Announcements 3/2/11 Prayer Term projects a. a.Proposals under review b. b.You can change your idea, but need to send me a new proposal My office hours.

Announcements 3/2/11 Prayer Term projects

a. Proposals under reviewb. You can change your idea, but need to

send me a new proposal My office hours this week: I’ll likely be in my

lab, room U130, just down the hall from normal office hour location. Find me there.

Exam 2 starts on Saturday Exam review session, results of voting:

a. Friday 3:30 – 5 pm. Room: C261 Next week: I’ll be out of town on Mon. You’ll

have Dr. Gus Hart as a substitute.

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?

Tone quality, cont. Spectrum Lab as frequency analyzer

From unknown website

Tone quality, cont. Odd-sounding instruments (“tonal

percussion”: bells, xylophone, tympani, etc.)

From http://web.telia.com/~u57011259/Bellspectra.htm

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

CD

EF

GA

BC

Image: http://www.music-for-music-teachers.com/piano-keyboard.html

2

12

5

7

4

3

C-sharp/D-flat

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

C E G

Image: http://www.music-for-music-teachers.com/piano-keyboard.html

Chords, cont.

Consonant chords: simple frequency ratios (small integers), many harmonics of each note overlap

Dissonant chords: not many harmonics match

Chord Freq. Ratios

Octave (C-C) 2:1

Major triad (C-E-G) 4:5:6

Minor triad (C-Eflat-G) 10:12:15

Major 7th (C-E-G-B) 8:10:12:15

Dominant 7th (C-E-G-Bflat)

4:5:6:7

Minor 7th (C-Eflat-G-Bflat)

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.)

Note Frequency Ratio to Fundamental

1st harmonic: Low C (with difficulty)

130.8 Hz(fundamental)

1:1

2nd harm: Middle C 261.6 2:1

3rd harm: G 392.4 3:1

4th harm: C above middle C

523.3 4:1

5th harm: E 654.1 5:1

6th harm: G 784.9 6:1

7th harm: B-flat?? 915.7 7:1

8th harm: High C 1046.5 Hz 8:1

B-flat on piano = 932.3 Hz

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

12 2

1312440 2 ?

12 2

(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 piano Frequency How calculated Ratio to Fundamental

Low C 130.8 Hz f1 = 21 half steps below A (440 Hz)

1:1

Middle C 261.6 f1 212/12 2:1

G 392.0 f1 219/12 2.997:1

C above middle C 523.3 f1 224/12 4:1

E 659.3 f1 228/12 5.040:1

G 783.9 f1 231/12 5.993:1

B-flat 932.3 f1 234/12 7.127:1

High C 1046.5 f1 236/12 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 Dflat = same as Bflat 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 : 5.993Creates 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. (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.

The wave nature of light

What is “waving”?http://stokes.byu.edu/emwave_flash.html

Medium?