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

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

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

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**Connections between Math and Music**Laura Harlow – HSPVA Rhodora Maligad – Austin HS**A village without music is a dead place.**African proverb**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.**Overview**• Historical Connections • Obvious Connections • Functional Connections • Geometric Connections**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**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**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**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**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.**The Obvious Connection: Rhythm**• Rhythm is the basis upon which music is built just as the concept of number is the basis of mathematics.**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.**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.**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**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.**What makes music different than noise?**The answer is in the mathematics.**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.**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)**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**Functional Connections**y = f(x) y = f(x) + 2 y = 2 f(x) y = -f(x) y = 2 f(x) + 4**Composers use math in subtle ways to create musical**compositions that are pleasing to hear.**Geometric Connections**Many geometric transformations have musical counterpart MusicMath repeat horizontal translation transposition horizontal and vertical translation**Many geometric transformations have musical counterpart**MusicMath inversion vertical reflection retrogression horizontal reflection retrograde inversion 180° rotation**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**“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)**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/>