The ear and perception of sound (Psychoacoustics) Updated 2013Aug05 1

Outline A.Structure of the Ear B.Perception of Pitch C.Perception of Loudness D.Timbre (quality of sound) E.References 2

Introduction Psychoacoustics is the study of subjective human perception of sounds. 3

A. The Structure of the Ear The length of the auditory canal has been greatly exaggerated 4

A.1 Outer Ear Amplifies Sound Auditory canal is a resonator at approximately 2000 to 5000 Hertz. 5

A.2 The Middle Ear The bones (ossicles) of the middle ear form a lever which “amplifies” the displacement by a factor of 3x. The stirrup transfers the force to the much smaller area of the oval window, resulting in 10 to 30 x increase in pressure level Overall the sound is amplified by as much as 1000x or 30 dB 6

A.3 Inner Ear Senses Sound Reference: 7 Over 20,000 hair cells!

B. Perception of Pitch 1.Range of Hearing 2.Pitch Discrimination and jnd 3.Combination tones 8

1a Range of Hearing Humans can hear from 16 to 20,000 Hertz (In terms of music, this is about 10 octaves) Piano only goes from 27.5 to 4186 Hertz 9

1b Test Hearing High Frequency Test Low Frequency Test 10

2a. Pitch Discrimination At 1000 Hz, the “jnd” is about 1 Hz (0.1%) At 4000 Hz, the “jnd” is about 10 Hz (0.25%) Above 10,000 Hz, our discrimination is terrible. (Most music is in range of 30 to 4000 Hertz) We can distinguish approximately 5000 different tones 11

2b. Beats Two tones closer than 15 Hertz we hear as a “fused” tone (average of frequencies) with a “beat”. 12 Demo:

3. Combination Tones When tones are far enough apart we hear them as two distinct tones We also hear difference and sum tones that are not really there (Tartini Tones 1714) 13 Demo:

C. Perception of Loudness 1.Fechner’s law and decibel scale 2.Discrimination (jnd) 3.Threshold of hearing 14

1. Which sounds half as loud as first? Reference: 15

1b. Decibels: Fechner’s Law 1860 Fechner’s Law As stimuli are increased by multiplication, sensations increase by addition (Sensation grows as the logarithm of the stimulus) Example: A 10x bigger intensity sound is “heard” as only 2x bigger by the ear 16 Gustav Theodor Fechner ( )

1c. Decibel Scale The decibel is a logarithmic scale A multiplicative factor of 10x in intensity is +10 db 0 dbis threshold of hearing 1 dbis just noticeable difference 15 dbis a whisper 60 dbis talking 120 dbis maximum safe level 150 dbis jet engine (ear damage) 180 dbstun grenade 17 ================== Power RatiodB ___________________ ==================

2a. JND: Just Noticeable Difference is 1dB Reference: 18

2b Discrimination of Loudness jnd = “just noticeable difference” The ear’s “jnd” for Loudness is approximately 1 dB Or, sound must be 30% louder in intensity for us to just notice that it is louder. This depends somewhat on frequency (pitch) and loudness (intensity). We have trouble distinguishing changes in loudness for very the very loud or the very soft sounds 19

2c. Smaller than JND (7% change) Reference: 20

3a. Threshold of Hearing & Age (Presbycusis) Note “Sound Pressure dB” (or SPLdB) is approximately half regular “energy” decibels (dB). 21

3b. Hearing Threshold The ear can hear as small as Watts/m 2 (one trillionth of a watt per square meter) ( 0.000,000,000,001 Watt/m 2 ) Example: you might be able to hear someone talking half a mile away under ideal circumstances Intensity is proportional to the square of the pressure amplitude Minimum ear can hear is 0.000,02 Pascals (Atmospheric pressure is 100,000 Pascal) 22

D. Timbre 1.Waveforms and Timbre 2.Fourier Theory 3.Ohm’s law of acoustics 23

1. Waveform Sounds Different “shape” of wave has different “timbre” quality 24 Sine Wave (flute) Square (clarinet) Triangular (violin) Sawtooth (brass)

1b. Waveforms of Instruments Helmholtz resonators (e.g. blowing on a bottle) make a sine wave As the reed of a Clarinet vibrates it open/closes the air pathway, so its either “on” or “off”, a square wave (aka “digital”). Bowing a violin makes a kink in the string, i.e. a triangular shape. Brass instruments have a “sawtooth” shape. 25

2a. Fourier’s Theorem Any periodic waveform can be constructed from harmonics. 26 Joseph Fourier

2b. FFT: Fast Fourier Transform A device which analyzes any (periodic) waveform shape, and immediately tells what harmonics are needed to make it Sample output: tells you its mostly 10 k Hertz, with a bit of 20k, 30k, 40k, etc. 27

2c. FFT of a Square Wave Amplitude “A” Contains only odd harmonics “n” Amplitude of “n” harmonic is: 28

2d. FFT of a Sawtooth Wave Amplitude “A” Contains all harmonics “n” Amplitude of “n” harmonic is: 29

2e. FFT of a triangular Wave Amplitude “A” Contains ODD harmonics “n” Amplitude of “n” harmonic is: 30

3a. Ohm’s Law of Acoustics Ohm's acoustic law a musical sound is perceived by the ear as a set of a number of constituent pure harmonic tones, i.e. acts as a “Fourier Analyzer” Georg Simon Ohm (1789 – 1854) For example:, the ear does not really “hear” the combined waveform (purple above), it “hears” both notes of the octave, the low and the high individually.

3b. Ohm’s Acoustic Phase Law 32 Hermann von Helmholtz elaborated the law (1863?) into what is often today known as Ohm's acoustic law, by adding that the quality of a tone depends solely on the number and relative strength of its partial simple tones, and not on their relative phases. Hermann von Helmholtz ( ) The combined waveform here looks completely different, but the ear hears it as the same, because the only difference is that the higher note was shifted in phase.

3c. Ohm’s Acoustic Phase Law 33 Hence Ohm’s acoustic law favors the “place” theory of hearing over the “telephone” theory. Review: –The “telephone theory” of hearing (Rutherford, 1886) would suggest that the ear is merely a microphone which transmits the total waveform to the brain where it is decoded. –The “place” theory” of hearing (Helmholtz 1863, Georg von Békésy’s Nobel Prize): different pitches stimulate different hairs on the basilar membrane of the cochlea.

E. Notes/References Fourier Applet (waveforms) Load Error on this page? FFT of waveforms: Demos: hearing-Shepard.htmhttp:// hearing-Shepard.htm 34