Field

Scales and modes

Which seven of the twelve, and what changes when the same seven start somewhere else.

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.

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.

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.

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.

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.

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.

Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.

A scale is not a set of pitches

Two ragas can have identical pitch sets and be different ragas. Two national theories of one maqam put its third degree thirty-five cents apart. Both facts are fatal to the idea that a mode is a collection of notes, and both are ordinary in the traditions concerned.

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.

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.

Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through 19.02 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 7/5, on 11/7, on 5/3, on 9/5, on 11/5, on 7/3, on 13/5, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

A scale without an octave, and the spectrum that asks for it

Every scale so far repeats at the 2:1. That looks like a law and it is a consequence — the octave fuses because every partial of the upper note lands on one of the lower. Take a spectrum with no even partials and the 2:1 stops doing that, while the 3:1 starts.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

What the engraver used here gives a note, measured off the page. The horizontal distance VexFlow allots each duration, read back off a formatted system rather than quoted from a manual. It is not a power of the duration: it is a constant of 58 points plus 15 points a crotchet, and the constant is 93 per cent of the width the shortest note here gets. A note four times as long as another is about 2 times as wide, not four. The floor is the notehead, its stem, its accidental and the space a reader needs to see them as separate events — which is a claim about legibility and not about time at all.

The axis that is not a time axis

Eight earlier essays have measured the staff's vertical axis to a position. Its horizontal one has never been asked about, and the answer is that it is proportional to nothing: measured off the typesetter used here, a note gets 58 points before its duration is considered at all and 15 points a crotchet after — so the constant is 93 per cent of what the shortest note gets, and a note four times as long is not four times as wide.

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.

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.

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.

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.

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 much music each notation fits on a page. The two axes multiplied. Vertically, a system is as tall as the staves the range needs; horizontally, a system holds as many notes as fit once the shortest is wide enough to read. the staff, seven to the octave: 3 staves to the system, 4 systems and 38 notes to a system, 152 notes to the page — 46 seconds at 100 beats a minute, so a page turn every 46 seconds; a chromatic staff, twelve to the octave: 4 staves to the system, 3 systems and 38 notes to a system, 114 notes to the page — 34 seconds at 100 beats a minute, so a page turn every 34 seconds; a whole-tone staff, six to the octave: 2 staves to the system, 7 systems and 38 notes to a system, 266 notes to the page — 80 seconds at 100 beats a minute, so a page turn every 80 seconds. a whole-tone staff, six to the octave holds 2.33 times what a chromatic staff, twelve to the octave does, which is a difference of 1.00 page turns a minute — and a page turn is a thing a player with two hands occupied cannot do.

How much music a page holds

Nine earlier essays have measured notations, and every one of them is a page — a two-dimensional object read in a fixed order by a reader who has to turn it. One measured the vertical axis and another the horizontal, and multiplying them gives the one design constraint on notation that is not about legibility at all: a chromatic staff turns pages a third more often than an ordinary one, and a proportional spacing rule turns them nearly twice as often as a columnar one.

How much of the reading comes from what has not happened yet. Every margin reported earlier is two-sided: the best path through a key at a bar is the best score into it plus the best score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line, because the dynamic program already had both halves separately. The mean margin falls from 3.90 bits with hindsight to 1.79 without it, so 54 per cent of this passage's certainty is retrospective. The two passes never disagree about which key is best here, so the hindsight buys confidence rather than a different answer. This is the quantity every earlier essay has assumed and none has measured.

How much of the reading arrives late

Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.

Every standard rastral size against the two bounds a reader imposes. Print the notes larger and the eye-hand span stops fitting inside one fixation, so the reader has to saccade ahead faster than the eye can move. Print them smaller and a notehead stops subtending enough angle to be identified. Both bounds come from the reader and neither from the music. At 100 beats a minute with 2 notes to the beat, the acuity bound sits at 1.63 millimetres and the saccade bound at 7.5 — so the saccade rate is nowhere near binding and acuity is doing all the work, which is the opposite of what the eye-hand span suggests. 5 of the 9 standard rastrals clear the acuity bound: rastral 4 and larger. Those are exactly the sizes used for parts, and the ones below are used for study scores — which are read at a desk rather than played from at a stand, and a shorter viewing distance moves the bound with them.

The page is read by an eye

A sight-reader's eye sits a fixed number of notes ahead of the sounding one and a fixation takes in a fixed number of millimetres, and the spacing rule converts between them. Two bounds follow, from the reader rather than from the music — and the one everybody would expect to bind does not. The saccade rate has enormous headroom at any playable tempo, and what decides is acuity.

The same eight notes are four times as much to read. How many bits each note of a line carries, taken as minus the log of the probability of the interval that reached it, under the distribution of melodic steps measured over the tunes used throughout. A scale costs 1.76 bits a note and a wide leaps costs 7.02 — a factor of 4.0 at the same number of notes on the page. Every quantity computed until now counts notes, and the page cannot tell these apart: eight quavers are eight quavers of horizontal space whichever line they spell.

A reader does not read notes

Eleven earlier essays count notes, and the page cannot tell one line of eight quavers from another. A reader can: a scale of eight is one object where eight leaps are eight. Measured against the melodic interval distribution, the same eight notes are four times as much to read — and the eye–hand span, the best-measured quantity in the reading literature, is four notes of a tune and one of a leaping line.

one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most.

The scale least committed to its own instrument

The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

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.

The term that was owed, and the corpus cannot hold it. For each of the three tunes everything here is measured on, how many of its notes have a duration that differs from the gap to the next onset. The answer is none, in 101 notes: these tunes are stored as a list of pitches and lengths with no rests in them, so a note's duration IS its inter-onset interval and conditioning one on the other leaves exactly zero bits. That is a fact about the representation rather than about music. The prediction was that the term would be small, and it could not have been known that the corpus would make it identically zero — which means the prediction cannot be tested here and the exceptions have to be priced directly.

A note lasts until the next one starts

Pricing where a note is against which note it is left duration as the term it had not, with a prediction that it would be small. Measured on the three tunes these readings are built on, it is exactly zero — and it is zero by construction, because those tunes are stored as pitches and lengths with no rests in them, so every duration is its own inter-onset interval. The prediction cannot be tested on the corpus that produced it. Priced directly, a rest costs 0.67 bits a note where a tenth of the notes have one, which is not well under half a bit.

The notehead that is not a note

Every quantity so far is charged per notehead, and a tie is the one mark on the staff that puts a notehead on the page carrying no event. Its cost is not the decision that identifies it — that is half a bit where a tenth of the noteheads are continuations. It is the decision plus the whole reading of a notehead that turns out to have been unnecessary, which is 1.05 bits, twice the decision and a fifth of what a note of music costs. Set beside a dot and a longer note value, the tie is five times the price of either and is the only one of the three that can cross a barline.

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.

One number a page, and what a hard rhythm buys against a hard tune. Every combination of six kinds of line and seven kinds of rhythm, placed by what each axis costs a reader. The duration term (0.67 bits) and the interaction (0.31) are the same for every cell, so the diagonals are pages of equal difficulty and the exchange rate between the two axes is the slope of one. The pitch axis spans 5.26 bits across the six lines and the position axis 4.46 across the seven rhythms, so a composer choosing between the hardest line and the hardest rhythm is choosing between quantities within 18 per cent of each other. The hardest page is wide leaps in off the beat at 14.0 bits a note and the easiest is a scale on the beat at 4.2.

One number for a page

Four terms and an interaction give a single bit rate per note, and with it the exchange rate a long run of essays has been pointing at. Six kinds of line span 5.26 bits and seven kinds of rhythm span 4.46, so a composer trading a harder tune against a harder rhythm is trading quantities within eighteen per cent of each other — and pages that look nothing alike sit on the same contour. The hardest page on the grid costs 13.96 bits a note and the easiest 4.24, a factor of three and a half, and the subject closes there.

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.

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

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.

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.

Two cues meet in a corner

The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.

A bass line is not a list of roots

Every bass note the key-finder has been given was its chord's root, and under that line a bass cue reads 89 per cent of scheme bars on the right degree. Give the same chords an economical bass that moves to the nearest chord tone, as a keyboard reduction would, and nearly half of them are inverted. The cue that rewards the triad rooted on the bass then reads 68 per cent at best and worse as it is trusted more; the cue that rewards any triad containing the bass cannot be fooled and stops at 61. The same inverted line does one thing the roots never did: it puts the leading note of each new key at the bottom, and finds the rondo's modulations.

Weighted by the pairs a melody sounds next to each other, every seven-note scale is rougher than most. Each scale placed by how smooth it is against 2000 random scales of the same size, under 4 spectra, with each pair of its degrees weighted by how often the plainest melody its ascent and descent permit sounds the two — here the pairs a melody sounds next to each other. Low is smooth. Raga Bhupali: 40.3 to 56.8, spread 16.5; Raga Deshkar: 25.1 to 37.5, spread 12.4; one measured slendro: 31.8 to 44.5, spread 12.7; Rast, Arabic theory: 92.0 to 98.6, spread 6.6; Rast, Turkish theory: 91.8 to 97.8, spread 6.0; the diatonic major, tempered: 90.7 to 97.0, spread 6.3; the diatonic major, five-limit just: 91.8 to 97.8, spread 6.0.

The smoothness is in the skips

Traditional scales come out smoother than random scales of their size because every pair of their degrees is counted once. Count only the pairs a melody on the scale actually sounds next to each other, and every seven-note scale here is rougher than ninety per cent of random ones, under every spectrum, and under a pure tone as well. The smoothness is carried by the pairs a melody reaches by skipping — its thirds and above all its fifths, which are smoother than all but two random scales in two thousand. Counted by their own ascents and descents, the two ragas that share one set of notes stand in different places at last: Deshkar's skipped Re buys a smoother ascent and costs it the fifths.

Weighted by the ring of the note before, a note every 0.6 seconds, Raga Deshkar moves least. Each scale placed by how smooth it is against 2000 random scales of its size under 4 instruments, each with its own spectrum and its own ring, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds. Low is smooth. Raga Bhupali: plucked string, 6 s 53.6, long-ringing string, 12 s 35.8, blown note, 2 s hall 41.3, free bar, 4 s 58.5; spread 22.8; Raga Deshkar: plucked string, 6 s 39.0, long-ringing string, 12 s 25.9, blown note, 2 s hall 26.1, free bar, 4 s 42.0; spread 16.1; one measured slendro: plucked string, 6 s 42.9, long-ringing string, 12 s 27.8, blown note, 2 s hall 32.6, free bar, 4 s 46.4; spread 18.6; Rast, Arabic theory: plucked string, 6 s 92.0, long-ringing string, 12 s 65.2, blown note, 2 s hall 92.3, free bar, 4 s 96.1; spread 30.9; Rast, Turkish theory: plucked string, 6 s 88.0, long-ringing string, 12 s 58.1, blown note, 2 s hall 90.8, free bar, 4 s 93.6; spread 35.5; the diatonic major, tempered: plucked string, 6 s 84.8, long-ringing string, 12 s 54.2, blown note, 2 s hall 89.5, free bar, 4 s 91.3; spread 37.0; the diatonic major, five-limit just: plucked string, 6 s 87.7, long-ringing string, 12 s 56.7, blown note, 2 s hall 90.3, free bar, 4 s 92.9; spread 36.2.

A scale is committed to how long its instrument rings

A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.

The tempo moves it further than the touch does. One measured slendro, scored among random scales of its size under a free bar, 4 s, at five tempi and under each touch. Left to ring it runs from 29 at 0.15 seconds a note to 83 at 2.4 — a span of 54 percentile points, where the two touches differ by at most 17. So the scale is smoother than most of its size when the music is fast and rougher than most when it is slow, and how the bar is damped is the smaller decision. The two touches converge at the slow end because a bar that has died before its successor is sounding against nothing whatever the player does.

The tempo moves a scale further than the touch

A gamelan is played two ways on the same bars: a saron's are damped as the next is struck and a gendèr's ring over their resonators. That decision moves a slendro's standing among random scales of its size by up to seventeen percentile points, which is real. Over the tempo levels a piece actually moves through it moves by fifty-four — from the twenty-ninth percentile at a fast elaboration to the eighty-third at a slow one. The same five pitches on the same bars are a smoother-than-average scale and a rougher-than-average one, and which depends on how fast they are played.

The scale's standing belongs to the ringing instrument. Where a measured slendro sits among random five-note scales, at three density ratios, scored three ways: the two instruments together, the fast ringing part on its own, and the slow damped part on its own. At one slow note to 2 fast ones the ensemble is at the 7th percentile, the ringing part alone at the 6th, and the damped part alone at the 95th; At one slow note to 4 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 7th, and the damped part alone at the 95th; At one slow note to 8 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 5th, and the damped part alone at the 95th. A low percentile is a scale smoother than most of its size. The damped part on its own is rougher than nineteen random scales in twenty, because it has almost no simultaneity for its intervals to be smooth in; the ensemble is at the ringing part's figure throughout. And the cross pairs, which are a quarter of the roughness, do not make the ensemble worse than either part alone.

The scale belongs to the ringing instrument

A gamelan plays two instruments on one scale at once — a saron damped at every stroke and a gendèr several times faster with its bars left to ring — and a quarter of the roughness a listener receives falls on pairs that cross between them, which no figure had computed. The cross pairs turn out to be no worse than either stream's own. What decides the scale's standing is the fast part: the ensemble sits at the sixth percentile among random five-note scales and so does the ringing instrument alone, while the damped one alone sits at the ninety-sixth.

Given the bar in octaves, the cue gets the degree back. How often the reading names the right scale degree, against how much the bass cue is worth, under three rules for what a bass note rewards. the bass is the root: 68 per cent at best; the bass is some chord tone: 67 per cent at best; the bar in octaves: 92 per cent at best; the bass is the root, on a root line: 89 per cent at best. The published cue rewards the triad rooted on the bass, which is right only when the chord is in root position; rewarding any triad containing the bass is right always and rewards three chords a bar instead of one. The third rule is not a bass cue at all — given the bar voiced in octaves a listener knows which three pitch classes are sounding, and that names the chord outright. It reaches 92 per cent on a real line against the published cue's 68 on the same line, and the dashed curve is what that cue manages on a line made entirely of roots — 89 per cent.

Given the bar in octaves, the degree comes back

A bass note is a bare pitch class in the usual figure, and a register in any realisation anybody plays. Voice each bar in octaves and a listener knows which three pitch classes are sounding and which is at the bottom, which names the chord outright — and the rule that uses it reads the right scale degree in 92 per cent of bars on a real bass line, against 68 for the published cue on the same line and 89 for that cue on a line made entirely of roots. The question was whether the register recovers the 89. It recovers it and passes it.

A sharper cue is worth nothing to a reading that will follow it anywhere. How often the reading names the right scale degree, against how much it costs to change key, at a bass worth 1.5 on a real bass line. the bass is the root: 20 per cent at a key cost of 0.5 and 68 at 4; the bass is some chord tone: 47 per cent at a key cost of 0.5 and 59 at 4; the bar in octaves: 46 per cent at a key cost of 0.5 and 91 at 4; roots, and a line of roots: 49 per cent at a key cost of 0.5 and 91 at 4. Where a key change is cheap the rule that names the chord outright reads no better than the rule that names three — 46 against 47 per cent — because a reading that will move key for one bar's evidence follows a sharp cue wherever it points. The sharper cue's whole advantage appears only once the reading is reluctant enough to stay put, and by a key cost of 2.2 it is 31 points ahead.

A sharper cue is worth nothing to a reading that moves

The rule that names the chord outright reads 92 per cent of scale degrees right where the published bass cue reads 68 — at the key cost these readings have always been run at. Sweep that cost and the advantage is not a property of the cue. Where a change of key is cheap the sharp rule reads 46 per cent and the vaguest rule 47, because a reading that will move key on one bar's evidence follows a sharp cue wherever it points. The cue's whole value is borrowed from the model's reluctance to be moved.

The two parameters turn out to have a ceiling between them

The essay before this one asked whether the bass cue's weight and the cost of changing key are one quantity with two names, and said the test was a contour: if they multiply, the curves of equal degree share are hyperbolae. They are not. Along the ninety per cent contour the product of the two runs from 2.7 to 15.2, because past a bass weight of about one and a half the reading saturates and more cue buys nothing. And the sweep finds something no single-parameter sweep here could: the key cost has a best value, and a reading that will never change key is worse than one that will.

The ceiling is thirteen bars, and every one of them has a name. Every bar of the six schemes read by the key-finder at a key cost of 3 and a bass weight of 3, with the chord in every bar given exactly — no segmentation is involved. 175 of 188 bars have both the key and the degree right, 93.1 per cent. The misses: 7 in the thirty-two-bar song's bridge, where the chain III7 is read in E major, III7 is read in E major, VI7 is read in E major, VI7 is read in D major, II7 is read in D major, II7 is read in D major, V7 is read in D major; 4 bars of the rondo's A minor episode read as C major; 2 next to a change of key; and 0 of any other kind.

The ceiling is thirteen bars with names

The key-finder's two cues stop buying anything at about 93 per cent of scheme bars read right, and the obvious suspect was the chord segmentation feeding it. The reading was never given a segmentation: every bar arrives with its true chord. What the ceiling is made of can be listed instead, and it is thirteen bars of 188 — seven in a bridge of secondary dominants, four in a minor episode whose chords C major also owns, and two at the edges of a modulation. No weight of any cue moves one of them.

A wrong bar costs the same whichever instrument it is on. What moving one degree of a measured slendro by 10 cents, flat or sharp, adds to the roughness per second of a two-instrument texture — a ringing part at 0.15 s a note over a damped one four times slower — on the ringing instrument and on the damped one. The ensemble in tune scores 7.47. Degree 1 (0¢): ringing 0.142 flat and 0.202 sharp, damped 0.163 and 0.185, of which beating 0.178; Degree 2 (231¢): ringing 0.119 flat and 0.068 sharp, damped 0.096 and 0.090, of which beating 0.093; Degree 3 (474¢): ringing 0.106 flat and 0.021 sharp, damped 0.069 and 0.057, of which beating 0.063; Degree 5 (717¢): ringing 0.079 flat and 0.072 sharp, damped 0.075 and 0.078, of which beating 0.077; Degree 6 (955¢): ringing 0.023 flat and 0.018 sharp, damped 0.019 and 0.022, of which beating 0.020. Over all ten errors the ringing instrument's cost 0.85 and the damped one's 0.85, and the wrong bar beating against the other instrument's right one comes to 0.86 on either — as much as the whole, because the intervals the error changes add as often as they save. One error costs about 1.1% of the texture's roughness.

A wrong bar beats the same on either instrument

The ringing instrument carries a gamelan scale's standing, so a tuning error on it was predicted to cost more than the same error on the damped instrument. Put in the beating the model lacked, and the prediction fails: moved ten cents, a degree costs the same 0.85 summed over the scale on either instrument, because almost all of the cost is the wrong bar beating against the right one on the other instrument, and a beat belongs to both of the bars that make it. Moving a whole instrument costs exactly what its five degrees cost separately, and it takes nothing from the scale's standing.

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