This is the second part of my musings on connections between music and color which started here.
I left off with the arithmetic progression. Now for the second of the three: Geometric. Again, this is a three-term proportion where a > b > c. In the language of ratios, a and c are the extremes and b is the mean. As with an arithmetic progression, begin with two of the differences in the numbers: a—b and b—c. In a geometric or harmonic progression, these differences are to each other in the same way as one of these numbers is to one of the other numbers, not as one of the numbers is to itself as in an arithmetic proportion.
The geometric proportion is expressed as:
a—b and b—c::a:b
The solution for a geometric proportion where the mean term is b is b2 = ac or b = √ac. Using this formula for the extremes of 4 and 16, the mean is 8. The geometric progression is 4, 8, 16. This proportion is also expressed as the golden mean:
If this were mapped to a color wheel, this could be two primaries (the two extremes) and their secondary (the mean): red is to orange as orange is to yellow.
With the Fourths and Fifths series, however, I am playing with the idea of mapping the musical proportion to color. The progression for the musical proportion, based on an octave, is 1 (the fundamental), 4/3 (a fourth), 3/2 (a fifth), 2 (octave above 1). That idea will be the third post in this series.