Announcements 10/24/11 Prayer Term project proposals under review, I’ve responded to

about 2/3 of them. I’ll send an email 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 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 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

C E G

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

1 0 2 0 3 0 4 0cos( ) cos(2 ) cos(3 ) cos(4 ) ...f a t a t a t a t

0 0 0cos(4 ) cos(5 ) cos(6 )f t t t (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

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

Major-minor, “dominant” 7th (C-E-G-Bflat)

4:5:6:7

Minor-minor, “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, 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”?http://stokes.byu.edu/emwave_flash.html

Medium?

Polarization: quick definition