The Mathematics Behind Indian and Western Music
By: Skanda Vivek
Tags: Writing, Language, Mathematics, Music, Science

Absolute vs Relative Scales

The fundamental difference between Western classical and Indian music, is that Western music has 12 absolute notes. Each note: C, C#, D, etc. has a specific frequency. Concert C is around 262 Hz for example. Ignore harmonics for now (i.e. the higher pitch or lower pitch version of C).

Western Music Notes | George Azzopardi

However, Indian music is relative. In Indian music, notes are marked Sa, Re, Ga, Ma, Pa, Da, Ni, Sa — or short form as below. Note that the end S is the higher pitch version of S.

Indian music notes | Skanda Vivek

Mathematical Permutations in Indian Music

Visualizing Raga Combinations | Skanda Vivek

The human ear can distinguish 12 semitones in a scale, the 13th is the harmonic of the first note. In South Indian music, the first and middle notes are fixed (S, P). But the others can be permuted over. Note there are 3 possible R, and three G, but two R and G overlap — the same story with D and N. To find the total combinations of distinct scales (also known as Ragas), we need to use combinations. From 4 possible R, G we can choose 2. From two M, we can choose 1. From 4 D, N we can choose 2.

From these combinations, the number of ragas comes to 4C2*2C1*4C2 = 72. Indian music has 72 distinct scales!

Modes in Western Music

Western music has 12 major scales, all starting from each of the 12 semitones (C, C#, D,….). In addition, there are 7 modes. 7 because a whole scale has 7 distinct notes e.g. C, D, E, F, G, A, B in the C major scale. And each mode starts from the next note. These modes are:

Ionian — C, D, E, F, G, A, B, C Dorian — D, E, F, G, A, B, C, D Phrygian — E, F, G, A, B, C, D, E Lydian — F, G, A, B, C, D, E, Mixolydian — G, A, B, C, D, E, F, G Aeolian — A, B, C, D, E, F, G, A Locrian — B, C, D, E, F, G, A, B

Thus, there are 7 distinct modes in Western music. We are not considering the 12 different scales, as they correspond to the same pattern, albeit starting from a different frequency. In addition, there are 3 different minor scales in Western music, natural, harmonic, and melodic minors. Same with the major scale, each of these minor scales has 7 intervals, and a mode can be built off of each interval. Therefore there are a total of 21+7= 28 modes in Western music.

Thus, Indian music has 72 distinct scales (ragas), in comparison Western music only has 28.

Creativity in Western and Indian Music

But don’t get me wrong, Western music also has similar diversity when it comes to the actual music. It’s just that Indian music is more structured when it comes to defining distinct scales.

An Indian music concert commonly goes for 2–3 hours, the artists typically include a singer and accompanying instruments (violin, Indian drums, etc.). Most artists stick to the same raga through a composition that can last more than half an hour to an hour sometimes. Artists carefully select ragas through the entire concert to keep the audience entertained, Some songs are lighter, and last a few minutes typically towards the beginning and end of a concert. Others typically during the middle, are heavier and longer pieces.

In Western music however, compositions do not typically stick to the same scale or mode. Especially in improvisation jazz pieces, the artist has more freedom to mix different scales together.

While mixing ragas is becoming more and more common in Indian music, usually the artist sticks to one raga, and the changes in ragas are very distinct. For music buffs, it is common practice to guess which raga the artist is performing in. Personally, me and my family used to love playing this game when I was young.

Conclusion

In summary, I’ve gone through the mathematical patterns behind whole scales in two fundamental musical forms: Western and Indian music. Indian music is more mathematically structured, and Indian ragas, the equivalent of Western modes are 72 in number. Whereas Western music has 28 modes. But keep in mind, Western music is more flexible in improvisation pieces where modes and scales often merge together. Whereas Indian music compositions are more structured or siloed.

Each has its own advantages, and I hope you find it interesting to breakdown the math and language of music. If you notice, I haven’t even mentioned incomplete scales — there’s a whole bunch more than the 72 major ragas in Indian music. And Western music has pentatonic, and blues scales. In fact, because of repetitions and combinations, I don’t think anyone has a clue on the total number of incomplete scales. It’s probably infinite or close to some unfathomable number!

Do let me know if you like this article, and would like me to cover any other topics on math, music, and language.