1. COMBINATION TONES The Science of Sound Chapter 8 MUSICAL ACOUSTICS

2. LINEAR SUPERPOSITION OF TWO SIMPLE HARMONIC MOTIONS AT THE SAME FREQUENCY SAME PHASEOPPOSITE PHASE

3. SIMPLE HARMONIC MOTION AS THE PROJECTION OF POINT P (MOVING IN A CIRCLE)

4. TWO POINTS P AND Q MOVE WITH THE SAME PERIOD T AND AMPLITUDE A AND MAINTAIN A PHASE DIFFERENCE Φ P – Φ Q

5. FREQUENCY BUT WITH A PHASE DIFFERENCE Φ B – Φ A = 90º LINEAR SUPERPOSITION OF TWO SHMs WITH THE SAME

6. ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45 O 90 O 135 O Practice problems:

7. ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45 O 90 O 135 O √ [16 + 9 + (1.4)(4)(3)] = 6.47

8. ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45 O 90 O 135 O √ [16 + 9 + (1.4)(4)(3)] = 6.47 √ [16 + 9] = 5

9. ESTIMATE THE RESULTANT AMPLITUDE OF TWO SINE WAVES HAVING AMPLITUDES OF 4 AND 3 IF THEIR PHASE DIFFERENCE IS 45 O 90 O 135 O √ [16 + 9 + (1.4)(4)(3)] = 6.47 √ [16 + 9] = 5 √ [16 + 9 - (1.4)(4)(3)] = 2.86

10. TWO TONES WITH FREQUENCIES ƒ 1 AND ƒ 2 : BEATS 32 Primary beats. Track 62 If f1 is fixed and f2 is variable, beats result for small differences between f1 and f2.

11. THE MUSICAL STAFF: MUSICIAN’S GRAPH PAPER

12. PROPOSAL FOR THREE NEW CLEFFS (See Appendix A.6)

13. COMBINATION TONES COMBINATION TONES ON A MUSICAL STAFF 34 Aural combination Tones, Tr 68-69

14. DIFFERENCE TONES MOST CLEARLY HEARD: DIFFERENCE TONE (f 2 – f 1 ) [TARTINI TONES] CUBIC DIFFERENCE TONE (2f 1 – f 2 ) OTHER DIFFERENCE TONES: (3f 1 – 2f 2 ), 4f 1 —3f 2 ), etc,

15. DIFFERENCE TONE MELODY

16. MODULATION OF ONE TONE (ƒ 2 ) BY ANOTHER (ƒ 1 ) Amplitude modulation results in “sidebands” as shown

17. AMPLITUDE MODULATION (AM) AND FREQUENCY MODULATION (FM) Both AM and FM are used in radio broadcasting Frequency modulation produces many sidebands, and these are useful in FM synthesis of music

18. ORIGIN OF DIFFERENCE TONES The simple difference tone with frequency f 2 -f 1 (properly called the quadratic difference tone) can be easily heard up to about 1500 Hz. The cubic difference tone with frequency 2f 1 -f 2 can be easily heard as well. The quartic difference tone (3f 1 -2f 2 ) and higher order difference tones are generally more difficult to hear.

19. OTHER NONLINEAR EFFECTS AURAL HARMONICS Fletcher suggested a simple power law for ear response x = a 0 + a 1 p + a 2 p 2 + a 3 p 3 +,,, This predicts that for a 1-dB increase in signal level, the 2 nd harmonic will increase by 2 dB, the 3 rd by 3 dB, etc. SUMMATION TONES If the ear has a nonlinear response, one might expect to hear summation tones (f 1 + f 2 ) as well as difference tones, but no one has presented convincing evidence for their existence.

20. CONSONANCE AND DISSONANCE: MUSICAL INTERVALS Plomp and Levelt (1965): If the frequency difference is greater than a critical band, they sound consonant. Maximum dissonance occurs when the difference is about ¼ of a critical band.

21. INTERACTIONS BETWEEN HARMONICS OF TWO TONES SEPARATED BY DIFFERENT INTERVALS

22. CONSONANCE EXPECTED BY COMBINING A TONE OF 250 Hz WITH ANOTHER TONE

23. STATISTICAL ANALYSIS OF CHORDS Bach “Trio Sonata for Organ” Dvorak “String Quartet op. 51” (Plomp and Levelt, 1965)

24. EFFECT OF PHASE ON TIMBRE Building up complex tones with the same spectrum of partials but with different phases results in totally different waveforms. Do they sound different The answer is “SOMETIMES” Plomp (1970): The maximum effect of phase on timbre is between a complex tone in which alternate harmonics differ in phase by 90 O 33 Distortion, Tracks 64-67

25. BEATS OF MISTUNED CONSONANCES A sensation of beats occurs when the frequencies of two tones f 1 and f 2 are nearly, but not quite, in a simple ratio If f 2 = 2f 1 + δ, beats are heard at a frequency δ. When f 2 = n/m f 1 + δ, mδ beats occur. These are called second order beats or beats of mistuned consonance, and they result in periodic changes in phase. The ear, which is a poor detector of stationary phase, appears to be sensitive to cyclical variations in phase BEATS OF MISTUNED CONSONANCES have long been used by piano tuners to tune fifths, fourths, and octaves. Violinists use them to tune their strings. 32 Secondary beats, Track 63 Tone pairs having intervals slightly greater than an octave, fifth, and fourth

26. PRACTICE PROBLEMS: IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND? IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND?

27. PRACTICE PROBLEMS: IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND? SOLUTION: 443 – 440 = 3 Hz IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND? SOLUTION: 442 = 4/3(330) + 2, SO mδ = (3)(2) = 6 Hz (THESE ARE BEATS OF A MISTUNED FOURTH)

28. PRACTICE PROBLEMS: IF TONES WITH FREQUENCIES OF 440 Hz AND 443 Hz ARE SOUNDED TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND? SOLUTION: 443 – 440 = 3 Hz IF TONES WITH FREQUENCIES OF 442 Hz AND 330 Hz ARE HEARD TOGETHER, HOW MANY BEATS ARE HEARD EACH SECOND? SOLUTION: 442 = 4/3(330) + 2, SO mδ = (3)(2) = 6 Hz (THESE ARE BEATS OF A MISTUNED FOURTH) Alternate solution: 3 rd harmonic of 442 is 1326 Hz; 4 th harmonic of 330 is 1320; if sounded together, beats would be heard at 1326-1320 = 6 Hz

29. CENTRAL NERVOUS SYSTEM Autocorrelation and cross correlation: Autocorrelation is a comparison of a signal with previous signals in order to pick out repetitive features (e.g., repetition pitch). Cross-correlation is a comparison between signals on two different nerve fibers (e.g., localization of sound) Cerebral dominance: The left side of the brain (in 97% of the population) is specialized for speech processing, and the right side for non-linguistic functions such as music. Speech processing requires analytic and serial processing; musical perception requires holistic processing.

30. BINAURAL EFFECTS Auditory demonstrations 36-39 illustrate binaural effects that can be heard through stereo headphones.

31. A BINAURAL AUDITORY ILLUSION Tones of 400 and 800 Hz alternate in both ears in opposite phase (i.e., when right ears hears 400 Hz, left ear hears 800 Hz). Most right-handed listeners hear the high tone in their right ear and the low tone in their left ear, regardless of how the headphone are oriented (even when they are exchanged). Left-handed listeners, on the other hand, are just as apt to hear the high tone in the right ear as in the left. That is because in right-handed people, the left hemisphere is dominant (and its primary auditory input is from the right ear), whereas in left-handed people either hemisphere may be dominant. High tones are perceived as being hear at the ear that feeds the dominant hemisphere (Deutsch, “Musical Illusions,” Scientific American, 1975). 39

32. ASSIGNMENT FOR MONDAY MIDTERM EXAM ON CHAPTERS 1-8 OPTIONAL REVIEW ON FRIDAY AT 200 EXERCISES 2,4,5,7,8,9 (p.172) Hand them in Thursday if you want them back to study for the exam ASSIGNMENT FOR WEDNESDAY: READ CHAPTER 10 OPTIONAL SESSION ON FRIDAY, FEB. 12 TO DISCUSS THE EXAM NO CLASS ON MONDAY, FEB. 15 (PRESIDENTS’ DAY)