Music 428
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Music 428 : Introduction to the Music of North India

THE HARMONIC/ACOUSTICAL METHOD ASCRIBED TO PYTHAGORAS.

            One of the basic problems that occupied the attention of musical theorists of the ancient world was the following : How can one communicate a precise idea of the musical scale then prevalent to a person who is far removed in time (or space), e.g. a person who lives, say, a hundred years later (or a thousand miles away)? As Aristoxenes is supposed to have asked : "How can we sing out across the chasm of time?"
            (It is worth noticing that this question has an impressive intellectual sweep. Indeed it is very similar in spirit to questions about extra-terrestrial communication that have been raised in this century by astronomers and cosmologists. It is only with the advent of very recent technology — which can create a digitized representation of musical sounds, and later recreate the original musical sounds from that digitized representation — that this problem can be said to have been solved in a somewhat satisfactory way.  That is, in a way that is not subject to the accidental features of the human condition, such as difficulties of transportation in space or time, vagaries of wear and tear on musical instruments and devices etc.)
            Ancient musical theorists attempted an answer to this question along scientific lines, describing an experiment that could be reproduced at another time or place, thus overcoming the limits imposed by the technology of their time. Their basic idea was to describe a physical experiment which could be replicated from that description at another place and/or time. The fundamental physical fact that they used was the empirically observed relation between the length of a string and the pitch of the musical note generated by that string, when stretched taut between two points and plucked.
            Thus, consider a string stretched between two points say P and Q as in the picture below :
                               C
|______________________________________________________________________|
P                                                                                                                                          Q

            When this string is plucked, it will produce a musical note. Let us call this note C. (it may be different from the note that is called C by present day musicians. This does not matter.) Now, ancient theorists noticed that if a fret is introduced half way in between P and Q, at R say, and if the string is stopped by depressing it on that fret and plucked with a plectrum, the resulting musical note is exactly an octave higher. We may label it as c (see below).

C                                                                     c                                                                     C
|___________________________________ | ___________________________________|
P                                                                     R                                                                     Q

 It is thus clear that it is possible to give a physically reproducible prescription by which the interval of one octave can be replicated by anyone who can perform this procedure. 

They also noticed that if frets are introduced at other points in between Q and R, other pitches result. Not all these pitches sound pleasant to our ears. But it is noticeably the case that certain pitches arrived at this way are pleasant sounding, or consonant. Two particular placements of frets was noticed to produce very pleasant consonances, namely: 

(1)  If a fret was introduced at a point S exactly in the middle between Q and R, one would get a note which when played in succession with the original C would produce a consonant (i.e. pleasant sounding) interval. This note was called the Mese in Greece, and Madhyama in India. Both terms mean: the middle one.  It is matter of observation that this is the note a fourth away from the original C, using our present day language. We may label it as F (see below).

(2) Similarly, a fret introduced at a point T located at two thirds of the distance between P and Q, another consonant interval resulted. The interval produced by this fret would be what we call a fifth today. We may label it as G (see below)

We now have arrived at the following situation :

C                                                                     c                     G         F                                C
|___________________________________ | ___________|_____ |_________________|
P                                                                     R                     T         S                                Q

           To summarize: the unstopped string of the full length PQ produces a note which we have called C. Stopping the string at S produces a note which is a fourth above our C, i.e. the note which we would call F. Stopping it at T would produce a note a fifth above C, i. e. the note G. Finally stopping it at R would produce the note c, a full octave above the original C.

            At this point, one has two "new" notes to work with, namely those produced at S and T. We can now generate further notes as follows: just as we produced the note G by stopping the string PQ at two-thirds of its length, we can produce another note (which would be a fifth above G) by stopping the string at a point U' which is two-thirds of the way between P and T, as in the picture below. However, this produces a note higher in pitch than c, i.e. it is in the next octave. It is in fact the note d. By doubling the distance PU', we arrive at the point U in the picture below, which produces the note D. Taking two-thirds of the distance between P and U, we get a fret at V, which produces the note A, a fifth above D.

C                                                             d      c           A        G         F                 D             C
|_______________________________ |___ | ______|_____|_____|_________ |_______ |
P                                                             U'     R           V        T        S                  U             Q

            For the next step, one can proceed with fifths as Pythagoras did, and arrive at e (a fifth above A) and transpose it down to E by putting frets at W' and W respectively, and finally at B (a fifth above E) by putting a fret at X, as in the picture below.

C                                                  e      d      c  B        A        G           F  E            D             C
|___________________________ |___ |___ |__|_____|_____|______|__I______ |_______ |
P                                                    W'     U'    R  X         V        T          S   W          U             Q

            This process produces the "Pythagorean major scale" as it were. However, it suffers from certain dissonances between the F and the notes E and B produced in the final go-around. To remedy this, there was an alternative procedure that was adopted for the last step. Namely, instead of introducing E and B as was done above, one introduced two new notes: the first one a fourth above F (and hence consonant with F), which gives us B^b, and the second one a perfect fifth below this B^b, giving us E^b. This is the scale which one may call the "Pythagorean minor scale" (C D E^b  F G A B^b  c). It has a larger number of internal consonances than the Pythagorean major scale arrived at initially.
            This was the scale most prevalent in ancient India, and is what we call the Kafi thāṭ today.