The Concept of Harmony

By S. R. Govindarajan, M.A., M.Sc.

“O, Mind! Betake thou to the Eternal Bliss of Nadavidya;
Know ye not that Shiva and Brahma, Indra and Vishnu
Perceive in Nadopasana the Everlasting?”¹

SHAKESPEARE spelt curses to him who is not moved by the concord of sweet sounds; we really wonder if he realised how he had unconsciously crystallised the purpose of creation in that couplet. Harmony is not merely the panacea of all human ailments, it is the keynote of cosmic existence. The apparently mad gyrations of the celestial spheres follow a mathematical law of harmony, difficult to conceive of, but nevertheless existing. The unceasing rising and setting of the sun, the almost taxing regularity of occurrence of the seasons, the innumerable cycles of existence that go on day after day, all make it clear that harmony is the sole purpose, not the bye-product of creation. Thus it is that our ancients symbolised the creator as a Dancer, rising to the tune of the Damaru, and sending forth quivering the message of harmony through the rhythm of Natya.

Is it surprising therefore, that music which lives on harmony, should have a greater appeal to the human, than any other form of aesthetic manifestation? Von Helmholtz once said: “It always struck me as wonderful and peculiarly interesting mystery, that in the physical and technical foundations of music, which above all other arts seems in its action on the mind most unmaterial, evanescent and tender creator of incalculable and indescribable states of consciousness, that here in especial the science of purest and strictest thought–mathematics–should prove pre-eminently fertile.”² Music and mathematics, which the unknowing place at antipodes, are nearer each other than any two other achievements of the human intellect.

Other forms of art draw from the world of experience for their delineations. A picturesque scenery might inspire painting, a sublime act, poetry, and a form of beauty, sculpture. But music is self-sustaining and “not attempting to describe and only exceptionally to imitate the outer world, necessarily withdraws from scientific considerations the chief points of attack which other arts present and hence seems to be as incomprehensible and wonderful as it is certainly powerful in its effects.”³ It is no wonder therefore that the enormous lot of written and spoken material on the science and art of music is both vague and inaccurate. No attempt shall therefore be made to give the reader a concept of music as an art and its claims to be classed a science, our purpose being solely to analyse the merest scientific principles of harmony in music.

Harmony is an intellectual necessity, not an emotion of luxury. The child, whose mental agony it is possible to understand, is transported to the realms of super-consciousness under the harmony of a lullaby. The venomous cobra subjects its wickedness to the supreme order brought about by the harmony of the flute. The hardest amongst us melts at the strains of meaningless verse doled out by the street singer. Was it the Mogul Emperor Aurangzeb who wanted music to be buried waist-deep? That is only an example of the rare mastery of the flesh over the mind. Harmony unlike a perfume, never satiates. A sweet dish may be unnecessary for body-building, but harmony is essential for the process of mind-building; nay it is one of the components of the mind. Disharmony is no doubt present on this good earth, but only for enhancing the value of the subsequent harmony.

Harmony is the result of one or more simple undulations in air, mathematically related to one another. But perception of harmony is a peculiar faculty of the ear. Paradoxical though it may appear, mere air vibrations do not become sound until they fall on a hearing eat. A sufficiently strong air vibration may even be felt by the skin, but it does not become sound. A deaf mute might feel it, but not hear it. Apart from this, the undulations have to take place within certain limits of rapidity, to be perceived as sound even by a good ear. These limits, for mere hearing, have been found to be 32 and 10,000 undulations per second. But harmony ceases after about 5,000. This depends on the compass of the particular instrument producing a note. The piano goes up to 4224, the violin 2640 and the normal human voice jars if it reaches beyond 1,200. The deepest C, called the contra-C in the piano goes down to 33. But our musicians realise that such deep tones become dull and indistinguishable drones and seldom pass an octave below the basic. Low notes are not gathered up as whole tones by the ear.

The undulations so limited go down the auditory canal, to the petrous bone out of which is hollowed out, the inner ear. This cavity accommodates the cochlea or snail-shell, a peculiar organ of hearing divided into three longitudinal sections. In the middle sections are the extraordinary formations of sound perception, the rods of corti. These microscopically tiny plates are arranged like the keys of a musical instrument and tuned to a particular modulation. The principle of resonance will show us how a particular undulation is picked up by a rod tuned to it. Hence the sensations are transmitted by the auditory nerves, to the centre of interpretation, the brain.

A harmonic impulse, single or composite, is seldom pure in a physical sense. It is made up of undulations of different rapidities, the individual components being pure no doubt. Experience and experiment show that the ear not only perceives, but also analyses the air waves into their elementary forms. Both the analysis of the “wave confusions” and their subsequent synthesis in the brain with a view to re-interpretation of harmony, have bewildered both physicists and physiologists. At any rate, this latter faculty seems to be a rare acquisition of man and the higher animals.

This takes us to the problem of harmonic perception by the brain. The material available at the disposal of the scientist is so subjective, that a lot of speculation has set in on the analysis of the mode of actual perception. To describe poetry and music, “two of the noblest arts,” as revealing “little to the sense and suggesting much for imagination” or to say that they “transcend all physical perceptions and take a glimpse of the unknown”⁴ is probably only an escape from the inexplicable. Perhaps this cannot be helped, or perhaps our scientific methods are not quite perfect for tackling the more of action of the little grey cells. Suffice it to say that art perceptions the result of mental training, something which cannot be written down in the form of a mathematical equation. As Tagore would have it “those of the audience who are appreciative are content to perfect the song in their own minds by the force of their own feeling.” ⁵ In saying this, we are not forgetting the classic contributions of Huxley and Darwin to the understanding of mental functioning. Nor do we discredit Huxley’s division of the mind into the “receptor” or sense receiver, the “effector” or motive organiser, and “the adjustor” or the central nervous system.⁶ We only express the fear that the mere physiological functioning of these regions alone might not result in harmony perception. The co-relation of physical facts to biological theories is fraught with dangerous speculation and we are not anxious to step into an unconscious error.

It is a matter of common observation that sounding bodies are in a state of vibration. Physics has established that if these vibrations are within the limits of rapidity referred to above, they will generate sound. For this sound to become a musical tone the impulses have to recur with perfect regularity and in precisely equal times. Irregular agitation leads to noise, like harmonious blending of colour leading to multifarious shades and a haphazard mixture becoming dark. Such regular impulses have two very distinct properties depending on two physical facts, the rapidity of the vibrations and the extent of vibrations. The former which is called “frequency” of undulation, controls the pitch of the note and the latter called “amplitude,” controls loudness. A veena string of constant length, plucked a little more vigorously, might produce a “louder” note but not one of different “pitch.” Pitch depends solely on the number of vibrations per second, whether the note is produced by the vibrating strings of the violin or veena, or the vocal chords of the larynx or the trembling lips of the trumpeter.

Quite apart from the pitch and loudness of a musical tone, there is the third characteristic quality which enables us to distinguish a tone equally high and equally loud, but produced by different musical instruments. We shall presently refer to the physical causes of varying musical quality, but it is here necessary to state that this is controlled by the nature of the vibrating mechanism (the string or plate or air colunm that vibrates), the nature of the exciting mechanism (the finger of the player or his bow or the hammer), and the general build of the musical instrument. There is, however, no doubt about the fact that these variations affect, not the quality of the individual notes uttered but the harmony of their blending. The study of musical quality was itself therefore called “harmonic analysis” by the pioneer in the field, Von Helmholtz.

Authorities on music hold that the eastern systems of music are of the melodic type and the western harmonic, distinctly different from each other.⁷ Actually, the physicist is compelled to argue that the two terms are much misused, from his point of view. Harmony and melody are so much allied that it is here necessary to state what they refer to from a purely physical analysis. When a single tone is emitted, its melody is controlled by other tones attendant on it and hence melody is what we might call “mono-tone harmony.” When an individual singer utters a series of tones in a ‘strain’ of music, we have one type of “multi- tone harmony,” leading thereon to the development of scales of music. When a number of singers perform simultaneously, we have the second type of multi-tone harmony, popularly called orchestral harmony or chorus harmony.

A single musical tone is always accompanied by a series of “overtones” weak in themselves, but nevertheless playing a very important part in determining the quality of the tone. Helmholtz, as a result of very laborius work, established that the nature, number, and relative intensities of these overtones determine musical quality.

By “nature” is meant the relationship between the overtone frequency and the fundamental or tone-frequency. Where the overtones happen to be exact multiples of the fundamental, they are called “harmonies” and the tone acquires musical harmony. In pronouncing vowels, or in what is termed articulation of speech, these overtones have no integral relation to the fundamental. We are not concerned with this, but with the problem of synthesis of harmonies with the fundamental, leading to melody. That the ear has the capacity to analyse and absorb even the weak harmonies has been established beyond any possible doubt.

A simple illustration might be relevant. Let us imagine a serene lake in which there is an incessant undulation due to the draughts of air. A bird suddenly pecking at a fish or a stone dropped in, might set up other undulations so that an observer would see a complicated wave- form floating along. But on the surface of water, the different individual waves travel with different velocities, so that in a short while we might observe the simpler vibrations left behind. In air, however, the undulations travel with the same velocity and the ear has to receive the waveform as a whole. It is the nature of such waveforms that produces the difference in harmonic blending. The more rounded off and smooth the resultant waveform, the softer is the resultant tone; the more angular or jerky, the harsher is the quality. That probably is a very unscientific manner of putting it. But that illustrates what we understand by monotone harmony. Other investigations have ventured to suggest that not merely harmonics, but overtones bearing ratios 5/4, 3/2, 4/3, etc., to the fundamental produce very harmonious combinations. They draw their inferences from multi-tone harmony, where these intervals are found to produce great concord.

The number of harmonics present controls the richness of the tone. If the nagaswaram music is rich as compared to the flute music, it is not because of the relative strength of its notes, but because of the full retinue of harmonics following them. No one would assert that melody in absent in flute music, for richness is not melody. The former depends on the number of harmonics present and the latter on their relative strength. For example, let us assume that the intensity associated with all the harmonics is only one per cent. If the greater part of this one per cent is associated with a harmonic seven times the fundamental, it is found that melody is lacking. But if the reinforced harmonic is five times, there is very good melody. All the while there would be richness, as in a harsh but rich violin note. The statements about the relative strength of the melodious and un-melodious harmonics should not be taken as theoretically sound. They are facts of experimental observation with quite a volume of corroborative evidence.

The overtones show themselves out more easily when they are not in tune with the fundamental, than when they are in harmony, like our becoming aware of an organism when it goes ill. The perfect artist is one who suppresses these inharmonious and out-of-tune overtones and reinforces the others. “The art of a bell-founder consists precisely in giving bells such a form that the deeper and stronger particles shall be in harmony with the fundamental tone, as otherwise the bell would be unmusical, tinkling like a kettles.⁸

The problem of multi-tone harmony would become simple once we analyse the physical nature of the combination of two tones. When two tones are of slightly varying rapidity of undulations, a remalleable phenomenon takes place. The elevations of one of the undulations might superimpose on the elevations of the other and then we hear the sum total effect. Very soon, one of the waves outstripping the other, the elevations of one might coincide with the depressions of the other. We have momentarily no sound at all. Thus we hear a sort of waxing and waning of sound which the physicist calls “beats”.

Beats are generally very unpleasant to experience, unless the rapidity of waxings and wanings itself gives the character of a “beat-tone” to the combination like flickering light which irritates the eye unless the flicker is very quick and by persistence of vision we feel to observe it. This leads us to the theory of concord and discord between musical tones.

Two tones of different pitches may mutually disturb each other and split up into very disagreeable beats. When the frequency ratio between the two tones is a semitone (about 1.07), in the normal scale, between 20 and 40 beats result, and the sensation is very harsh. When the difference is a whole tone, the roughness is less and when it is a third, it is altogether absent. Even when the fundamentals are very widely separated and as such may not beat, their overtones may interfere and produce beats. For example, if two tones bearing ratio 2:3 are sounded together, there is one harmonic in each which is exactly six times the fundamental. These two harmonics will therefore be in great concord. If one of the basic tones be slightly out of tune, beats are produced between the harmonics and discord sets in. Actually the tones referred to above are the shadja and panchama of the scale of music. We know by experience how harsh it would be if these two important tones are slightly out of tune. The shadja and madhyama have a ratio 3:4 and will have concord of the twelfth harmonic. It is possible to work on like this and with a given tone as fundamental, a precisely determinate number of other degrees of tones which can be sounded at the same time with it without producing any want of uniformity, could be obtained. Such a sequence would be called a musical scale. In the very nature of things, a musical scale could only employ musical intervals or frequency ratios which are best calculated to avoid discordant beats of any of the harmonics.

It becomes imperative to a single singer performing a strain of music, to employ only the intervals of any particular scale of music he chooses to exhibit. The ear can never forget an earlier tone in a sequence, so that even if at any instant, he emits only a single tone, the ear cannot fail to recognise its relationship to its predecessor. We in India, employ a drone (Tambura) to aid the ear in the process of recognition of the fundamental, fifth, and octave. Sruti, to us, is the life and soul of harmony as we understand.”

In an orchestra also, the need for avoiding discordant beats is very paramount. The fundamentals of the different members of the orchestra or choir have got to bear concordant intervals to one another and at no time could any two notes uttered produce discord. Harmonious blending is disturbed if the fundamentals do not agree or any of the other tones of the scales do not agree, or even if their overtones do not agree. The problem gets even more complicated when we realise that the interference of two loud waves results in the production of what are called “combinational tones”. These tones again have subjective existence and have frequencies, the sum and difference of the component tones. The combinational tones are capable of producing beats with the harmonics of the individual tones. For example, if three tones, c, e, g, having frequency ratios 4:5:6 are sounded together, they produce a combinational tone c which does not beat with any of the tones or their harmonics. But if the tones are not thus exactly tuned, this combinational tone beats and produces disharmony. Only experience could really help us in determining the good companions in this kind of multi-tone building.

Harmony is thus a physical process capable of a physical analysis and comprehension. But while we are able to determine the “causes” of concord and discord, the actual “feeling” of compatibility or incompatibility is a matter of intellectual experience. “For the attainment of the higher beauty which appeals to the intellect, harmony and inharmony are only the means, although essential and powerful means.”⁹ Reluctantly we will be compelled to admit that however accurate our analysis of pleasurable sensations, they suffer from the limitations of the scientific method and its utter incapacity to describe an emotion. But at the same time what matters is what we aim at, and when achievement comes “it is no more emotionless than it must have been to Dalton when he reduced the untidy pile of facts about chemical composition to the law of constant proportions and atomic theory; or to Mendel when he swept away a whole rubbish heap of nonsense about heredity and replaced it by his simple notion of the heredity factor.” ¹⁰

¹ Saint Thyagayya’s Kalyanavasanta Kirtana “Nadaloludai,”–a broad rendering.
² Helmhotz: Address at Bonn (1857).
³ Helmholtz:Ibid.
⁴ Quoted by K. Chandrasekharan, “An approach to Indian Art.” Page 7.
⁵ Ibid, page 12.
⁶ For further details see “Essays of A biologist”: Julian Huxley, page 30.
⁷ Prof. P. Sambamurti, South Indian Music Series, Book I, page 3.
⁸ Helmholtz: Address at Bonn(1857).
⁹ Helmholtz: Address at Bonn(1857).
¹⁰ Waddington: “The Scientific Attitude.” Page 47.