Harmonic Division and Sims' Scale
This diagram represents the extent of Sims’ 24 note scale that is generated by pure harmonic division. You can easily see that it is really just a matter of showing the new tones that show up in each new octave up the overtone series. Take the section that has 9:8,11:8,13:8,15:8 and look at the doublings of the previously attained notes as they fit in the spaces. 5x2=10, (3x2)x2=12, 7x2=14. This shows that, in this octave, all of the harmonics are present. 10,12, and 14 are old, 9,11,13, and 15 are new. This process continues for every new octave up the overtone series. 9 now doubles(that is, rises an octave) to 18 to fill the gap of the two new tones 17 and 19. So how is this harmonic division? Let’s look at 5:4 and 11:8. First we’ll put them in the same octave by doubling the 5:4 so it becomes 10:8. The 4 and 8s represent the fundamental, while the 5(10) and 11 represent the harmonic number. So we have 10 and 11 interacting with each other. If we look at the summation tone(as described by Sims) we get 10+11=21, which you will notice is the tone in the next octave between the 5 and 11. Another way to see it is: (10x2)=20 and (11x2)=22. The 21 which fills the gap is the harmonic mean of 20 and 22. By harmonic mean, we mean that numerically each side is equivalent (1) but in terms of space, they are different(the bottom is larger). Such as the interval of the fifth(3) is numerically right between 2 and 4, but we know that a fifth is certainly larger than a fourth. This pattern of division that characterizes the progression of the overtone series is very similar to the division pattern of the golden ratio, expressed in a continued fraction as: 1 plus 1 over 1 plus 1 over 1 plus 1....
How does harmonic division relate to resultant tones? Let’s say now we want to use 5:4 and 9:8 to give us 7:4. First the resultant tone method. 5+9=14 summation and 9-5=4 difference tone. The difference tone fills its name in more than one way. In the harmonic division method, (2x5=10, 2x9=18) the difference tone is the difference from each of these extremities to the harmonic mean or summation tone of 14. This approach, it seems to me, works for all the examples I could think of, including tones that are not in clear harmonic relation to one another. For example: 445 Hz and 764 Hz produce resultants of 319Hz and 1209Hz. (445x2=890, 764x2=1528). 890Hz and 1528Hz are both 319Hz (the difference) away from 1209Hz, the summation and harmonic mean.
To fill out the scale that Sims has used since 1970, we take the upper tetrachord divisions and superimpose them over the lower pentachord. This is clear even at glancing at his full scale (see below) because many of the notes that appear early on(25:24, 13:12, 7:6, 29:24) are ratios based on the 3rd harmonic. By doing this, he takes full advantage of the interval space generated by the upper tetrachord, the space that supplies the perfect evolutionary step for intervallic structure. In this way, he sidesteps the obvious problem of the Fibonacci sequence (being sent too quickly to the harmonic stratosphere) while still preserving and amplifying the characteristic growth structure of the Fibonacci and Golden Section. One of the aspects of the Fibonacci/Golden Section that I find very important and pervasive is the centrifugal force created by the smaller and smaller spiraling of the Fibonacci around the irrational Ratio. I feel that this principle is very much in effect in musical harmony, and that resultant/harmonic mean tones are a potent way to guide the listener’s perceptions into the vortex. The difference being, in a working musical system such as Sims’, there are many focal points that create many vortexes. For a visual on the diagram, look at the zigzag of 5:4, 9:8, 19:16, 37:32. This is the level that I feel the Fibonacci is active.
Sims’ 24 note scale (in ratios):
1:1 – 33:32 – (25:24) – 17:16 – (13:12) – 35:32 – 9:8 – 37:32 – (7:6) – 19:16 – (29:24) – 39:32 – 5:4 – 21:16 – 11:8 – 23:16 – 3:2 – 25:16 – 13:8 – 27:16 – 7:4 – 29:16 – 15:8 – 31:16 – 2:1
Note: The ratios in ( ) are the ones produced from the upper tetrachord of the octave, but superimposed over the bottom to stabilize upper tones with a fifth below and to flesh out the scale with more simple relationships than could be attained purely through harmonic division. You can tell that they all relate to the 3rd harmonic because all of the denominators are doublings of 3. Because all of the ratios ultimately relate back to the fundamental, what this means is ‘make an interval above the fundamental that is the same size as the interval of the 24 harmonic to the 25’.