1. Announcements 10/25/10 Prayer Change to TA’s office hours: Monday will now be 5-6 pm (to match Wed and Fri schedule). Project proposals: in process of reading through & giving feedback Exam 2: starts Thursday, goes through next Thurs a. a.Covers through today’s lecture, Wed’s HW – – Is cumulative! – – Specifically, there are 3 multiple choice questions very similar to ones from Exam 1. – – There is 1 worked problem very similar to one from Exam 1. No partial credit on this one! b. b.Exam review session: Wed Oct 27, 5:30 – 7 pm. Room: C295 (next door) c. c.Can bring handwritten 3x5 card (both sides) – – But just one! d. d.Time: took me 35 minutes (just like last exam)

2. Tone “quality” Why does a trumpet playing 440 Hz sound different than when I whistle the same frequency? The wave: Spectrum Lab as oscilloscope The sounds have different ____________ … but both sounds have the same ____________ What does that imply about their Fourier frequency components?

3. Tone quality, cont. Spectrum Lab as frequency analyzer From unknown website

4. Tone quality, cont. Odd-sounding instruments (“tonal percussion”: bells, xylophone, tympani, etc. ) From http://web.telia.com/~u57011259/Bellspectra.htmhttp://web.telia.com/~u57011259/Bellspectra.htm

5. Piano keyboard layout Half step: C to C-sharp (or, e.g. E to F) Whole step (C to D): ___ half steps Octave (C to C): ___ half steps Fifth (C to G): ___ half steps Fourth (C to F): ___ half steps Major Third (C to E): ___ half steps Minor Third (C to E-flat): ___ half steps C D E F G A B C Image: http://www.music-for-music-teachers.com/piano-keyboard.html 2 12 5 7 4 3 C-sharp/D-flat

6. Chords Why does this sound “good”? Because they are all harmonics (aka overtones) of the same note!  Their fundamental frequencies are all integer multiples of the same number  What happens when you add frequencies that are multiples of the same number? What is the note? – – It’s actually a C, two octaves below the C that’s being played! – – The frequencies of the three notes are 4:5:6 CEG Image: http://www.music-for-music-teachers.com/piano-keyboard.html

7. Chords, cont. Consonant chords: simple frequency ratios (small integers), many harmonics of each note overlap Dissonant chords: not many harmonics match ChordFreq. Ratios Octave (C-C)2:1 Major triad (C-E-G)4:5:6 Minor triad (C-E flat -G)10:12:15 Major 7 th (C-E-G-B)8:10:12:15 Dominant 7th (C-E-G-B flat ) 4:5:6:7 Minor 7th (C-E flat -G-B flat ) 10:12:15:18

8. Trumpets The notes you can play with no valves pushed in: (Lets suppose a “C trumpet” instead of a regular “B-flat” trumpet, so we don’t have to worry about the usual whole-step shift between piano and trumpet scales.) NoteFrequencyRatio to Fundamental 1 st harmonic: Low C (with difficulty) 130.8 Hz (fundamental) 1:1 2 nd harm: Middle C261.62:1 3 rd harm: G392.43:1 4 th harm: C above middle C 523.34:1 5 th harm: E654.15:1 6 th harm: G784.96:1 7 th harm: B-flat??915.77:1 8 th harm: High C 1046.5 Hz8:1 B-flat on piano = 932.3 Hz

9. Back to Pianos Why is a high B-flat on a piano 932.3 Hz? How many half steps is it? How many half steps in an octave? How much frequency change in an octave? Each half step = increase freq by a factor of ______ A = 440 Hz (defined as reference) high B-flat (middle C)

10. So, why are there 12 half-steps in an octave? Smallest number of tones that can give you close to the right ratios needed for harmonics and chords  Fewer equally-spaced tones in a scale wouldn’t get close enough  More equally-spaced tones in a scale adds unnecessary complexity Note on pianoFrequencyHow calculatedRatio to Fundamental Low C130.8 Hzf 1 = 21 half steps below A (440 Hz) 1:1 Middle C261.6 f 1  2 12/12 2:1 G392.0 f 1  2 19/12 2.997:1 C above middle C523.3 f 1  2 24/12 4:1 E659.3 f 1  2 28/12 5.040:1 G783.9 f 1  2 31/12 5.993:1 B-flat932.3 f 1  2 34/12 7.127:1 High C1046.5 f 1  2 36/12 8:1

11. Which is better? The debate “Equal-tempered”“Just-intonation” Advocated by Galileo’s father, 1581; Extremely influential work by J.S. Bach, 1782: “The Well- Tempered Clavier” Still used in many instruments, without even thinking about it (just not piano) Same ratio between successive notes: all halfsteps are the same. C to D flat = same as B flat to B All halfsteps are not equal. In fact, what’s a halfstep? Makes key changes possible without retuning instrument Key changes sound very bad unless you re-tune Chords are a little off (not exact integer ratios), e.g. C-E-G = 4.000 : 5.040 : 5.993 Creates beats (see PpP Fig 7.1) Chords are precise (integer ratios exact), e.g. C-E-G = 4:5:6 No beats Disclaimer: In actuality, piano tuners don’t use a strict equal-tempered scale

12. The Exam Handout: “What’s on the exam?”

13. Light Textbook: “Sometimes light acts like a wave, and other times it acts like a particle.” Colton: Light is made up of quantum-mechanical particles. (Same with electrons, protons, etc.) Quantum-mechanical particles are neither waves nor particles in the macroscopic sense, but rather we should think of the converse: “waves” and “particles” as we typically use the words are based on our observations of large-scale effects of these quantum-mechanical particles. Advertisement for grad school.