Since no natural music/colour correlation is apparent, I have adopted a system of colour-coding in this project which is purely functional but utilises certain structural features of the visible spectrum.
All ratios of unity as well as their doublings (2/1, 4/1 etc.) and halvings (1/2, 1/4 etc.) are coded white. As these ratios are perceived as the most consonant so the lightness of the colour informs us of their relative lack of tension. The movement from light to dark, from white to black through the coloured spectrum, is an attempt to reflect the increasing dissonance of ratios as the Lambdoma radiates outwards, giving a general measure of tension.
The dominant consideration when assigning colours to the Lambdoma, should be to reflect the simple number relationships between certain musical ratios. In figure 18 we saw that subtle gradations of spectral hue clouded this issue due to their indistinctness. Here, in choosing a limited palette, the more consonant ratios are more easily identified while coding ratios with terms greater than 8 as black, adds a clarity to the figure.
Within the 8 limit, the colour scheme derives from bisecting the spectrum. Thus, on one side of the central white, the overtone series 1/1, 2/1, 3/1 etc. moves through yellow and orange to red within each octave. Similarly, the reciprocal undertone series 1/1, 1/2, 1/3 etc. moves through light and dark green to blue.
The assignment of the other colours within the top quadrant is based on numerical affinity allied with aesthetic considerations . The coding of the rows with a 7 identity works well - the overtone series based on 7 starts from blue and progressively darkens to purple as the tension of dissonance increases; likewise, the undertone series based on 7 starts from red and progressively darkens to deep brown for similar reasons. Some of the other overtone and undertone series are less successful since they do not clearly indicate the relationships within each series, each diagonal row.
Whatever its failings, this colour system serves its purpose as a tool for investigatiing and notating the principles of Harmonicism. It may be a rather leaky boat to set sail in, but it is seaworthy, and familiarity breeds a certain fondness for its idiosyncracy.
Figure 21 shows the lambdoma matrix mapped in three dimensions. 21a and 21c show proportional frequencies while 21b shows proportional wavelengths. Note the difference in nature of the linear arithmetic progression and the logarithmic harmonic progression.
Figure 22 is the decimal equivalent of the ratios in figure 20.
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In figure 23, 256 hertz is taken as unity 1/1. This frequency is often called mathematical C (middle C on the piano is 261.625 hertz).
Figure 24 shows the corresponding metrical wavelengths in air of the frequencies in figure 20 (the wavelength of 256 Hz is 1.34 metres).
Figure 25 shows the lambdoma matrix expanded from a 16 to a 32 limit.
Figure 26 shows the major consonances in figure 22 in the form of vectors.