Review

Section IV

Chapter 13

The Loudness of Single and Combined Sounds

Useful Relationships

Energy and Amplitude, E A2

Intensity and Energy, I E

Energy and Amplitude, E A2

Decibels Defined When the energy (intensity) of the

sound increases by a factor of 10, the loudness increases by 10 dB = 10 log(I/Io) = 10 log(E/Eo) = 20 log(A/Ao)

Loudness always compared to the threshold of hearing

Decibels and Amplitude

Amplitude vs. Loudness

0

2

4

6

8

10

0 5 10 15 20

Loudness (decibels)

Am

plit

ud

e R

ati

o

Single and Multiple Sources Doubling the amplitude of a single speaker

gives an increased loudness of 6 dB (see arrows on last graph)

Two speakers of the same loudness give an increase of 3 dB over a single speaker

For sources with pressure amplitudes of pa, pb, pc, etc. the net pressure amplitude is

... p p p net

p 2c

2b

2a

Threshold of Hearing Depends on frequency

Require louder source at low and high frequencies

Perceived Loudness

One sone when a source at 1000 Hz produces an SPL of 40 dB

Broad peak (almost a level plateau) from 250 - 500 Hz

Dips a bit at 1000 Hz before rising dramatically at 3000 Hz

Drops quickly at high frequency

Adding Loudness atDifferent Frequency

As the pitch separation grows less, the combined loudness grows less.

Critcal Bandwidth

Note critical bandwidth plateau for small pitch separation, growing for lower frequencies.

The sudden upswing in loudness at very small pitch separation caused by beats.

Upward Masking Tendency for the loudness of the upper tone to be

decreased when played with a lower tone.

Frequency Apparent Loudness1200 13 sones1500 4 sones

17 sones

900 13 sones1200 6 sones

19 sones 

600 13 sones900 6.5 sones

19.5 sones

Notice that upward masking is greater at higher frequencies.

Upward Masking ArithmeticMultiple Tones

Let S1, S2, S3, … stand for the loudness of the individual tones. The loudness of the total noise partials is…

)S S 0.2(S 0.3S 0.5S 0.5S 0.75S S S 87654321tnp

Closely Spaced Frequencies Produce Beats

Audible Beats

Notes on Beats Beat Frequency =

Difference between the individual fre-quencies = f2 - f1

When the two are in phase the amplitude is momentarily doubled that of either component

Adding Sinusoids

Masking (one tone reducing the amplitude of another) is greatly reduced in a room

Stsp = S1 + S2 + S3 + ….

Total sinusoidal partials (tsp versus tnp)

Notes Noise is more effective at upward masking

in room listening conditions Upward masking plays little role when

sinusoidal components are played in a room

The presence of beats adds to the perceived loudness

Beats are also possible for components that vary in frequency by over 100 Hz.

Chapter 14

The Acoustical Phenomena Governing the Musical Relationships of Pitch

Other Ways Of Producing And Using Beats

Introduce a strong, single frequency (say, 400 Hz) source and a much weaker, adjustable frequency sound (the search tone) into a single ear. Vary the search tone from 400 Hz up. We hear beats at multiples of 400 Hz.

A Variation in the Experiment

Produce search tones of equal amplitude but 180° out of phase. Search tone now completely cancels

single tone. Result is silence at that harmonic Each harmonic is silenced in the same

way. How loud does each harmonic need to be

to get silence of all harmonics?

Waves Out of Phase

Waves Out of Phase

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Dis

pla

cem

ent Superposition

of these waves produces zero.

Loudness Required for Complete Cancellation

400 Hz 95 SPL Source Frequency 800 Hz 75 SPL 1200 Hz 75 SPL 1600 Hz 75 SPL

Harmonics are 20 dB or 100 times fainter than source (10% as loud)

Start with a Fainter Source

400 Hz 89 SPL Source – ½ loudness 800 Hz 63 SPL ¼ as loud as above 1200 Hz 57 SPL 1/8 as loud as above 1600 Hz 51 SPL 1/16 as loud as above

…And Still Fainter Source 400 Hz 75 SPL Source 800 Hz 55 SPL 1200 Hz 35 SPL Too faint 1600 Hz 15 SPL Too faint

This example is appropriate to music. Where do the extra tones come from?

They are not real but are produced in the ear/brain

Heterodyne Components Consider two tones (call them P and Q)

From above we see that the ear/brain will produce harmonics at (2P), (3P), (4P), etc.

Other components will also appears as combinations of P and Q

OriginalComponents

Simplest HeterodyneComponents

Next-AppearingHeterodyneComponents

P (2P) (3P)

(P + Q), (P – Q)(2P + Q), (2P – Q)(2Q + P), (2Q – P)

Q (2Q) (3Q)

Heterodyne Beats

Beats can occur between closely space heterodyne components, or between a main frequency and a heterodyne component.

See the vibrating clamped bar example in text.

Driven System ResponseNatural

Frequency, fo

2nd Harmonic is fo

3rd Harmonic is fo

Other Systems

More than one driving source We get higher amplitudes anytime

heterodyne components approach the natural frequency.

Non-linear systems Load vs. Deflection curve is curved Heterodyne components always exist

Harmonic and Almost Harmonic Series

Harmonic Series composed of integer multiples of the fundamental

Partial frequencies are close to being integer multiples of the fundamental Always produce heterodyne components The components tend to clump around

the harmonic partials. May sound like an harmonic series but

“unclear”

Frequency - Pitch

Frequency is a physical quantity Pitch is a perceived quantity Pitch may be affected by whether…

the tone is a single sinusoid or a group of partials

heterodyne components are present, or noise is a contributor

The Equal-Tempered Scale Each octave is divided into 12 equal parts

(semitones) Since each octave is a doubling of the

frequency, each semitone increases frequency by 12 2

Each semitone is further divided into 100 equal parts called Cents

The cent size varies across the keyboard (1200 cents/octave)

Calculating Cents The fact that one octave is equal to 1200

cents leads one to the power of 2 relationship:

ln(2)

ff

ln

1200 cents 1

2

Or,

Frequency Value of CentThrough the Keyboard

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 1000 2000 3000 4000 5000

Frequency

Hz/cent

The Unison and Pitch Matching Consider two tones made up of the

following partials

Harmonic 1 2 3 4

Tone J 250 500 750 1000

Tone K 252 504 756 1008

Beat Frequency 2 4 6 8

Adjust the tone K until we are close to a match

Notes on Pitch Matching

As tone K is adjusted to tone J, the beat frequency between the fundamentals becomes so slow that it can not easily be heard.

We now pay attention to the beats of the higher harmonics. Notice that a beat frequency of ¼ Hz in

the fundamental is a beat frequency of 1 Hz in the fourth harmonic.

Add the Heterodyne Components

In the vicinity of the original partials, clumps of beats are heard, which tends to muddy the sound. Eight frequencies near 250 Hz Seven near 500 Hz Six near 750 Hz Five near 1000 Hz.

Results A collection of beats may be heard.

Here are the eight components near 250 Hz sounded together.

The Octave Relationship

Tone P 200 400 600 800

Tone Q 401 802 1203 1604

As the second tone is tuned to match the first, we get harmonics of tone P, separated by 200 Hz.

Only tone P is heard

The Musical Fifth A musical fifth has two tones whose

fundamentals have the ratio 3:2.

Tone M 200 400 600 800

Tone N 301 602 903 1204

Now every third harmonic of M is close to a harmonic of N

Results

We get clusters of frequencies separated by 100 Hz.

When the two are in tune, we will have the partials…

200 300 400 600 800 900 1200

This is very close to a harmonic series of 100 Hz The heterodyne components will fill in the

missing frequencies. The ear will invariably hear a single 100 Hz tone

(called the implied tone).

Chapter 15

Successive Tones: Reverberations, Melodic Relationships,

and Musical Scales

Audibility Time Use a stopwatch to measure how long

the sound is audible after the source is cut off

Agrees well with reverberation time Time for a sound to decay to 1/1000th

original level or 60 dB It is constant, independent of

frequency, and unaffected by background noise

Advantages of Audibility Time

Only simple equipment required Many sound level meters can only

measure a decay of 40-50 dB, not the 60 dB required by the definition

Sound level meters assume uniform decay of the sound, which may not be the case

Successive Tones We can set intervals easily for successive

tones (even in dead rooms) so long as the tones are sounded close in time.

Setting intervals for pure sinusoids (no partials) is difficult if the loudness is small enough to avoid exciting room modes.

At high loudness levels there are enough harmonics generated in the room and ear to permit good interval setting.

Intervals set at low loudness with large gaps between the tones tend to be too wide in frequency.

The Beat-Free Chromatic(or Just) Scale

  Chromatic Scales  

  Interval Name Interval Ratio Frequency (beat-free)

C     261

E 3rd 5/4 327

F 4th 4/3 349

G 5th 3/2 392

A Major 6th 5/3 436

C octave 2/1 523

Harmonically Related Steps

CGDC E F A B

Notice the B and D are not harmonically related to C

Intervals with B and D

5th

CGDC E F A B

4th

5th

3rd

Filling in the Scale

3rd

3rd

3rd 3rd

4th

Minor 6

G CDC E F A B

Notice that C#, Eb, and Bb come into the scheme, but Ab/G# is another problem.

Finding F#

3rd

3rd

min3

CDC E F A BG

Equal Temperament An octave represents a doubling of the frequency

and we recognize 12 intervals in the octave. The octave is the only harmonic interval.

Make the interval 1.059463 212

Using equal intervals makes the cents division more meaningful

The following table uses

Complete Scale Comparison

IntervalRatio to Tonic

Just ScaleRatio to Tonic

Equal Temperament

Unison 1.0000 1.0000

Minor Second 25/24 = 1.0417 1.05946

Major Second 9/8 = 1.1250 1.12246

Minor Third 6/5 = 1.2000 1.18921

Major Third 5/4 = 1.2500 1.25992

Fourth 4/3 = 1.3333 1.33483

Diminished Fifth 45/32 = 1.4063 1.41421

Fifth 3/2 = 1.5000 1.49831

Minor Sixth 8/5 = 1.6000 1.58740

Major Sixth 5/3 = 1.6667 1.68179

Minor Seventh 9/5 = 1.8000 1.78180

Major Seventh 15/8 = 1.8750 1.88775

Octave 2.0000 2.0000

Chapter 16

Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano

Notes on the Just Scale

Major Scale

The D corresponds to the upper D in the pair found in Chapter 15. Also, the tones here (except D and B) were the same found in the beat-free Chromatic scale in Chapter 15.

Here we use the lower D from chapter 15 and the upper Ab.

Minor Scale

Notes on the Equal-Tempered Scale

The fifth interval is close to the just fifth = 1.49831 whereas the just fifth is 1.5 712 2

Only fifths and octaves are used for tuning Perfect fifth is…

Three times the frequency of the tonic reduced by an octave – f5th = 1.5 fo

3*fo = 2*f5th

Equal-tempered fifth is reduce 2 cents from the perfect fifth

Tuning by Fifths Recall that the tonic contains the perfect

fifth as one of the partials We tune by listening for beats Ex. The equal-tempered G4 is 392 Hz

*C4 712 2

Use perfect fifth rule 3(261.63) – 2(392.00) = 0.89 Hz

This difference would be zero for a perfect fifth We tune listening for a beat frequency of slightly less

than 1 Hz

Just and Equal-Tempered

 Interval Just

Equal-Tempered

Cent Diff.

Tonic 1.00 261.63 261.63  

Major 2nd 1.13 294.33 293.67 -4

Major 3rd 1.25 327.04 329.63 14

Major 4th 1.33 348.84 349.23 2

Major 5th 1.50 392.45 392.00 -2

Major 6th 1.67 436.05 440.01 16

Major 7th 1.88 490.56 493.89 12

Octave 2.00 523.26 523.26 0 

         

Minor 3rd 1.20 313.96 311.13 -16

Minor 6th 1.60 418.61 415.31 -14

Use of the Previous Table It is there to compare the two scales, not to

memorize. Know how to generate the frequencies in the

table Just frequencies come by multiplying by whole

number ratios Equal-tempered frequencies come by

multiplying by a power of 12 2

Notes

Certain intervals sound smoother (or rougher) than others. Notice particularly the Major 3rd, 6th, and

7th and the minor intervals

The Circle of Fifths

Pythagorean Comma Start from C and tune perfect 5ths all the

way around to B#. C and B# are not in tune.

A perfect 5th is 702 cents. 702+702+702+702+702+702+702+702+702+

702+702+702= 8424 cents An octave is 1200 cents.

1200+1200+1200+1200+1200+1200+1200= 8400 cents

8424 - 8400 = 24 cents = Pythagorean Comma

Well-Tempered Tuning

Many attempts distribute the Pythagorean Comma problem around the circle of Fifths in different ways to make the problem less obvious Werckmeister III – shrunken fifths

The interval is found by dividing the Pythagorean Comma into four equal parts (23.46/4 = 5.865). So instead of the perfect fifths being 702 cents, they are 696.1 cents.

Werckmeister Circle of Fifths Numbers in the

intervals refer to differences from the perfect interval. The ¼ refers to ¼

of the Pythagorean Comma.

Result is that transposing yields different moods

Physics of Vibrating Strings Flexible Strings

Frequencies of the harmonics depend directly on the tension and inversely on the length, density, and thickness of the string.

fn = nf1

Hinged Bars Frequencies of the harmonics depend directly on

thickness of the bar and inversely on the length and density.

fn = n2f1

Real Strings We need to combine the string and bar

dependencies

(bar)f (flexible)f ion)under tens string (stifff 2n

2nn

Physics of Vibrating Strings The Termination

Strings act more like clamped connections to the end points rather than hinged connections.

The clamp has the effect of shortening the string length to Lc. The effect of the termination is small.

Physics of Vibrating Strings The Bridge and Sounding Board

We use a model where the string is firmly anchored at one end and can move freely on a vertical rod at the other end between springs

FS is the string natural frequency

FM is the natural frequency of the block and spring to which the string is connected.

The string + mass acts as a simple string would that is elongated by a length C.The slightly longer length of the string gives a slightly lower frequency compared to what we would have gotten if the string were firmly anchored.

Piano and harpsichord tuning is not marked by beat-free relationships, but rather minimum roughness relationships.

The intervals not longer are simple numerical values.

Larger sounding boards have overlapping resonances, which tend to dilute the irregularities. Thus grand pianos have a better harmonic

sequence than studio pianos.