Acoustics & Music Theory
why it sounds like that
Series

Masking — the series

15 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    One sound hides another, and it hides upward

    A tone can be made completely inaudible by a second tone that is nowhere near it in frequency, and the region it disappears into is lopsided. Masking spreads up the spectrum and barely down it, and the reach grows with the masker's level — so which line in an arrangement vanishes is a prediction, not a matter of taste.

    part 1 · perception
  2. The threshold before, during and after a burst. The level a brief probe needs in order to be heard, plotted against when it happens relative to a 70 dB burst that occupies the shaded band. To the right is forward masking, which decays over about 200 milliseconds. To the left is backward masking: the threshold is raised for a probe that has already finished before the masker begins.

    A sound hides what came before it

    Masking does not stop when the masker does. A loud sound raises the threshold for about a fifth of a second after it ends, which is unremarkable, and for several milliseconds before it begins, which is not. The auditory present is a window rather than an instant, and inside the window the order of events is not the order they arrived.

    part 2 · perception
  3. A loud chord is a smaller chord. The share of a voicing's partials that stand above what the rest of it masks, and the share of its computed roughness that is between partials a listener actually has, from 30 decibels to 100. Both fall: 83 per cent of the partials survive at 30 decibels and 38 at 100, and the roughness share goes from 88 per cent to 67. The direction is the upward spread of masking, which grows faster than linearly with level: a loud partial masks a band above itself much wider than a quiet one does, so the chord's own top disappears into its own bottom. Two earlier essays are drawn at one level, and this is what they were holding.

    A loud chord is a smaller chord

    Two earlier essays hold the level fixed, and the level decides how much of a chord a listener is given. At thirty decibels twenty of a triad's twenty-four partials stand above what the rest of it masks; at a hundred, nine do. Every roughness figure until now counts partials that are in the score, and a partial the chord masks is not a partial the listener has.

    part 3 · harmony
  4. Nothing at all until fifteen decibels, and then it depends on the tempo. The fraction of a line's partials that stay above threshold, over how fast the line moves and how much louder everything before each note is. Darker is more lost. The whole left-hand side is white: at equal levels a note cannot be masked by its predecessor at any tempo, and that is a proof rather than a measurement — forward masking leaves a threshold at most ten decibels below the masker, and a note's own partials mask each other from the same components at full level. The boundary is between twelve and eighteen decibels, and beyond it the loss grows with the tempo: at 280 to the crotchet and 36 decibels of contrast, 26 per cent of the line's partials are gone. Fifteen decibels is about the gap between a forte and a piano.

    An equal note cannot be masked

    Three earlier essays are about one instant, and forward masking lasts two hundred milliseconds — longer than a note at any brisk tempo. So a fast line should be a sequence of events hiding each other, and it is not: a note masks itself ten decibels harder than its predecessor can, at any speed. What does hide a line is dynamic contrast, and the boundary is fifteen decibels.

    part 4 · perception
  5. The top voice arrives whole and the bottom one arrives as a sine. 4 parts sounding together, each of 8 partials, with every partial tested against the summed masked threshold of every component in the texture. A filled mark is a partial the listener receives and an open one is a partial the part would have had alone and does not have here. The bass at C3 keeps 1 of 8, the tenor at C4 keeps 2 of 8, the alto at E4 keeps 5 of 8, the soprano at G5 keeps 8 of 8, every one of them at 70 decibels. Every part is at the same level and the difference is entirely where each one sits: masking spreads upward, so the part at the top of the texture has nothing above it to be masked by and the part at the bottom has everything.

    The listener is given the top voice, and the bass as a sine

    Four earlier essays put the masker and the probe in the same voice. Put them in different voices — a four-part texture at one level — and the soprano arrives with all eight of its partials, the alto with five, the tenor with two and the bass with one. Balancing the loudness, which is the constraint a scoring is solved under, changes none of that: equal loudness is not equal spectrum and cannot be made so.

    part 5 · perception
  6. The roughest chord on the page is at C1 and the roughest one heard is at A♭2. One close triad at 70 decibels through 17 registers, with its roughness computed twice: over every partial in the score, and over only the partials that stand above what the chord itself masks. The written curve rises all the way down and its maximum is the lowest register drawn, C1, which is the low-interval rule as it has always been computed here. The delivered curve turns over at A♭2 and falls to nothing below A♭1: a close triad down there is not rough, because it is not arriving as a chord — 1 of its 24 partials survives at C1 and there is almost nothing left to beat against anything.

    A low chord stops being rough by stopping being a chord

    Every count of audible partials until now is of a chord at middle C, and the three registers it did compare span C3 to C5 — a third of the range a chord is written in. Move the same triad down and the count collapses: 79 per cent of its partials arrive at E3 and 4 per cent at C1. So the roughest chord on the page is the lowest one and the roughest chord a listener receives is at G2, and where that maximum sits moves nearly two octaves with the dynamic.

    part 6 · perception
  7. Scored the way these figures score it, a clarinet is the worst of the six. One close triad at 70 decibels through 17 registers, drawn once for each of the 6 spectra to hand. The score is the share of every partial written, which is the quantity the register figure published. pure 100 per cent at best, string 79 per cent at best, clarinet 54 per cent at best, reed 71 per cent at best, bell 52 per cent at best, organ 72 per cent at best. A clarinet's four even partials are twenty-eight decibels below its odd ones and are inaudible beside their own neighbours before any chord is built, so counting them in the denominator makes the spectrum that survives its own masking best look like the one that survives it worst.

    A clarinet keeps what a string loses

    Every masker, probe, chord, line and texture until now is eight partials falling as 1/n, and it was not even an option a placement could pass. Sweeping the six spectra to hand says the clarinet is the worst of them — 54 per cent of itself at best against a string's 79 — and that answer is an artefact of the score. Counted against what each note keeps on its own, the clarinet keeps 100 per cent where the string keeps 79, because its components stand a twelfth apart rather than an octave. The missing parameter was the spectrum; the second missing parameter was the denominator.

    part 7 · perception
  8. The arch belongs to hearing, and the spacing only moves it. The share of a close major triad's twenty-four components that stand above what the rest of the chord masks, at 70 dB, with the root from C1 to C7, for three spectra given the same amplitude law and different frequencies: the harmonic series, a founder's bell, and a stiff string with B = 0.01. harmonic series: 0.04 at C1, peaking at 0.79 on E3, 0.42 at C7; a founder's bell: 0.04 at C1, peaking at 0.75 on C4, 0.38 at C7; a stiff string: 0.04 at C1, peaking at 0.71 on E3, 0.46 at C7. Only one of the three is a harmonic series, and all three rise out of the bass, peak in the middle of the compass and fall in the treble.

    The arch belongs to hearing, not to the series

    A chord delivers most of its partials in the middle of the compass and loses them in the bass and the treble, and every spectrum that showed that arch was built on whole multiples of a fundamental. Give the same amplitudes to a bell's eight modes and to a stiff string's stretched partials and the arch is still there, peaking within a major third of where the harmonic series peaks. What the spacing changes is the detail: a bell crowds its tierce and quint into a quarter of a critical band in the bass and loses them, and a stiff string's stretch buys the bass back.

    part 8 · perception
  9. A bell's tierce does not arrive late; it leaves early. The tierce's margin over what the bell's other modes put across it, in decibels, against time since the strike, for four bells, with the hum's sixty-decibel time set to 40 seconds and each mode's decay time that divided by its frequency ratio to the hum raised to 1. G1: 0 s -3.5 dB, 2 s -5.3 dB, 3 s -7.4 dB, 5 s -9.8 dB, 6 s -12.4 dB, 8 s -15.2 dB; C2: 0 s -2.2 dB, 2 s -3.7 dB, 3 s -5.6 dB, 5 s -7.7 dB, 6 s -10.1 dB, 8 s -12.6 dB; G2: 0 s 0.1 dB, 2 s -1.0 dB, 3 s -2.4 dB, 5 s -3.9 dB, 6 s -5.8 dB, 8 s -7.9 dB; C3: 0 s 1.9 dB, 2 s 1.1 dB, 3 s 0.2 dB, 5 s -0.9 dB, 6 s -2.2 dB, 8 s -3.8 dB. Above the line the tierce is heard. The earlier prediction was that the quint's faster decay would lift the tierce clear of its masking a second into the note; every curve here falls instead.

    A bell's tierce does not arrive late, it leaves early

    The essay before this one found a low bell's tierce masked at the strike and predicted it would emerge a second into the note, because the quint above it decays faster than the prime and tierce below. It does not emerge, and the reason is that the quint was never masking it: removing the quint moves the tierce's threshold by a fifth of a decibel and removing the prime moves it by three to five and a half. The tierce is masked from below, by two modes that outlive it under any law that falls with frequency — so a bell that has a tierce at the strike loses it within a few seconds, and a bell that has none never gains one. The minor third a carillon's bass is known for is a property of its attack.

    part 9 · perception
  10. A bell's tierce is best placed at the first instant of the strike, and only for an instant. The tierce's margin above its masked threshold, in decibels, through the first 400 milliseconds after the strike, for bells on G1, C2, E2, G2, C3, with every mode starting at full amplitude and built up in the listener's own auditory filters. G1: -1.3 at 0.5 ms, -3.9 at 400 ms, never above threshold; C2: 0.3 at 0.5 ms, -2.5 at 400 ms, above threshold from 0.5 to 0.7 ms; E2: 1.6 at 0.5 ms, -1.2 at 400 ms, above threshold from 0.5 to 4.3 ms; G2: 2.7 at 0.5 ms, -0.2 at 400 ms, above threshold from 0.5 to 104.2 ms; C3: 4.7 at 0.5 ms, 1.7 at 400 ms, above threshold from 0.5 to 400.0 ms. Every curve falls from its first moment: there is no peak inside the window.

    A bell's strike gives its tierce a millisecond

    A low bell's tierce is masked from below by its prime and hum, so it is present at the strike or never — and the strike was the one moment the count treated as a steady state. Give the modes a rise as well as a fall and the prediction was that the tierce's margin would peak somewhere inside the first two hundred milliseconds. It peaks at the very first instant and falls from there, because the shadow a masker casts upward grows faster than the masker. On a C2 bell the tierce is above its threshold for less than a millisecond, and under the most generous rise anybody could propose for twenty-five: ten to thirty decibels too brief to be heard. What the onset visits for an instant, a softer stroke keeps for the whole note.

    part 10 · perception
  11. A bell is never minor alone. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels: whether the masked-threshold count delivers the tierce, the minor third above the prime, and the deciem, the major third above the nominal. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MBBBMMMMMMMMMM····; E2: ·BBBBBBBBMMMMMMM···; G2: ·BBBBBBBBBBMMMMMM··; C3: BBBBBBBBBBBBBMMMMM·; E3: BBBBBBBBBBBBBBBMMM·; G3: BBBBBBBBBBBBBBBBMMM; C4: BBBBBBBBBBBBBBBBBMM (B both thirds, M the major only, m the minor only, a dot neither; levels in steps of five from 20). Of 190 cells, 82 deliver both thirds, 59 the major third only, 49 neither, and 0 the minor third alone. A dot in a cell marks a stroke at which fewer than two of the three modes that imply the strike note survive.

    A bell is never minor alone

    A tuned bell carries two thirds: the tierce, a minor third above the prime, which is why a bell is called minor, and the deciem, a major third above the nominal. Run the masking count across every register and every stroke and the minor one is never delivered without the major one. Struck softly a bell delivers both; struck harder, only the major; and below C2 the tierce is never delivered at all. The deciem survives because it is masked by one mode barely louder than itself, the tierce by two that outweigh it.

    part 11 · perception
  12. A bell whose deciem is the tierce's octave is minor at every stroke. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels, with the deciem tuned from 2.5 times the prime to 2.4 — a minor third above the nominal and exactly an octave above the tierce: whether the masked-threshold count delivers a third at all. C1: ···mm··············; E1: ···mmmmmm··········; G1: ··mmmmmmmmm········; C2: ·mmmmmmmmmmmmm·····; E2: ·mmmmmmmmmmmmmm····; G2: ·mmmmmmmmmmmmmmm···; C3: mmmmmmmmmmmmmmmmm··; E3: mmmmmmmmmmmmmmmmmm·; G3: mmmmmmmmmmmmmmmmmm·; C4: mmmmmmmmmmmmmmmmmmm (m a minor third, a dot none; levels in steps of five from 20). 131 of 190 cells read minor and none major. Of the 59 cells in which the bell as cast delivered only its major third, 49 now deliver a minor one.

    A bell can be made minor by the third nobody tunes

    A low bell struck hard delivers its major third, the deciem, and loses its minor one, the tierce, to masking. A founder could flatten the deciem from a major to a minor third above the nominal, which puts it exactly an octave above the tierce. The masking count says that move costs the deciem only one or two decibels of stroke at the top of its window, so the flattened deciem still outlasts the tierce in every register. A bell tuned that way reads minor in every cell of register and stroke that delivers a third at all. In-between tunings read as neutral thirds, so the move only works if it goes all the way.

    part 12 · perception
  13. A bell with a major tierce is major wherever it delivers a third. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the deciem as cast at 2.5 times the prime and the tierce at 1.25, 386 cents above the prime: which third the masked-threshold count delivers. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MMMMMMMMMMMMMM····; E2: ·MMMMMMMMMMMMMMM···; G2: ·MMMMMMMMMMMMMMMM··; C3: MMMMMMMMMMMMMMMMMM·; E3: MMMMMMMMMMMMMMMMMM·; G3: MMMMMMMMMMMMMMMMMMM; C4: MMMMMMMMMMMMMMMMMMM (M major only, b a minor tierce with the major deciem, n a neutral tierce, a dot none; levels in steps of five from 20). 141 cells major, 0 with thirds that disagree, 49 with none. Dots mark the 10 cells in which this bell delivers a third and the minor bell with its deciem flattened to the tierce's octave delivers none.

    The major-third bell has no mirror problem

    A minor bell struck hard loses its tierce to masking and delivers its major deciem instead. Raise the tierce to a major third and the two thirds agree — but the raised tierce sits closer to the quint, and it was fair to fear that the quint would bury it as the prime buried the minor one. It does not. The prime is what masks a tierce, and moving away from it gains four to twelve decibels of stroke while the quint stays nine decibels too weak to matter. Tuned to agree, the major bell delivers its colour in 141 of 190 cells; the minor bell, made consistent, in 131.

    part 13 · perception
  14. With the tierce at 1.25 times the prime, the lower modes point only at the strike note's own name. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the tierce at 1.25 times the prime, 386 cents: the series the delivered lower modes — hum, prime, tierce and quint — fit best, within thirty cents. C1: ···················; E1: ·······SSS·········; G1: ·····SSSSSSS·······; C2: ··SSSSSSSSSSSS·····; E2: ·SSSSSSSSSSSSSSS···; G2: ·SSSSSSSSSSSSSSS···; C3: SSSSSSSSSSSSSSSSS··; E3: SSSSSSSSSSSSSSSSSS·; G3: SSSSSSSSSSSSSSSSSS·; C4: SSSSSSSSSSSSSSSSSSS (S the strike note's name, 3 the note a major third below it, a dot fewer than two lower modes delivered; levels in steps of five from 20). 124 cells point at the strike's name, 0 a major third below, 66 at nothing.

    The raised tierce points the bell at its own note

    A bell's four lowest modes — hum, prime, tierce and quint — can imply a pitch of their own. With a major tierce they are the second, fourth, fifth and sixth harmonics of the prime two octaves down: the strike note's own name, with one empty slot. It was the minor tierce that pointed elsewhere all along. Its lower modes are the fifth, tenth, twelfth and fifteenth harmonics of the note a major third below the strike, and in 82 of 190 struck bells that is the series the delivered modes fit. Raising the tierce removes a competitor rather than adding one.

    part 14 · perception
  15. In a ring of minor bells, five of the eight hidden pitches fall off the scale the ring rings. A ring of eight bells with minor tierces tuned to a major scale, the tenor's strike note C4: for each bell and each stroke with the hum at 40 to 90 decibels, what the delivered lower modes — hum, prime, tierce and quint — imply. C4 (hidden A♭, off it): 333333333SS; D4 (hidden B♭, off it): 3333333333S; E4 (hidden C, on the scale): 33333333333; F4 (hidden D♭, off it): 33333333333; G4 (hidden E♭, off it): 33333333333; A4 (hidden F, on the scale): 33333333333; B4 (hidden G, on the scale): 33333333333; C5 (hidden A♭, off it): 33333333333 (3 the note a major third below the strike, S the strike's own name, a dot too few lower modes delivered). 33 strokes imply a note on the ring's scale and 52 a note off it.

    A ring of minor bells rings a second scale a major third below

    A minor-tierce bell's four lowest modes, read as a residue, imply a note a major third below the bell's strike note. Put eight such bells in a ring tuned to a major scale and the implied notes are that scale moved down a major third — a ring in C carries the scale of A♭ major underneath it. Only three of the eight land on notes the ring rings, from the mediant, the submediant and the leading note; the tonic, the octave, the supertonic, the subdominant and the dominant all imply notes the ring never sounds, each a semitone from two that it does. Above a tenor on C3 the lower modes deliver the implied note at nearly every stroke. A reading that weights the whole spectrum still hears none of it.

    part 15 · perception

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