PHYS 536 R. J. Wilkes

Session 12 Musical acoustics;

Musical instruments: strings 2/19/2019

Course syllabus and schedule –second part

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See : http://courses.washington.edu/phys536/syllabus.htm

Tonight

9 12-FebK.Ch.8,9,

10 H:Chs.7,13,pistons,dipoles;Nearfield,farfield;RadiaConimpedance;Helmholtzresonator,wavesinpipes

10 14-Feb K.Ch11,12 H:Chs.2,27Decibels,soundlevel,dBexamples,acousCcintensity;noise,detecConthresholds;EnvironmentalacousCcsandnoisecriteria;noiseregulaCons

12 19-Feb H:Chs.16-19 Musicalinstruments:wind,string,percussion

13 21-FebK.Ch.11,12,13 H:Ch.21--19

Theear,hearinganddetecCon;ReverberaCon,roomacousCcs.REPORT1DUEby7PM;REPORT2PROPOSEDTOPICDUE

14 26-Feb K.Ch.14,15 H:Ch.7Transducersforuseinair:Microphonesandloudspeakers;UnderwateracousCcs;soundspeedinseawater;underseasoundpropagaCon.

15 K.Ch.15 SonarequaCons,underseanoise;transducersforuseinwater(hydrophonesandpingers),sonarandposiConingsystems

16 28-Feb Notes ApplicaCons:acousCcalposiConing,seafloorimaging,sub-bo`omprofiling;

17 5-Mar GuestLecturer

18 7-Mar GuestLecturer

19 12-Mar Studentreport2presentaCons

20 14-Mar Studentreport2presentaCons.TAKE-HOMEFINALEXAMISSUED

-- 18-Mar FINALEXAMANSWERSDUEby5PM

Announcements

•  Paper 1 absolute deadline is now! •  Don’t forget to submit your proposed topic for paper 2 •  TA Jared Dziurgot will not be able to hold office hours Thursday,

will have extended hours (5:30--) next week •  PCE would like your comments on Zoom technology:

“This short (3 question) survey provides important feedback about your course technology:

The survey will be available through Monday, March 4th. Thanks!”

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Interference à standing waves

•  Two waves propagating in opposite directions with same l and amplitude superpose to form a standing wave

y(x, t) = Asin(kx −ωt)+ Asin(kx +ωt) = 2Asin(kx)cos(ωt)Forward wave Backward wave Standing wave Trig

identity

Notice: Amplitude vs x is fixed, but at each x position, y vs t oscillates Where sin(kx) =0: Minima = nodes Where sin(kx) = 1: Maxima = antinodes

Musical acoustics

•  Musical terminology and scales –  In addition to loudness, human perception of sound (other than simple

monofrequency tones) is complex –  Musical acoustics includes special terminology for factors such as

•  Pitch: perceived tone (not just frequency) •  Timbre – tone quality •  Temperament: definition of musical scales, relation to frequencies

–  Physics of human ears affects perception (more later) –  Brain software creates “aural illusions” (more later)

•  Analogy to optical illusions caused by brain interpreting vision –  Ohm’s law of acoustics (1843): as interpreted by Helmholtz

•  All musical tones are periodic functions, but only sinusoidal vibrations are perceived as pure tones; other qualities are due to mixing (Fourier sum) of different sinusoids

•  Misleading: brain is not simply a Fourier analyzer à Musicians’ distrust of physicists’ analyses of music

Georg Ohm

Pitch: not just frequency

•  Pitch = characteristic of sound that determines position on scale –  A. Seebeck’s siren experiments (c. 1840)

•  Hole spacing + rotation speed à perceived pitch –  Siren (b) sounds 1 octave higher than (a), but… –  Siren (c) sounds ~ the same as (a)

“Moaning Minnie” (London, 1940)

Brain “fills in” missing lower tones because their effect on Fourier spectrum is not critical

Similarity of waveforms with low f’s missing

Fundamental missing Full spectrum

Signal + FT of a square pulse train, similar to Seebeck’s siren

Another example: small loudspeaker in phone has poor response to 100 Hz pure tone (sinusoid) but creates 100 Hz sound nicely with complex signal

More on perceived pitch

•  Interference à perceived tone –  White noise reflected from wall –  Perceived tone has f=1/T=c/L

L

Basset and Eastman, JASA 36:912, 1964

Delay 1 ms 2 ms

4 ms 8 ms

Whi

te n

oise

Notice peaks at 1KHz, 500Hz, 250Hz, 125Hz intervals

More on perceived pitch

•  Perceived pitch also depends on SPL –  cent = logarithmic unit for musical intervals 100 cents = 1 semitone (adjacent piano keys), 1200 cents = octave

Shepard tone illusion

•  Shepard tones are heard as pitch rising from one semitone to the next –  After 12 semitones=1 octave, perceived pitch returns to beginning

•  Normal scale (“equal-tempered”, more later) rises by factor 21/12 for each semitone, each note with full set of partials at fn = n f1

•  Shepard includes only those that are powers of 2 times f1 , fn = 2n f1 , n = 0,1,2,... So frequencies in 1st run up the scale are fm,n = 2m/12 2n f0 ; m =0,1,...11 (note in sequence); n = 0,1,2,.... (repetition of sequence = octave)

–  Modulating the amplitudes of successive octaves à repeated perceived pitches: for m=0: f0, 2f0, 4f0, 8f0,…, when m=12, repeat the same set!

Another way to display weights: Each set of squares in a vertical line composes one Shepard tone. Color indicates increasing loudness of the note, purple to green.

Shepard tone amplitudes, Heller, ch. 23

Shepard tone illusion diagram

Each clockwise step is a semitone higher in pitch, shifts the autocorrelation peaks left, with small changes in their shape. They appear earlier in time and correspond to higher pitch. When the first two tall peaks are about equal in height, they are an octave apart, but the peak closer to t = 0 starts to diminish in height, gradually making the lower pitch more dominant. New peaks arrive after 12 steps, to exactly reproduce the first autocorrelation function.

Pitch and autocorrelation

•  Seebeck’s sirens show perceived pitch may be frequency of missing fundamental component –  “Missing fundamental”

effect – Heller calls it “residue pitch”

•  Signal with period T has autocorrelation peaks at t=nT, n=0, 1, 2,… –  Same true for (signal)2

A.  Power spectrum of sound with f1 =100 Hz and several partials

B.  Fundamental removed à same autocorrelation peaks

C.  Increase power in f2 =100 Hz à autocorrelation peaks appear at half-intervals à perceived as sound with 200 Hz fundamental

Pitch and autocorrelation

•  Example in Heller textbook (p. 448)

•  Initial signal has 3 partials, 200/400/600 Hz –  Perceived as 200 Hz

sound

•  Second signal has additional partials at 50 Hz intervals –  Perceived as 50 Hz tone

Pitch standards for music

•  Today: A440 or A4 (A above middle C ), with f = 440 Hz is the general tuning standard for musical pitch

•  Not so until 20th century! Pitch standard was subject of bitter fights… –  Handel’s tuning fork was 422.5 Hz –  1859 French government commission (Berlioz, Rossini et al) chose 435 Hz –  Verdi wanted to stop “creeping pitch” rise, suggested 432 Hz, based on… –  “Scientific pitch” definition had all C’s powers of 2 (128 Hz, 256, 512, etc)

•  So A4 ~ 431 Hz Note f, (Hz) C0 16 C1 32 C2 64 C3 128 C4 256 C5 512 C6 1024 C7 2048 C8 4096 C9 8192

Lyndon LaRouche, leader of “socialist worker party” cult (died last week), had his followers lobby for “Verdi tuning” and proposed a law in Italy “to impose scientific notation on state-sponsored musicians that included provisions for fines and confiscation of all other tuning forks.” (Wikipedia, Scientific pitch)

Timbre – tone quality

•  Timbre – tone quality or “color” –  Criterion: “attribute that allows listener to judge two sounds dissimilar by

criteria other than pitch, loudness and duration” –  Depends primarily on spectrum but also on waveform, SPL, frequency

range, and envelope shape •  Descriptor scales used for timbre :

Dull … Brilliant Cold … Warm Pure … Rich

–  Attack = onset of the sound (eg, bow on string) •  If attack is deleted leaving only the sustained tone, it is difficult to

identify the instrument •  Attack-decay envelope = shape of amplitude envelope of sound

–  Beating: modulations of amplitude due to summation of partials with similar f’s

•  Interference pattern like this shows locations of nodes and antinodes in space.

•  We can also create moving interference patterns, so a stationary observer hears cyclic intensity changes as maxima pass: This is called beating, or a beat frequency

y1 t( ) = Acos 2π f1t( ), y2 t( ) = Acos 2π f2t( )y1 + y2 = Acos 2π f1t( ) + Acos 2π f2t( )

Trig fact: cos(a) + cos(b) = 2 cos a − b2

⎛⎝⎜

⎞⎠⎟

cos a + b2

⎛⎝⎜

⎞⎠⎟

y1 + y2 = 2Acos 2πf1 − f2

2⎡

⎣⎢⎤

⎦⎥t

⎛⎝⎜

⎞⎠⎟

cos 2πf1 + f2

2⎡

⎣⎢⎤

⎦⎥t

⎛⎝⎜

⎞⎠⎟

fBEAT = f1 − f2 fosc =f1 + f2

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Beats Anti-node

Node

•  Beats are heard if waves with similar frequencies overlap at the observer s location, x: then at that spot, amplitude vs time is

The sum has a base frequency fOSC , modulated by an envelope of frequency fBEAT

fOSC

fBEAT

Notice: fBEAT is twice the f in the cosine function: Envelope goes from max to min in ½ cycle, so frequency of pulsation is 2

f1 − f22

⎡

⎣⎢⎤

⎦⎥= f1 − f2

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fBEAT = f1 − f2

fOSC =f1 + f2

2

Beats •  Below: 2 waves with slightly different f’s are travelling to the right. •  The waves are in the same medium, so have the same speed. •  Superposition sum wave has the same direction and speed as the

two component waves, but its local amplitude depends on their relative phase.

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Beat wave oscillates with the average frequency, and its amplitude envelope varies with the difference frequency. The dots show how y vs t varies at two fixed x positions

fBEAT = f1 − f2 = 392 Hz − 261 Hz( ) = 131 Hz

fOSC =f1 + f2

2= 326.5 Hz

Beats •  Example: pluck 2 strings on a guitar: middle C (261 Hz) and G (392 Hz)

base frequency modulated by an envelope:

fOSC=327Hz

fBEAT=131 Hz

fOSC=327Hz = ~ E in next octave

fBEAT=131 Hz = C an octave below middle C

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Musical scales and temperament

•  “Just intonation”: tuning system that involves intervals limited to integer ratios of fundamental f – examples: –  “5-limit tuning”: notes multiply f1 (the base note) by products

of integer powers of 2, 3, or 5 •  Powers of 2 = octaves (f ratio 2:1), powers of 3 = intervals

of perfect fifths (3:2), powers of 5 = intervals of major thirds (5:4)

–  Pythagorean scale: only pure octaves and perfect fifths allowed

•  If only integer ratios are used, chords sound in tune only if based on the same fundamental frequency –  changing musical keys is not possible.

major third

Frequencies vs temperament

•  Temperaments allow fixed-pitch instruments (eg, piano, harpsichord) to be played in different keys without dissonance –  “Well-tempered” (eg Bach) = some keys are more in tune than others, but

all can be used –  Equal-tempered (modern standard) = every pair of adjacent notes has the

same ratio of f’s: 21/12 à pitch is perceived as ~ log(f)

Acoustics of stringed instruments

•  Guitars: –  Nomenclature

–  Lumped-parameter model for guitar body

We’ve studied vibrating strings, and resonators…

Recall: Guitar strings

•  Excite waves by plucking guitar string –  Waves with λ such that L = multiple of λ/2 are reinforced

(resonate) •  L = λ/2, 2(λ/2), 3(λ/2)… : harmonics

–  What determines λ? Speed of wave on string depends on •  Tension in string (force stretching it)

–  Taut = higher speed, slack = lower speed •  Inversely on mass per unit length of string material

–  Heavy string = slower speed, light string = faster So v = √ F/µ , F is in newtons and µ = kg/meter

www.physicsclassroom.com/

Vibrating strings

•  Example: First string of guitar (thinnest) is tuned to E above middle C, f = 330 Hz

•  String has length L=0.65m and mass 2 grams. What tension is needed?

•  Frequency f=v/λ–  to get f=330 Hz for λ =L=0.65m, we need

v=fλ=(330Hz)(0.65m) = 214 m/s

•  Mass density of string is µ = 0.002kg / 0.7m = 0.0028 kg/m

•  v 2 = F/µ , so F= v 2 µ = (214m/s)2 0.0028kg/m = 128 kg-m/s2 (notice: units=newtons) 4.5N=1lb, so about 28 lbs tension needed

L

L/2 L/4

Nut

Bridge

Frets

Examples •  Guitar string is tuned to E, at f = 330 Hz when open (L= nut to bridge) •  Where should a fret be placed to make it resonate at E one octave

higher, at f = 660 Hz? In general,

In this case,

•  Where should the fret be to make it resonate at E two octaves higher, at f = 1320 Hz?

44 141

LLff =→=

L1 =λ12→ λ1 = 2L1→ f1 =

cλ 1=

c2L1

λn =λ1n=2L1n

→ Ln =L1n, fn =

cλ n

= n f1

21

2LL =

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Guitar acoustics

•  Body serves as Helmholtz resonator –  Typically tuned (vary hole, body volume, bracing) to

•  A2 (55.0 Hz) for steel-string guitars •  G#2 (103.8 Hz) for classical guitars •  F#2 ~G2 (92.5 ~ 98.0 Hz) for Flamenco guitars

For guitar with circular hole radius r, body volume V

Helmholtz resonant freq f0 =c

2ππr2

LEFFV, LEFF =1.7r

Vibrations at low f’s

Vibrations at high f’s

Guitar acoustics

•  Body serves as Helmholtz resonator –  Typically tuned (vary hole, body volume, bracing) to

•  A2 (55.0 Hz) for steel-string guitars •  G#2 (103.8 Hz) for classical guitars •  F#2 ~G2 (92.5 ~ 98.0 Hz) for Flamenco guitars

For guitar with circular hole radius r, body volume V

Helmholtz resonant freq f0 =c

2ππr2

LEFFV, LEFF =1.7r

Bracing in body is critical for tuning resonances, and to support forces

Spectrum of resonances in bass guitar playing 110 Hz A

Violins

•  Nomenclature Many figs from newt.phys.unsw.edu.au/music/

Acoustics of violins

•  Bow drives strings à bridge serves as driver for body –  Bow drags short distance, then releases: sawtooth motion

•  Constant speed of bow, then rapid return to equilibrium •  Vibrations of bowed vs plucked string

–  Violin body = resonator to amplify the vibration of the bridge from air in body to surroundings

•  Need large surface area to push a reasonable amount of air •  The belly and back plates have a number of resonances

–  Recall Chladni patterns of vibration •  Shaping of plates and wood used

are critical factors

Efficiency vs f, for 2 real violins: solid=mass produced; dashed=Guarneri; note resonances

Bowing: displacement and velocity of string vs time

Violin components

•  Bowing a string

Plucked string

Violin bridge

Violin acoustics

•  Lumped parameter model of violin body –  h=f-holes, p=top plate, b=bottom plate

•  Bow force vs position vs timbre

•  Wolf tone: –  At some point in range, the tone varies

strongly and harshly, pulsating at ~5 Hz –  Due to overlapping resonances between

string and body –  Defeat by adding mass near bridge

Wolf tone: oscillogram by C.V. Raman, 1916 (yes, the Raman)