1. IF you are doing the homework religiously, and doing clicker quizzes in every class, then a 28/40 is the cutoff for an A- and a 15/40 is the cutoff for a B- If you are missing some homework &/or classes and want to get extra credit (or if you just want extra credit), consider signing up to do an “end of chapter” project and possibly present your work to the class.

2. Quick review of Tuesday 1. Place & Periodicity Theories of hearing 2. Just Noticeable Differences 3. Loudness - a psychological variable In a forced two-choice experiment with two tones, one at 70 dB and one at 70.8 dB, 90% of the subjects correctly identify the louder tone. How many actually KNEW that the 70.8 dB tone was louder? A] 40%B] 45%C] 70%D] 80%E] 90% http://webphysics.davidson.edu/faculty/dmb/JavaSounddemos/harmonics.htm

3. What is the loudness in phons of an 80 dB sound at 20 Hz? A] 20 B] 80 C] 100 D] 1600

4. What is the loudness in phons of an 80 dB sound at 20 Hz? A] 20 phons Note that phons = dB at 1000 Hz

5. What is the loudness in phons of an 80 dB sound at 20 Hz? A] 20 phons How loud in sones is this sound? A] 0.1 sone B] 1 sone C] 20 sones

6. Which is louder, 80 dB @ 70 Hz -or- 70 dB @ 3500 Hz? A] 80 dB B] 70 dB C] both are the same loudness

7. Musical Intervals An interval is a fixed frequency RATIO

8. To combine two intervals, we multiply their ratios. For example, C - D is C - C# combined with C# - D (Two successive semitones.) The frequency ratio of D:C is thus 1.05946 x 1.05946 = 1.1224 A just intonation fifth is a frequency ratio of 3:2 (or 1.5.) What is the frequency of a just fifth above A440? A] 445 HzB] 587.3 Hz C] 660 Hz D] 880 Hz

9. A just intonation fifth is a ratio of 3:2. A just intonation fourth is a ratio of 4:3. If you start at A440 and go up a fifth and then up a fourth, what is the total interval ratio? A] 7:5 B] 2:1 C] 6:9 D] 8:9 Answer: 3:2 (or 3/2) times 4:3 (or 4/3) is 12:6 or 2:1. This is an OCTAVE. A fifth plus a fourth is an octave.

10. Harmonic series (of notes): A fundamental frequency and all its integer multiples Example: A110, 220, 330, 440, 550, 660, 770, 880… A110, A220, ~E329.6, A440, ~C#554.4, E659.3, ??, A880…

11. A harmonic series can be built on any fundamental frequency A note blends well with its harmonic overtones. It sometimes blends so well that you can’t tell there’s more than one note! (Pythagoras: intervals with frequency ratios that are small integers sound nice. 2:1, 3:2, 4:3) In fact, whenever an instrument plays a “note”, it also sounds overtones, which are not heard as distinct notes. They just change the timbre of the note. The equal tempered scale does not exactly include the notes of a harmonic series. However, it comes close for the low overtones. 3:2 ratio is nearly a fifth in the equal tempered scale. 1.05946 7 = 1.4982 ~ 1.5

12. A “note” of caution: Intervals are generally related to scales, but we sometimes talk about intervals that are present in the harmonic series. I’ll try to call these “just intonation intervals”. A just intonation fifth is a ratio of 3:2 = 1.5 An equal tempered fifth is a ratio of 1.05946 7 = 1.4982 (there are 7 semitone intervals in a perfect fifth.)

13. Why don’t we just tune the piano fifths to be ratios of 3:2? That would sound better! Circle of fifths: F C G D A E B F # C # G # D # A # E # B # Fx Cx … Fx = F double sharp. We need an infinite number of keys per octave! Perhaps we can compromise and use the same key for (for example) E # and F. Then we could get by with only 12 keys per octave.

14. Circle of fifths: F C G D A E B F # C # G # D # A # F C G D A … So: if we go around the circle of fifths 12 times, starting on F, we need to end on a higher F, several octaves up in pitch. Alas, 1.5 12 = 129.746… NOT 128 (= 2 7 ). IF WE TUNE OUR FIFTHS PERFECTLY (JUSTLY), THEN OUR OCTAVES WILL BE OFF. THAT’S INTOLERABLE!

15. Cents Intervals are frequency ratios. To combine two intervals, we multiply the ratios. Example: go up a fifth (3:2) and then up a fourth (4:3). The total interval is 3/2 x 4/3 = 2 = octave. If we were to take the log of the frequency ratios, we could just ADD the logs. 100¢ = 1 semitone so 1200¢ = 1 octave. Thus, MI/¢ = 1200 log 2 (f 1 /f 2 ) = 3986 log 10 (f 1 /f 2 )

16. How many cents in an equal tempered fourth? A] 300 B] 400 C] 500 E] none of these

17. How many cents in an equal tempered fourth? C] 500 ¢ How many cents in a just (Pythagorean) fourth? A] 500 B] not 500

18. When an instrument plays a note, overtones are sounded. If the overtones are harmonic (integer multiples of the fundamental), we call them harmonic overtones. We generally don’t hear the overtones clearly. Rather, their presence is sensed as the timbre of the note. We sense the pitch of the note as the pitch of the fundamental. In fact, if you play 200 Hz, 300 Hz, 400 Hz, 500 Hz together (in steady phase), you will “hear” 100 Hz!!! (even though it is completely ABSENT!)

19. Even though we don’t sense the overtones individually, we CAN hear beats between overtones. Consider tuning a piano. If you play A220 and E329.6, the third harmonic of the A (660) will beat against the second harmonic of E (659.2). How many beats should you hear? A] 0.4 per second B] 0.8 per second C] 1.6 per second

20. If you play A220 and E329.6, the third harmonic of the A (660) will beat against the second harmonic of E (659.2). How many beats should you hear? Answer B] 0.8 per second The E should be a little flat from Pythagorean pitch. Tune it to a Pythagorean fifth (the most pleasant, beatless sound), and then detune it flat until you hear 0.8 beats per second. Note: before you run off to become a piano tuner, be sure to dampen the extra strings!! You can’t hear beats if both “notes” are from striking 3 strings!

21. Harmonic series are important in all of mathematics and the physical sciences! Here’s why: ANY periodic waveform with frequency f is the SUM of purely sinusoidal waves with frequencies f, 2f, 3f, 4f, 5f, 6f, 7f, ….. Fourier series

22. These three sine waves add to give this complex wave Fundamental, 2nd harmonic, 4th harmonic

23. The frequency of the complex wave is the frequency of the fundamental… even if the fundamental is absent! Sum of 2nd and 3rd harmonics. Fundamental absent.

24. A complex wave is formed by adding 400 Hz and 800 Hz sine waves. What is the frequency of the complex wave? A] 200 Hz B] 400 Hz C] 600 Hz D] 800 Hz Note: the “fundamental” is the highest frequency that divides evenly (without remainder) into all frequencies given.

25. A complex wave is formed by adding 400 Hz and 600 Hz sine waves. What is the frequency of the complex wave? A] 200 Hz B] 400 Hz C] 600 Hz D] 800 Hz

26. We know that different instruments produce different complex waveforms. We now know that, regardless of how complex the waveform is, if it is steady and periodic, it is simply a SUM of harmonics. So: perhaps we can describe our complex waveform in terms of a RECIPE: how much of each harmonic do we need to “mix in” to bake up an oboe? Or a trumpet?

27. Professor Bruce Dalby Tuning & Temperament

28. Acoustic beats can occur between a.two fundamentals b.a fundamanental and a harmonic c.two harmonics d.all of the above

29. Equal temperament means that all fifths must be tuned about 2¢ flat. When a piano tuner plays fifths, there should be beats. The fifth A 1 - E 1 should have A] fewer beats per second B] the same beats per second C] more beats per second than the fifth A 4 - E 4. A 1 = 55 Hz. A 4 = 440 Hz.

30. What is a “wolf fifth”? A] the natural interval that mexican grey wolves howl when in the wild B] a fifth of a gallon of bad whiskey C] a bad-sounding fifth that is unavoidable if all other fifths are justly tuned.

31. Prof. Dalby finished up by tuning a scale this way. Let’s figure out how bad the D-A fifth is…

32. The Well-Tempered Clavier J.S. Bach Well tempered ≠ equal tempered There are other ways to skin this cat. Well tempered means a “circulating” temperament… can play in all keys. But if it is not equal temperament, the keys will sound different! Bach probably used a circulating variant of sixth-comma meantone temperament.

33. “The modern practice of tuning all organs to equal temperament has been a fearful detriment to their quality of tone. Under the old tuning, an organ made harmonious and attractive music…. Now, the harsh thirds, applied to the whole instrument indiscriminately, give it a cacophonous and repulsive effect.” -William Pole, 1879

34. Complex periodic waveforms are simply a sum of components (sine waves) of the harmonic series. Melodic instruments (including voice) give complex waveforms. Hence, these sounds include harmonics or “overtones”. We can think about a “recipe” for a tone… how much of each harmonic do we add?

35. For now, let’s express the amount of each harmonic in terms of its amplitude. (Later, we’ll switch to intensity, which is more common.) For the example we did, for this complex waveform consisting of the sum of 2nd and 3rd harmonics, how much of the 2nd harmonic is in the recipe? A] 0B] 0.4 C] 0.5D] 1 time Change in Pressure

36. The recipe for this complex wave is 0.5 of the 2nd harmonic, + 0.4 of the 3rd harmonic. If we wanted to say what the recipe was in terms of intensities, we could say I 2 = 0.25, I 3 = 0.16* These are just relative numbers. If we doubled both amplitudes, or doubled both intensities (which is different!), the complex wave shape would be the same, only bigger. * If you wanted to use absolute numbers, you could multiply by 0.0012 (W/m 2 )/(N/m 2 ) 2 Change in Pressure time

37. The complex wave at the bottom is the sum of the two sine waves at the top. The black line is the fundamental. What harmonic is added to the fundamental? (Fundamental = 1st harmonic…) A] 2nd B] 3rd C] 4th D] 6th time Change in Pressure

38. The 3rd harmonic was added. So the recipe will consist of the 1st and 3rd harmonics. Recall that intensity is proportional to pressure squared, if the intensity of the fundamental is 0.0012 W/m 2, what is the intensity of the 3rd harmonic (in W/m 2 ) ? A] 0.0012 B] 0.0006 (half as much) C] 0.0003 (one quarter) D] 0.0001 (1/12th as much) time Change in Pressure

39. Intensity is proportional to pressure squared, if the intensity of the fundamental is 0.0012 W/m 2, what is the intensity of the 3rd harmonic (in W/m 2 ) ? Answer: since the amplitude is half, the intensity will be one- quarter as much… 0.0003 W/m 2 time Change in Pressure

40. I 1 = 0.0012 W/m 2 I 3 = 0.0003 W/m 2 So for this sound, SIL 1 = 90 dB = 10log 10 (I 1 /I ref ). What is SIL 3 ? A] 22.5 dB B] 84 dB C] 87 dB D] 90 dB time Change in Pressure

41. So for this sound, SIL 1 =90 dB. SIL 3 = 84 dB (down 3 dB for each factor of 2.) Suppose the fundamental is A440. We could graphically show the recipe for this sound like this: Such a graph, showing the intensity of each frequency in a sound, is called a spectrum

42. This recipe isn’t complete! Both these waves have the same recipe, but they sure look different! ? ?

43. Although these waves look different, they SOUND THE SAME!! So the spectrum of a sound does tell us what it sounds like! ? ?

44. Recall the PLACE THEORY of pitch perception. Different places on the basilar membrane respond to different frequencies. With complex waveforms, the different harmonics will stimulate different places and different hair cells. To some extent, the ear IS a spectrum analyzer.

45. Summary so far: Any periodic wave is a sum of harmonic sinusoidal waves The period of a complex wave is the period of the fundamental (even if it is missing). A spectrum is the amount of energy (intensity) at each frequency, i.e. in each harmonic. Usually, spectra are displayed in dB. The relative phases of harmonics (i.e. whether we add in sines or cosines or a combination) does not affect timbre; it is also absent from the spectrum.

46. What about non-periodic waves? Any non-periodic waveform is also the sum of sinusoidal waves… but the frequencies of those waves do NOT fall into a harmonic series. Conversely, if you add sinusoidal waves that are not in a harmonic series, you will get a non-periodic wave. For example, adding together a 100 Hz, and a 323.2365… Hz wave will give you a non-periodic wave.

47. Why is this so? For the complex wave to be periodic, the component sine waves must “line up” evenly after the period, T complex. With harmonic series, the period of harmonic n is just T fundamental /n. So after T fundamental, all the component sine waves have had an integer number of periods. The graphs at right show this for the 1st & 3rd harmonics. Notice how the 3rd harmonic is back to its start state after T fun

48. With two waves of arbitrary frequency, when the slower one is back to its start state, the faster one is in essentially a random state. For arbitrary frequencies, they will never line up perfectly again! *For the mathematically-inclined: waves with rational frequencies will line up again eventually. But if you choose a frequency “at random”, the probability of choosing a rational number is ZERO! Black - 100 Hz Red - 323.2365… Hz time Change in Pressure

49. Nonperiodic waves are also a sum of sinusoidal waves. They may be just a few frequencies, which are not in a harmonic series, Or, there may be gazillions of different sinusoidal waves… essentially a continuous spectrum of frequencies. An example of the former is the sound from a drum, for which f 2 = 1.59 f 1, f 3 = 2.14 f 1, etc. An example of the latter is the noise from a detuned radio, which is “white” - equal energy at all frequencies. (Pink noise has equal energy in each octave; since higher octaves span a larger frequency, the energy per Hz decreases.)

50. Lastly, real spectrum analyzers looking at real (finite duration) sounds, don’t give infinitely sharp peaks. A spikelike peak means a pure sine wave… a pure sine wave goes on forever and has been going forever.

51. We know that Purely periodic waves are just a sum of harmonic sine waves Aperiodic waves are just a sum of anharmonic sine waves From whence cometh these waves? Sound waves come from the motion of their sources. So… The motion of a source of periodic waves is a sum of harmonic sinusoidal motions The motion of a source of aperiodic waves is a sum of anharmonic sinusoidal motions.

52. What is a “sinusoidal motion” of a complex object? (A simple mass & spring does only sinusoidal motion.) Classic example: coupled pendulums. Let’s graph motions on the board. A sinusoidal motion is one in which each part of the object is in simple harmonic motion at the same frequency These motions are called NORMAL or NATURAL MODES Which mode has a higher frequency? A B Or C: both same frequency

53. Different natural modes have different frequencies. (But, for a given mode, all parts of the object are moving at the same frequency.) If the coupling between the pendulums is weak, the frequencies of the two modes will be very close. Let’s graph the motion of the left mass for mode A and mode B. Can the left mass execute a motion that is the sum of mode A and mode B? What would that motion look like? -Demonstration… beats-

54. We said that every periodic wave is a sum of harmonic sinusoidal waves. There is a similar true statement about objects that are vibrating: Every possible free vibration of an object is a sum of its natural modes.

55. Mode 1 Mode 2 time An object has only two natural modes. The motion of a point on the object in each mode is shown… Which graph(s) here could NOT be the motion of that point in a free vibration? Or choose D: neither A nor B Or E: all could be ABCABC