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Lecture 14

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.

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Lecture 14

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  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 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

  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 l=2L 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 L Increasing Frequency Higher Tones

  6. The String Frequencies

  7. String Frequencies fn = (n+1) 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 (f+f5)

  14. f+1.4f (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. 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

  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)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

  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 don’t 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

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