All the things which can be known have number; for it is not possible that we can conceive anything or know anything without number.
The grammar of western music’s language is based upon the principles of diatonicism and chromaticism. Speaking the language of proportional ratios will more often than not summon up images in a musician’s mind of inscrutable equations on dusty blackboards. In truth, the underlying syntax of Harmonicism is a good deal simpler than that of Equal Temperament and makes more coherent sense, as will become apparent. This unothodox language is only daunting in its unfamiliarity.
Let’s turn our attention to the alphabet of this language. Its basic symbol is the ratio, a measure of the relative size of two values a/b. We are all conversant with divisions, like 2/3, but less familiar with multiples in the form of inverted divisions, like 3/2. However they both emanate from unity, 1/1, and diverge from it in mutally reflective patterns.
By way of introduction to the world of ratios, as well as the world as ratios, let’s consider the overtone or harmonic series, which we encountered above with reference to woodwind instruments.
The musical overtone series can be defined as a series of frequencies which are simple whole number multiples of a fundamental. The further along this series we proceed, the higher the pitch, the weaker the tone and the more distant the harmonic relationship with the fundamental. In mathematical terminology this is called the arithmetic progression, in which the difference between successive terms is constant.
Now if consonance is understood in terms of the relative numerical simplicity of the ratio, then it would seem logical that the inverse of the overtone series is of equal harmonic importance.
This is known as the undertone or subharmonic series, although ‘known’ is probably the wrong word since this term is another of those often passed over by dictionaries, whether musical or scientific. Acoustically, it is a descending sequence of tones emanating from unity, a precise reflection of overtones; mathematically, it is referred to as the harmonic progression. While most acousticians question its empirical existence, it can, nevertheless, be demonstrated acoustically, although it’s presence as a natural phenomenon is a good deal more veiled than the omnipresent overtone series. When the vibrating prong of a tuning fork is held against a sheet of paper, the paper buzzes in accordance with the structuring of the undertone series. What do these undertones sound like? This is best answered with reference to theoretical explanations of major and minor tonality which state that the major triad (c e g ) derives from the terms 4/1, 5/1, and 6/1 of the overtone series while the minor triad (f ab c ) originates from the corresponding reflected undertones, 1/6, 1/5 and 1/4 (Figure 11).
Whether or not one subscribes to this view, the minor tonality in the buzzing of the paper alerts us to the presence of undertones.
Evidence of undertones as a musical resource can be found in the music systems of older cultures, where the equidistant positioning of holes on simple wind instruments, like the aulos of ancient Greece, produce a descending scale derived from terms eight through to sixteen of the undertone series (figure 12). Whether this is due to conscious design or the accidental result of boring holes in this simple manner is debatable.
What I hope to make clear in what follows is the capacity of these two distinct progressions, the arithmetic and the harmonic (overtones and undertones), to generate the entire gamut of whole number ratios, when combined. This realization was first recorded by the Pythagoreans but the idea no doubt predates them. A figure similiar to the one below evidently appears in a footnote to a treatise by the neo-Pythagorean Nicomachus, entitled Introduction to Arithmetic, written in the second century A.D. It is referred to as a Lambdoid, after the Greek letter to which it owes its shape (λ), and in form is clearly a skeleton waiting to be fleshed out.
Before doing so, however, I want to draw attention to another structuring principle, inherent in both the arithmetic and harmonic progressions, which forms the main vertical arteries within the Lambdoid, and offers insight into the generative process at the root of Harmonicism.
So far in analysing the overtone series, we have considered each term in relation to unity (1/1). Let us now examine the relationship between successive terms.
Contrary to both the arithmetic and harmonic progressions, this series does not tend to infinity but to unity, in logarithmic steps. Thus, if these musical tones were played in relation to an implied fundamental, they would begin at an octave doubling of frequency and proceed downward in ever smaller steps towards, but in theory never reaching, unison. This is termed the supernumary series, and, like the overtone and undertone series, is characterized by a progressive movement away from consonance.
Figure 14 highlights the polarities within the matrix, the magnetism of both unity and infinity, and it is from the interplay of these polar attractions that the dynamic symmetry of the figure derives. The American architect, Gyorgy Doczi, has coined the term Dinergy to express this pattern-forming process of the union of opposites, integrating the terms Synergy (combined action) and Dichotomy (division into two opposed parts).
It was not until the nineteenth century that the Pythagorean Lambdoid re-emerged in its completed form (Figure 15). As the Lambdoma, it appears in the writings of Albert Von Thimus (1806-1878), a German polymathic researcher into ancient harmonic theories. It was subsequently taken as the cornerstone for the research of Hans Kayser and Rudolf Haase, who saw in it the manifestation of cosmic design, very much within the spirit of German idealism. Nowadays the Lambdoma is encountered, if at all, in the work of Barbara Hero, who single-mindedly champions it in the field of holistic music therapy.
How the Lambdoma can be developed as a practical creative resource in music - how this glorified multiplication and division table can be realised as an alternative structuring principal in musical harmony and rhythm, needs further investigation. This will involve reconsidering some commonly held assumptions about the nature of music and its practice, for the process of unlearning goes hand in hand with with that of learning.