Music and mathematics

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On music, mathematics and tuning

If you tension a suitable string between two bridging points, perhaps connected to an object that acts as a resonator or a soundboard, you will have created a simple chordophone. Striking, plucking, or bowing the tensioned string, can produce a musical note. Such a simple musical instrument can be further developed by providing some means for stopping the string, or dividing its length.

Pythagoras is said to have experimented in this way, and to have discovered that dividing the length a string in the ratio 3:2, so that one part is twice as long as the other, produces two different notes, one for each part of the string. The part of the string that is twice as long as the other part, sounds a note one octave below the shorter part. Stopping a string (as, for example, on a guitar fret) at half its speaking length, produces a note one octave higher than the open string. Similarly, if a string is stopped so that the stopped length is 2/3 of the open length, the stopped note will be a perfect fifth above the open string. If the ratio is 3/4, the musical interval will be a perfect fourth.

Pythagoras is said to have discovered this around 500 BC, and whilst we might today regard this as a "scientific discovery", the evidence points to an alleged Pythagorean view of these relationships between musical intervals and simple ratios, in which Number was mystically or at least philosophically linked to musical intervals and to harmony. The Pythagoreans are reported to have engaged in what we would perhaps today call numerology, an obsession, so to speak, with numbers, or with the idea of Number, but not what we would now regard as a proper science. The Pythagoreans certainly had no rational explanation (in the modern sense) for the relationships.

I would not want to give the impression from this, that Pythagoreanism, whatever that was for the original sect, was nothing more than pseudo-science. Firstly, Pythagoras is accredited with legitimate mathematical discovery, such as the well known Pythagoras' theorem for right-angled triangles. Also, the relationships of the simple ratios to musical intervals on the tensioned string, are to a high degree of practical accuracy, a "scientifically correct" approximate model, for most strings. Secondly, and arguably more importantly, whilst Pythagoras has often been viewed as an "early scientists" by those modern interpreters interested in the development of science, much of content of the reports of what would dismissed as pseudo-science, is actually important in that it illustrates the gulf between the modern "world view" and the ancient. There are, equally, attestations to the nature of Pythagoreanism in which the thrust of it is overtly mystical and metaphysical. Combined with these attestations, the "scientific" discoveries attributed to Pythagoras can be viewed almost as an epiphenomenon of the philosophy, mysticism or quasi-religious belief system in which they were engaging. Unless we ourselves are engaging in scientism, then there may be as much importance in the latter, at the very least from the point of view of understanding cultures and beliefs.

Whether or not Pythagoras was actually the first to discover these relationships, we do at least know today (from extant cuneiform tablets and the work of Anne Kilmer,) that very much earlier than Pythagoras, in the oldest civilisations in ancient Mesopotamia, proper consideration was being given to issues of musical intervals and tuning. It is of course natural laws themselves (the laws of physics or acoustics), which, being independent of the Age or the culture, both the civilisations of ancient Mesopotamia and ancient Greece were confronting when experimenting with tensioned strings, or when faced with the task of constructing musical scales (or modes) or tuning musical instruments. It is these same laws that we confront today, when attempting the same tasks.

What does change with Age or culture, is the understanding of natural laws, and their representation in concepts and ideas. The beginning of the recognition of a connection between what we would now call mathematics and music, took place in an Age earlier than our own, so we should expect the earlier ideas and concepts relating to music and mathematics, tuning, or "harmony", to reflect something of the pre-scientific "world view" or "world views" associated with that Age. Ancient Greece appears to have been somewhat of a boiling pot of different "world views", and the "world view" of the original Pythagoreans is in any case not revealed in any extant documentation other than as hearsay. Nevertheless, the idea of a mystical or Divine connection between the ratios and the intervals, whether truly originating from the Pythagoreans, or whether inherited and adapted by them from cultures further East, is an idea that enjoyed a long and influential tradition in the West. It is behind Plato's thinking in Timeaus, and is connected with his depiction of the Myth of Er in the Republic. It is what lies behind Ptolemaic astronomy and the long tradition of the harmony of the spheres.

Modern Western civilisation is no exception from the connection of culture and the understanding of natural laws. We may think that we have now successfully separated what we would call "scientific understanding" from cultural influence, owing to principles of "scientific objectivity", and certainly to have separated science from "cultural myth", but there are nonetheless still certain "grey areas" that exist between scientific fact, and what could be argued to be part of cultural belief. Nevertheless, the separation of our understanding of natural laws and cultural (or subcultural) beliefs is to say the least, more pronounced now than at any time prior to the so-called "scientific revolution" of the seventeenth century. Today, as far as music and mathematics is concerned, music is mostly a cultural activity, whilst mathematics is mostly a scientific pursuit. In this context, to talk about a connection between mathematics and music is to speak of a connection between the science of mathematics and human culture.

This is often overlooked. The "connection between mathematics and music", where by "music" we mean recognisable musical intervals on a tensioned string, is relatively simple to see. The connection between mathematics and music as a cultural activity, will exist, because there is mathematics in everything, but it is likely to get immensely complex, and no more especially "significant" than the connection between mathematics and anything else.

On the one hand, mathematical structure in music, in the part of music that is our cultural and artistic creation, may indeed be present, but it is not really a simple, salient demonstration of either natural laws or Divine enigma. Mathematical structure in the raw medium of music, on the other hand, that is, in musical sound itself, can indeed be a salient display of natural laws, or even of Divine enigma, if you prefer. This is where the simple ratios and their connection with perceived harmony arises.

The extension of this connection, into music itself, i.e. music as a cultural activity, involves the construction of musical scales, and the use of harmony in conjunction with them. It is here that the connection of mathematics and music becomes important. Where, by "music", we refer to an activity in which there is no particular attention paid to the precise tuning of scales, and the precise tuning of musical intervals, then the connection of music and mathematics is itself not something demanding any special attention above a relatively elementary level, in which it may be used, perhaps, as an educational tool.

The idea of a significant philosophical or actual connection between music and mathematics can therefore be misleading. The salient connection between music and mathematics arises in the first instance in the careful construction of scales (or modes, or ragas, etc.), and the use of harmony in conjunction with scales. There can be a great deal of mathematics in music without scales, for example, in textural synthesis, but this again arises either in the musical design, which in effect is a question of culture, or in the acoustical structure of the sound, which exists quite independently of its use in music. By far the larger part of music as a cultural activity, does use scales and/or harmony.

On scales, harmony, tone, pitch (and piano tuning)

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