Not only your form but all your members will charm
If they are composed of harmonious numbers.
Charm resides in the whole compound of the members,
Harmony comes from the viewing of the whole.
The relation of music to geometry was not such a strange idea to our medieval forebears. The two disciplines, along with arithmetic and astronomy formed the quadrivium, the higher division of the seven liberal arts in medieval learning, and underpinning all four was the pervasive primacy of number.
To the modern mind, while number is clearly intrinsic to geometry, its relevance to the structuring and aesthetic quality of music seems dryly obscure. However, if we strip away the layers of enculturation and trace music to its vibrational core, the essence of musical consonance, what we find pleasing to the ear, lies in proportional relationships of number.
I have developed this interest in the physical and psychological aspects of musical consonance through playing woodwind instruments for the last 30 years and more recently by making a variety of both string and wind instruments. In doing so I have learnt that there are inherent symmetries in both the physical production and sensory perception of musical tone, which have become distorted in the development of modern musical theory.
This is not the place for a lengthy excursion into the history of intonation, which is well covered elsewhere, but a few observations will hopefully clarify how and why my approach to musical harmony deviates from the well-beaten track.
Music education leads us blindfold into the eighteenth century musical world of Bach, with the keyboard instrument firmly ensconced as the harmonic law-giver. We practise scales and undergo aural and keyboard harmony tests based exclusively on the architecture of the piano, without ever really questioning why certain notes sound more consonant when played together than others. It is all very well learning that c and g is a purer interval than c and f# because one is a fifth and the other an augmented fourth, but such terminology itself betrays our complete reliance on the twelve equal steps of a keyboard octave, the musical construct known as Equal Temperament. We develop little sense of music’s underlying acoustic foundations. Furthermore, with the blanket proliferation of music written on or for equally tempered instruments such as the piano and the fretted guitar over the past three hundred years, we tend to lose sight of the fact that the structuring of Equal Temperament has harmonic inconsistencies, being an ingenious result of fudged compromise.
The problem started with Pythagoras, when music theory was in its formative years. While he is hailed as the first person to explore the relationship between the aesthetic quality of music and the simple logic of arithmetic, nevertheless, the manner in which he proceeded musically from first principles engendered many of the theoretical disputes which have plagued music over the intervening centuries.
First, Pythagoras stated that the relative consonance of two notes depends upon on the relative simplicity of the ratio of the string lengths which produce them. This principle forms the bedrock upon which all tuning systems are founded. So the octave was considered consonant since it is produced by plucking 1/2 of the string length, as was the fifth, produced by sounding 2/3 of the whole string. Here Pythagoras rationally drew a line, denying the consonance of less simple string ratios such as 3/4 and 4/5 which produce the natural fourth and major third respectively. This aesthetic limitation shaped both the way he developed his own musical theory and by implication the evolution of future Western music.
Obviously two notes are a rather limited vocabulary for musical expression, and so the next issue to be addressed was how to string more than two notes together consonantly. Since music in ancient Greece music was melodic rather than harmonic in character, so the consonance of consecutive notes in a phrase was more important than that of notes played together in a chord. This led Pythagoras to generate further harmonious notes through a cyclic process; that is, he constructed a series of seven fifths which were seqentially consonant and then transposed the resultant notes so that they fell within the span of one octave. The scale which emerged from this process was the diatonic scale, the template for all Greek modes as well as for the mediaeval Ecclesiastical modes and the modern major scale.
While this tuning system works well for monophonic music, nevertheless, with the advent of polyphony in the 16th century serious flaws emerged. Through the influence of secular vocal music, natural thirds and sixths were now considered consonant as they were ideally suited to harmonize lead lines. This harmonic tendency, however, conflicted with diatonicism because the tuning of natural thirds and sixths sounded excruciating alongside thirds and sixths derived from the cycle of fifths. Equal Temperament was the culmination of various schemes adopted to remedy this impasse; a compromise between two grudging bedfellows: the cyclic, closed system of diatonicism as detailed above and the radial, open system of natural tuning which is termed Just Intonation.
The words of Andreas Werckmeister, quoted above, read well as an endorsement of Just Intonation in which notes are proportionally related. In fact, Werckmeister was instrumental in the introduction of Equal Temperament which only goes to show the relief the musical community must have felt at finally adopting a tuning system which, if not entirely harmonic, at least had a sensible internal logic, unlike some of its half-baked predecessors.
It is not my intention to dismiss three hundred years of Western music on the grounds of harmonic imprecision but rather to stress the validity of alternative approaches to structuring tonal space. Our hearing has a generous tolerance to imprecise tuning, quantizing it to the various nodes of consonance, which explains why Equal Temperament is musically viable. Moreover, as a culture we have been earwashed into hearing the twelve semitones of the piano octave as the exclusive musical reality. But, as Harry Partch wrote, a distinction should be made “between what the ear will accept and what the ear prefers, if given a choice” and surely it is worthwhile developing a music system with firm acoustic foundations, particularly if this system is to be used as a tool for cymatic enquiry. If I want to take advantage of the creative cross-fertilisation between musical harmony and physical patterning then it helps if they talk the same language.