The way upward and downward are one and the same.
After the previous theoretical detour, let’s return to the practical consideration of the architecture of a vibrating string, armed with a coloured Lambdoma.
The colour-coded diagram in figure 27 shows that a string behaves in a similar harmonic fashion to the baritone saxophone air column shown in figure 6. It vibrates in various distinct modes in accordance with the overtone series. If the string is lightly touched at a nodal point the respective overtone is produced. So, if the contact is at a third of the total length, then the third harmonic sounds (coded yellow in figure 27). The further along the series you progress, the weaker the tone.
By itself, one vibrating open string obviously has limited musical potential; but if we alter its sounding length then the proportional architecture of Harmonicism emerges.
Much of what follows is a good deal easier to understand if the proportions can be heard, so I recommend using a monochord. If you don’t have one conveniently to hand, it’s worthwhile making one; or better still a two string harmonic canon (figure 29), which is a double monochord, so allowing different tones to be compared. There’s a wide margin of error in constructing such an instrument, so don’t be put off by woodworking incompetence. The one illustrated was adapted from an old drawer.
Figure 29 below shows how the overtone and undertone series are relatable to string length. If a moveable bridge is placed under the string at points marking the 1/2 through to the 1/8 of the whole length, as in figure 29(a), then the inversely proportional tone sounds when the length to the right of the bridge is plucked . So placing the bridge at 1/3 of the length, produces the ratio 3/1, the third harmonic (coded yellow). The vesicas () in the diagram highlight this process and the resultant overtones. The harmonic progression (1/1, 1/2, 1/3 etc.) produces the arithmetic progression (1/1, 2/1, 3/1 etc.).
Conversely, in figure 29(b), the undertone series arises from a hypothetical extension of the open string length. Thus, if the string length is multiplied three times (3/1), then the resultant tone will be the third undertone (1/3). Here the arithmetic progression engenders the harmonic.
Now the process of string division in figure 29(a) also gives rise to the supernumerary series (the vertical progression within the Lambdoma shown in figure 12). This is brought about by plucking the string segment to the left of the bridge, as opposed to the right. Figure 30 demonstrates this process of increasingly smaller proportional steps towards unity (1/1).
So far I have talked about consonance in terms of the relationship between a tone (or string length) and unity (1/1), but obviously if we want a more diverse musical resource, we must examine the interrelationships between these consonant string divisions.
Figure 31 represents the nodal points and resultant tonal consonances of half a string length since acoustically, one half of a string is an exact reflection of the other.
With the bridge placed at the midpoint of the string the two segments are identical, both sounding an octave above unity (2/1). When the bridge is moved a third of the whole length (from the right), the segments produce 3/2 and 3/1. Having considered the relationship of both these ratios with unity (coloured type in figure 31), let’s now examine their mutual relationship (background colour in figure 31). The denominators of these ratios tell us that 3/1 is one octave above 3/2. So in modern parlance, if the whole string length sounds a c, then two thirds will sound a fifth above (g) while one third will sound an octave and a fifth above (g1).
If we now move the bridge leftwards to the point which is 3/5s from the left and 2/5s from the right of the total length, the tones produced are 5/3 and 5/2 respectively. Following the same reasoning as above, the relationship between the two will be 3/2, what we nowadays call a fifth. Along with the fundamental c, we now have the notes a (major sixth above) and e1 (an octave and a major third above). Roughly, that is, because the sixth will be appreciably flatter than the equally tempered interval, while the third will be sharper. Indeed, from this point on, thinking in terms of Equal Temperament becomes a hindrance rather than an aid, since the method of distortion which facilitates cyclic twelve-tonality is at odds with this proportional approach to harmony.
Music based on natural tuning has in the past been seen as rather limited in terms of harmonic movement. Intricate subtleties of harmonic progression which are possible in equally-tempered music are not a feature of, say, overtone chanting which is fundamentally grounded and harmonically static. This need not be the case. One of the main aims of this project is to develop a music system founded on natural proportion which possesses literally infinite harmonic nuance.
Figure 32 (click to enlarge and play)
The process by which three note chords where produced in figure 31 can be expanded to produce the thicket of ratios in figure 32. Here, the split string chords are shown to the sixteen limit and are graded by consonance. All are interrelated through their relationship with unity 1/1.
While the right -hand side of the diagram is clearly an extension of figure 31, the left-hand side needs some explanation. Physically, one string half exactly reflects the other, but we can generate a hypothetical balancing group of concords by reciprocal reflection, in the same manner as ratios in the Lambdoma can be inverted by reflecting them across the central unity. For example, the right-hand ascending concord 3/2 3/1, produced by 2/3 and 1/3 of the string, is balanced by the left-hand descending concord 1/3 2/3, which is produced by sounding lengths 3/1 and 3/2 of the open string.
With such potential inherent within one vibrating string, any accusations of harmonic limitation are certainly unfounded. Indeed, the opposite is true. For, as a system which tends radially towards infinity, the scope for harmonic exploration here is boundless, particularly when the investigation is broadened by examining instruments with more than one string.
However, to do so we need suitable instruments.