Connections between Math and Music Laura Harlow – HSPVA Rhodora Maligad – Austin HS

1. Connections between Math and Music Laura Harlow – HSPVA Rhodora Maligad – Austin HS

2. A village without music is a dead place. African proverb

3. Our Goal • To understand some of the mathematics found in music • To make mathematical functions and geometric transformations better understood through music • To be able to differentiate the curriculum for the auditory learners and/or the musically-inclined.

4. Overview • Historical Connections • Obvious Connections • Functional Connections • Geometric Connections

5. Connections between the two disciplines have been studied since ancient times. • Pythagoras (580 BC) • Plato (424 – 347 BC) elaborated on • “music of the spheres” • Archytas (estimated 430-350) On Music • Nicomachus (100 AD) Introduction to Music • Ptolemy (100-165 AD) Harmonics

6. More Historical Connections… • Boethius (500 AD) Principles of Music • Kepler (1571 – 1630 AD) refined • “music of the spheres” • Galileo (1600) some combinations of tones are more pleasing than others • Euler (1707 - 1785) A New Theory of Music • Bernoulli (1700 – 1782) extended Euler’s work

7. Pythagorean music • Identified music with numbers • Music was defined and restricted by the math that dictated its theory • Pythagoras used only whole number ratios of string length and the frequencies of notes • If you divide an octave into 12 equal parts, we get the irrational number 2 1/12

8. Pythagorean music • Proved the existence of irrational numbers but chose to ignore numbers that could not be written as a fraction • The omission of irrational numbers resulted in scale known as a minor scale • Speculations arise about the effect on Greek play (tragedies) since the music is much more sinister

9. Cultural Differences • Cultures have developed their music in various ways, among them differences in the ways they divided an octave into notes. • Western music uses a pattern of 5 - 7 notes in a scale. • African cultures also use 7 notes with the 3rd and 7th notes slightly flattened, these are now known as “blue notes” • Most Asian music uses a pattern of 12 notes in a scale.

10. The Obvious Connection: Rhythm • Rhythm is the basis upon which music is built just as the concept of number is the basis of mathematics.

11. Measures of Time • Time signature is a fraction whose numerator tells us how many beats make up a measure and whose denominator tells what note is assigned to that beat.

12. GCD in Music • The concept of Greatest Common Denominator and Addition of Fractions can be used to determine if a musician is working within the given time signature or rhythm.

13. Note Combinations That Work • ½ note + ¼ note + ¼ note = 4/4 = 1 • ¼ note + ½ note + 1/8 note + 1/8 note = 8/8 =1 • Note Combinations That DON’T WORK • ¼ note + ½ note = ¾ < 1 • ¼ note + ⅛ note + ⅛ note + ½ note + ⅛ note = 1⅛ > 1

14. LCM in Music • The math concept of Least Common Multiple can be used to determine where the second note will fall in relation to the three-note rhythmic scale.

15. What makes music different than noise? The answer is in the mathematics.

16. We need some definitions • Frequency – number of vibrations per second • Pitch – a listener’s evaluation of frequency • Tone – a sound that lasts long enough and is steady enough to have pitch, quality and loudness • Octave – same note (tone), frequency doubled • Scale – The pattern used to travel an octave.

17. Some other interesting definitions used in music: • Amplitude – distance between max and min • Wavelength – distance traveled in a cycle • Period – time to complete a wavelength • Loudness – listener’s evaluation of amplitude • Pure tone – constant frequency and amplitude (creates the sine wave)

18. So, what is music and what is noise • Music is an organization of sounds with some degree of rhythm, melody, and harmony • Music is said to be an art and often defined by contrast with noise • Noise is a mixture of different frequencies • White noise – equal amounts of sound power from each spectrum of available frequencies

19. Functional Connections y = f(x) y = f(x) + 2 y = 2 f(x) y = -f(x) y = 2 f(x) + 4

20. Composers use math in subtle ways to create musical compositions that are pleasing to hear.

21. Geometric Connections Many geometric transformations have musical counterpart MusicMath repeat horizontal translation transposition horizontal and vertical translation

22. Many geometric transformations have musical counterpart MusicMath inversion vertical reflection retrogression horizontal reflection retrograde inversion 180° rotation

23. Some Miscellaneous Information • Golden proportion • Fractal music • More child prodigies in math and music than any other disciplines • Music and math do not require much experience and interpretation on manipulation of symbols is significant • Mozart’s Melody Dice – Use 2 6 sided dice rolled to determine what was played in each of 16 bars of music to create a waltz

24. “That person is a musician, who, through careful rational contemplation, has gained the knowledge of making music, not through the slavery of labor, but through the sovereignty of reason.” Boethius (A.D. 480)

25. Bibliography • Garland, Trudi Hammel and Kahn, Charity Vaughan. Math and Music: Harmonious Connections. Palo Alto: Dale Seymour Pulbications, 1995. • Beall, Scott. Functional Melodies. Key Curriculum Press, 2000. • Peterson, Ivars. “Circles of Dissonance" MAA Online. November 24, 1997. June 15, 2005 <http://www.maa.org/mathland/mathtrek_11_24.html>. • Peterson, Ivars. “Medieval Harmony" MAA Online. 1999. June 15, 2005 <http://www.maa.org/mathland/mathtrek_1_25_99.html>. • Mathematics and Music. Rusin, David.2004. June 15, 2005. <http://www.math.niu.edu/~rusin/uses-math/music/>