Announcements 10/24/11 Prayer Term project proposals under review, I’ve responded to about 2/3 of them. I’ll send an when I’m done, and when scores should show up in computer. You can change your project idea, but if so you’ll need to send me a new proposal Due today: HW 22 and HW 23 Exam 2 review session: Tuesday 5-6 pm. Room: C255 (not C460 where we were last time) Exam 2 starts on Thursday morning, goes until next Tuesday evening Frank & Ernest

Tone “quality” Why does a trumpet playing 440 Hz sound different than when I whistle or sing 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?

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

Tone quality, cont. Odd-sounding instruments (“tonal percussion”: bells, xylophone, tympani, etc. ) From

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: C-sharp/D-flat

Chords Why does this sound “good”? Because they are all harmonics of the same note! 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: (plus higher harmonics of each term)

C, E, G

G combined with G#

Chords, cont. “nice” chords: simple frequency ratios (small integers), many harmonics of each note overlap “ugly” 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 Major-minor, “dominant” 7th (C-E-G-B flat ) 4:5:6:7 Minor-minor, “minor” 7th (C-E flat -G-B flat ) 10:12:15:18

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) Hz (fundamental) 1:1 2 nd harm: Middle C261.62:1 3 rd harm: G392.43:1 4 th harm: C above middle C :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 Hz8:1 B-flat on piano = Hz

Back to Pianos Why is a high B-flat on a piano 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)

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/ :1 C above middle C523.3 f 1  2 24/12 4:1 E659.3 f 1  2 28/ :1 G783.9 f 1  2 31/ :1 B-flat932.3 f 1  2 34/ :1 High C f 1  2 36/12 8:1

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 = : : 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

The Exam “What’s on the exam?” (you ask)

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, called “photons”. Electrons, protons, etc., are also quantum mechanical particles. 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. Colloquium speaker a few weeks ago: “Photons don’t exist. They are only quantized oscillations of electro-magnetic fields.” Advertisement for grad school.

The wave nature of light What is “waving”? Medium? Polarization: quick definition