M & M David Rhee SASMS, summer 2011

Music & Math David Rhee SASMS, summer 2011

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

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

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

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

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

1. Music Goal: approximate well with a rational number

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

2. Math Continued fraction

2. Math

Example

2. Math Lemma 1

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

2. Math Proof (continued)

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

2. Math

3. Back to music

● 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)

3. Back to music 19 ET keyboard

3. Back to music Adjustable microtonal guitar

3. Back to music 31 ET organ

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

References ● - Main source for the talk ● - retrieved on June 8, ● - Piano keyboard picture ● - Microtonal guitar ● ET organ ● ET piano music by Jeff Harrington