Musical Intervals

Musical Interval is the distance of frequency between any two swars measured with a numerical value, as a difference or a ratio. Right from ancient times, the concept of interval was treated as very important. It was only with the understanding of musical intervals the various swars were identified. 

Intervals in music are expressed in various units such as Ratios, units like Savart, Cents.

Right from ancient times, the concept of interval was mentioned in texts like Natya Shastra. The word Shruthi was a measure of interval in ancient musical scales called Gramas. We find the terms Dwishruthi, Trishruthi, Chaturshruthi all of which refer to the distance between two swars in terms of the unit shruthi. For a given Grama, the Shruthi interval between the seven swars was fixed. Any changes in the interval between swar would result in a new Grama. 

The word Swar was used in two ways, one was to refer to a particular stable frequency or note and in the other in as an interval between two swars such as Sa-Pa, Sa-Ma etc.

If we analyze the current Swar Sapak, we get the below intervals in Ratios from the Tonic Sa.

Here we notice that there are three types of intervals - 9/8, 10/9, 16/15. In terms of actual value, they are arranged in descending order of value.

When we want to get an interval of any two notes Say Sa-Ga, we can get it by multiplying or dividing relevant intervals. Since the values are in ratios or fractions, it cannot be added directly. It has to be either multiplied or divided. Multiplication is used to add subsequent intervals to get a larger interval.

Interval of Sa-Re x Re-Ga = interval of Sa-Ga

9/8 x 10/9 = 10/8 or 5/4 which is the same as 300/240

Division is used to subtract an interval from a larger interval. When we want to get an interval of any two notes Say Ga-Dha. We can get it by dividing the interval of Sa-Dha with interval of Sa-Ga and also by multiplying the smaller intervals inbetween such ass Ga-Ma, Ma-Pa, Pa-Dha

Interval of Sa-Dha ÷ Sa-Ga = interval of Ga-Dha

        27/16 ÷ 5/4 = 27/20

Interval of Ga-Ma x Ma-Pa x Pa- Dha = interval of Ga-Dha

        16/15 x 9/8 x 9/8 = 27/20

French physicist and doctor Félix Savart (1791–1841) developed the unit Savart based on Logarithm. The advantage of this system was one can add the intervals instead of multiplying.

1000 x Log(2) = 301.029 ~ 301. 

Hence the saptak can be said to have 301 savarts between Madhya Sa and Taar Sa.

With these units, we can simply add the intervals to get the new interval. For e.g the Interval Sa-Ga = sum of Guru swar and Laghu swar which is 51.1 + 45.8 = 96.9 which is the same as log(5/4).

In western equally tempered scales, the total saptak or octave is divided into 12 equal parts and the unit is taken as 300 for ease of division.

Another measure of interval was the cent also based on logarithm. In the Twelve-tone equal tempered Scale, the octave is divided into 12 semitones of 100 cents each. Hence 301 Savarts is equal to 1200 cents. the ratio of frequencies one cent apart is precisely equal to 21⁄1200 = 1200√2, the 1200th root of 2, which is approximately 1.0005777895.