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## Chapter 14

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**Chapter 14**The Acoustical Phenomena Governing the Musical Relationships of Pitch**Use of Beats for Tuning**• Produce instrument tone and standard • Tuning fork or concert master • Download NCH Tone Generator from Study Tools page and try it • Open two instances of Tone Generator • Set one for 440 Hz and the other for 442 Hz • Adjust instrument until beat frequency is zero • Here we examine other ways of producing and using beats**Beat Experiment**In the other ear introduce a strong, single frequency (say, 400 Hz) source and a much weaker, adjustable frequency sound (the search tone). Mask one ear of a subject so nothing can be heard. Vary the search tone from 400 Hz up. We hear beats at multiples of 400 Hz.**Alteration of 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 of the Beat Harmonics**• 400 Hz 95 SPL Source Frequency • 800 Hz 75 SPL • 1200 Hz 75 SPL • 1600 Hz 75 SPL Note: 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 Component Example**So the ear hears (200), 400, 600, (800), (1000), (1200), (1400), (1600), (1800).**Producing Beats**• Beats can occur between closely space heterodyne components, or between a main frequency and a heterodyne component. • Ex. Consider three tones P at 200, Q at 396, and R at 605 Hz. • Two of the many heterodyne components are (Q – P) = 196 Hz and (R – Q) = 209 Hz. • Also (Q – P) will beat with P at 4 Hz.**Mechanical Analogy toHeterodyne Components**For small oscillations of the tip, we have simple harmonic motion. The bar never loses contact at A or comes into contact at B. The graph of the motion of the tip is a pure sine wave. Make the natural frequency 20 Hz.**Higher Amplitudes**• Bar loses contact at A on upward swing • Bar is momentarily longer and less stiff • Amplitude is greater than the pure sine wave. • Bar touches clamp at B • Bar is momentarily shorter and more stiff • Amplitude is less than the pure sine wave. • The red curve on the next slide describes the situation • But the red curve is the superposition of the two sine waves shown.**Driven System**• Now add the spring and drive the system at a variety of frequencies. • We expect large amplitudes when the driver frequency matches the natural frequency of 20 Hz. • We also get increases in amplitude at ⅓ and ½ the natural frequency (6⅔ Hz and 10 Hz) • See the response graph on the next slide**Natural Frequency, fo**3rd Harmonic is fo 2nd Harmonic is fo Driven System Response**Response Curve Explained**• When the driver frequency becomes 6⅔ Hz, the heterodyne component (third harmonic) is also excited. 3 X 6⅔ Hz = 20 Hz, the natural frequency. • When the driver frequency is 10 Hz, the second harmonic (2 X 10 Hz = 20 Hz) is also stimulated as a heterodyne component. • The 20 Hz frequency is self-generated**More than One Driving Source**• We should expect high amplitude whenever a heterodyne component is close to 20 Hz. • EX: Suppose two frequencies are used at P = 9 Hz and Q = 30 Hz. • We get a heterodyne component at (Q-P) = 21 Hz, which is close to the natural 20 Hz frequency.**Non-Linear Response**• At small amplitude the system acts like a Hooke’s Law spring (deflection [x] load [F]) • A graph of F vs. x will give a straight line (linear) • At higher amplitude the F vs. x curve becomes curved (non-linear) • See graphs below.**Black is linear (Hooke’s Law)**Colored is non-linear Load vs. Deflection**Notes on Non-linear Systems**• In a non-linear system, the whole response is not simply the sum of its parts. • Non-linear systems subject to sinusoidal driving forces generate heterodyne components, no matter what the nature of the non-linearity. • The amplitudes of the heterodyne components depend on the nature of the non-linearity and the amplitude of the driver.**The Musical Tone**• Special Properties of Sounds Having Harmonic Components • Imagine a single sinusoidal frequency produced from a speaker • At low volume the single tone is all you hear. • At higher volumes the room and our hearing system may produce harmonics.**Change the Source**• Now have the source composed of the same frequency, a weak second harmonic, and a still weaker third harmonic. • The added harmonics will probably not be noticed, but the listener may say the tone is louder. • Reason is that the additional harmonics is exactly what happens with the single tone at higher volume.**Almost Harmonic Components**• Suppose the tones introduced are at 250 Hz (X), a second partial at 502 Hz (Y), and a third at 747 Hz (Z). • Heterodyne components include: • (Y-X) (252) • (Z-Y) (245) • (Z-X) (497) • (X+Y) (752) • 2X (500) I have color-coded frequencies which form “clumps.” These are heard as musical tones, but may be called “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**Frequency Assignments**• 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 • Ex. G4 has a frequency of 392 Hz • G4# has a frequency of 415.3 Hz**Cents**• Each semitone is further divided into 100 parts called cents. • The difference between G4 andG4# above is 23.3 Hz and thus in this part of the scale each cent is 0.233 Hz. • A tone of 400 Hz can be called [G4 + (400-392)/0.233] cents, or (G4 + 34 cents). • 500 Hz falls between B4 (493.88 Hz) and C5 (525.25 Hz). We could label 500 Hz as (B4 + 20 cents)**Calculating Cents**• The fact that one octave is equal to 1200 cents leads one to the power of 2 relationship: • Or,**Advantage of the Cents Notation**The same interval in different octaves will be difference frequency differences, but the interval in cents is always the same.**Frequencies (Hz) for Equal-Tempered Scale("Middle C" is C4 )****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**Frequency Matching vs.Pitch Matching**• Most cases these give the same result • Can use frequency standards to match pitch • May produce different results • Recall the difficulty of assigning pitch with bell tones from Chapter 5.**Buzz Tone Made from Harmonic Partials**• Consider forming a “buzz” sound by adding 25 partials of equal amplitude and a fundamental of 261.6 Hz (C4).**Compare the Buzz Tone to a Pure Sine Wave of Same Frequency**• Present the two alternately • Pitch match occurs if the sine wave is made sharp. • Present the two together • No frequency changes required • The physicist’s idea of matching frequency by achieving a zero beat condition agrees with the musician’s idea of matching pitch when the tones are presented together, as long as the tones are harmonic partials.**Practical Application**• In music only the first few partials have appreciable amplitude • Pitch matching for tones presented alternately and together gives the same result.**Almost Unison Tones**• Consider two tones constructed from partials as below. Neglect heterodyne effects for the time being.**Matching Pitch**• As the second tone is adjusted to the first, 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.**Now Add Heterodyne Components**• (J2 – K1) = (500 – 252) = 248 Hz • (K2 – J1) = (504 – 250) = 254 Hz • (J3 – K1) = (750 – 252) = 498 Hz • (K3 – J1) = (756 – 250) = 506 Hz • Now we have frequencies near the fundamentals and the second harmonic • Recall that heterodyne components arise from differences between the harmonics of the two tones**Complete set of Heterodyne Components**Can you find the differences and sums that result in these frequencies?**Results**• 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 (cont’d)**• The multitude of beats produced by tones having only a few partials makes a departure from equal frequencies very noticeable. • The clumping of heterodyne beats near the harmonic frequencies may make the beat unclear and confuse the ear. • These two conclusions are contradictory and either may happen depending on the relative amplitudes of the partials.**Next - Separate the Tones More**The spread of the clumps is quite large and the resulting sound is “nondescript.”**Results**• A collection of beats may be heard. • Here are the eight components near 250 Hz sounded together. • Achieving unison is well-defined.**The Octave Relationship**• We can make two tones separated by close to one octave. • Tone P has a fundamental at 200 Hz and three harmonic partials. • Tone Q has a fundamental at 401 Hz and three harmonic partials**Results**• 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. • Again consider an almost tuned fifth and look at the heterodyne components produced. • Now every third harmonic of M is close to a harmonic of N