Musical Notes (स्वरs)

Baji J. Ram Rao
12:31 +0530 Thu. 08-Jan-2015

In Western music, a C note is an absolute pitch, whose fundamental sine wave has the following frequencies.

C0 = 16.35 Hz. (most humans can't hear below 20 Hz)
C1 = 32.70 Hz
C2 = 65.41 Hz
C3 = 130.81 Hz
C4 = 261.63 Hz
C5 = 523.25 Hz
C6 = 1,046.50 Hz
C7 = 2,093.00 Hz
C8 = 4,186.01Hz
C9 = 8,372.02 Hz
C10 = 16,744.04 Hz (most humans can't hear above 20,000 Hz)

keys on a keyboard instrument
Keys on an 88-key piano and 61-key keyboard

However, unlike in Western music where a C note has an absolute pitch, when we speak of the स्वरs in Hindustani music, the pitch of षड्ज or सा is variable and can be set to any note frequency.

There onwards, instead of being fixed at absolute frequencies, the pitches of notes in Hindustani music can be computed, based on their relationship with the tonic षड्ज(सा). It is called षड्ज - the creator of six notes, because the frequencies of the following six notes (रे, ग, म, प, ध and नी) depend on it.

Since the octave is equally tempered into 12 chromatic semitones, each semitone can be computed by multiplying the earlier semitone by .

By convention, for the 12 chromatic notes:

  1. षड्ज(सा)-शुद्ध,
  2. ऋषभ(रे)-कोमल,
  3. ऋषभ(रे)-शुद्ध,
  4. गान्धार(ग)-कोमल,
  5. गान्धार(ग)-शुद्ध,
  6. मध्यम(म)-शुद्ध,
  7. मध्यम(म)-तीव्र,
  8. पञ्चम(प)-शुद्ध
  9. धैवत(ध)-कोमल,
  10. धैवत(ध)-शुद्ध,
  11. निषाद(नी)-कोमल,
  12. निषाद(नी)-शुद्ध,

we use the Latin notation:
S r R g G M m P d D n N.
Each chromatic note (a.k.a. semitone) is a factor of away from the earlier note. It is conventional to also use a logarithmic unit of measure for musical intervals. This is called the cent. There are 12 semitones separated 100 cents from each other. In fact the interval of one cent is much too small to be discerned between successive notes. Humans can distinguish a difference in pitch of about 5-6 cents.(Loeffler 2006).

Let the chosen pitch of the tonic षड्ज(सा), be x.

kaanton-se-khiinch-ke

On the one hand, the Just Scale (a.k.a. von Helmholtz’s harmonic tuned scale) is meaningful for simple systems such as vibrating strings or air columns.
The frequencies of notes in the scale are related by ratios of small whole numbers. However, and quite unfortunately, tuning (in the case of Just Tuning) depends on the scale you use.
The tuning for C Major is not the same as for D Major. The players must match pitch with each other “by ear”.

On the other hand, the Equally Tempered Scale was developed for keyboard instruments, such as the piano, to enable them to be played equally well (or equally badly) in any key.
This scale is a compromise tuning scheme. It uses a constant frequency multiple between the notes of the chromatic scale.

Temperaments other than Just and Equal exist, such as the Pythagorean Scale, Mean-tone Scale, and the Werckmeister Scale. (Berg and Stork 1995)

References

1) Berg, R. E., & Stork, D. G. The physics of sound. 1995.
2) D.B. Loeffler, Instrument Timbres and Pitch Estimation in Polyphonic Music. M.S. thesis, Dept. of Electrical and Computer Engineering, Georgia Tech. April (2006)