Review
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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 AmplitudeAmplitude vs. Loudness
0
2
4
6
8
10
0 5 10 15 20
Loudness (decibels)
Ampl
itude
Rat
io
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 netp 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 PhaseWaves Out of Phase
-1-0.8
-0.6-0.4
-0.20
0.2
0.40.6
0.81
Time
Dis
plac
emen
t 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)ffln
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 partialsHarmonic 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 ComparisonInterval Ratio to Tonic
Just ScaleRatio to Tonic
Equal TemperamentUnison 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 -4Major 3rd 1.25 327.04 329.63 14Major 4th 1.33 348.84 349.23 2Major 5th 1.50 392.45 392.00 -2Major 6th 1.67 436.05 440.01 16Major 7th 1.88 490.56 493.89 12Octave 2.00 523.26 523.26 0 Minor 3rd 1.20 313.96 311.13 -16Minor 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.
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