Acoustics & Music Theory
why it sounds like that
Theme

How many of the twelve

A second family of small integers, and they are not ratios. Seven notes chosen from twelve, three of those seven stacked, six semitones that split the octave evenly, three beats laid against two. These arguments never mention a frequency: the object is a set of positions on a circle, and the answers are counted rather than measured.
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.

 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.

 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.

 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.

 Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use. Rhythm and metre

As evenly as possible, which turns out to be a famous rhythm

Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba.

 3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things. Rhythm and metre

Two clocks at once, and where they agree

Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

 All 55 chords of 3 notes, on both counts. Every 3-note subset of the twelve pitch classes containing the root — 55 of them — with summed Plomp–Levelt roughness on a string spectrum across, and the largest whole number needed to write the chord in just intonation up. The smoothest is 0–5–10, 15:20:27, which is stacked fourths. The simplest is 0–5–9, 3:4:5, which is the major triad. Exactly one chord is in the best 18% on both axes: 3:4:5, the major triad. Intervals and chords

Three is the largest agreeable number

Take every chord of every size that can be built from the twelve, score each one for roughness and for the size of the whole numbers it needs, and ask how many are good on both counts. At two notes the answer is one. At three notes it is still one. At four it is eight, and the question stops having an answer.

 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.

 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.

 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.

 How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire. Harmony and voice leading

The chord that did not come

A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.

 Which traditions make the requirements that produce a comma. Four requirements and five tuning traditions. A comma is what is left over when a scale is generated by fifths, the octave is kept pure, and the chain has to close so that any degree can be a tonic — so a tradition that declines any one of them has no comma to hide. The best fifth each system actually contains is printed beside it, because declining a requirement has its own price. Scales and modes

Where the chain was never closed

Eleven earlier essays treat the comma as a fact about music. It is a fact about three requirements — that the scale is built from fifths, that the octave stays pure, and that the chain closes so any degree can be a tonic — and most of the world's tuning traditions decline at least one of them. What they pay instead is computable, and it is paid somewhere else entirely.

 The 7-note sets in which every interval occurs a different number of times. Each set of 7 notes containing C whose six interval counts are all different, with the counts printed. Every one of them is a rotation of one of two shapes, and only one of the two has steps a scale could use — the other is a run of semitones with the gap at the end. Scales and modes

Every interval a different number of times

Count the intervals inside a major scale and the six answers are 2, 5, 4, 3, 6 and 1 — six different numbers, no two alike. That is not decoration. It means the number of notes a key shares with a transposition of itself identifies the distance uniquely, so a listener who can only count common tones can still tell exactly how far a modulation went.

 One construction, a scale on one side and a rhythm on the other. Both rings are the same computation: k things placed as evenly as possible in n positions. On the left it is 7 notes in 12 semitones, which is the major scale; on the right 5 strikes in 8 steps, which is the cinquillo. The filled positions come from the J-function and the open marks from Bjorklund's algorithm, and they are the same set of positions turned by 9 and 6. Scales and modes

The same algorithm made a Cuban rhythm

Ask Bjorklund's algorithm for seven onsets in twelve steps and it returns 101101011010, which read as pitch classes is the natural minor scale. That function has been producing the tresillo and the cinquillo here from the beginning, and nobody has said that the scale field's central object comes out of it unchanged.

 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.

 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.

 Every triad, by how far it is from G7. All 24 chords ranked by the smallest total motion that takes G7 onto each, every note of the target sounded and one voice doubled where the sizes differ. The spread is narrow — 2 semitones at the nearest and 6 at the furthest — and the ranking is flat in the middle: 12 chords tie at four semitones. Harmony and voice leading

The resolution the metric cannot find

Rank every triad by how far its voices are from the dominant seventh and the tonic is not at the minimum. It sits in a twelve-way tie in the middle, the chord a deceptive cadence goes to is the furthest away of all twenty-four, and the two nearest are chords no cadence ever uses. What selects the two resolutions is not a distance — it is a direction, and it picks out exactly two triads from the twenty-four.

 The sizes a chain of a pure fifth will make. Chains of a pure fifth of two to 12 notes, each folded into one octave and drawn on a 1200-cent line. A chain has exactly two step sizes at 2, 3, 5, 7, 12 notes and three or more at every other size in this range. The five-note and seven-note cases are the chain's pentatonic and its diatonic set, and they are neighbours in a series rather than two separate facts. Scales and modes

The only sizes a fifth will make

Stack pure fifths and fold them into an octave, and at almost every number of notes the result has three or more different step sizes. At 2, 3, 5, 7, 12, 17 and 29 it has exactly two — and at no other size below thirty. The pentatonic and the diatonic set are not two discoveries. They are neighbours in one series, and it is the comma's series.

 The same census, in every universe from four to thirty. How many sets have all four properties, in a universe of n equal steps. None at all when n is two more than a multiple of four — 6, 10, 14, 18, 22, 26, 30 — exactly one when n is a multiple of four, and exactly two when n is odd. The multiples of four each have their survivor at n/2 + 1 notes generated by n/2 − 1 steps, so twelve's seven notes generated by the fifth is the general answer with n put at twelve rather than a fact about twelve. Pitch and tuning

Every universe has one, or none

Run the census that found the diatonic set in a universe of nineteen equal steps, or twenty-four, or fifty-three. The answer is completely regular and nobody appears to have written it down — none at all when the universe is two more than a multiple of four, exactly one when it is a multiple of four, and exactly two when it is odd.

 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

Nothing in the census knows which note is home

All four properties six earlier essays are about are invariant under rotation and transposition — one property tuple over all eighty-four rotations and transpositions of the diatonic set. So the census that separates 349 shapes cannot separate a major scale from its own Aeolian mode, and everything that makes one note a tonic is outside it.

 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 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.

 Every 3-note stack of one interval. Each chord built by repeating a single interval 2 times from the root, scored for summed roughness on a string spectrum and asked whether one low fundamental accounts for all its notes within 30 cents. 8 of the 11 are smoother than the major triad, which scores 0.288: stacked major thirds (an augmented triad) at 0.279, stacked fourths at 0.218, stacked tritones at 0.144, stacked fifths at 0.140. And 3 of them have no fundamental at all — stacked minor sixths, stacked major sixths, stacked major sevenths — meaning no series of harmonics up to the 16th accounts for their notes to within a 30-cent tolerance. Harmony and voice leading

A stack that is not thirds

Build every chord that repeats a single interval and score them all. Eight of the eleven three-note stacks are smoother than the major triad; the four-note stack of fourths is smoother than a dominant seventh by a third and has no fundamental any matcher can find. Two absences, both computable, and both are exactly what the chord was adopted for.

 3 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. diatonic (ditonic) puts them at 90 and 294 cents, giving steps of 90, 204, 204; intense chromatic puts them at 81 and 231 cents, giving steps of 81, 151, 267; enharmonic puts them at 63 and 112 cents, giving steps of 63, 49, 386. Against the twelve equal steps below, 1 of 3 land within twenty-five cents of a semitone at every degree; the worst miss is 37 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it. Scales and modes

A scale built downward from a fourth

A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.

 Ode to Joy as a path. Ode to Joy plotted as 30 notes against the 8 scale degrees it uses, one column per note. Its largest melodic interval is 2 semitones and it spans 7; the mean absolute step is 1.24 semitones. Beethoven, Ninth Symphony, finale, 1824 — the theme as first stated, eight bars. Scales and modes

A melody is a walk, not a set

Nine essays here are about which seven of the twelve a scale takes, and every one of them describes a set. A tune is not a set; it is a path across one, and the path is nearly all small steps. That is not a matter of taste. Above about eight notes a second the ear stops being able to hold a large interval and a small one in the same line, and at sixteen the choice disappears altogether — so a fast passage is scalar because a fast passage that leaps is two pieces of music.

 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.

 Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once. Scales and modes

A degree is where it goes next

The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.

 Euclidean up to the one thing a timeline is for. Five named timelines, each drawn above the Euclidean pattern with the same number of onsets in the same number of steps, with the rotation between them found by search. 3 of 5 are rotations of the Euclidean pattern and 2 are not Euclidean at any rotation — son clave, 3–2 and rumba clave, 3–2, whose gap sequences are 3·3·4·2·4 and 3·4·3·2·4 against the algorithm's 3·3·3·3·4. Where the match holds it holds only up to rotation, and a rotation is not a small difference: the algorithm has no way to produce a starting position, and a starting position is what a timeline is. Rhythm and metre

The rotation the necklace cannot see

Ask Bjorklund's algorithm for the world's timelines and the usual answer is that it produces them. Search every rotation of each Euclidean pattern for a match and the answer is more interesting: the tresillo is E(3,8) exactly, the bossa-nova and the standard bell pattern are rotations of theirs, and the son and rumba claves — the two best-known timelines in the world — are not Euclidean at any rotation whatever. Where the match does hold it holds up to a starting position, and a starting position is the one thing a timeline is.

 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 much of the writing the prohibitions forbid. I – vi – ii – V – I in C major, written in 3, 4, 5, 6 parts, with every ensemble covering the same total compass. A triad has three pitch classes, so n parts double n − 3 of them and a duet cannot state one at all. Of every ordered pair of complete voicings of V and I, the share with no parallel fifth or octave between any pair of voices: 91.2% at 3, 75.0% at 4, 44.3% at 5, 17.9% at 6. Harmony and voice leading

Why the exercise is in four parts

Eight earlier essays move four voices, and nothing here ever chose four. Cut one choir's compass into three parts instead and the two most famous prohibitions in music cost exactly nothing — the cheapest realisation already obeys them. Cut it into six and they cost more than half a semitone per voice per chord change, because a parallel octave needs a doubled note and six voices sharing three notes can hardly avoid one. Four is the smallest number of parts at which the rules have a price at all, and it is the largest at which the price is small.

 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.

 How many notes an octave can hold, asked twice. The share of trials on which a category is named correctly, against how many equal categories the octave is cut into, for a listener whose internal estimate carries 11 cents of noise — the logistic scale this site's identification figures already use, which is the thirty-cent transition the studies report. At 95 per cent accuracy the ceiling is 6 categories, and seven scores 94.9 per cent — on the line. The other ceiling is resolution: 151 to 356 difference limens fit in an octave depending on register, which is a factor of forty larger. Every system marked below sits between the two, and the marks separate: the number of degrees a mode uses clears the criterion, and the size of the gamut it chooses them from does not. Intervals and chords

How many boxes an octave holds

An identification model of pitch is usually handed twelve categories, and nothing ever asked how many an octave can hold. There are two answers and they are a factor of forty apart. Resolution allows between 151 and 356 — a listener can tell that many pitches apart in a direct comparison. Naming one of them without a comparison is a different faculty and it runs out at six or seven, which is where every mode in every tradition compared here sits. Turkish theory names fifty-three commas to the octave and a makam uses seven of them, and the gap between those two numbers is the whole of the argument.

 How unequal a beat is allowed to be. Five bars — 2+3, 2+2+3, 2+2+2+3, 2+5, 3+4 — with the range of subdivision durations at which every one of the bar's beats stays inside the window where a series of events can be a beat — a tenth of a second to two seconds. The metres made of twos and threes have the widest bands, and they have them for a reason that is not about the window: a group of four subdivisions decomposes into two twos and a group of five into a two and a three, so neither is a beat at all. The only indivisible lengths are two and three, which fixes the ratio between an aksak metre's long and short beats at 3:2 by arithmetic. Rhythm and metre

How unequal a beat is allowed to be

A Balkan bar of nine is four beats, three short and one long, and the long one is always three subdivisions against two. Never four against two, never five against three. The usual explanation is that this is what the tradition does; there is a better one and it needs no perceptual measurement at all. A group of four subdivisions decomposes into two twos and a group of five into a two and a three, so neither is a beat — it is two. The only indivisible lengths are two and three, which fixes the ratio at three to two before anybody listens, and leaves the tempo window to decide only how fast the bar can go.

 Three ways a category boundary could move, and how far each moves it. The predicted shift of one boundary against how strong the context is, for three mechanisms. Expectation alone — a listener who thinks one category 20 times more likely than the other — moves the optimal boundary by σ²·ln(odds)/Δ, which with the eleven-cent noise used here is 2.8 cents at ten to one and 3.6 at 20. Re-learning the centres from a context 30 cents away moves it by half of that, 15 cents. Selective adaptation moves it the OTHER way. The two directions are what an experiment would separate, and no absolute calibration is needed to do it. Perception and the listener

The boundary that barely moves

Every identification figure here has fixed category centres, and the essay before this one ended by admitting that real boundaries are supposed to move with context. Three mechanisms could move one, and their predictions are an order of magnitude apart and in two different directions. Expectation on its own — a listener who thinks one interval twenty times more likely than the other — is worth three and a half cents.

 The staff holds 11 positions and nothing fits in it. Each clef's eleven staff positions — five lines, four spaces and the space either side — as a bar on an axis that counts letters, with eight ranges laid underneath. The clefs step through the axis in thirds and cover fifteen positions of offset between them. Every range drawn is wider than eleven positions: the four voices span 13, 12, 13, 13 and the four instruments 24, 24, 25, 23, so the best clef for each still leaves 1 to 7 positions off the staff. A clef is a choice of which end sticks out. Scales and modes

The clef is an integer

The first essay on notation found that the staff's vertical axis counts letters rather than pitch, and named the clef as a question it was leaving open. Paid, it is arithmetic: a staff holds eleven letters, no voice or instrument is that narrow, and the eight clefs of European practice step through the axis in thirds — a spacing that buys everything a set of fifteen would buy on a wide range, for eight.

 Every result so far, against the listener's own noise. Three findings drawn against the one parameter all of them assume: how finely the listener resolves a pitch. At 11 cents — a trained listener, and the value every earlier essay used — the octave holds 6 nameable categories, twelve equal ones are named right 91 per cent of the time, and a 20-to-one expectation moves a boundary by 3.6 cents. At 35 cents it is 2 categories, 72 per cent, and 37 cents. The capacity falls roughly as one over sigma and the shift rises as its square, so the three curves separate rather than moving together. Scales and modes

The listener the model was never run for

Five earlier essays rest on one number — how finely a listener resolves a pitch — and every one of them used a trained listener's eleven cents. The model's dependence on it is not gentle: the capacity goes as its reciprocal and the expectation shift as its square, so an untrained listener at thirty-five cents has two nameable categories per octave rather than six, and a foreign tuning system is not mis-transcribed by them but absorbed.

 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.

 What eleven positions cover, at seven to the octave and at twelve. A clef's 11 positions, read as a range, in three systems. At seven positions to the octave they cover 1.57 octaves — an octave and a fourth, which is the seventh rung's own number. At twelve they cover 0.92. Writing a 44-semitone range then takes 3 staves and about 15 ledger positions on the staff, and 4 staves and 33 on a chromatic one. The accidentals a chromatic staff removes are paid for in vertical space, at a rate the two integers fix. Scales and modes

The notations invented for the overflow

Every proposal to replace the staff since the seventeenth century is a response to a specific overflow, and the commonest one — a chromatic staff with twelve positions per octave instead of seven — makes a trade computable from the same two integers the clef essay counted. It buys the accidentals outright and pays 40 per cent of the range, three ledger positions for one, and twenty-three enharmonic distinctions that are not merely absent from the page but unrecoverable.

 Every resolution claim here, against the number it rests on. How many equal steps of the octave can be named at 95 per cent accuracy, against the internal noise the model gives a listener. The laboratory value this collection quotes everywhere is 11 cents, which gives 6 nameable categories — the "about seven per octave" every claim here has been repeating. The laboratory measures it on isolated intervals and music never presents one, so the effective value inside a piece is smaller by an amount nobody has measured: at 4 cents it is 18, which is the chromatic scale, and the conclusion changes from "the ear has fewer boxes than the notation" to "it has exactly as many". Nothing here measures it. This is what it is worth if it moves. Scales and modes

The number every claim here has been quoting

Sigma is the internal noise a listener's pitch judgements carry, it is measured in a laboratory on isolated intervals, and music never presents an isolated interval. Every resolution claim here rests on the laboratory value. Sweep it and the headline finding moves: at eleven cents the ear has about six nameable categories per octave, and at six it has twelve — which is the chromatic scale, and turns 'the ear has fewer boxes than the notation' into 'it has exactly as many'.

 The one number the ordered key-finder was tuned on. For each rate of alternation between two keys, the cost of changing key at which the model stops hearing two keys and starts hearing borrowed chords in one. The threshold rises with the period — 0.95 at 1 bar, 0.95 at 2 bars, 0.95 at 4 bars, 2.00 at 8 bars, 3.50 at 16 bars — so the parameter and the rate trade off against each other exactly. The value tuned earlier, 2.2, sits above every threshold on this axis, which means its verdict about fast alternation was a consequence of the tuning rather than a finding about the music. Filled means the model names two keys; hollow means it names one and calls the rest borrowings. Scales and modes

A modulation and a borrowing are one number apart

The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning.

 Two kinds of evidence, each measured against its own chance. Every key as a point: across, how many standard deviations its cadence count is above what the same bars resampled would give; up, the same for its pitch-class profile correlation. The winner on cadences is key C at 6.2 standard deviations and on profile C at 1.1, and they agree. Standard deviations above chance are the same unit whatever produced them, which is the commensuration this figure is for — and it costs something: it assumes the two chances are equally interesting, which is a weighting in disguise. The obvious null does not work at all for one of the two: shuffling the bars leaves a pitch-class histogram exactly as it was, so its spread is zero and every key scores nothing against it. Harmony and voice leading

A count and a correlation

Cadence evidence is a count of ordered pairs and profile evidence is a correlation with a template, and the cadence essay refused to total them because they are not in the same units. Score each against its own chance and they are — standard deviations above chance are the same unit whatever produced them. Then the trouble moves: the obvious null does nothing at all to one of the two, because shuffling the bars leaves a pitch-class histogram exactly as it was.

 The passage built so the two statistics disagree. A body of chords diatonic to C major, of growing length, ended by a ii–V–I in G. The bag of notes says one key and the ordered pair says the other, which is the case the earlier figures never contained. The line is the cadence evidence for G, and under the axis is what each reading actually names at each length. The cadence reading holds G up to a body of 6 bars and is overturned at 8, so one explicit cadence is worth about that many bars of profile evidence — and it is overturned by the body's OWN incidental root motions rather than by the histogram at all. That is the finding the earlier essay could not have: on real material the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key. Harmony and voice leading

The passage built to make them disagree

A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

 How sure the reading is, bar by bar. Every earlier essay reports one best reading. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed and has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07. Scales and modes

The margin the dynamic program already had

Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.

 How long a bar of unequal beats can be, at a present of 3.5 seconds. Two quantities against the number of subdivisions in the bar. The bars are how many genuinely distinct unequal metres that length admits — groupings of twos and threes, up to rotation, discarding any that repeats a shorter grouping — and they run from one at 5 to 28 at 23. The line is how good a beat the best of those metres can manage once the whole bar is required to fit inside a psychological present of 3.5 seconds. It is flat at 0.865 up to a bar of 16 units, which is where the bar at the best subdivision first overruns the present, and falls after it: 17 at 0.830, 18 at 0.797, 19 at 0.765, 20 at 0.735. The supply of metres is still growing where the quality has begun to fall, so the lengths a tradition can use are a bounded prefix of an unbounded list. Rhythm and metre

How long a limping bar can be

The bound on an unequal beat turned out to be arithmetic, and the tempo window was left with only the tempo to decide. It decides nothing: every metre built from twos and threes gets the same answer, because the window is asked a yes-or-no question. Graded instead, an unequal beat costs 0.135 of the window's own preference at every bar length — and the constraint that does depend on length is the one nobody applied, that the whole bar has to fit inside the psychological present. At the subdivision that suits both beats best, a bar of sixteen units just fits and a bar of seventeen does not, which is where the supply of distinct metres has only started to grow.

 The twelve keys a key-finder has never had. Every scheme the key-finder reads, read twice: over the twelve major collections the model has always used, and over twenty-four with the harmonic minor added. The pale bar is the first and the dark one the second. On 5 of the 6 the extra twelve states change nothing a reader would see — the largest loss of certainty is 3.2 per cent, on the rondo — and no bar of any of them is renamed. The exception is the ostinato, every bar of which the twelve-collection model calls E♭ and the twenty-four-collection model calls C minor — the same seven notes, the right name. So the missing states were not costing the key-finder its answers. What they were costing is the ability to say which of a collection's seven degrees is home, and that is a different repair. Harmony and voice leading

The key-finder with no tonic

Thirteen essays of key-finding have run over twelve major collections and not twenty-four keys, so a passage in A minor is read as C. Adding the missing twelve costs almost nothing and fixes almost nothing — because the model has no tonic in it at all, and below three raised sevenths in a passage the extra states are worth exactly zero.

 6 step patterns, 2 category counts, and only the count matters. Each scale drawn as its own steps across the octave, with the share of trials a listener whose internal noise is 11 cents names correctly — and, beside it, the same figure for a scale of equally spaced degrees of the same size. The two agree to three decimal places on every row, though the narrowest step here is 100 cents on the diatonic major, tempered and the widest is 300. Naming is lost at boundaries, an n-degree scale has n of them wherever they are put, and each costs the same as long as no category is narrow enough for the noise to carry an estimate clean across it — 9.1 standard deviations, at the worst here. Scales and modes

A boundary costs the same wherever it is put

Every capacity figure drawn until now cuts the octave into equal categories, and no scale in the world is equal. Putting the real step patterns through the same model returns exactly the same accuracy to four decimal places — because naming is lost at boundaries, an n-degree scale has n of them wherever they are, and each costs the mean absolute value of the noise. That turns the most-quoted result about hearing from a search into one line: 1504 times one minus the criterion, over sigma.

 Which intervals in a key can be mistuned invisibly, and from which end. Every interval between two degrees of the major scale, ranked by how differently its two ends treat a 25-cent departure on notes of 0.25 seconds. The C to B is the most lopsided, at 47 points: a mistuning on its upper note reaches the listener nearly 2.5 times as strongly as the same mistuning on its lower one. The D to E is the most even, at -0. A negative bar is an interval whose LOWER note is the one that carries a mistuning into the listener, which happens whenever the lower degree is the less specified of the two. No account of interval perception predicts a table like this, because an interval is usually treated as one quantity rather than as a difference of two estimates. Intervals and chords

Which end the mistuning is on

Twenty-one intervals in the major scale, each with two ends, and the same twenty-five cents reaches a listener at anywhere between 32 and 79 per cent of its size depending on which of the two notes carries it. The most lopsided is the tonic to the leading note, where a departure on the upper note arrives two and a half times as strongly as the same departure on the lower. Two mechanisms produce it and they can be separated by one flag: two thirds of the asymmetry is register and one third is the key.

 One noise everywhere, and the noise each boundary actually gets. The seven boundaries of the tempered diatonic scale, with the noise the earlier model gives each of them — 11 cents, the same everywhere — and the noise the harmonicity model gives them instead, which runs from 11.0 cents to 35.6. The error rate is the sum of these over the octave rather than seven copies of one, so it rises from 5.1 per cent to 11.8. And the moment the boundaries differ, where they are put matters: a scale that moved its degrees would move its boundaries onto different intervals and would pay a different sum. That is the earlier null broken by the assumption its own last section named. Scales and modes

A boundary beside a fifth

A closed form established earlier says an n-category division costs n·σ·√(2/π) whatever the widths are, so a boundary costs the same wherever it is put and the step pattern cannot matter. Its own last section named the assumption that produces the null: one σ, applied to every boundary in the octave. Let σ follow how securely each interval is held and the formula becomes a sum over boundaries rather than n copies of one — the diatonic's error rises from 5.1 per cent to 11.8, and where the degrees are put matters again.

 The best seven of the twelve is a scale nobody has ever used. All 462 ways of choosing seven of the twelve semitones with the tonic fixed, ranked by the identification error the harmonicity model gives them. The best is C C♯ F♯ G A♭ B♭ B at 8.8 per cent and the worst is 13.4; the diatonic major sits at rank 376, in the worse fifth of the ranking, at 11.8. The optimum is a cluster of semitones around the tonic and around the fifth, and the reason is visible in the criterion rather than in music: a boundary next to the unison or the fifth is a boundary with very little noise on it, so the cheapest way to satisfy this measure is to crowd the degrees where the model says the ear is sharpest. A criterion whose optimum is a scale nobody plays is a criterion that is not what scales are chosen for, and the useful reading of this drawing is that rather than its winner. Scales and modes

The best seven of the twelve

Once the noise is allowed to differ from boundary to boundary, a scale can be chosen to minimise identification error — and the choice is a search over four hundred and sixty-two sets rather than an argument. Run, it returns a cluster of semitones around the tonic and around the fifth, and puts the diatonic major at rank 376 of 462, in the worse fifth of the ranking. A criterion whose optimum is a scale nobody has ever played is a criterion that is not what scales are chosen for, and the reason it fails is legible in the model rather than in the music.

 Each tradition's own steps against the equal division of the same size. Six scales, each drawn against the equal division into the same number of degrees, under both models of how securely an interval is held. On the harmonicity model the tempered diatonic is 11 per cent better than seven equal steps; on the profile model the same comparison is 0.9 per cent, which is nothing. The two models disagree about the one comparison anybody would want the measure for, and only one of them is free of circularity: the probe-tone profile was measured on listeners raised inside the diatonic tradition, so using it to explain why the diatonic is well chosen assumes the answer. The harmonicity model assumes only that a simple ratio is easier to hold than a complicated one. Scales and modes

The unequal scale that is easier to name

The whole of the earlier result was that a scale's step pattern cannot matter, and every tradition it drew agreed with the equal division of its own size to three decimal places. With one noise per boundary the comparison is live again, and the tempered diatonic beats seven equal steps by eleven per cent — while the pentatonics gain nothing and the maqam scales gain two tenths of one per cent. Only one of the two security models produces the effect, and it is the one that was not measured on listeners raised inside the tradition it is being used to explain.

All themes