Acoustics & Music Theory
why it sounds like that
Depth

Series — page 2

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.
E(3, 8) as a cycle. A rhythm drawn round a circle, one equal arc per step, with the struck steps filled. On a circle the pattern has no beginning, which is why the same necklace of onsets is several different named rhythms depending on where a listener decides bar one is.

Cyclic-rhythm

  1. 1 Rhythm is a circle, and the bar line is a choice
  2. 2 A cycle cannot cadence
  3. 3 A cycle that says where it is
  4. 4 The cycle that outruns the memory
  5. 5 Repetition buys least where it is needed most
  6. +7 more
12 essays · rhythm Three envelopes. How loudness changes over the life of a note, for plucked, bowed and struck. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum.

Envelope

  1. 1 The shape of a note, which is most of what an instrument is
  2. 2 The first fifty milliseconds
  3. 3 The note that gets duller as it dies
  4. 4 The middle nobody could have guessed
  5. 5 The collapse belongs to the bass
  6. +7 more
12 essays · timbre 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. +7 more
12 essays · instruments Every point on one curve sounds equally loud. The equal-loudness contours of ISO 226:2003, evaluated from the standard's own parameters. The lowest curve is the threshold of hearing. Because the curves are not parallel — they crowd together in the bass and spread apart in the middle — the same change in decibels is a different change in loudness at every frequency, and a spectrum that was balanced at one level is not balanced at another.

Loudness

  1. 1 The quietest thing audible, and why the volume knob is a tone control
  2. 2 A chord is not as loud as its notes
  3. 3 Loud is relative, and it comes down slowly
  4. 4 The dynamics are in the score already
  5. 5 A page has two decibels
  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 4 bores of one length, and the series each supports. cylinder, cone, exponential horn, Bessel horn — every one of them 148 cm long, 5.5 mm at the throat and 62 mm at the mouth, so the only thing that differs is the shape between the two. Beside each is the series of resonances Webster's equation gives it, and the number is how far that series is from evenly spaced, in cents. cylinder: 0.0 cents, with each mode sitting at minus 0.50 of a spacing off a whole multiple; cone: 9.6 cents, with each mode sitting at minus 0.12 of a spacing off a whole multiple; exponential horn: 22.8 cents, with each mode sitting at minus 0.16 of a spacing off a whole multiple; Bessel horn: 3.7 cents, with each mode sitting at minus 0.11 of a spacing off a whole multiple.

Bore profile

  1. 1 Only two shapes make a series
  2. 2 What the mouthpiece is actually for
  3. 3 A horn has one length per partial
  4. 4 The cutoff a maker can actually measure
  5. 5 A family resemblance in the heights
  6. +6 more
11 essays · instruments 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 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 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.

Melody

  1. 1 A melody is a walk, not a set
  2. 2 The leap that pays itself back
  3. 3 The shape that survives everything else
  4. 4 The register where the series becomes a scale
  5. 5 The note that is sharp because of where it goes
  6. +5 more
10 essays · scales A 7-semitone sequence at 120 ms a tone. Tones drawn as pitch against time, one bar per tone. The events are the same in both readings of this pattern; what changes is whether a listener assigns them to one line that leaps back and forth or to two lines that each stay put. Nothing in the drawing decides which, and nothing in the sound does either.

Auditory scene

  1. 1 The ear builds objects, and sometimes offers a choice
  2. 2 What makes two partials one note
  3. 3 The spectrum that will not fuse
  4. 4 The partials that do not die together
  5. 5 The exchange rate nobody has
  6. +4 more
9 essays · perception One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

Metre induction

  1. 1 The beat is inferred, and sometimes wrongly
  2. 2 The beat has a preferred rate, and it is not the notation's
  3. 3 The bar above the bar
  4. 4 The right period at the wrong phase
  5. 5 Syncopation is a number about the metre
  6. +4 more
9 essays · rhythm How much swing there is, against how fast the music goes. The ratio of the long note to the short note of a swung pair, as a function of tempo, derived from the finding that the short note holds a roughly constant hundred milliseconds. The notated readings are horizontal lines, and each is correct at exactly one tempo. Above about three hundred beats a minute the ratio reaches one and the swing has gone.

Microtiming

  1. 1 Swing is a ratio, and it is not two to one
  2. 2 The milliseconds that are the groove
  3. 3 The deviations are not noise
  4. 4 Two players and no clock
  5. 5 How late is a different note
  6. +4 more
9 essays · rhythm A note with its first partial removed. The spectrum of a 220 Hz tone with the lowest partial deleted, and the wave that remains. The wave still repeats 220 times a second, because the repeat rate of a sum of harmonics is fixed by the spacing between them and the spacing has not changed. The pitch heard is the one that is no longer in the sound.

Missing fundamental

  1. 1 The note that is not there
  2. 2 The ear makes its own sound, and it is not the missing fundamental
  3. 3 A bell has no fundamental
  4. 4 What a drum is doing instead
  5. 5 Which harmonics carry the pitch
  6. +4 more
9 essays · intervals 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.

Modes

  1. 1 The same seven, started later
  2. 2 What a raised seventh is for
  3. 3 What a tonic costs in seconds
  4. 4 The mode with no fifth
  5. 5 One dominant among seven
  6. +4 more
9 essays · scales One envelope, and the three places a listener might be said to hear it. The amplitude envelope of a note with a 90 millisecond exponential attack, with the three criteria the literature offers drawn across it. The heard moment is 8.0 ms at the detection criterion, 28 ms at the perceptual-onset criterion and 94 ms at the perceptual-attack criterion. The physical onset is at zero on this axis and no criterion puts the heard moment there. The buttons play this attack against a two-millisecond one, started at the same instant.

Perceptual-centre

  1. 1 A note is heard after it starts
  2. 2 The players who have to be early
  3. 3 Playing louder is playing earlier
  4. 4 A low note cannot start on time
  5. 5 Which notes have to be played early
  6. +4 more
9 essays · rhythm A phrase is a number of seconds, and the bars follow the tempo. Phrase durations for 1, 2, 4, 8, 16-bar phrases at seven tempos, on a logarithmic seconds axis, with the 2 to 8 second window shaded. The window is a property of the listener and does not move; which bar count falls inside it is decided entirely by the tempo.

Phrase

  1. 1 A phrase is a number of seconds
  2. 2 One of these eight-bar phrases accelerates
  3. 3 Where a phrase ends
  4. 4 A boundary at a stated level
  5. 5 The level the tempo chooses
  6. +4 more
9 essays · form 3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things.

Polyrhythm

  1. 1 Two clocks at once, and where they agree
  2. 2 A rhythm fast enough to be a chord
  3. 3 A process that enumerates its own form
  4. 4 The third pattern nobody played
  5. 5 Which of the two is the beat
  6. +4 more
9 essays · rhythm thirty-two-bar AABA, every bar against every other bar. A self-similarity matrix of 32 bars of thirty-two-bar AABA. Each cell is the cosine similarity of two bars' pitch-class vectors, with the chord's own notes weighted 1 and the rest of the key 0.5. Similarity is quantised into four bands for drawing and anything under 0.35 is left as paper. Nothing in the computation knows what a section is; the blocks and stripes are what the arithmetic returns.

Repetition

  1. 1 A piece is mostly itself again
  2. 2 The boundary is where the neighbourhood changes
  3. 3 How much of this is new
  4. 4 How long until it comes back
  5. 5 The form a first hearing cannot have
  6. +4 more
9 essays · form

All essays