October 20, 2004

1
Lecture 14

• October 20, 2004
• Stringing You Along

2
Countdown

• Exam is on Friday.
• A combination of multiple choice questions, short
  calculations and short three sentence essay
  amswers.
• If I cant read what you write or understand what
  you write you are in deep doo doo.
• All of the lectures 5-13 are on the website.
• There is a study guide as well as a copy of the
  equation sheet that you will be supplied with.
• If you dont see it, you probably need to KNOW it
  but feel free to ask.
• Good Luck

3
Today Probably Monday

• Tie up some loose ends of waves on strings (no
  pun intended).
• Discuss why our music system is very much based
  upon the standing waves of a string.
• Take another look at consonance.
• Build the Ptolemaic of Just Scale
• Explain the Equal Tempered Scale that we use
  today.
• Move on to other types of instrument.

4
Bring on the String!
L
Anti-Nodes
Nodes
NOTICE l2L
Rocking forces are transmitted to the Face of
the instrument Which forces the pressure
waves Out into the room.
5
All Waves Like This Work
Increasing Frequency Higher Tones
6
The String Frequencies
7
String Frequencies
fn (n1) f0
Overtone Number
8
Sine Waves Good Match
reflected
9
Bottom Lines

• Only waves with nodes at the support points will
  be stable on the string.
• Others will die out quickly because they fight
  with the wall and lose!
• The string can sound ALL of its resonant
  frequencies at the same time.
• The frequencies that ARE established on the
  string are all multiples of the fundamental
  frequency and will form a repetitive pattern that
  will be ok with our ears.

10
Adding the Tones and Overtones
11
What happens when we play two strings of
differing lengths together?
f0 f
L1 L2
12
Tones From the String
OCTAVE 2f0
13
The Fifth (ff5)
14
f1.4f (not quite a fifth)
15
What about..
?
16
Real People
BAD
17
A Real Instrument
18
Claudius Ptolemy (75AD)

• Planets travel in circles whose centers move
  around the crystal sphere (epi-cycles).
• Sun goes around the Earth
• Harmony is based upon the interval of the fifth.

19
Lets make a scale based on the fifth

• We start with f0 and add the octave 2f0
• As before, we add the fifth which we created from
  a string of (2/3)L.
• Our scale is based upon 7 tones whose names
  are
• C D E F G A B C
• f0 1.5 2

20
Ptolemy Then Said

• Next take the fifth of the fifth
• L(2/3) x (2/3) L (4/9)L

NEW TONE OF E
21
We now have C E G C

• Ptolemy Then Said
• Hey If we can divide the string into three
  parts and use 2/3 of the string ..
• We should be able to divide the string into 4
  parts and use ¾ of the string.
• This would give us fnew(4/3)f01.33f0
• This is the tone of F and provides a new interval
  of the fourth.
• What about 4/5 of a string 1.25 f0

22
And Wah Lah ..

• We have a scale called the JUST scale.

23
And LooK!
24
Definition
Semitone
Tone
25
Houston we have a problem

• The scale starting on C is pretty good.
• If you start the scale on A, you get a set of
  frequencies that dont quite match those of the C
  starting scale.
• This led to real problems with instruments that
  are of fixed tunings like strings.

26
This led to things like this
27
The Answer A CompromiseThe Well-Tempered
Scale