Every essay — page 6
Pitch and tuning Intervals and chords Scales and modes Harmony and voice leading Rhythm and metre Timbre and acoustics Perception and the listener Instruments and their design Form and structure Series Objects Sounds Search
Scales and modes
Which seven of the twelve, and what changes when the same seven start somewhere else.
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.
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 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.
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.
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 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 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 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.
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 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.
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.
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'.
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.
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 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 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.
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.
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.
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.
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.