Mathematics and Music What? Archytas, Pythagoras Other Pythagorean Philosophers/Educators: The Quadrivium Mathematics ( study o the unchangeable ) Number Magnitude Arithmetic numbers at rest Music numbers in motion Geometry magnitudes at rest Astronomy magnitudes in motion
Physics o Sound and Musical Tone y Pressure 0 - -. 0. x Time - - Pitch: requency o wave = number o cycles per second (Hz) higher pitch more cycles per second skinnier waves on graph Volume: amplitude o wave = dierence between maximum pressure and average pressure higher volume taller waves on graph Timbre: quality o tone = shape o wave
Notes between: Musical Scale: (increasing pitch = requency) C D E F G A B C C # D # F # G # A # or D b E b G b A b -note Musical Scale (Chromatic Scale): C C # D E b E F F# G G # A B b B C B b Interval Name Musical Intervals: Examples nd hal steps C D, E F # rd hal steps G B, B D # th 7 hal steps G D, B F # Octave hal steps C C, F # F # s ( Partials ) Multiply Frequency by Interval Produced Example C=undamental octave C octave + perect th (% sharp) G octaves C octaves + major rd (% lat) 6 octaves + perect th (% sharp) 7 octaves + dominant 7th (% lat) E G B b 8 octaves C 9 octaves + whole step (% sharp) D 0 octaves + major rd (% lat) E
When a musical instrument is played, the harmonics appear at dierent amplitudes --- this creates the dierent timbres. A branch o mathematics Fourier analysis deals with decomposing a wave o a certain requency into its harmonic components. Joseph Fourier (768 80) discovered these methods and utilized them to solve heat low problems. All o this mathematics uses Calculus in an essential way (discovered by Newton and Leibniz independently in late 600s). Creating new musical tones using harmonics rom just one musical tone: Given tone: C (tuned to requency 6 Hz) How can we make E? Let me count the ways: Method : Just tuning Multiply requency by C E (0 Hz) Multiply requency by E E (7. Hz) Corresponds to actor o or Major rd interval Method : Pythagorean tuning Multiply requency by C G (786 Hz) Multiply requency by G G (9 Hz) Multiply requency by G D (79 Hz) Multiply requency by D D (9.7 Hz) Multiply requency by D A (88. Hz) Multiply requency by A A (. Hz) Multiply requency by A E (6.7 Hz) Multiply requency by E E (.6 Hz) Corresponds to actor o or Major rd 6 interval
Yet another example: the Pythagorean comma The cycle o perect iths: Starting tone: C (tuned to requency 6 Hz) Multiply requency by C G (9 Hz) Multiply requency by G G (97. Hz) Corresponds to actor o =. or Perect th interval Keep doing that: G D (6. Hz) A (9.8 Hz) E (9.06 Hz) B (9.9 Hz) F # (70.9 Hz) C # (0.6 Hz) G # (66.9 Hz) E b 6 (98.8 Hz) B b 6 (78. Hz) F 7 (6. Hz) C 8 (8. Hz) (go down 7 octaves) C (6.8868 Hz) Wait a minute!!!! 9 The requency (pitch) is high by a actor o.06... (the Pythagorean comma). Evenly-spaced intervals between octaves: Equal tempering Pythagorean tuning (popular through 6 th century): C to G is a perect ith actor o =. 8 F# to C# is a perect ith actor o. =.8 comma Thus music with C s and G s sounds good. Music with F# s and C# s sounds a little weird. Equal-tempered tuning (introduced by Simon Stevin (mathematician) in 96; in 60s Father Mersenne ormulated rules or tuning by beats; became popular in 8 th century): All intervals are the same in all keys. All keys sound roughly the same. A hal-step interval is a actor o = / Thus, there are even hal-steps between octaves.
Interval Example Frequency Multiple (Pythagorean) 8 hal step C - C# 6 =.0 whole step C - D 9 = =. 8 minor third C - E =.8 7 major third C - E 8 =.66 6 6 perect ourth C - F. perect ith C - G =. octave C - C Frequency Multiple (Even-tempered) / / / / / 7 / / =.09..89.60..98 From W. W. Norton Catalog: A captivating look at how musical temperament evolved, and how we could (and perhaps should) be tuning dierently today. Ross W. Duin presents an engaging and elegantly reasoned exposé o musical temperament and its impact on the way in which we experience music. A historical narrative, a music theory lesson, and, above all, an impassioned letter to musicians and listeners everywhere, How Equal Temperament Ruined Harmony possesses the power to redeine the very nature o our interactions with music today. For nearly a century, equal temperament the practice o dividing an octave into twelve equally proportioned hal-steps has held a virtual monopoly on the way in which instruments are tuned and played. In his new book, Duin explains how we came to rely exclusively on equal temperament by charting the ascinating evolution o tuning through the ages. Along the way, he challenges the widely held belie that equal temperament is a perect, naturally selected musical system, and proposes a radical reevaluation o how we play and hear music. Ross W. Duin, author o Shakespeare s Songbook (winner o the Claude V. Palisca Award), is Fynette H. Kulas Proessor o Music at Case Western Reserve University. He lives in Shaker Heights, Ohio. 6
Pairs o Harmonious Tones: tones sound harmonious when they are played together i they share common harmonics with nearly the same requencies. Octave Interval Lower Pitch 6 Higher Pitch 6 8 0 Pairs o Harmonious Tones Pythagorean Perect th Lower Pitch 6 Higher Pitch 8 6 0 8 7
Pairs o Harmonious Tones Just Major rd Lower Pitch 6 Higher Pitch 0 0 The intervals in harmonious order (mathematically determined) Interval Rational approx. denominator octave / Perect th / Perect th / Major 6 th / Major rd / Minor rd 6/ Tritone 7/ Augmented th 8/ Minor 7 th 9/ Major nd 9/8 8 Major 7 th /8 8 Minor nd 6/ Note: Hindemith 90s: The Crat o Musical Composition 8
The -note chords in harmonious order (consonant to dissonant) Chord ratios LCD, sum o degrees C-F-A /,/,/ LCD=, sum=0 C-E-G /,/,6/ LCD=, sum= C-E b -A b 6/,8/,/ LCD=, sum= C-F-G /,/,9/8 LCD=6, sum= C-E b -G 6/,/,/ LCD=0, sum=...... C-C # -B 7/6,/8,/ LCD=6, sum=8 C-C # -D 6/,9/8,/8 LCD=0, sum= Would Space Aliens want to listen to Mozart? To construct an even-tempered scale that includes the irst a nontrivial harmonic, we need to ind a raction b such that b a / The resulting chromatic scale would have b distinct notes. We would have a log ( ) =.896007... b Possible a/b Decimal Error in th /.6.0% /7.7 0.9% 7/.8 0.% 7/9.86 0.086% 6/.86 0.0% 9