October 20, 2004 Stringing You Along Lecture 14 October 20, 2004 Stringing You Along

Countdown Exam is on Friday. A combination of multiple choice questions, short calculations and short three sentence essay amswers. If I can’t 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 don’t see it, you probably need to KNOW it but feel free to ask. Good Luck

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.

Which forces the pressure waves Bring on the String! L Anti-Nodes Nodes NOTICE l=2L Rocking forces are transmitted to the Face of the instrument Which forces the pressure waves Out into the room.

All Waves Like This Work Increasing Frequency Higher Tones

The String Frequencies

String Frequencies fn = (n+1) f0 Overtone Number

“Sine” Waves – Good Match reflected

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.

Adding the Tones and Overtones

What happens when we play two strings of differing lengths together? L1 L2

Tones From the String OCTAVE 2f0

The Fifth (f+f5)

f+1.4f (not quite a fifth)

What about….. ?

Real People BAD

A Real Instrument

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.

Let’s 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

E Ptolemy Then Said Next take the “fifth” of the “fifth” L=(2/3) x (2/3) L = (4/9)L NEW TONE OF E

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)f0=1.33f0 This is the tone of F and provides a new interval of the fourth. What about 4/5 of a string … 1.25 f0

And Wah Lah ….. We have a scale called the “JUST” scale.

And LooK!

Definition Semitone Tone

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 don’t quite match those of the C starting scale. This led to real problems with instruments that are of fixed tunings … like strings.

This led to things like this:

A Compromise The Well-Tempered The Answer – A Compromise The Well-Tempered Scale