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Physics of Sound & Music: Day 14 Place Theory of Hearing & Amplitude Response. WarmUp: Ear ↔ Piano. You read about the "Place Theory of Hearing." In this theory there is a good analogy to be made between the inner ear and a piano keyboard. In what way are they similar ?

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Physics of Sound & Music:Day 14

Place Theory of Hearing & Amplitude Response

Warmup ear piano
WarmUp: Ear ↔ Piano

You read about the "Place Theory of Hearing."

In this theory there is a good analogy to be made between the inner ear and a piano keyboard. In what way are they similar?

~33%→ Locations correspond to frequencies

~50%→ Octaves are equally spaced

~17%→ Strings in piano are like hair cells in cochlea

Warmup ear piano1
WarmUp: Ear ↔ Piano

“the biggest way I could compare them would be the mechanisms used to produce the sound. Inside the ear, the three bones work together to communicate the sounds heard to the inner ear. Inside the piano, the same thing happens with the hammers hitting against the strings.”

Warmup ear piano2
WarmUp: Ear ↔ Piano

“different frequencies cause different spots along the basilar membrane to vibrate causing your brain to perceive a different tone for each, much like each key on a piano causes a different string to vibrate making a different tone for each.”

“There appears to be equal spacing along the basilar membrane that correlates octave intervals.”

Place theory of hearing
Place Theory of Hearing

Different frequencies of sound cause movement in different locations on the basilar membrane.

The membrane is about 3.5 cm long and humans can hear roughly 10 octaves.

Octaves are detected at equal intervals, thus they are about 3.5 mm apart on the basilar membrane.

Warmup jnd
WarmUp: JND

  • What is a frequency JND? How is it measured?

    ~75%→ “Just Noticeable Difference”

    ~33%→ Sketchy/incorrect or no description of how to measure frequency JND.

Warmup response jnd
WarmUp Response: JND

“It is the "just noticeable difference" between two pitches played simultaneously.”

“It can be measured by playing two identical sine waves simultaneously, then increasing one of them very slowly.”

“The JND is the smallest threshold that a listener can discern between similar tones. It can be tested by playing two tones in succession and asking the listener if they could tell them apart.”

Measuring jnd
Measuring JND

How do we measure the Just Noticeable Difference between two tones?

Simultaneous or in sequence?

Pure or complex tones?

When should we call it noticeable?

Two methods:

Vary the frequency of one tone.

Play two steady tones in sequence.

Warmup response jnd1
WarmUp Response: JND

b) If I played an A4 tone (440 Hz) and then played another tone at 444 Hz would most people be able to tell which one is higher? Would you?

~33%→ Most people: NO. Me: NO

~8%→ Most people: NO. Me: YES!

~0%→ Most people: YES! Me: NO

~50%→ Most people: YES! Me: YES!

Warmup response jnd2
WarmUp Response: JND

“b. yes, yes. I went to and tried it, and after several plays of each I could tell them apart.”

“I tried it at home and believe I was able to tell the difference, though I suppose there could be a bias since I knew I was chaining the pitch on the computer.”

“The difference is about (0.5Hz-0.6Hz) called the limit of frequency discrimination. B)Yes, most people would be able to hear the difference because it is beyond the limit of frequency discrimination.”

Frequency jnd
Frequency JND

An important question in all kinds of perception: What is the minimum discernable difference between two stimuli?

One way to describe this is the idea of just noticeable difference. How close do two successive tones have to be before we can't tell them apart?

For frequency the JND is around 1 Hz for sounds up to 1000 Hz, and then rises quickly.

An easy shorthand is that the JND is generally 0.5% - 0.6% of the frequency, but increases at lower frequencies.

Warmup response jnd3
WarmUp Response: JND

“I found this YouTube video to test and see if I could: I could tell a very slight difference in pitch going from 444 Hz to 440 Hz but it was very subtle.”

Just noticeable difference
Just Noticeable Difference

~0.6% in frequency, greater at lower frequencies

From: Juan G. Roederer, The Physics and Psychophysics of Music

Limit of frequency discrimination
Limit of Frequency Discrimination

Another way to look at how well we can tell apart two pure tones is to listen to them simultaneously.

When the tones get too close together, we no longer hear them as separate, but instead hear the average of their frequencies, modified by a beat frequency.

For lower frequencies this limit is about ~7% (just more than a half-step).

For higher frequencies it is ~15%.

Non sinusoidal sounds are much easier to discriminate, owing to the harmonic content that aids in their sharpening.

Displacement vs pressure
Displacement vs. Pressure

When measuring sound we have to choose between measuring displacement of air molecules and measuring pressure fluctuations.

But… even a loud sound might cause a wiggle of only a millionth of a meter (~100 times smaller than a human hair).

Luckily, the pressure changes associated with sound are much more to measureable.

Remember that sound is a fluctuation of air pressure above and below atmospheric pressure.

Amplitude intensity
Amplitude & Intensity

Using pressure to measure the strength of sound leads us back to the units for pressure, the pascal.

For all waves the quantity called intensity tells us about the amount of energy in the wave.

The intensity is proportional to the amplitude squared.

For example, if you have one sound that is 10 times the amplitude of another, the intensity will be 100 times as large (102 = 100).

Intensity is measured in watts per square meter or W/m2. This is energy, per second, per unit area.

Limits of hearing intensity
Limits of Hearing (Intensity)

  • Humans can generally hear sounds with intensities as low as 10-12 W/m2.

  • This is called the Threshold of Hearing.

    • Corresponds to a movement of the eardrum of less than 1/10 of the size of an atom!

  • At the loud end, the Threshold of Pain is a sound with an intensity of 1 W/m2 (1012 times larger!).

  • Huge range of sensitivity, you can sense sounds that are a trillion times different in strength!

  • But would you ever say one sound was a million times stronger than another?

Perception of sound strength
Perception of Sound Strength

The problem with intensities is this: We don't hear equal jumps in intensity as equal jumps in the strength of a sound!

Instead (roughly) we hear equal ratios of intensity as being equal jumps in the strength of a sound.

Sound Intensity Level (SIL) is used as a physical scale that approximates the way humans perceive sound strength.

SIL is measured in decibels, as in a deci-bel, meaning 1/10th of a bel, and is abbreviated dB (just like kHz).


Tuesday (10/7) → 6.4


WarmUp due Monday night by 10 PM

Homework #6 (Due Thursday the 9th): Ch 6: Q: 2, 3, 5, 7, 14, 18  P: 1, 2, 4

Exam I…

Exam 1
Exam 1

Average: 73% (High C on my scale)

In-progress grades will be posted online next week.

This exam is 10% of your overall grade.

Come see me if you want to check in.