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Harmonicism I

Music, “good” or “not good”, has only two ingredients that might be called God-given: the capacity of a body to vibrate and produce sound and the mechanism of the human ear that registers it. These two ingredients can be studied and analysed, but they cannot be changed; they are the comparative constants. All else is the art of music, which may also be studied and analysed, was created by man or is implicit in human acts and is therefore subject to the fiercest scrutiny - and ultimately to approval, indifference, or contempt. In other words, all else is subject to change.

Harry Partch (1901-1974)

What do I mean by Harmonicism and how can its basic principles of consonance be expanded as a practicable musical resource?

Let’s start by examining how a woodwind instrument physically responds when blown.

Simply put, a woodwind instrument consists of a tube with holes spaced along its length which can be opened and closed. To make a sound, an initial vibration is produced by exciting a reed (as in the case of oboes and saxophones) or by edgetones (in the case of flutes). The air column inside the tube then resonates in sympathy with this initial vibration. What note sounds is largely dependant upon the length of the air column, which can be shortened or lengthened by opening and closing toneholes sequentially. The shorter the column the higher the pitch of the resulting tone, for in acoustics a tone’s wavelength is inversely proportional to its frequency. From penny whistle to sub-contra bassoon, this is the guiding principle behind woodwind design, but the positioning of toneholes to create scales, and the resultant different lengths of air column, are predominantly cultural constructs.

The direction taken in woodwind instrument development has been towards an ever increasing sophistication in the superstructure of the keywork, enabling an ever smoother and faster manipulation of these resonating air columns; witness the evolution of flutes from the beautiful simplicity of the five holed shakuhachi to the technical ingenuity of the Boehm system used on the modern western flute. This in turn has encouraged a rather mechanistic approach to playing, where pressing certain keys produces certain equal-tempered pitches.

All this tends to overshadow the acoustic properties and possibilities inherent within the instrument. Pressing a key on a piano will play one note and no other, but any fingering pattern on a woodwind instrument can potentially produce a wealth of different pitches, depending on how it is blown. In its simplest form this principle allows a player to use the same fingering patterns when playing in higher octaves by a process of overblowing. For an air column in a flute, as in a brass instrument, can be incited to resonate in various, clearly defined modes of vibration, known as the overtone series.

The simplified figures below clarify what’s happening acoustically when most woodwind instruments are overblown.

Figure 3 represents the fundamental frequency of the air column when all holes on the body of the instrument are closed. The wavelength occupies the whole length of the tube. For the sake of argument let’s assume that the tone sounded is c.

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figure 3

In figure 4, the instrument is overblown an octave, to c1 . The air column vibrates in its second stable mode, producing the second harmonic of the overtone series. The frequency of the tone has been doubled as the wavelength has been halved.

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figure 4

As shown in figure 5, continuing the process of overblowing produces the third and the fourth harmonic, g1 and c2 respectively. The wavelength of the fourth harmonic is clearly a quarter the length of the fundamental in figure 1, just as the frequency of the tone is four times that of the fundamental, that is to say two octaves above.

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figure 5

While in theory the overtone series is infinite, for higher woodwind instruments their size and resultant pitch range makes such an approach to playing somewhat limited. However, for low instruments such as the baritone saxophone, the harmonic potential is more promising, particularly when this overtone approach is pursued in tandem with varying the length of the air column.

Thus, in figure 6, we can see a harmonic correlation between overtones and air column length. The same tone (g1) can be produced by two different sized air columns; the third harmonic of the whole length coincides with the second harmonic of an air column two thirds the length.

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figure 6

Conversely, in figure 7, the fourth harmonic of the whole (c2) is identical in pitch with the third harmonic of an air column three quarters the length.

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figure 7

So why make life difficult for myself, playing odd fingerings for notes that are more easily produced simply by changing the length of the tube?

Firstly because the notes are not in fact identical. Every tone produced by an acoustic instrument is not an isolated frequency but a rich mixture of different partials derived from the overtone series. What gives a tone its character, its timbre, is the different emphasis accorded to each partial. So in figure 5, the fourth harmonic tone will have a distinct character from that of the third harmonic tone since they are stressed partials deriving from different fundamentals. This can be demonstrated clearly on the baritone saxophone through its capacity to play multiphonics, or chords, by resonating in different vibrational modes simultaneously. Figure 8 represents a multiphonic of the whole tube to the fourth harmonic (c2).

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figure 8

The wavelength of this harmonic can also be recognised in figure 9, but is both visually and acoustically distinct in its relationship with its neighbours.

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figure 9

My second reason for pursuing this approach stems from a general attitude towards playing music, which as a physical process works best in cooperation with, rather than domination of, an instrument. As we have seen above, there is an inherent natural structuring in the way an instrument responds when played; a clear patterning which deserves fuller investigation - an exploration which as an instrumentalist I find more fulfilling than simply viewing an instrument as a tool to realise an extraneous musical system imposed upon it.

The irony of this is the fact that woodwind instruments, while intimating this natural Harmonicism, are not best suited for extending this investigation. Due to the limitations of its structure an ideal saxophone, for instance, is a physical impossibility: it can never be perfectly conical because no mouthpiece tapers to a point. Furthermore, wavelengths behave in a very complex fashion when holes are opened on the body of an instrument - the response of the tube in figure 7 will be different from that of a tube sawn off at the three quarter point since the overblown partials of a tube with tone holes become increasing inharmonic the shorter the air column becomes. Therefore, trying to coax a musicallly meaningful and varied harmonic language from woodwind instruments, as envisaged above, is like trying to play a hornpipe on a kettle. It can probably be done but there are more fruitful and responsive means.

Changing my line of enquiry from the complex acoustics of woodwind instruments to the relatively simple physics of a vibrating string offers a far more rewarding resource, both aurally and visually, for examining and developing a system of proportional Just Intonation which from now on I shall refer to as Harmonicism. Throughout history the manner in which a single string vibrates has been the major inspiration behind both music theory and cosmological concerns on a larger scale. The Pythagoreans laid the foundations of modern music by meditating on a monochord and it is surely more than coincidence that the Egyptian hieroglyph for the supreme creator, vesica, takes the form of a vesica, like a vibrating string.