Soothing Math

- Arghadeep Mukherjee

UG1

Isn’t it surprising to imagine those verbose formulas of integration juxtaposed with soothing Beethoven’s Fur Elise or Chopin’s Nocturne?

It is interesting to note that all fields of music including the Western melodic patterns, the Hindu raga, the Japanese pentatonic scale, etc. conform to a mathematically derived code. This is especially true of Indian classical music where the concept of ‘taal’ or meter, is intrinsically linked to numbers.

The mathematics of music is really a child’s play provided you’ve attended class, 7 Math class, carefully. Keeping all jokes aside,

In staff notation, this means the start of a segment of music. Now that initial symbol is a Treble-clef (Symbolises the octaves where one must play those notes). We have all been taught the basics of any scale,

{Sa, Re, Ga, Ma, Pa, Dha, Ni} as in Indian Classical Music or

{Do, Re, Mi, Fa, So, La, Ti} as in Western Classical Music. These lines here, are a collection of pitches brought to life by this G shaped symbol. It says that the second line from the bottom denotes the pitch (G) or {Pa} in the Chromatic Scale of (C) or {Sa}. This means that all the notes on this line are (G) or {Pa}. As defined in Music-Theory101 the lines from the bottom are (EGBDF) {note: remembered as Every Good Boy Does Fine}. Whenever one sees a black dot as shown below, we know two things,

1. The pitch to be played

2. The period for which the note is to be played

Surprisingly we must credit Pythagoras, the mathematician, for the majority of the music we hear because he is the very first Music Theorist. Pythagoras first developed what modern-day musicians call a Chromatic Scale using a raw and unprofessional but interesting method. Sadly back then the science behind this was yet to be discovered in other words the concepts of fundamental frequency overtone and nth harmony were unknown. Pythagoras devised a musical interval which is the ratio of the frequency of the sound waves of two tones, a fundamental and a second tone that is either a step lower or higher in pitch.

He used experimentation and mathematics.

1. He listened to the pitch produced by hanging different weights to strings of the same length.

2. He listened to the pitch of strings, of different lengths stretched end to end like an instrument.

3. He listened to the pitch of notes played on wind instruments.

4. Using a collection of vases, each filled to different quantities of the same liquid. He observed the “rapidity and slowness of movements of air vibrations”. Then, he hit the vases in pairs and listened to the harmonies produced. Associating numbers to consonances Pythagoras concluded that the Octave, Fifth, and Fourth correspond respectively to the ratios 2/1, 3/2, 4/3 in terms of quotients of levels of liquid.

All these experiments agreed with Pythagoras’ hypothesis that musical intervals correspond to defined ratios of integers in an immutable way, whether the integers were the length of pipes, strings or weights.

Working out one of The Seventh (for our jazz enthusiasts) here:

In the Chromatic Scale of C i.e.

C D E F G A B C

the fifth pitch or note is G. And as mentioned earlier a Fifth is the ratio 3:2

(G: C = 3: 2) or (C: G = 2: 3). Now in the Chromatic Scale of D i.e.

D E F# G A B C# D

which also has a G but as a Fourth (D: G = 3: 4).

Now it’s just a simple maths question from class 7,

If (C: G = 2: 3) and (D: G = 3: 4) what is (C: D)?

Just mental maths isn’t it? C: D = 8: 9

If the ratio between two frequencies is 8:9 then it is a Seventh

This study of musical intervals led to the discovery of semitones. And as time flew, the

Pythagorean Scale was replaced with newer and better models.

Just for a fact: We all know that the ratio of the terms of the Fibonacci sequence approaches the Golden Ratio (0.618). Needless to say many composers including Mozart, Beethoven and Debussy have used the Fibonacci sequence in their pieces. According to a study most songs climax at

(0.618 * Song_Length), this can be seen in the first movement of Bartok’s piece Music for Strings, Percussion, and Celeste where the climax is at the fifty-fifth bar of an eighty-nine bar composition.

Bartok

At the end of this tour, we can’t help but touch upon the very basis of composing and how that too has a bit of math in it. The Circle of Fifths is a musical theory tool that has its roots firmly in mathematics. It explores the relationships between those musical intervals that are most pleasing to the ear, based on discoveries of Pythagoras two and a half thousand years ago.

Starting with the note of C at 12 o’clock and moving clockwise around the Circle of Fifths, each new note (or step) is a perfect fifth above the previous one. The whole Circle of Fifths progression is simply perfect fifths stacked on top of each other, going step-wise around the circle clockwise, eventually coming back to C from F. Note that it includes all 12 tones of the chromatic scale – just not in order.

Here is a quick DIY as to how you can use the Circle of Fifth:

Looking at the Circle of Fifth, one can easily figure out the Blues Progression, the Blues chords are the dominant seventh of the keys written on the left and right to the chosen key.

For example, if you choose B♭, the Blues Chords will be, D#7 and F7

If you choose C, the Blues Chords will be F7 and G7

In a four-bar Chords Progression, the keys to the left and right of a chosen key are seen in all possible chord progressions. Starting from Classic Rock and Roll to Pop Music, musicians come across what is called the Holy Grail of Rock or the Doo-Wop progression verse, the infamous (I-V-vi-IV) progression.

If we pick D as the (I) then the other chords fall in place as follows, (D-A-Am-G) with A as the Fifth(V), G as the Fourth(IV), and Am as the minor Sixth(vi). That’s the hack to Let it Be by The Beatles. One can go about noodling about and come up with better progressions using this tool.

“There is geometry in the humming of the strings, there is music in the spacing of the spheres.”

---- Pythagoras