Physics 1251 The Science and Technology of Musical Sound

1
Physics 1251The Science and Technology of
Musical Sound

• Unit 2
• Session 24 MWF
• Review of Perception
• and Strings

2
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Why do the characters in the film clip concern
  themselves so much with the tones produced by
  tapping on the plate?

The vibration modes of the plates determine the
timbre of the finished instrument.
3
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Wolf Tones

String Harmonics
Body Harmonics
Beats !_at_!!!
4
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Fighting the Wolf in a Famous Violin

Stradivarius Violin
The solution a reflector
5
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 1' Lecture (Unit 2 Review)
• The function of the Human Ear determines the
  essential features of acoustics.
• Sound Intensity Level (in dB) is a logarithmic
  measure of intensity.
• Loudness is a subjective measure of the dynamic
  of sound.
• Pitch, which is determined by frequency, is a
  measure of the highness or lowness of a musical
  sound.

6
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 1' Lecture (continued)
• Waves are characterized by
• Frequency
• Wavelength
• Velocity
• Reflection
• Specular
• Diffuse
• Refraction
• Difraction

7
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 1' Lecture (continued)
• Doppler effect
• Beats
• Interference
• Room acoustics are determined by the volume,
  surface area, shape and absorption properties of
  its contents.
• The reverberation time of a room can be
  calculated using the Sabine Equation.

8
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 1' Lecture (continued)
• Scales are based on tones whose frequencies are
  the ratio of whole numbers.
• Strings produce a harmonic series of vibrations
  described by the Marsenne equation.
• The vibration modes of the body and air cavity of
  a stringed instrument strongly modify the
  harmonic recipe produced by the string vibrations.

9
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• How to organize so much information?

• By Topic
• Physiology
• Perception
• Room Acoustics
• Wave Properties
• Strings

- the ear is a traducer from pressure
fluctuations to neural stimulation
- non-linear in intensity and frequency
sensitivity
- properties determine quality of sound
- reflection, refraction, diffraction, Doppler
effect, beats and inference
- f determined by standing waves
10
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• By Equation
• SIL
• Pitch
• Inverse Square Law
• Reverberation Time
• Harmonics
• Law of Reflection
• Beats
• Marsenne Equation

SIL10?Log(I / Ithreshold ) Pitch
Interval3986?Log(f1 /f2 ) I /I0 (r0
/r ) 2 TR 0.16 V/Se fn
nf1 Angle in Angle
out fmean (f1 f 2)/2 fbeat ?f1
f2? fn n/(2L) ? v(T/µ)
11
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• By Chronological facts
• Anatomy of the Ear
• 80/20Middle Ear The Ossicles (little bones)

• Malleus -- (the hammer) moved by Tympanium.

2. Incus -- (the anvil) supported by ligaments
that protect against loud percussion.
3. Stapes (the stirrup) force multiplied by
1.3 because of lever action.
12
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Anatomy of the Ear
• 80/20Inner Ear

Cochlea (the Snail) converts displacement into
neural impulses.
Auditory Nerve neural impulses to brain
Semicircular canals detect motion and
orientation
13
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Frequency Discrimination in Cochlea
• 20 Hz to 20 kHz (typical in Humans)
• Resonances in Basilar membrane and in HC cause
  spatial separation by frequency.
• Differential movement of membranes stimulate HC.
• Minimum stimulation required for response.
  Inhibition of neighbors causes non-linear
  response.

14
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The frequency range of detectability for
  humans is approximately 20 Hz to
• 20 kHz.
• 80/20Humans are most sensitive in the frequency
  range 2 kHz to 5 kHz.

15
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The objective relative intensity level of sound
  is quantified as the Sound Intensity Level (SIL)
  and is measured in deciBel (dB), where
• SIL 10 Log( I / Ithreshold )
• 80/20The Intensity of a sound wave is the energy
  radiated per unit time per unit area. W/m2

16
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The lowest detectable intensity (the
  threshold of hearing Ithreshold ) is about
  1 pW/m 2 or 1x10 -12 W/m 2 SIL 0dB.
• 80/20The intensity at which one experiences pain
  (the threshold of pain) is about 1 W/m
  2 120 dB.

17
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Sound Intensity Level
• SIL 10 Log ( I / I threshold ).
• The Sound Intensity Level is 10 times the
  logarithm of the ratio of the intensity of a
  sound and the threshold of hearing.
• The units of SIL are deciBel or dB.
• I I threshold 10 SIL/10

18
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Application
• What is the Sound Intensity Level of a tone that
  has an intensity of I 1.26 mW/m2 ?

SIL 10 Log ( I / Ithreshold)( I / Ithreshold)
(1.26 x10 -3 W/m2/1 x10 12 W/m2) 1.26 x 10
9 Log ( I / Ithreshold) Log(1.26 x 10 9)
9.1SIL 10 Log ( I / Ithreshold ) 10 (9.1)
91 dB
19
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Application
• What is the Intensity Level of a tone that has
  an SIL of 35 dB?

I Ithreshold 10SIL/10SIL/10 35/10
3.510SIL/1010 3.5 3.2 x 10 3 I Ithreshold
10SIL/10 (1 x 10-12 W/m 2 )( 3.2 x 10 3 )
3.2 x 10 -9 W/m 2
20
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Just Noticeable Difference (JND) is the
  limen of difference that elicts 75 in a Two
  Alternative Forced-Choise test (2AFC test).

The limen of intensity is a ratio of about 1.26
which corresponds to a SIL difference of 1
dB. 10 Log( 1.26 ) 1.0
21
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Musical Dynamics
• Pianissimo pp very soft 50 dB
• Piano p soft 60 dB
• Mezzopiano mp medium soft 66 dB
• Mezzoforte mf medium loud 76 dB
• Forte f loud 80 dB
• Fortissimo ff very loud 90 dB
• Fortississimo fff Very, very loud 100 dB

22
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The Fletcher-Munson Diagram is a plot of
  the SIL (in dB) versus frequency for the SIL
  required to produce an equal sensation as that
  produced at 1000 Hz.
• 80/20The contours are of equal loudness level.
• 80/20The unit of loudness level is the phon.

23
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Fletcher- Munson Diagram
• Fletcher and Munson (1933) J. Acoust. Soc. Am. 5,
  82-108

130
120
110
100
90
SIL (dB)
80
70
60
50
40
30
20
10
Loudness (phon)
Frequency (Hz)
24
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Loudness
• 80/20 A subjective measure of the magnitude of
  auditory sensation is called Loudness and is
  measured in sone. In this system, one listens to
  two sounds and judges how much louder or softer a
  test sound is compared to the reference.
• For example, a sound of 2 sone sounds twice as
  loud as a tone of 1 sone.

25
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Loudness Scaling
• Loudness ?? I
• 80/20Thus, an eight (8) singer ensemble sounds
  about twice (2x) as loud as a soloist. Because
  ?8 2.
• Likewise, a choir of sixty-four (64) sounds
  about four (4x) times louder than a soloist. ?64
  4.

26
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The Just Noticeable Difference (JND) or
  difference limen is the difference in pitch (or
  loudness) that will elicit 75 correct responses
  in a Two-Alternative Forced-Choice test (2AFC)
  test.
• 80/20The Difference limen for pitch is about 1/30
  the critical band width and varies from 40 at
  low frequencies (lt62 Hz, C2) to 6 at high
  frequencies (gt8kHz, C9 ).

27
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (SIL and Loudness)
• Loudness is the magnitude of the sensation
  produced by a sound it is measured in sone.
• Loudness Level (in phon) is equal to the SIL at
  1000 Hz that produces the same magnitude of
  sensation.
• Loudness increases approximately with the cube
  root of intensity.

28
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• The loudness depends on frequency.

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Pitch Interval
• 80/20An equal ratio of frequencies sounds like an
  equal difference or interval of Pitch.
• 80/20An octave is the pitch interval
  corresponding to a frequency ratio of 21.

30
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Pitch Interval
• 80/20A semitone is 1/12 of an octave.
• 80/20A cent () is 1/100 of a semitone or 1/1200
  of an octave. 1 octave 1200 .
• 80/20Pitch Interval corresponding to f2 and f1
• P 1200/Log 2 ? Log (f2 /f1 )
• P 3986 ? Log (f2 /f1 )

31
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Pitch Interval Application
• What is the pitch interval (in ) that
  corresponds to a perfect 5th?

An interval of a perfect 5th corresponds to a
ratio of f2 /f1 3/2 1.5. P 3986 ? Log (f2
/f1 ) 3986 ? Log(1.5) 3986 ? (0.176)
702
32
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The sensation of pitch is a property of
  human auditory perception that infers pitch from
  the repeat period.
• 80/20Harmonics (including the fundamental) may be
  missing but we hear the difference frequency as
  well as the harmonics.

33
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Critical Band Width wcritical
• 80/20If ?f lt wcritical , sounds like one sound.
• If ?f gt wcritical , sounds like separate
  sounds.
• 80/20If ?f lt wcritical , sounds softer than
• if ?f gt wcritical .
• For example, a choir singing in prime unison
  (exactly the same pitch) sounds softer than when
  singing parts.

34
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (Pitch)
• Tones are indistinguishable if they are closer
  than a critical band width.
• Pitch interval is proportional to the logarithm
  of the ratio of the frequencies.
• An octave corresponds to a ratio of 2/1.
• A semitone is 1/12 of an octave.
• A cent is 1/100 of a semitone.

35
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• The sensation of pitch is a property of human
  auditory perception that infers pitch from the
  repeat period.

36
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The intensity of a spherical sound wave decreases
  as the distance from the source increases.
  Inverse

d m I W/m2
1.0 1.0
12 /12 12/2 2 12 /3 2 12 /4 2 12 /5 2 As
1/r 2
2.0 0.25
3.0 0.11
4.0 0.068
5.0 0.040
37
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Inverse Square Law
• The intensity of sound (originating from a point
  source in an open environment) diminishes as the
  square of the inverse ratio of the distances from
  a source.
• I / I0 (r0 /r) 2

38
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Inverse Square Law
• The Sound Intensity Level (SIL) decreases by 20
  dB for every 10x increase in distance.

SIL SIL0 20 Log (r0 / r)
39
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• ApplicationInverse Square Law
• What is the intensity of the sound of an
  explosion heard 100 m away when the intensity is
  1 W/m2 at a distance of 10 m?

I I0 (r0 /r) 2 I (1 W/m2)(10m/100m)2 (1
W/m2)(.01) 0.01 W/m2 10. mW/m2
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Reflection
• 80/20Reflection, a bouncing back, occurs
  whenever there is an abrupt change in the medium.

41
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• What happens when a wave hits a change in the
  medium?

Reflection
Tin Tout
Tout
Tout
Tin
Tin
Transmission
Medium 1
Medium 2
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• When the surface is smooth we have specular
  (mirror-like) reflection.

Specular Reflection
Smooth Surface Roughness ? ?
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• What if the surface is rough?

Diffuse Reflection
Rough Surface Roughness gt ?
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Refraction
• 80/20Refraction, a bending aside, occurs
  whenever there is a change in velocity along the
  wavefront.

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Refraction occurs when a wave enters a medium
  that has a different velocity?

Refraction
V1 lt V2
46
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• In the second medium the wavefront races ahead,
  changing the direction.

Refraction
V1
V2
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Wind Speed Refraction
• velocity varies along wavefront

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Cold Aloft
• v - slower Refraction
• velocity varies with
  temperature
• v - faster
• Warm below

silence
49
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Diffraction
• 80/20Diffraction, a bending around obstacles
  because every point on a wave is a source waves
  cannot terminate abruptly.

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Huygens Principle every point on the wave
  is the source of a new (spherical) wavelet.

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• What happens when a wave is partially obstructed?

Diffraction
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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Doppler Shift
• 80/20The Doppler shift is the change in the
  frequency of a source fs to that observed fo due
  to the relative velocity of the source vs , or
  observer vo ,where the velocity of sound in the
  medium is v.
• f o f s ? v - vo / v - vs,
• sign is in direction of v, e.g. for observer
  moving toward source vo lt0, source moving
  toward observer vs gt0.

53
Physics 1251 Unit 2 Session 18 Room Acoustics
Doppler Shift Moving source
Higher f
Lower f
fobserver fsource v - vobserver / v
vsource
54
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Addition of Waves
• 80/20Coherent addition of waves is the addition
  of the amplitudes of waves of the same or nearly
  the same frequency the intensity then is
  proportional to the square of the the combined
  amplitude.
• 80/20Incoherent addition of waves is the addition
  of their independent intensities if they are not
  within the critical bandwidth.

55
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Beats
• 80/20Beats are a modulation of intensity due to
  the coherent addition of two sound waves whose
  frequency lie with a critical band width of each
  other but differ slightly in frequency.

56
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Beat Frequency
• 80/20Two tones of frequency f1 and f2 sound like
  one tone of mean frequency
• f mean (f1 f 2)/2
• that beats at a beat frequency of
• f beat f1 f2 .

57
Physics 1251 Unit 2 Session 18 Room Acoustics
Beats f1 f2
In phase
Out of phase
fmean
fbeat
58
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Interference
• 80/20Interference is the coherent addition of
  waves at various points in space leading to
  spatial modulation of intensity.

59
Physics 1251 Unit 2 Session 18 Room Acoustics
Interference
Constructive
Destructive
Softer
Louder
60
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (Propagation)
• Reflection is the return of a wave from a change
  in a medium.
• Refraction is bending of the wave due to velocity
  variation along the wavefront.
• Diffraction is the bending of the wave around
  obstacles because of Huygens Principle.
• Doppler Shift the change in frequency due to
  the relative motion

61
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• Beats are intensity modulations in time due to
  coherent addition of waves of near equal
  frequency.
• Interference is spatial modulation of intensity
  due to the coherent addition of nearly equal
  frequency waves at different points in space.

62
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Standards for Good Acoustics
• Clarity
• Uniformity
• Envelopment
• Smoothness
• Reverberation
• Performer satisfaction
• Freedom from noise

little overlap of sounds
everywhere the same
sound from all directions
no echoes
appropriate length of time
reflected to stage
no competition
63
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20 Haas or Precedence Effect
• The earliest sound that arrives determines the
  sense of the origin of a sound, even if the later
  (lt100 ms) reflections are louder.
• The direct sound should arrive first.

64
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Room Acoustics Reverberation

Speaker
Hearer
65
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Energy Lost in Reflections
• The sound reflects many times, each time losing
  energy to the reflecting surfaces.
• The quantity a is the absorptivity of the
  surface.
• The intensity of the sound that is lost in a
  reflection is ?Ilost a Iin.

66
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The intensity of the reflected wave is
• Ireflected (1-a) Iin.
• Values for a, the absorptivity, for many types of
  surfaces have been measured and appear in
  extensive tables.

67
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Absorptivity
• Material a (at 500 Hz)
• Acoustic tile 0.6
• Plaster wall 0.1
• Concrete 0.02
• Person 0.8 (x1 m2)

68
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Wallace Sabine
• (Harvard professor 1868-1919)
• Asked How long will it take for the
• sound to die down to 1 millionth (-60 dB)
• of the initial value?
• 80/20The reverberation time is the time for the
  intensity to decay by a factor of 10 6 (-
  60dB) of its initial value.

Wallace Sabine
69
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The Sabine Equation
• I Io ? 10 6 (t/TR)
• TR 0.16 V/Se
• V is the volume of the room.
• Se is the effective surface area of the walls
  S1 , floor S2 and ceiling S3 (in sabin) etc.
• a is the absorptivity of the surface (in table)
• Se a1 S1 a2 S2 a3 S3 a4 S4

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Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (Reverberation)
• Direct sound should come first
• Haas or precedence effect
• Reverberant sound
• TR 0.16 V/Se
• 0.8 sec for clear speech, 1-2 for music
• Freedom from echo and interference
• Use diffuse and random reflectors
• Background noise level
• Assure good acoustic isolation

71
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Sabine EquationApplication
• What will be the relative intensity of a sound
  after 1.0 second in room with a reverberation
  time of 3.0 seconds?

Sound decays exponentially in a room I Io ? 10
6 (t/TR) I /Io 10 6 (t/TR) 10 6 (1.0
sec/3.0 sec)10 2 I /Io 0.01 1
72
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Sabine EquationApplication
• What is the reverberation time of a room that is
  1000 m3 in volume with a surface area of 125 m2
  of highly absorbent acoustical material with an a
  of 0.95?

Sabine Equation TR 0.16 V/Se Se a S 0.95
(125 m 2) 119 sabineTR 0.16 (1000/119) 1.3
seconds
73
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Build a Scale from Whole Number Ratios of
  Frequencies!
• 80/20A scale is a series of tones arranged in
  ascending pitch.
• 80/20In Pythagorean (or Just) intonation the
  frequencies of the tones of the scale are integer
  ratios of each other.
• 80/20The beginning pitch of the scale is called
  the tonic.

74
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Musical Notation

?
?
?
?
?
?
?
?
?
?
?
D5
A5
C6
G4
G5
F4
C5
F5
?
B4
B5
E4
?
E5
D4
A4
?
?
?
?
?
?
?
?
?
?
C4
F2
B3
E2
A4
D2
G2
C2
75
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Musical Notation

?
?
?
?
??
?
??
??
??
?
??
C5?
G4?
B4
F4?
?
E4
A4?
??
??
D4?
??
?
??
??
?
?
?
?
?
?
A3?
E3?
C4?
D3?
F3
G3?
B3?
C3
76
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20With Just Temperament it is impossible to
  tune all notes or tones of the chromatic scale so
  that they are all in tune, that is, the correct
  ratio for all keys.
• 80/20Equal Temperament intonation is a compromise
  in which the semitones are precisely 100 or a
  ratio of 1.05946 of its neighbor.

77
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (scales)
• A scale is a progression of tones arranged in an
  ascending order of pitch.
• The tonic of a scale is the pitch that is the
  basis of the scale.
• The chromatic scale is a series of tones, each
  separated from the next by a pitch interval of a
  semitone.
• The diatonic scale contains a series of (whole)
  tone and semitone intervals.

78
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• Just Temperament uses integer ratios of adjacent
  notes in the chromatic scale.
• Equal Temperament uses a constant ratio of
  adjacent notes in the chromatic scale.

79
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• What is a scale?
• Gamut Note G-Clef

?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
Do Re Mi Fa So La Ti Do
Do Re Mi Fa So La Ti Do
C-major
G-major
Guido dArezzo gamma ut?gamut Solfeggio G is
Do in the G-scale
80
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• What is a scale?
• Solfege

Do
Ti
La
So
Fa
Mi
Re
Zoltan Kodaly
Do
81
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Circle of Fifths

45
56
23
82
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20It is impossible to tune perfectly scales in
  all keys using the same frequencies and just
  temperament.
• 80/20The Equal Tempered Scale sets each semitone
  exactly 100 apart or at a ratio of 1.05946.

83
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Musical Notation

??
?
??
??
??
?
??
C5?
G4?
B4
F4?
?
E4
A4?
??
??
D4?
??
?
??
??
?
A3?
E3?
C4?
D3?
F3
G3?
B3?
C3
84
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Velocity of a wave on a string
• vstring vT/ µ
• T is the tension in the string N.
• µ is the mass per unit length kg/m

85
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20f1 1/(2L) ? vstring
• f1 1/(2L) ? v(T/ µ)

Tuning pegs
More Tension Raises pitch
86
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20f1 1/(2L) ? v(T/ µ)

Denser wire lower pitch
Less dense wire higher pitch
87
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (scales and strings)
• A Just Tempered Scale sets the frequencies of the
  notes in the scale at precise ratios of whole
  numbers.
• The Equal Tempered Scale is the compromise that
  sets all notes an equal interval apart (100,
  frequency ratio1.05946).
• f1 1/(2L) ? v(T/ µ)

88
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• ApplicationMarsenne Equation
• What is the fundamental frequency of a 1.00 m
  long string stretched with a tension of 700 N
  that has a mass density of 0.005 kg/m.

f 1/(2L) v (T/ µ) 1/(2(1.00 m)) v (700/
0.005) 1/ ( 2.00) (374.) 187. Hz
89
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A musical instrument comprises four
  important components
• (1) a mechanical energy source
• (2) a frequency generator
• (3) a frequency filter
• (4) an antenna.

90
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A stringed instrument uses standing waves on
  a string to provide the frequency generation.

String Standing Waves
91
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The distance between neighboring nodes
  antinodes is ¼ ?. N-A d ¼ ?

Second Harmonic
f2 vstring / ?2 vstring / L
Node
Node
Node
Antinode
Antinode
?2 /4
?2 /4
?2 /4
?2 /4
92
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20Frequency of nth harmonic
• Marsenne Equation
• fn n/(2L) ? v(T/ µ)

93
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• ApplicationMarsenne Equation
• What are the harmonics of the string discussed
  earlier? (L 1.00 m, T 700 N, µ 0.005 kg/m)

f1 1/(2L) v (T/ µ) 1/(2(1.00 m)) v (700/
0.005) 1/ ( 2.00) (374.) 187. Hz f2 2
f1 374 Hzf3 3 f1 561 Hz Etc
94
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The modes that will be present are those
  that are excited.

Energy source can excite selected modes
95
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The modes that will be present are those
  that are excited.

96
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Guitar is a fretted stringed instrument
• that is plucked or strummed.

Tuning pegs
Neck
Frets
Strings (6)
Body
Bridge
97
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The coupling of the string to the sound
  board will actuate or attenuate various
  frequencies.

Sound Board and Acoustics filter harmonics
98
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A Harp is a stringed instrument that is
  plucked. Its strings are of various lengths.

Strings
Traditional Irish Harp
Sound Board
99
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The principal pitch of each string in a Pedal
  Harp is determined by its length.

Pedal Harp
Strings
Pedals
However, depressing a pedal shortens or lengthens
the string length, raising (or lowering) the
pitch incrementally.
100
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A Harpsichord is a keyboard stringed
  instrument whose strings are plucked.
• Each string is used for one pitch only.

101
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A Harpsichord is a keyboard stringed
  instrument whose strings are plucked.
• Each string is used for one pitch only.

Harpsichord Action
Damper
Plectrum
String
Jack
Key
102
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A Piano is a keyboard stringed instrument
  whose strings are hammered. Each string is used
  for one pitch only.

Fortepiano
Piano
Strings
Key Board
Sound Board
103
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20A Piano has multiple strings (in the treble)
  to provide a more powerful sound.

Multiple Strings
Sound Board
Bridge
Frame
104
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Piano strings are tensioned by tuning pins in a
  strong frame.

Frame
Bass Strings
Tuning Pins
105
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Piano hammers are made of felt and wood.

Hammer Head
Back Check
Wood
Felt
Pivot hinge
106
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The timbre of the piano sound is affected by the
  hardness of the hammer felt.

Felt hard or soft?
107
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The timbre of a musical instrument is
  determined by the excitation, frequency
  generation, filtering and broadcast of the
  harmonics.

Excitation
f Response
f Generation
Broadcast
108
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (strings)
• fn n/(2L) ? v(T/ µ)
• The node-antinode distance is ¼ ?.
• The timbre of a stringed instrument is determined
  by its harmonic recipe.
• The harmonic recipe of a stringed instrument is
  determined by its mode of excitation, the
  harmonics of a string, and the natural frequency
  response of the instrument.

109
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• A Guitar is a stringed instruments with frets
  played by plucking.
• A Harp is a fretless stringed instrument played
  by plucking.
• The Harpsichord is a stringed instrument with
  keys which when depressed lift a plectrum that
  plucks one or more strings.
• The Piano is a stringed instrument with a
  keyboard which causes a felt hammer to strike
  the string.

110
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The timbre of bowed strings is affected by
  the properties of strings, by bowing and by the
  frequency resonances of the instrument.

111
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Bowed String Instruments
• Violin
• G3 , D4 , A4 , E5 (5ths)
• Viola
• C3 , G3 , D4 , A4
• (5th below violin)
• Cello
• C2 , G2 , D3 , A3
• (8vo below viola)
• Bass (Viol)
• E2 , A2 , D3 , G3 (4ths)

Bridge
Bow
Strings
112
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Pitch is changed by shortening the speaking
  length of the strings by fingering

113
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings
The strings are tensioned by means of tuning pegs
in a scroll.
114
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Density of strings sets range of each string

Denser wire lower pitch
Viola Strings Metal over catgut
C3
G3
D4
Less dense wire higher pitch
A4
115
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Different pitches can
• be played on different
• strings or on the same
• string.

Example Air on the G-String J.S. Bach
116
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Anatomy of a String

Stradivarius Violin
Scroll
Fingerboard
Body
Bridge
f Holes
Tail piece
Guarneri Viola
117
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The sound post transfers vibration from
  front plate to back plate and supports bridge.

Front plate
Sound post
Back plate
118
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• A rosined horse hair bow rubs the string.

119
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Action of the Bow
• The rubbing of the bow
• alternately pulls the string
• forward, then releases
• it when the string wave
• reaches the bow.

120
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The Stick-Slip Mechanism causes the string to
  vibrate when rubbed by the bow.

Horse Hair of Bow
Slip point
String
121
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• A complicated wave shoots down the string when
  the string slips from the bow.

Moving Bow
Scroll
Bridge
Waveform envelope
122
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• 80/20The coupling of the string to the sound
  board will accentuate or attenuate various
  frequencies.

Sound Board and Acoustics filter harmonics
123
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Cross Section of Violin

Finger board
f-hole
Scroll with Tuning pegs
Body
Sound Post
124
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Plates are of varying thickness and are hinged
  at edge (Purfling)

Purfling
Thicker
Thinner
125
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• The bridge is essential to transforming the
  lateral motion of the string into vibrations of
  the top plate.

Bridge
126
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Bridge Action

Bass bar
Sound post
127
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Vibration is transmitted to the body and air
  cavity by the action of the Bridge.

Rocking motion
Helmholtz Resonance
Sound post
128
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Chladni plates simulate the vibrational modes of
  Violin Plates.

129
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary (Bowed Strings)
• fn n/(2L) ? v(T/ µ)
• The bow works by the Slip-slick mechanism.
• The timbre of a bowed stringed instrument is
  determined by its harmonic recipe.
• The harmonic recipe of a bowed stringed
  instrument is determined by its modes of
  excitation, the harmonics of a string, and the
  natural frequency response of the instrument.

130
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary
• A violin, viola, cello and bass are stringed
  instruments with a finger board played by bowing.
• The string vibrates the bridge which vibrates the
  top plate.
• The sound post transmits the vibration to the
  back plate.
• The timbre of the violin family is strongly
  affected by the modes of vibration of the
  instrument.

131
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary of Unit
• Human sound perception sets the priorities in
  musical acoustics
• SIL in dBA is more like human hearing intensity.
• Pitch is inferred from the place that resonance
  occurs in the Basilar membrane of the Cochlea.
• The range of human hearing is 20 Hz to 20 kHz.
• The softest sound that is audible has an SIL of
  0 dB and an Intensity of 1 x 10 -12 W/m2.
• Waves have the following properties
• Reflection specular and diffuse, a bouncing back.

132
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary of Unit
• Refraction bending due to velocity changes.
• Diffraction bending around obstructions.
• Doppler shift frequency change due to motion of
  the source or observer.
• Beats amplitude modulation due to the coherent
  addition of two tones of nearly equal frequency.
• Interference spatial modulation of intensity due
  to coherent addition of two or more waves of same
  f.
• The acoustics of a room depend on its shape and
  size.
• The reverberation time is proportional to the
  ratio of the volume and its effective surface
  area.

133
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary of Unit
• A Scale is the arrangement of tones in a
  progression lower to higher pitch.
• The frequency of notes in the scale is determined
  by the ratio of whole numbers in Just Intonation
  or temperament.
• The ratios of frequencies of notes in a scale
  using Equal Intonation or Temperament are
  constant Semitone ratio 1.05946.
• The pitch interval is proportional to the
  logarithm of the ratio of the fundamental
  frequencies.
• A semitone corresponds to 100.

134
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary of Unit
• Strings obey the Marsenne Equation.
• In Standing Waves the distance from a node to an
  adjacent antinode is ¼ wavelength.
• The frequency of a bowed instrument is determined
  by the feedback from the string.
• The harmonic recipe of a stringed instrument is
  determined by
• The means of excitation of the instrument.
• The normal modes of vibration of a string.
• The filtering by the resonances of the
  instrument.
• The coupling efficiencies of the instrument to
  the air.

135
Physics 1251 Unit 2 Session 24 Review of
Perception and Strings

• Summary of Unit
• One can distinguish the instruments
• Guitar, Harp, Harpsichord and Piano by their
  respective acoustic construction and excitation
  and by their resulting timbre.