A Modern Introduction to Music – 9

By Anjum Altaf

In the last installment we went back to the origins of instrumental music tracing it to the sound that resulted from the draw and release of a hunting bow. This was presumed to have led to experimentation with more strings being stretched across a bow-like frame – a precursor of the harp. Since the shape of the frame mandated strings of unequal length, we asked the natural question: Did there need to be any kind of relationship between the lengths of the various strings in order for the harp to produce music rather than noise?

Recall from an earlier istallment that the frequency generated when a string stretched between two points is plucked depends upon at least four characteristics of the string: its material, its thickness (or gauge), the tension with which it is stretched, and its length. These can be easily verified by actual or imaginary experiments. Thus a slack string, a very thick rod, or a very long string (stretched say between two electricity poles) would produce no sound. In actual fact, the frequencies would be so low (because the wavelengths would be very long) that the ear would not be able to register the sound. As these dimensions are varied (more tension, finer gauge, shorter length), the frequency of the sound would increase and fall within the audible range mentioned in the last installment.

We can now simplify the discussion and assume that all the strings of our prototype harp are of the same material and gauge and are stretched across the frame with the same tension. The only characteristic of the string that we can vary is its length.

We can conclude from the experiments mentioned above that frequency is inversely related to the length of the string. Thus, as the length decreases, the frequency of the sound produced by plucking the string would increase.

Now assume that the first string (of length L1) tied across the bow, the longest one, vibrates at any arbitrary frequency (F1) – say 200 Hz. This arbitrariness arises from the fact that the material, gauge, tension and length were all chosen for the purpose of hunting, not for making music. As far as music is concerned, they were the result of random choice.

Now suppose, the second string (of length L2) is just slightly shorter and on plucking this string vibrates at F2 = 201 Hz. It turns out that the ear is not sensitive enough to pick up this minute difference – the frequencies would sound the same. If L2 is continuously shortened in small increments, there would come a point where the ear would pick up the difference in the sounds.

Now here is the critical part. While the ear would pick up the difference, the composite sound produced when L1 and L2 are plucked in succession might or might not be pleasant to the ear – it might tend to music (harmony) or it might tend to noise (dissonance).

A crucial discovery resulted from continuous experimentation – for some reason musical sounds were found to be related to the ratios of small integers. Thus if the ratio F2/F1 is 2/1, the sound would be pleasant. The other ratios for F2/F1 that were found to be consonant were 3/2, 4/3, 5/4, 8/5, 6/5 and 5/3. Thus in our example where F1 was 200 Hz, frequencies of 240 Hz, 250 Hz, 300 Hz, and 320 Hz would be among the frequencies that would be consonant with 200 Hz.

Note also that given our assumption that all other characteristics of the strings were kept constant except the length, the ratios of the lengths of the strings would vary exactly in inverse to the ratios of the frequencies, i.e., L1/L2 = F2/F1. Thus if L1 produces 200 Hz and L2 produces 300 Hz then L1 must be one-and-a-half times as long as L2. The bottom line to take away is that to produce pleasant sounds when different strings are plucked together, the lengths of the different strings cannot be random. They need to have a very precise relationship to each other. Jumping ahead a little bit (because we have not defined Sa and Re yet) while Sa can be any arbitrary frequency, the relationship of Re to Sa cannot be arbitrary – they are locked in a very precise embrace.

The ancients discovered this fact but did not have an explanation for it. Physicists were able to relate this to the phenomenon of harmonics. Only mechanical devices can produce a pure frequency. All natural sounds comprise of a fundamental frequency accompanied by related frequencies called its harmonics. When a string is plucked, the fundamental frequency that is heard results from the entire string vibrating back and forth. But the string also vibrates in halves, in thirds, in fourths, and so on. The vibration in halves produces the second harmonic which has half the wavelength and twice the frequency of the fundamental, and so on.

A very simple explanation of consonant sounds is as follows: If you pluck a string and generate the fundamental F1 you simultaneously generate its second harmonic that has the frequency twice that of F1. Now if you pluck another string with the fundamental F2 such that F2 = 2*F1, then F2 and the second harmonic of F1 are equal and reinforce each other making the resultant sound pleasant to the ear. If, on the other hand, F2 is slightly greater or less than 2*F1, this reinforcement would not occur and the resulting sound would not be pleasant.

This explanation can seem tedious but without it you cannot understand a very important characteristic of sound. You can generate the exact same frequency from two different musical instruments (say a guitar and a violin) but you would still be able to tell them apart. This is possible because the shapes and materials of the two instruments emphasize different combinations of harmonics of the fundamental frequency. Musicians call this the color or timbre (pronounced tam-ber) of sound. And it is this color that is in play when a sound is described, say, as mellow or thin or hollow or heavy. Thus, along with amplitude and pitch, timbre is a key characteristic of musical sound.

We are at the point where we can now define the sargam and the relationship of its various components to each other. Once we do so, we will be able to talk about the language of music and its alphabet. Then we would be able to learn music just as we learn any other language.

More soon.

You can introduce yourself to the timbre of sound by listening to the same raga on the sitar, the surbahar, the rudra veena, the violin, and the guitar. Hopefully, you will find the sound of each instrument to have a distinct color.

 

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9 Responses to “A Modern Introduction to Music – 9”

  1. Arun Pillai Says:

    Anjum,

    You say: “If you pluck a string and generate the fundamental F1 you simultaneously generate its second harmonic that has the frequency twice that of F1.”

    I have the following questions:

    1. Why does the second harmonic get generated? Presumably, you have plucked the given string of fixed length and thickness and material with some definite force. Why does it create frequencies beyond the fundamental frequency?

    2. How does one determine how many harmonics get generated?

    3. How can two or more frequencies co-exist physically in the same vibrating string? A given point on the string can move up and down f times per second. How can it be moving both f1 times per second and, say, 2f1 times per second?

    A different question: isn’t it more plausible that singing (chanting) preceded the harp?

    • Anjum Altaf Says:

      Arun: This is just a property of the vibration of strings. The theory is too complex to explain in this series. This Wikipedia entry has a schematic that shows how a string actually vibrates as a whole and also in halves, thirds, etc. These integer multiples of the fundamental frequency give rise to the ratios of small integers that are a property of consonant sounds. Let me know if you come across a simpler way to explain harmonics.

      Yes, singing/chanting would definitely have preceded the harp but conscious experimentation could be better associated with the deliberate construction of a string (or wind) instrument. I was simulating that presumed experimentation to motivate the discussion of frequency and the relationship between consonant frequencies. It would be much harder to do that using voice as the model.

    • Anil Kala Says:

      I think if you pluck the string not at the middle, may be it will have tendency to vibrate at second or third harmonics. Actually string can vibrate at other frequencies too if forced but then there will be a phase lag. For instance, if a string is vibrating at its fundamental frequency then the next string will also vibrate at the same frequency albeit at very thin amplitude and in delayed phase.

      • Anjum Altaf Says:

        Anil: You are quite right. Most Indian string instruments have sympathetic strings underneath the main strings. The sympathetic strings are rarely plucked directly. Rather, they resonate in harmony with the main strings due to the phenomenon you describe thus creating a pleasing envelope of sound. Often the widow panes in houses begin to vibrate because of resonance with the sounds generated by airplanes. There was the very famous case of the London Millennium Bridge that began to vibrate in sympathy with the lock-step marching of pedestrians.

  2. Sohail Kizilbash Says:

    Looking forward to more.

  3. Vinod Says:

    I wonder what is a Raaga? It is a tune? If so, why did the surbahar sound boringly slow while the sitar have bursts of rapid music?
    I hope you will address this sometime later in this series.

    Very enlightening, I must say. Now I too can sound as if I know something about music and hide my black hole of ignorance with a thin veil.

    • Anjum Altaf Says:

      Vinod: I am sorry that was my fault. I have now replaced the Surbahar and Rudra Veena samples so that they are in a faster tempo. My objective was only for readers to listen to the different sound of the various instruments all playing the same raaga. Yes, I will address the raaga and its various components in a forthcoming installment.

  4. sakuntala Says:

    Anjum,
    There is one more aspect of the origin of instrumental music that we were taught in musicology class — about wind instruments (you have mentioned stringed instruments, their development from a taut bow string etc) When jungle wasps bore through a bamboo shaft in the forest to get at the sap, the holes that result produce a whistling sound when the wind blows through them, and primitive man, noticing that some of these notes sounded pleasant, developed the bamboo flute- and subsequently the class of wind instruments…..

    • Anjum Altaf Says:

      Sakuntala: I agree completely. While the principle of placing the holes in the flute is exactly the same, the concepts are just easier to explain using strings as illustration. The shape of the harp motivates the discussion nicely. Once the concepts are grasped via strings, it is easy to apply them to wind instruments. It is harder going the other way.

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