1. Introduction

Like every other field of human endeavour from medicine to the arts, music has also been significantly impacted by the arrival of the digital age. On the one hand, the creation, distribution and rendition of music have been made enormously easy by digital technologies; on the other, these advantages are not totally without negatives. This article examines the convergence of digital technologies on Karnatic music and some of the difficulties that arise therein.

2. The Physics of Music

It was recognized, even from the times of ancient Greeks, that harmony had a mathematical base. Pythagorus explored the connection between the length of a vibrating string and its frequency, and also about two strings vibrating together. He discovered that when two strings vibrated together, some combination of lengths sounded more pleasing; in fact, the most pleasing—or harmonious—combinations happened when the length of the vibrating strings had simple, integral numbers as ratios of their lengths. In other words, two tones, when sounded simultaneously, sounded most harmonious when their frequencies (expressed in Hertz) and lengths where related in simple ratios such as 2:1, 3:2 or 5:3.

An enumeration of the most harmonious combinations of two tones clearly brings out this fact. Considering two vibrating strings, the most concordant or harmonious combinations are usually considered as the following:

• Unison (when both tones have the same frequency), 1:1
• Octave (when one tone sounds ‘higher’ but otherwise exactly similar) 2:1
• The Perfect Fifth (Panchamam in Karnatic music or G in Western music) 3:2
• The Fourth (Sudhamadhyamam in Karnatic music or F in Western music) 4:3
• The Major Third (Antaragandharam in Karnatic music or E in Western music) 5:4

On the other hand, some of the most discordant tones had more complex ratios (these ratios are not consistently agreed upon, and several variants are used in practice):

• The major 7 th (Sa-Kakali Nishadam in Karnatic, and C:B in Western music) 15:8
• The minor second (Sa-Sudha Rishabham in Karnatic, and C:Db in Western Music) 16:15

The ancient Greeks used these numbers to underscore the relation between tonal harmony and nature. However, on closer inspection, this scheme reveals an interesting problem.

3. Comma of Pythagorus

Consider an instrument such as a piano that has 7 octaves of keys. The first key corresponds to a low C (C1), whereas the last key corresponds to a C that is 7 octaves higher (C8). Suppose a piano tuner wants to tune the piano using the arithmetic formula as described above. He tunes the first key at the commonly accepted frequency 32.7 Hz, and C2 at double that. He successively doubles the frequency of C at every octave, and reaches the last key, C8, at a frequency of 32.7 times 2 7, or 4185.6 Hz.

Now, suppose another tuner is tuning the piano, and this tuner doesn’t want go octave to octave, and wants to tune the fifth interval. Thus, he tunes C1 at 32.7 Hz. Now he tunes the fifth or G1, which lies at a 3:2 interval, at 49.05 Hz. Next, he tunes the next fifth, E2, at 73.575 Hz, and proceeds this way all the way up to C8. The frequency that he arrives at is 32.7 times (3/2) 12, since it takes 12 steps to reach C8 if you go by fifths. This frequency turns out to be 4242.7 Hz, a considerably different frequency for the same key…!

Mathematically, the difference corresponds to the difference between 2 7 and (3/2) 12, or an error of about 1.746 intervals, otherwise called the Comma of Pythagorus. In other words, it thus turns out that there is an inherently irreconcilable imperfection harmonic music.

4. Tuning of fixed pitch instruments

The discussion above brings out the fact that no matter which way fixed pitch instruments are tuned, there is bound to be an imprecision which has to be contended with. This fact has been historically recognized and has been dealt differently by different groups of musicians in different periods of time. Essentially, the error arising out of the Comma of Pythagorus has to be distributed in a way without affecting the harmony of simultaneously sounded notes (or chords).

Two most popular approaches include:

• Load the errors into the ‘black’ keys (ie., the black keys on the piano), which are less used than the white keys. This way, melodies & harmonies that use only the white keys sound perfect, but anything that uses the black keys will sound dissonant
• Distribute the error equally among the 12 intervals in an octave. In other words, since the frequency doubles every octave, each interval becomes the 12 th root of 2

The latter approach, called equal temperament, is the most frequently used approach today. However, in this approach, every interval has a small error component. In other words, not a single note is perfectly correct….!  

5. Karnatic music and equal temperament

Karnatic music has always used microtonality (ie., using intervals less than a semitone) for its gamakams. In its theory, Karnatic music talks about 22 srutis in an octave, but since these have not been calibrated to universal standards, the 22 srutis only remain theoretical. However, since different ragas use different positions for a given note, most Karnatic ragas remain outside the scope of fixed pitch instruments (such as the piano, which has only a single frequency for any key). This is the reason why Karnatic music considers the harmonium as an apaswara-vadya (a dissonant instrument).

While most notes have fuzzy positions in Karnatic music (as opposed to a single frequency), there are two invariant tones: Shadja (Sa) and Panchama (Pa). These two canonical tones form the foundation of Karnatic music. The Sa-Pa interval (the Fifth) is sacrosanct in Karnatic music, and has been carried over from generation to generation by intensive aural training.

The advent of electronic musical instruments, whether the simple sruti-box or the complex synthesizer, challenges this foundation. Given the equal-temperament tuning of most of these instruments, the Sa-Pa interval now has an error component loaded into it. Any Karnatic musician who uses these instruments to calibrate her own sense of Sa-Pa, is liable to go wrong. This also applies every other note in the octave.

With the coming of cheap and ubiquitous electronic musical instruments, there is a real risk of losing the subtle tones of Karnatic music. It is important that music practitioners and teachers recognize this limitation of electronic musical instruments.

 Satish Babu.

The author is the  President and Chief Technology Evangelist of  InApp, a software services company based in Palo Alto, USA.