Acoustics & Music Theory
why it sounds like that
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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 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. +29 more
34 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. +22 more
27 essays · intervals 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. +22 more
27 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. +21 more
26 essays · timbre A string quartet's open strings are five keys of a Pythagorean keyboard. The five pitch classes a string quartet's open strings sound — C, G, D, A and E — laid out as the chain of fifths they are tuned along, outward from the A the ensemble is given, with each fifth pure. The bars give each string's departure from the same note on an equal-tempered keyboard: C −5.87 cents, G −3.91 cents, D −1.96 cents, A 0.00 cents, E +1.96 cents. Above, the strings each instrument owns: the violin G, D, A, E; the viola C, G, D, A; the cello the same four an octave lower. The cello's C2 is 0.221 hertz below the keyboard's, and the widest span of the chain, from the cello's C to the violin's E, is a Pythagorean third and two octaves, 21.5 cents wider than a just one.

Open strings

  1. 1 The tuning a string quartet cannot change
  2. 2 The cello cannot hear its own tempering
  3. 3 An open string pulls the quartet flat
  4. 4 A guitar tuned by harmonics hides a comma
  5. 5 One tuning has no comma to place
  6. +21 more
26 essays · tuning 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. +19 more
24 essays · scales 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. +17 more
22 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. +15 more
20 essays · timbre 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 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. +13 more
18 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. +11 more
16 essays · perception Three metres as trees. Three metres — 4/4, 2 + 2 + 2 + 2; 6/8, 3 + 3; 5/8, 3 + 2 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

Additive metre

  1. 1 Beats of unequal length
  2. 2 How unequal a beat is allowed to be
  3. 3 How long a limping bar can be
  4. 4 A twenty-five is a nine until its last unit
  5. 5 The accent buys two units, however loud it is
  6. +10 more
15 essays · rhythm 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. +10 more
15 essays · form What one tone hides, and in which direction. The masked threshold beside a tone masker: any probe below one of these curves is inaudible while the masker sounds. The frequency axis is in Bark, the scale on which the ear's filters are evenly spaced, so the pattern is a pair of straight lines. The upper slope is much shallower than the lower one and gets shallower still as the masker gets louder — masking spreads upward, not downward.

Masking

  1. 1 One sound hides another, and it hides upward
  2. 2 A sound hides what came before it
  3. 3 A loud chord is a smaller chord
  4. 4 An equal note cannot be masked
  5. 5 The listener is given the top voice, and the bass as a sine
  6. +10 more
15 essays · perception 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 A tempered interval moves its difference tone several times further than itself. For every interval inside the octave tuned to twelve equal steps, how far the difference tone f₂ − f₁ lands from where the just interval would put it, in cents, with the interval's own departure from just drawn as the thin bar beside it. minor second −198.0 (the interval −11.7); major second −35.5 (the interval −3.9); minor third −96.0 (the interval −15.6); major third +67.4 (the interval +13.7); fourth +7.8 (the interval +2.0); fifth −5.9 (the interval −2.0); minor sixth −36.7 (the interval −13.7); major sixth +38.8 (the interval +15.6); minor seventh −39.8 (the interval −17.6); major seventh +25.0 (the interval +11.7). The major third's product is +67.4 cents out and the minor third's −96.0, and the largest error is the minor second's, at −198: a product moves p/(p − q) times as far as the interval p:q that made it.

Combination tone

  1. 1 The third sound magnifies cents, not hertz
  2. 2 The tone on the root changes hands at the fifth
  3. 3 A major triad's combination tones are its own notes
  4. 4 The bass line under a passage in thirds
  5. 5 A combination-tone bass needs a forte
  6. +8 more
13 essays · intervals 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.

Euclidean rhythm

  1. 1 As evenly as possible, which turns out to be a famous rhythm
  2. 2 The rotation the necklace cannot see
  3. 3 What the clave buys with its unevenness
  4. 4 The frontier and the ruler
  5. 5 The longest silence is not a third axis
  6. +8 more
13 essays · rhythm Six named proportions, as blurred as the durations that make them. Six proportions between two parts of a piece — 1 : 1, 4 : 3, 3 : 2, golden section, 2 : 1, 3 : 1 — placed on one axis by the logarithm of the ratio of the longer part to the shorter, and drawn as bars one criterion wide (d′ = 1) for a listener timing both parts with a Weber fraction of 7%, 15%, 35%. Bars that overlap are proportions that listener cannot tell apart. At 7%, 4 of 5 neighbouring pairs stay apart; at 15%, 3 of 5 neighbouring pairs stay apart; at 35%, 0 of 5 neighbouring pairs stay apart.

Proportion

  1. 1 A proportion is only as fine as its two durations
  2. 2 A return is shorter than its first hearing
  3. 3 A count is not an estimate
  4. 4 A form is sharp at the bottom and vague at the top
  5. 5 A golden section is a coin toss with six coins
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13 essays · form 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. +8 more
13 essays · perception

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