with its inversion — the interval left when the lower note is raised by an octave. The two always add to twelve semitones, so exactly one interval can be its own inversion, and it is the one at six. Harmony and voice leading

The interval that inverts to itself

Six is the only number that divides twelve into two equal halves, so the tritone is the only interval unchanged by being turned over. That symmetry is not a curiosity — it is why one tritone belongs to two dominant chords, and why they resolve to keys a tritone apart.

 The twenty-four triads, and the three moves between them. Every major and minor triad, placed on the single cycle that alternating two of the three transformations produces. Neighbours round the ring differ by one voice; the chords across the middle are the third transformation. Every edge's cost is the measured voice-leading distance, and none of them is more than two semitones. Harmony and voice leading

Chords as a space

Three operations turn any triad into another by moving one voice. They generate all twenty-four major and minor triads in a single cycle, their costs are one, one and two semitones, and the map that results is a geometry rather than a list of rules.

 How many sizes each interval comes in. Each generic interval of the diatonic scale and of a seven-note set that is not a mode of it, with the specific sizes it takes as the starting degree moves round. The first gives exactly two sizes for every one of them; the second gives 3, 4, 4, 4, 4, 3. Of the 462 seven-note selections from the twelve that contain the tonic, 14 have two sizes for every generic interval — 3.0% of them, and they are the rotations of just 2 step patterns: 1·1·1·1·1·1·6 and 1·2·2·1·2·2·2. Scales and modes

Two sizes of every step, which is why the names work

A third is three semitones or four, and every musician learns to call both of them thirds without being told why that is allowed. It is allowed because of a property the diatonic scale has and 448 of the other 461 seven-note selections from the twelve do not.

 thirty-two-bar AABA, every bar against every other bar. A self-similarity matrix of 32 bars of thirty-two-bar AABA. Each cell is the cosine similarity of two bars' pitch-class vectors, with the chord's own notes weighted 1 and the rest of the key 0.5. Similarity is quantised into four bands for drawing and anything under 0.35 is left as paper. Nothing in the computation knows what a section is; the blocks and stripes are what the arithmetic returns. Form and structure

A piece is mostly itself again

Take a piece of music, encode each bar as the notes sounding in it, and compare every bar with every other bar. The picture that comes out has blocks and stripes in it, and those blocks and stripes are the form — arrived at by arithmetic that has never heard of an exposition, a chorus or a refrain.

 Two ways to fill eight bars, and only one of them accelerates. The period against the sentence, drawn as the lengths of their constituent units against position on a grid of eight bars. The ratio beside each row is the mean unit length in its second half divided by the mean in its first: the period at 1.00, the sentence at 0.67. A ratio below one is an acceleration — the unit shortening as the phrase approaches its arrival — and a ratio of one is a plan whose unit never changes length. Form and structure

One of these eight-bar phrases accelerates

The sentence and the period both occupy eight bars, both end with a cadence, and both are recognised by ear rather than counted. What separates them is arithmetic. One halves its unit halfway through and the other does not, and the difference comes out as a single ratio — 0.67 against 1.00 — computed from nothing but the lengths of the parts.

 Five signals, computed separately, and no total. The five components of closure for 6 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted. Form and structure

What makes an ending an ending

Cadences are ranked. The authentic one is strong, the plagal weaker, the deceptive weaker still — and the solver here measured the quantity that ranking is usually explained by and found it says something else entirely. What survives is not a weaker version of the ranking but a different kind of object, with five components and no total.

 Five signals, computed separately, and no total. The five components of closure for 4 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted. Form and structure

An ending that exists so a bigger one can

Half of the cadences in tonal music are built to fail. A phrase that stopped convincingly at bar four would be a piece four bars long, so the ending at bar four is engineered to arrive and not to settle — and the components it withholds are exactly the ones its partner at bar eight supplies. Closure is nested, and the nesting is what turns two phrases into one thing.

 The key plan is the shape. The key of a classical sonata-form movement against position in the movement, measured in steps along the chain of fifths from the home key. The positions are the proportions such a movement is described by rather than bar numbers from any one score. The furthest point is 4 steps out, sharing 3 of seven notes with home. Form and structure

The key plan is the form

The large shape of a classical movement is not a shape at all, it is a journey — out to one key and back. Which key is not a matter of taste. Of the two keys that share six of their seven notes with home, only one introduces a note the home key does not use in any of its chords, and that note is the arriving key's own leading note. The departure is audible because of one accidental.

 Twelve stages, and the composer chose the rule. Every rotation of a 12-step pattern with 8 onsets against the unrotated original. Each row is one stage of a phase piece: the filled cells are what is heard when the two parts sound together, and the count beside it is how many of the two parts' onsets coincide. The number of stages is the length of the pattern, so the length of the piece is arithmetic. Rhythm and metre

A process that enumerates its own form

Take a twelve-step pattern, play it against itself, and move one copy along by one step at a time. The piece is over when the copy returns to where it started, so its length is twelve — arithmetic, not a decision. What is heard at each stage is the union of the two parts, and nobody composed any of it.

 Every delay, scored — a round and a tune that is not one. Mean Plomp-Levelt roughness of the simultaneities a tune makes against a copy of itself entering a whole number of bars later, for Frere Jacques and Twinkle, twinkle. The two do not overlap: the worst delay of the first is smoother than the best delay of the second. The model sees roughness and nothing else — no voice leading, no parallels, no distinction between a passing dissonance and a structural one. Harmony and voice leading

A melody that can accompany itself

Whether a tune works as a round is decidable before anyone sings it. Score every simultaneity it makes against a delayed copy of itself, at every delay, with the roughness model already in use, and two nursery tunes separate completely — every delay of one is smoother than every delay of the other. What the scan does not do is pick out the entry the tradition uses, and that refusal is the most informative thing in it.

 The least rough 7 notes of the twelve, at 262 Hz. Every 7-note selection from the twelve that contains C, scored for total Plomp–Levelt roughness at a root of 262 Hz, ranked. The best 8 are shown with the spread between them. The major scale ranks 5 of 462; the first selection that is a mode of the diatonic set ranks 1. Change the root and the ranking changes, because roughness is a fact about frequencies and a scale is not. Intervals and chords

Roughness cannot choose a scale

Score all four hundred and sixty-two seven-note selections from the twelve for roughness and the answer at middle C is a mode of the diatonic set, first out of four hundred and sixty-two. Ask the same question an octave lower and the same selection ranks a hundred and forty-first. The model is not wrong; it is answering a question about frequencies, and a scale is not one.

 Four properties, ten sizes, one survivor. Every subset of the twelve pitch classes, counted by shape, at each size from two notes to eleven, and how many shapes of each size have each property. Exactly one shape in the whole table has all four at once: the 7-note set with steps 1221222, which is the diatonic scale. Scales and modes

Why seven

Four properties, each shared with some other set. Run all four over every subset of the twelve, at every size from two notes to eleven — three hundred and forty-nine shapes in all — and exactly one has all four at once. It is the diatonic set, and it is the only survivor in the whole twelve-note universe.

 Sensitivity and specificity on one dial. Aligned similarity — bar i against bar i+L, which is what a return is — for 4 eight-bar comparisons, as the key-invariance dial turns. One comparison is constructed: a literal repeat in the encoding, moved up a fifth, which is a stated manipulation because no scheme encoded here repeats a section in a new key. The shaded band is the margin between the two named comparisons, and it runs from 0.021 to 0.106. Form and structure

The same thing somewhere else

A measure built on which notes are sounding calls a passage that comes back a fifth higher a stranger. There is a dial that fixes this, and turning it is supposed to be a trade — more sensitivity to a transposed return, less specificity against a coincidental one. It is not that trade. Two different statistics answer opposite ways, and the setting that would compromise between them is the worst one available.

 Both criteria, measured from C major. The 23 chords other than C major placed by the smallest total voice motion and by how many notes are held in common. Agreement between the two criteria would be a staircase; the cells drawn in the discrepancy colour are the ones that break it, and over the whole set of 24 chords the two criteria order a pair of destinations oppositely in 72 of 6072 comparisons. A minimal assignment drops a common tone it could have held in 0 of the 552 pairs. Harmony and voice leading

Two criteria that are taught as one

Move the least and keep the common tones arrive in one breath, as though they were one instruction. They are two different functions, and enumerating every pair of chords settles exactly where they agree: as a way of connecting two given chords they never once disagree, and as a way of saying which chord is nearer they disagree about one comparison in eighty.

 Which rotations have a fifth above their own tonic. Each rotation drawn by scale degree, with the degree a perfect fifth above the tonic marked. 1 of the 7 does not have one: Locrian. A tonic without a fifth above it has no triad of its own, which is a stronger disqualification than any preference. Intervals and chords

The mode with no fifth

Six of the seven rotations have a perfect fifth above their own first degree and one does not, which is a structural disqualification rather than a preference. And the set's single tritone sits on a different pair of degrees in every rotation — on the fourth and seventh in Ionian, on the tonic itself in two others — which decides most of what each mode can do.

 The triad on every degree of every rotation. The quality of the triad built on each degree of each rotation, computed from the set rather than tabulated. The triad on the fifth degree is major in 3 of the 7: Lydian, Ionian, Locrian. Of those, Locrian has no triad on its own first degree to resolve to, and Lydian has its fifth degree on the note the parent collection is built from, so the cadence would confirm the parent rather than the mode. What is left is Ionian — one rotation of the 7, and the others have to end some other way. Harmony and voice leading

One dominant among seven

The triad on the fifth degree is major in three of the seven rotations. One of those three has no triad on its own first degree, and one has its fifth degree on the note the parent collection is built from — so exactly one rotation of the seven has a dominant, and the other six end by moves that cost twice as much in voice leading.

 The rotations of the diatonic set, brought to one tonic. The same rotations started on the same note rather than on their own, ordered by how many of their degrees are raised. Every step lowers exactly one note by a semitone, and the notes it lowers, in order, are F♯, B, E, A, D, G — which is the chain of fifths read backwards. A rotation is a rotation whatever the scale; the chain is a property of a set generated by a single interval, and it disappears the moment the set is not. Scales and modes

Seven rotations that are not seven modes

Rotate harmonic minor and the result is not a family the way the diatonic modes are a family. Four of its six generic intervals come in three specific sizes rather than two, so a fourth might be four, five or six semitones; three of its seven rotations have no fifth above their own tonic; and four have a tonic inside a tritone. The word mode has been doing two different jobs.

 The rotations of the diatonic set, brought to one tonic. The same rotations started on the same note rather than on their own, ordered by how many of their degrees are raised. Every step lowers exactly one note by a semitone, and the notes it lowers, in order, are F♯, B, E, A, D, G — which is the chain of fifths read backwards. A rotation is a rotation whatever the scale; the chain is a property of a set generated by a single interval, and it disappears the moment the set is not. Scales and modes

Parallel and relative are two different maps

Bring the seven modes to one tonic and order them by brightness, and each step lowers exactly one note by a semitone — and the notes it lowers, in order, are F♯, B, E, A, D, G, the chain of fifths read backwards. Do the same to harmonic minor's rotations and each step moves two, three or four notes at once. The famous chain belongs to the generator, not to rotation.

 How far every key is from C major. The 11 other major keys under three measurements. Notes in common and steps round the circle are the same measurement — the first is seven minus the second until it bottoms out at two — and voice-leading distance between the tonic triads is not. E major shares 3 notes and is 2 semitones away; A♭ major shares 3 notes and is 2 semitones away, against the dominant's 6 and 3. Harmony and voice leading

Three ways to measure how far a key is

Notes in common and steps round the circle of fifths are the same measurement — among the twelve major keys the first is seven minus the second until it bottoms out. Voice-leading distance between the tonic triads is a different one, and it disagrees in exactly one place: E and A♭ are four steps from C, share three of its seven notes, and are two semitones away, which is nearer than the dominant.

 The map is two-dimensional and the circle is one axis of it. Major keys along the top, each a fifth from the last; underneath each, its relative minor, which shares all seven of its notes. Moving sideways changes one note; moving down changes none at all and changes the tonic. Among the twelve major keys alone there is only the sideways move, which is why that map really is a circle — the second axis needs the minor keys to exist. Scales and modes

The circle is a circle, and the map is not

Among the twelve major keys, notes in common is a strict function of distance round the circle of fifths — one value for each step count, no exceptions — so there is nothing else to measure and the map really is one-dimensional. A second axis appears only when the minor keys are added, and it is a different kind of move: the relative shares all seven notes and the parallel is one semitone away.

 The arch is not a preference. Every sequence of 6 notes over 8 scale degrees — 262,144 of them, enumerated rather than sampled — classified by contour, under three constraints. With none, the nine classes are spread. Requiring the sequence to return to its starting degree leaves only the arch, the valley and the flat, at 42.2 per cent each for the first two. Requiring it to begin and end on the LOWEST degree leaves the arch alone, at 99.6 per cent. Nothing here prefers a rise followed by a fall; the constraint is that the melody comes home, and a melody that comes home from below has nowhere to go but up first. Form and structure

The shape that survives everything else

Throw away a melody's key, its tuning, its instrument and the sizes of its intervals, and what is left is a string of pluses and minuses. That string is what a listener who cannot name a note still has, and it costs 37 per cent of the tune to keep. The arch that melodic shape is famous for is not in it as a preference: enumerate every six-note sequence that begins and ends on the lowest degree it uses and 99.6 per cent of them are arches, because a melody that comes home from below has nowhere to go first but up.

 The chain the page counts on. Every way of writing a pitch with up to one accidental, laid out along the chain of fifths and grouped by the pitch class each one sounds as. The vertical axis is what a keyboard has and the horizontal axis is what the page has: the spellings of one pitch class are seven steps of a fifth apart, which is exactly the distance a comma is measured over. Notation did not choose an arbitrary redundancy; it kept the coordinate a tuning system is built in, and equal temperament is the projection that loses it. Scales and modes

The stave is not a ruler

A hundred and eighty essays here draw pitch against an axis somebody computed. The one axis every reader already owns is the five lines, and it is not a pitch axis at all: it counts letters. Seven positions carry twelve pitches, so the same vertical distance is two intervals before an accidental is allowed and six after — and the accidental is not an extra symbol on a complete scale but the repair for a scale with five values missing.

 What one progression leaves open. Realisations of I–IV–V–I in four parts with no parallel fifths or parallel octaves, counted exactly by a dynamic programme over the voicings rather than sampled. The chord symbols admit 16,100,352,296; the Roman numerals 59,418,496; the figured bass 2,042,672. The three notations differ by four orders of magnitude, and every one of them was in daily professional use. Harmony and voice leading

Three notations, one progression

A figured bass, a Roman numeral and a chord symbol are three professional notations for the same four chords, and the number of four-part realisations each of them admits can be counted exactly rather than argued about. With no parallel fifths or octaves the counts are sixteen billion, fifty-nine million and two million: a factor of eight thousand between the loosest and the tightest. What each one collapses is what its tradition thought a chord was, and the three do not agree.

 The mode is its tritone. Every set of degrees that lies inside at least one of the seven modes — 510 of them — scored by how many modes it leaves standing, with the 15 minimal sets that decide printed in full. All 15 have two members and every one of them either is a tritone or contains the tritone above the tonic. The only degree every mode has is the tonic itself; there is no degree belonging to just one mode. Averaged over every order the degrees could arrive in, 5.67 of the seven have to have been heard before the mode is settled — most of the scale, because the deciding pair arrives when it arrives. Scales and modes

The one note that decides the mode

A key signature names the seven and not the rotation, so the mode has to be heard. Enumerate every set of degrees that lies inside at least one of the seven modes — 510 of them — and count how many modes each leaves standing. Fifteen minimal sets decide, every one has two members, and every one is the mode's own tritone. The only degree all seven share is the tonic; no degree belongs to a single mode; and averaged over the orders the degrees could arrive in, 5.67 of the seven have to have been heard first.

 How many modes a set has, and why some have fewer than notes. Every non-empty subset of the twelve — 4095 of them — sorted by size, with how many have a transposition that returns the same set. 75 do, which is 1.8 per cent, and they reduce to 16 distinct step patterns. A set of size k with a symmetry of order s has exactly k/s distinct rotations, so the number of modes is arithmetic rather than musical. Sets of five, seven and eleven notes have none at all, because those sizes share no factor with twelve — which is why every seven-note scale has seven modes before any musical question is asked. Scales and modes

The set with fewer modes than notes

Eight earlier essays rotate one collection, and the seven modes were never a result — they are arithmetic. A set has as many modes as it has notes divided by the order of its own transposition symmetry, so the whole-tone scale has six notes and one mode, the octatonic has eight and two, and the question asked just before — how much has to be heard before the mode is settled — is not merely hard for those collections but undefined. Seventy-five of the twelve-note universe's 4,095 subsets are built this way and they reduce to sixteen step patterns. Five, seven and eleven notes cannot be among them, which is why every seven-note scale has seven modes before anybody plays one.

 The test a histogram cannot pass. Two passages built from the same two keys: one takes them in turn and the other sounds them together. Over a window long enough to hold both, their pitch-class histograms are 98.6 per cent alike, so they are very nearly the same object to a profile model — and it names two keys in both. The ordered model names up to four for the alternation and one for the simultaneity, because a simultaneity has no sequence in it that any single key's grammar will not fit. Harmony and voice leading

A key-finder that keeps the order

Every key-finding model until now begins by throwing order away — a histogram over a window, correlated against twenty-four profiles — and the thing that most obviously declares a key is an ordered pair of chords. A model that keeps the order costs 84 states and 7,056 transitions against 24 hypotheses and none, and what it buys is the one test a histogram was said not to pass: two keys in turn and two keys at once have histograms 98 per cent alike, and are two different sequences.

 Two passages, one note apart, and opposite cadences. Two passages that share eight bars drawn from the six pitch classes C major and G major have in common, and differ only in their last three bars: one cadences in C and one in G, and the single note that separates them is an F against an F sharp. The cadence evidence names C for the first and G for the second, by 9 to 3 and 6 to 4. The profile model names E minor for both, with correlations differing in the third decimal. Harmony and voice leading

The cadence as evidence

Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.

 A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players. Rhythm and metre

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

 Leaps do not fall where offbeats do, and a reader gets the difference free. Where each size of melodic move actually lands in the bar, over the 101 moves of the three tunes measured here. The two axes are priced separately everywhere and they are not independent: the mutual information between them is 0.31 bits a note, which is 20 per cent of the smaller of the two. That is the amount the sum over-charges. A reader who has seen where a note falls already knows something about how far it moved, so the joint cost is 3.47 bits rather than the 3.79 the two axes add to — and every reading load computed so far is high by the difference. Scales and modes

Leaps do not fall where offbeats do

Every reading load computed so far is a sum of two terms priced as though the axes were independent, and an earlier essay named the interaction it could not reach. Measured on the same hundred and one notes every other essay uses, the mutual information between how far a note moves and where it falls in the bar is 0.31 bits — a fifth of the smaller axis, and a sixth of a note's total load. Every reading load published so far is high by that amount, and the quantity saturates at exactly the grid the tunes are notated on, which is the check that it is measuring the music rather than the grid.

 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. 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.

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