1. M & M David Rhee SASMS, summer 2011

2. Music & Math David Rhee SASMS, summer 2011

3. 1. Music Question: Why does our musical scale have 12 notes?

4. 1. Music Music: The art of combining sound Sound: Vibration that people can hear

5. 1. Music Humans like simple ratios 1:2 Octave 2:3 Perfect 5th 3:4 Perfect 4th 4:5:6 Major triad

6. 1. Music Humans hate complex ratios 15:16 Semitone 8:15 Major 7th 32:45 Tritone 15:25:27 ???

7. 1. Music Notes we use will be equally spaced geometrically. Want: such that and

8. 1. Music Goal: approximate well with a rational number

9. 2. Math Diophantine Approximation – Number theory Approximation of real numbers using rational numbers Example error: ~0.0024

10. 2. Math Continued fraction

11. 2. Math

16. Example

17. 2. Math Lemma 1

18. 2. Math Lemma 2 Proof I claim that. x is the limit of so we are done if the claim is correct.

19. 2. Math Proof (continued)

20. 2. Math Theorem Proof is an alternating series and the (n+1)th term is

21. 2. Math

22. 3. Back to music

23. ● Pentatonic – common all over the world ● 12 equal temperament – what everyone uses ● 41 ET – some modern musicians use it ● 19 ET – approximates major third well (4:5) ● 31 ET – approximates all basic intervals well except for major 5th (2:3)

24. 3. Back to music 19 ET keyboard

25. 3. Back to music Adjustable microtonal guitar

26. 3. Back to music 31 ET organ

27. 3. Back to music A piano music written in 19 ET Part of Prelude 3 for 19 ET piano Composed by Jeff Harrington

28. References ● http://thinkzone.wlonk.com/Music/12Tone.htm - Main source for the talk http://thinkzone.wlonk.com/Music/12Tone.htm ● http://en.wikipedia.org/wiki/Continued_fraction - retrieved on June 8, 2011 http://en.wikipedia.org/wiki/Continued_fraction ● http://squeezehead.com/uniform-keyboard/ - Piano keyboard picture http://squeezehead.com/uniform-keyboard/ ● http://www.erhanbirol.info/tolgahan/EN/microtonal_en.html - Microtonal guitar http://www.erhanbirol.info/tolgahan/EN/microtonal_en.html ● http://www.huygens-fokker.org/instruments/fokkerorgan.html - 31ET organ http://www.huygens-fokker.org/instruments/fokkerorgan.html ● http://parnasse.com/jh/blog/ - 19ET piano music by Jeff Harrington http://parnasse.com/jh/blog/