Announcements 10/22/12 Prayer Term project proposals: I responded to all I received,

and everyone should have a score. a.You can change your project idea, but if so you’ll

need to send me a new proposal Careful: Lab 6 due Wed night Exam 2 starts Thurs morning, goes until next Tues

eveninga.Review session: this Tues, 5:30 – 7 pm, C255

Frank & Ernest

Demos

f(x,t) tests with Slinky (results from last lecture)

From warmup

Extra time on?a. (nothing in particular)

Other comments?a. Liked it!b. If i did the lab on time but forgot to turn it

in...is it still late?

From warmup

Why does a trumpet playing a 440 Hz note sound qualitatively different than when a violin plays the same note? Both are producing 440 Hz waves, aren't they?

a. Yes they are, but they have different timbres- meaning that they are both playing the fundamental and then certain levels of the harmonics above the fundamental. Due to the different levels of harmonics played there is a different feel to the final tone. wave shape

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. Frequency analysis

From unknown website

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

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

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

From warmup What is a chapter on musical scales doing in a physics

textbook? What's the connection to physics? a. It is acoustics and AWESOME.b. Relation between notes of frequencies. Types of

scales used were mathematical relations between frequencies.

c. Everything's connected to physics. Physics defines how things happen. Music created by sound waves of certain frequencies, and as we've been learning in class, waves, frequencies, and harmonics have a lot of physics applications.

d. Just cause we are physicists doesnt mean we are incapable of doing anything else...like art or music. This statement is not reflexive however, If you are a musician....you probably cant do physics.

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, aka “dominant 7th” (C-E-G-Bflat)

4:5:6:7

Minor-minor, aka “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

Start with 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)

The wave nature of light

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

Medium?

Polarization: quick definition