1. “In ancient times, music was considered a part of mathematics. Mathematics is the music of the mind; music is the mathematics of the soul.” Harald Ness, Jr. in “Mathematics; an Integral Part of Our Culture” in Humanistic Mathematics

2. An Infusion of the Fine Arts Barbra Steinhurst Pennsylvania College of Technology AMATYC 2007

3. Why Use the Fine Arts? Students can relate Fun for both teachers and students Employ mathematics at both deep and superficial levels Expand student notions of mathematics as a field

4. How to Design Activities Use the art to motivate; extend the math Use the art to illustrate; find the math Use the art to teach the math explicitly Use the art to assess understanding; students search out the math Use the art to answer “when will I ever use this?”

5. Topics Number Theoretic Ideas LCM and Polyrhythms GCF, relatively prime, Euclidean algorithm, and plaited mat sona Functions

6. Polyrhythms A polyrhythm is the co-occurrence of two or more rhythmic patterns. Jazz syncopation African and Indian drumming Caribbean music many others

7. Let’s Try It. represents a beat x represents a clap x..x..x..x..x..x..x. x...x...x...x...x...

8. Tschokwe sona The sona are part of a rich storytelling tradition among the Tschokwe tribe in Africa. Let’s Draw Some!

9. Topics Number Theoretic Ideas Functions Inverse Functions and Lewis Carroll Function Transformations and Bach

10. Hunting Snarks Read the excerpt Examine Stanza 8. What happens to the number three? Suppose rather than the number 3, we use the variable x. Write a function f(x) that describes what operations occur in stanza 8.

11. Hunting Snarks Examine Stanza 9. What is the input? What happens to that input? Write a function g(x) that describes the operations that occur in stanza 9.

12. Hunting Snarks What is the relationship between f(x) and g(x)? (Hint: Find g(f(x)) and f(g(x)).) How does that relationship tell us that the butcher will end with the number he starts with? Did the Butcher show what he set out to show?

13. A Function Transformations Activity based on The Art of Fugue by J.S. Bach

14. Set Up and Start Up Define dependent and independent variables of this function Translate musical notation into numerical function notation Sketch the melodic contour of the function Listen to the function

15. Variations on a Function f(x)f(x)-f(x)-4 f(x)+5f(x/2) -f(x)-f(x)-f(x+1)

16. Listening in Context Contrapunctus I Contrapunctus III Contrapunctus IX Contrapunctus XIV

17. Just for Fun 2f(x)2f(x) 0.5f(x)

18. Lessons Learned Warm students up to the idea Careful demonstration Did it bomb? Think quick and adapt. Don’t push it if not working. The more you try it, the easier it gets.

19. Thank you! Barbra Steinhurst barbra@musicandmath.com http://www.musicandmath.com