Acoustics & Music Theory
why it sounds like that
Depth

Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
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.

Key-relations

  1. 1 Keys are neighbours, and the map is computed
  2. 2 The key plan is the form
  3. 3 Three ways to measure how far a key is
  4. 4 How much evidence a modulation needs
  5. 5 The circle is a circle, and the map is not
  6. +16 more
21 essays · scales 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.

Notation

  1. 1 The stave is not a ruler
  2. 2 Two names for one key
  3. 3 The time signature is a claim
  4. 4 The mark that is not a level
  5. 5 What a tablature keeps
  6. +13 more
18 essays · scales The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

Progression

  1. 1 A progression is a path, and the map can be drawn
  2. 2 The progression that never comes home
  3. 3 How often the chord changes
  4. 4 Which notes are the chord
  5. 5 Three decisions that constrain each other
  6. +12 more
17 essays · harmony 220 Hz against 223 Hz. Two tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear.

Beating

  1. 1 Beats are arithmetic that anybody can hear
  2. 2 A tuner counts beats, and that is the whole method
  3. 3 The other wolf
  4. 4 Three strings, and the note that comes back
  5. 5 The pair tuned apart on purpose
  6. +11 more
16 essays · intervals 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.

Beyond twelve

  1. 1 A scale is not a set of pitches
  2. 2 A scale without an octave, and the spectrum that asks for it
  3. 3 The third the model has no opinion about
  4. 4 Three answers to how finely a pitch can be heard
  5. 5 The spectrum that was supposed to explain the gamelan
  6. +9 more
14 essays · scales The first eight partials of a string. A string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning.

Harmonic series

  1. 1 A string does everything at once
  2. 2 The piano is tuned wrong on purpose
  3. 3 The series is not a chord
  4. 4 The series has three tops
  5. 5 The fourth top is the maker's
  6. +9 more
14 essays · timbre Every assignment, at equal levels and at its own best balance. The 6 ways of putting 3 players on a chord, each drawn twice: hollow at equal levels, which is what an assignment ranking sees, and filled at the levels that balance the parts and then minimise roughness. Every scoring here is at the same total loudness, 23.3 sones, so two points are comparable. Solving the discrete problem first picks violin · clarinet · oboe; solving both at once picks violin · oboe · clarinet, and the two-stage answer costs 9.0 per cent more roughness. The orderings do not keep their places between the two columns, which is the whole of the argument: a ranking taken at equal levels is not a ranking.

Orchestration

  1. 1 Who plays what and how loud is one question
  2. 2 The ranking survives the dynamic and the chord does not
  3. 3 A dynamic mark changes what a note is
  4. 4 A subito piano is a rate, not a level
  5. 5 The dissonance arrives and the dynamic does not
  6. +9 more
14 essays · form Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

Pitch standard

  1. 1 The pitch nobody agreed on, for four hundred years
  2. 2 A memory for the note itself, and it is dated
  3. 3 The note that moved by a minor third
  4. 4 A standard is a specification
  5. 5 What the climb was a search for
  6. +9 more
14 essays · tuning What a 60 cm tube supports, by how its ends are closed. The first 6 modes of an open cylinder and a stopped cylinder, all of the same acoustic length. An open tube supports every whole multiple of the fundamental and reaches the second mode an octave up. A cylinder stopped at one end supports only the odd multiples and reaches its second mode 1902 cents up, which is a twelfth. Its fundamental is also an octave below the others, because it fits a quarter of a wavelength where they fit a half.

Air column

  1. 1 A tube that skips every other partial
  2. 2 A cone is not a cylinder
  3. 3 The tube ends after it ends
  4. 4 The reed is a valve, not a vibrator
  5. 5 A wind instrument is a thermometer
  6. +8 more
13 essays · instruments 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.

Closure

  1. 1 What makes an ending an ending
  2. 2 An ending that exists so a bigger one can
  3. 3 An ending is a deceleration
  4. 4 An ending that can be heard coming
  5. 5 A silence long enough to be an ending
  6. +8 more
13 essays · form Four spectra of the same note. The amplitude of each partial for 4 timbres at the same pitch — pure, string, clarinet, bell. These are the exact lists the sound buttons here synthesise from, so the picture and the sound are the same data.

Spectrum

  1. 1 The ear hears the list, not the shape
  2. 2 A vowel is two resonances
  3. 3 The body is the filter
  4. 4 An instrument is not one timbre
  5. 5 Which instrument is underneath
  6. +8 more
13 essays · timbre The flow through the larynx, over two periods of a 110 Hz note. Volume flow against time, in Rosenberg's two-half-cosine model of the glottal pulse — a slow opening, a faster closing, and a closed phase during which no air passes at all. M1 — chest is open for 50 per cent of each period and opens 2.4 times as slowly as it closes. Nothing here is a displacement: the folds are a valve on a steady stream of air, and the flat stretches are the moments they are shut. At 110 Hz each period lasts 9.1 milliseconds, of which 4.5 is silence.

The voice

  1. 1 The other instrument with a reed
  2. 2 Two mechanisms, and the seam between them
  3. 3 One voice over ninety players
  4. 4 A note that is never at its pitch
  5. 5 The sound a listener knows best
  6. +8 more
13 essays · instruments A source 45° off centre, and the path difference it makes. A head from above with a source to one side. The near ear is reached first; the far ear's path runs round the head, and the difference between the two is 13.1 centimetres, which at 343 metres a second is 381 microseconds. That number, and the level difference the head's shadow produces, are the whole of what the ear has to work with.

Localisation

  1. 1 Two ears, and the whole of the difference is 655 microseconds
  2. 2 The first wavefront wins
  3. 3 The beat that is not in the air
  4. 4 Where the two ears stop agreeing
  5. 5 A room with directions in it
  6. +7 more
12 essays · perception The smallest audible difference, and what has to clear it. The difference limen for frequency, converted from Wier, Jesteadt and Green's 1977 fit into cents, against the intervals and commas the rest of these essays argue about. Anything drawn below the curve is a quantity nobody can hear as a change of pitch; anything well above it is a quantity a listener can be asked about. The limen is for pure tones, successive, with trained listeners — the most favourable case there is, and therefore the right one to test a claim against.

Pitch-acuity

  1. 1 How small a difference is audible
  2. 2 The octave that is not two to one
  3. 3 How long a note has to be
  4. 4 An interval is two errors
  5. 5 How much an anchor would have to be worth
  6. +7 more
12 essays · perception Twelve fifths do not make seven octaves. Pitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round 7 times and a little further, finishing 23.46 cents past seven octaves — the Pythagorean comma. Nothing in the chain closes that gap; it is what a tuning has to absorb, hide or spread.

The comma

  1. 1 Twelve fifths and seven octaves, which are not the same thing
  2. 2 Where to hide the comma, which is the only real question
  3. 3 The wolf at the end of the chain
  4. 4 A second comma, arriving by a different road
  5. 5 Keys that had characters, and could be measured
  6. +7 more
12 essays · tuning Where one interval stops being itself. Identification as a function of interval size: the probability that a listener names each category, modelled as a logistic with the boundary positions and sharpness a study reports. One category's share falls from three-quarters to one-quarter over 24 cents, against the 100 that separate adjacent categories — so the change of mind happens in 24% of the gap and the rest of it is not in doubt at all. That is what makes a mistuned third a third that is out, rather than a different interval.

Categorical-hearing

  1. 1 The ear sorts into boxes, and the boxes are the theory
  2. 2 The chord is still major, and that is why temperament works
  3. 3 The same distance, under two names
  4. 4 How many boxes an octave holds
  5. 5 The boundary that barely moves
  6. +6 more
11 essays · intervals A string struck at one 7th of its length. The amplitude of each partial of an ideal string excited at 0.1429 of its length. The mode shape is a sine, so a partial with a node at the excitation point cannot be set moving at all: partials 7, 14 are silent here. The envelope over the rest is one over n, a struck string's.

Excitation point

  1. 1 Where the hammer lands
  2. 2 A hammer is not an impulse
  3. 3 The hammer is not a point either
  4. 4 The hammer that is heavier than its string
  5. 5 The pulse that was assumed
  6. +6 more
11 essays · instruments The probe-tone profile, major key. How well each of the twelve pitch classes was rated as fitting, after a context establishing the key — Krumhansl and Kessler, 1982. The shading is not part of the measurement: it is the tonic, the rest of the tonic triad, the rest of the scale and the remaining five notes, which are categories this subject had before anybody ran the experiment. The profile separates all four without overlap.

Tonal-expectation

  1. 1 Counting produced the hierarchy
  2. 2 The chord that did not come
  3. 3 Surprise is a number
  4. 4 A chord, given a key and a predecessor
  5. 5 Expectation is a curve, not a list
  6. +6 more
11 essays · perception Helmholtz motion, bowed at 9% of the way from the bridge. Above: the string at 5 instants of one period. It is two straight lines meeting at a corner, and the corner travels round the string rather than the string swinging. Below: the resulting force on the bridge, a sawtooth whose two segments are in the ratio 0.09 to 0.91 — the bow's own position. A sawtooth contains every harmonic at exactly one over n, so the spectrum barely changes with bow position even though the waveform plainly does.

Bowed string

  1. 1 The bow makes a corner
  2. 2 How much bow is allowed
  3. 3 The note has to start somewhere
  4. 4 The note the body will not let start
  5. 5 The same note is a different width
  6. +5 more
10 essays · instruments Two tones in the ratio 3 to 2. Two sine tones whose frequencies are in the ratio 3 to 2, and their sum. The combined pattern repeats every time both waves return to the start together, which for a simple ratio is soon — here after 2 cycles of the lower tone and 3 of the upper.

Consonance

  1. 1 Two notes and a ratio, which is the whole of consonance
  2. 2 Roughness can be computed, and the answer looks like a scale
  3. 3 A third is rougher in the bass
  4. 4 Consonance is half learned, and this is the half
  5. 5 A spectrum chooses its own scale
  6. +5 more
10 essays · intervals

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