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 !#@%!!!

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 SIL=10‧Log(I / Ithreshold ) Pitch Interval=3986¢‧Log(f1 /f2 ) I /I0 = (r0 /r ) 2TR = 0.16 V/Se fn = nf1 Angle in = Angle out fmean= (f1 +f 2)/2; fbeat =⃒f1 –f2⃒fn = n/(2L) ‧ √(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/10=10 3.5= 3.2 x 10 3I = 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 (<62 Hz, C2) to ~6¢ at high frequencies (>8kHz, 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.

29. 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 2:1.

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 : ₧ = [1200¢/Log 2] ‧ Log (f2 /f1 ) ₧ = 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. ₧ = 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 Δf < wcritical , sounds like one sound. If Δf > wcritical , sounds like separate sounds. 80/20If Δf < wcritical , sounds softer than if Δf > 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 Application—Inverse 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

40. 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 Θin = Θout Θout Θout Θin Θin Transmission Medium 1 Medium 2

42. 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 ≲ λ

43. Physics 1251 Unit 2 Session 24 Review of Perception and Strings What if the surface is rough? Diffuse Reflection Rough Surface Roughness > λ

44. 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.

45. 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 < 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

47. Physics 1251 Unit 2 Session 24 Review of Perception and Strings Wind Speed Refraction velocity varies along wavefront

48. 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.

50. 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.”