Mathematics and Music Sunil K. Chebolu Undergraduate Colloquium in Mathematics

1.Introduction 2.Pythagorean Scales 3.Equal Temperament Problem 4.Diatonic Scales 5.Rhythms in Sanskrit Poetry 6.The Music of Numbers Overview

Introduction

The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic. - Gottfried Wilhelm Leibniz What is Music?

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. MusicMathematics ScalesModular arithmetic IntervalsLogarithms ToneTrignometry ChordsGroup theory TimbreHarmonic Analysis CounterpointGeometry/Topology RhythmCombinatorics

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.

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). Scales

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 In this manner we get: 440, 495, , , 660, 742.5, , 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 r n = 2 or r = 2 1/n (An irrational number!) Find n and k, such 2 k/n closest to 3/2. The magic numbers are 12 and /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? 2 k/n ~ 3/2 2 k/n+1 ~ 3 k/n+1 ~ log /53 ~ So divide octave into 53 equal steps! Such keyboard were designed in the 19 th 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 18 th century.

Circle of 5 th

Diatonic Scales

A diatonic Major scale is a scale which partitions the octave into seven steps: W W H W W W H H = 2 1/12 and W = 2 2/12

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 NoteHalf 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 = = = = = 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) 8 th 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. H n = H n-1 +H n-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 {F n } = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …. {F n mod 3} = 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0 … {F n mod 7} = 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0,1,1,2,... Proof: F n = F n-1 + F n-2. The number of possible values for two consecutive terms mod n is n 2. Pigeonhole principle shows that the sequence is eventually periodic with period at most n^2+1.

Music of Fibonnaci sequence 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. π is an irrational number ….. (base 10) ……(base 7) Listen to the music of pi

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!) 2 74,207,281 − 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