Transcript Mathematics and Music - Department of Mathematics | Illinois State

Mathematics and Music
Sunil K. Chebolu
Undergraduate Colloquium in Mathematics
Overview
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6.
Introduction
Pythagorean Scales
Equal Temperament Problem
Diatonic Scales
Rhythms in Sanskrit Poetry
The Music of Numbers
Introduction
What is Music?
The pleasure we obtain from music comes from counting,
but counting unconsciously.
Music is nothing but unconscious arithmetic.
- Gottfried Wilhelm Leibniz
• Mathematics - the most abstract of the sciences.
Music - the most abstract of the arts.
• Since ancient times mathematicians and musicians revealed
multitude of fascinating connections between these two
abstract worlds.
• Mathematics helps describe, analyze, and create musical
structure: rhythm, scales, chords, and melodies.
• Mathematics also helps understand the nature of sound
• Musicologists have used math to solve musical problems for
centuries.
Music
Mathematics
Scales
Modular arithmetic
Intervals
Logarithms
Tone
Trignometry
Chords
Group theory
Timbre
Harmonic Analysis
Counterpoint
Geometry/Topology
Rhythm
Combinatorics
Musical Mathematicians
1. Pythagoras: constructed consonant intervals based on simple
ratios.
2. Plato: Music gives a soul to the universe, wings to the mind,
flight to the imagination and life to everything.
3. Johannes Kepler : music was central to his search for
planetary laws of motion in his Harmonices mundi. The
ratios of the maximum and minimum speeds of planets on
neighboring orbits approximate musical harmonies.
4. Rene Descartes: his first work is on Compendium musicae
5. Mersenne considered music the central science, and
explored in his encyclopedic Harmonie universelle
6. Isaac Newton’s notes show his interest in musical ratios. He
tried to impose the musical octave on the color spectrum.
5. Euler Tentamen novae theorae musicae ex certissimis
harmoniae principiis dilucide expositae
Essay on a New Theory of Music Based on the Most Certain
Principles of Harmony Clearly Expounded -Too mathematical for musicians and too musical for
mathematicians.
Modern Text Books
1. Mathematics and Music – David Wright (Undergraduate)
2. Music : A mathematical offering – Dave Benson (Graduate)
Pythagorean Scale
• Why are some combinations of notes consonant (pleasing to
the human ear), while others are dissonant?
• The Greek mathematician Pythagoras discovered that notes
which are consonant obey certain mathematical regularity.
• Pythagoras’ discovery is equivalent to saying that two notes
played together will be pleasing to the ear if the ratio between
their frequencies is 1:2 (octave) or 2:3 (perfect fifth).
• Grand extrapolation: All is number. They developed an entire
theory that connects numbers, musical notes, and the motion
of planets.
• While their planetary theory has been flawed, their work on
music has had a great influence on western music.
Scales
A musical scale is a collection of notes which from a partition of
an octave.
Definition of a Pythagorean scale.
• MATH: A partition in which the ratio of the frequencies of any
two notes involves only primes 2 and 3.
• MUSIC: A partition that is obtained by stacking perfect fifths
(2:3).
Construction: (later formed the basis of Euclidean Algorithm)
• Let us consider octave [440, 880] Hz.
• A perfect fifth above 440 is 440 x 3/2 = 660.
• A perfect fifth above 660 is 660 x 3/2 = 990.
This frequency is outside our interval, but an octave
below it is 495.
• Another perfect fifth above 495 is 742.5
• In this manner we get:
440, 495, 556.875, 586.667, 660, 742.5, 835.3125, 880
• Divide all these numbers by 440 and reduce to lowest terms:
1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2
do re
mi
fa so la
ti
do
Pythagorean Ratios for guitar frets
Equal temperament Problem
• Iterations of perfect fifth will never “close the loop”
(Let us see this mathematically!)
• So what? This results in the Transposition problem/Equal
Temperament Problem. In ancient keyboard instruments
(before 1700) it was not possible to play the same melody in
different keys.
• An annoying problem. Generations of musicologists debated
and proposed multitude of tuning systems.
• The modern solution is slick and elegant: divide the octave
into a certain number of equal musical intervals.
• This solves the transposition problem.
• A new problem arises: we can play only one Pythagorean
interval exactly. (which one?)
• The rescue comes from the limitations of the human ear.
• Into how many equal intervals should we partition the octave?
•
n = number of equal steps r = the common ratio
rn = 2 or r = 21/n (An irrational number!)
• Find n and k, such 2k/n closest to 3/2.
• The magic numbers are 12 and 7. 27/12 ~ 1.498,
3/2 = 1.5
• Equal tempered fifth
Pure perfect fifth
Mathematical formulation of the Equal Temperament problem:
What is a good rational approximation to log 2 3?
2k/n ~ 3/2
2k/n+1 ~ 3
k/n+1 ~ log 2 3
231/53 ~ 1.49994 So divide octave into 53 equal steps!
Such keyboard were designed in the 19th century.
• The tuning method in which the octave is divided into 12 equal
musical intervals is called equal temperament – widely
accepted tuning system.
• Bach was very excited about this possibility! He wrote his
masterpiece “Well-tempered clavier” in early 18th century.
Circle of 5th
Diatonic Scales
Diatonic Scales
A diatonic Major scale is a scale which partitions the octave
into seven steps: W W H W W W H
H = 21/12 and W = 22/12
Let us understand this mathematically:
x = number of whole steps
y = number of half steps
2x + y = 12 (Scale is partition of the octave)
x + y = 7 (Diatonic scale)
Solving these equations simultaneously gives
x = 5 and y = 2
The 2 half steps have to be maximally separated for the
scale to sound good.
So what are the possibilities?
In how many ways can 5 boys and 2 girls be seated in a
round table if the girls have be maximally separated?
Only 1 circular arrangement but 7 linear arrangements!
• These correspond to the 7 modes of the major scale.
Rhythms in Sanskrit poetry
• Sanskrit poetry consists of two kinds of syllables, short and
long.
• Long syllables are Stressed (guru)
• Short syllables are Unstressed (laghu)
• A long last two beats (say, half note)
• A short last one beat (say, quarter note)
Quarter Note
Half Note
Problem: How many rhythms can one construct of say 8
beats consist of long and short syllables?
In how many ways can one write a number 8 say as sums
of 1 and 2s?
8 = 1+1+1+1+1+1+1+1
= 2+2+2+2
= 1+ 1+ 2 + 2 + 2
= 1+1+1+1+ 2+2
= 1+ 1+ 1+ 1+ 1 + 1 + 2 etc.
Ingenious answer given in ancient India
• Write down 1 and 2.
• Each subsequent number is the sum of the previous two
• The nth number we write down in the number of rhythms
on n beats.
• 1,2,3,5,8,13,21,34 ,… - Hemachandra numbers (c.1050
AD)
• 8th number in this sequence is 34. So there are 34
rhythms of 8 beats consisting of long and short syllables.
• Manjul Bhargava (Fields Medal Winner)
• Hemachandra’s proof: Every rhythm on n beats ends in
a long or a short beat. Hn = Hn-1 +Hn-2 (QED).
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,….
In the west, these numbers are called Fibonacci numbers
after the Italian mathematician Fibonacci.
They appear in nature’s art: Fruitlets of a pineapple,
flowering of artichoke.
Fibonacci numbers were invented by scholars in ancient
India (200 years before Fibonacci) when they were
analyzing rhythms in Sanskrit poetry!
Music of Numbers
A mathematical result.
Theorem: For any integer n, the Fibonacci sequence modulo n is
a periodic sequence.
Examples
• {Fn}
= 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ….
• {Fn mod 3} = 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0 …
• {Fn mod 7} = 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0,1,1,2,...
Proof: Fn = Fn-1 + Fn-2. The number of possible values for two
consecutive terms mod n is n2. Pigeonhole principle shows that
the sequence is eventually periodic with period at most n^2+1.
Music of Fibonnaci sequence
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•
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Fibonacci sequence mod 7 on treble clef
Fibonacci sequence mod 3 on the bass clef.
The treble in quavers (period of sixteen).
The bass in crotchets (period of eight).
• We just turned a mathematical theorem into a nice melody!
Music of π
• π = ratio of the circumference of a circle to its diameter.
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π is an irrational number.
3.14159265358979323846264338327950288….. (base 10)
3.06636514320361341102634022446522266……(base 7)
Listen to the music of pi
Music of Primes
The most fascinating sequence of numbers:
• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61
67, 71,…
• There are infinitely many primes (Euclid 300 BC)
• Largest known prime number (found earlier this month!)
274,207,281 − 1, a number with 22,338,618 digits.
• Music of Primes
Why do rhythms and melodies, which are composed of sound,
resemble the feelings, while this is not the case for tastes,
colours or smells?
- Aristotle. Prob xix. 29
Thank you