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## 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 • b = 10 log(I/Io) • b = 10 log(E/Eo) • b = 20 log(A/Ao) • Loudness always compared to the threshold of hearing**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**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**Critcal Bandwidth**Adding Loudness atDifferent Frequency As the pitch separation grows less, the combined loudness grows less. 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 Loudness 1200 13 sones 1500 4 sones 17 sones 900 13 sones 1200 6 sones 19 sones 600 13 sones 900 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…**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**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**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.**Natural Frequency, fo**3rd Harmonic is fo 2nd Harmonic is fo Driven System Response**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 • 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: • Or,**3.0**2.5 2.0 1.5 1.0 Hz/cent 0.5 0.0 Frequency 0 1000 2000 3000 4000 5000 Frequency Value of CentThrough the Keyboard**The Unison and Pitch Matching**• Consider two tones made up of the following partials • 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**• 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. • 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… • 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.**F**G A B C C E D Harmonically Related Steps Notice the B and D are not harmonically related to C**5th**3rd F G A B C C E D 4th 5th Intervals with B and D**G**3rd 3rd 3rd 3rd F A B C C E D 4th Minor 6 Filling in the Scale Notice that C#, Eb, and Bb come into the scheme, but Ab/G# is another problem.**G**A B C C D E F min3 3rd 3rd Finding F#**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 • Using equal intervals makes the cents division more meaningful • The following table uses**Chapter 16**Keyboard Temperaments and Tuning: Organ, Harpsichord, Piano