By: Reyanmallayas Taken from www.medium.com

Have you ever wondered why the note ‘C’(‘Sa’ in classical music) repeats itself in the end of an octave(C to next C forms an octave — Sa(C), Re(D), Ga(E), Ma(F)…. Ni(B), Sa(C))? Have you ever wondered why the same song can be played in multiple scales? We know that the pitch of the music seems to increase with notes as we go up from C, D, E…. upto B. But after ‘B’, a similar increase in pitch, causes the note ‘C’ to repeat again. Eventually all the notes repeat again. Keep reading, you will be able to demystify this without any prior music knowledge. (Spoiler alert: Physics!)

My tryst with music started when my uncle gifted me a keyboard after 10th grade. Played around with it, trying to reproduce some famous songs. I underwent classical vocal training for nearly a year to understand these notes more. I was always curious to know the logic or math behind it. Little did I know the answer lies in the intersection of maths and physics.

Every sound that we hear are nothing but sound waves that travel in air. Every sound wave have their own frequency(number of cycles per second). The first pic with low frequency has 3 cycles:

The frequency gets increased with each note from ‘C’(Sa) to ‘B’(Ni). This increase in frequency follows a pattern. Consider this piano image.

We also need to consider the sharp notes(Black keys are sharp notes). Thus the order of notes are C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C(‘#’ denotes a sharp note). If we consider this as the second octave in a piano the frequency of the first ‘C’ would be 65.4 Hz(Frequency is measured in hertz)(Verified all these frequencies with a keyboard and frequency sensor).

The frequency of C — 65.4 Hz
The frequency of C#— 69.3 Hz
The frequency of D— 73.4 Hz
The frequency of D# — 77.7 Hz
.
.
(next octave)
The frequency of C — 130.8 Hz
The frequency of C# — 138.6 Hz
The frequency of D— 146.8 Hz

If we look closely, we will find a pattern in this. The next C’s frequency(130.8) is twice that of first ‘C’(65.4). Similarly the next D’s frequency(146.8) is twice that of first ‘D’(73.4). So, every note in the higher octave, will have a frequency twice that of its same note in previous octave. Since frequency of similar notes in both the octaves share a same LCM(Least Common Multiple), they sound similar to us.

Further digging into it, we can easily find the ratio with these findings. A note needs to jump 12 notes to reach its same note in next octave and the frequency becomes doubled(2X) at that point. Thus at each jump the frequency gets multiplied to itself by 12th root of 2 (12√2) (Have added the derivation / proof for this at the end of this article). Value of 12√2 is 1.059 approx. Thus,
frequency of C — 65.4 * 1.059 = 69.3 approx. (frequency of C#)
frequency of C# — 69.3 * 1.059 = 73.4 approx. (frequency of D)
[The values here are approximate since the decimal part is rounded for simplicity]

The jump from C to C#(no in-between notes) is called a ‘semitone’ and C to D(C# skipped in-between) is called ‘tone’

Each semitone jump has a factor of 12√2(1.059). Now extending the same concept to a tone jump: Since there are 6 whole notes (excluding sharp notes) within an octave jump, the factor would be 6√2. Thus C — D transition multiplies the C’s frequency by 6√2(=1.122). So, every whole note jump(i.e. C-D, D-E) would multiply its frequency with 1.122. Eg: frequency of C — 65.4 * 1.122 = 73.4 approx. (frequency of D)

All these numbers look beautiful right? So, simply said, the pitch increase in notes from ‘C’ to ‘B’ corresponds to the frequency of the waves and the volume of the note corresponds to the amplitude(height) of the wave.

Hope you understood the maths and physics behind music! Next time when you try to understand music or wonder why a same song can be played in multiple scales, I am sure that this would be of a solid foundation.

If you have any doubts in this, please contact me, I will be happy to clarify them. Please share your suggestion and feedback to me. Let me know if you had learnt something new and what you think of this. Also, checkout my arts on Instagram. Feel free to contact me if you want to have a conversation regarding number theory, discrete math concepts and proofs, drawings, music or literally anything.

Optional / Bonus topic(Adding it as an optional topic in the last, since I wanted this article to be as simple as possible. But I guarantee that this is easy.):
Derivation / proof of 12th root of 2(12√2):
C * r = C#
(Let C be the frequency of the note ‘C’ and r be the common ratio between all notes.)
C * r * r = D
C * (r)³= D#
Similarly,
C * (r)¹² = nC , let ‘nC’ be the frequency of next ‘C’.
We know that frequency of second ‘C’ is twice that of first ‘C’ note. Thus by substituting this(nC = 2 * C), we get,
C * (r)¹² = 2 * C
Now cancel out C.
(r)¹² = 2
r = 12√2
Thus we have derived that the common ratio is 12√2. Now, the dots are connecting right? This is one of my favourite derivations / proofs.

So what are you waiting for? Time to start playing!

Come to our Teora Music School, a music school near you. (near me? yes, near you)

Teora Music School was established in Tomball, TX in December 2018 as a music school endorsed by the Japanese firm. Teora Music School serves the community of The Woodlands, Augusta Pines, Springs, Tomball, and surrounding areas.

We offer music lessons or music classes for kids and adults such as piano lessons, guitar lessons, drums lessons, vocal and voice or singing lessons, violin lessons, bass guitar lessons, . At Teora Music School we offer music production services such as composition, song writing, recording, mixing, and mastering, as well as rehearsal and recording studio rental services.

At Teora Music School, we also offer Music Therapy, employing musical interventions to achieve personalized goals led by a qualified professional. Teora Music School also offers a program specifically designed for preschoolers. In addition to musical instruction, this program aims to introduce young children to the fundamentals of music in a fun and engaging way. Teora Music School serves the community of The Woodlands, Augusta Pines, Springs, Tomball, and surrounding areas.

Reyan Mallaya S
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