833 Cent Golden Scale (Bohlen)
Heinz Bohlen's 833 Cents Scale is a combination-tone-based scale which repeats at the interval of the golden ratio (φ = 1.618...).
Theory
There are seven unequal steps in the Golden Scale, for starters:
- 99.27 - 136.50 - 131.14 - 99.27 - 131.14 - 136.50 - 99.27
Which means these are the cent-intervals in the scale before it repeats:
- 99.27 - 235.77 - 366.91 - 466.18 - 597.32 - 733.82 - 833.09
833.09 is the cents value of the golden ratio, on which the scale is based and at which point the scale step pattern repeats. Octaves do occur in this scale but only incidentally; many octaves are out of tune, and the scale is not an octave-repeating scale.
The scale is designed to take advantage of the naturally occurring mathematical concept of combination tones in order to create a whole different breed of consonant harmony. The interval of the Golden Ratio has the peculiar quality of combining harmoniously with itself when stacked. Consider the Fibonacci Sequence, and say we're starting on A = 55 Hz. You have 55, 89, 144, 233, 377, 610, 987, 1597 Hz and so on. These frequencies are all related intrinsically by combination, and the ratios of each adjacent pitch are extremely close to the Golden Ratio. So likewise, when any number of Golden intervals are stacked you get a stable, sonorous sound which could be considered "consonant." For this reason the scale repeats at the GR and not the octave; the harmony is built around the GR and its properties. The scale is designed so that no matter what step of the scale you're on, if you go up seven steps you'll arrive at a perfect GR.
The scale has its own harmonious triads, based on three crucial intervals:
- a. the Golden Ratio or GR, 833.09 cents
- b. the Inversion of the GR, or GRI, 366.91 cents
- c. the difference of these two, or GRD, 466.18 cents
A Root - GRI - GR chord has a more "minor" sound, whereas a Root - GRD - GR chord has a more "major" sound.
A combination of these is the GRID chord, with a dissonance in the middle, which uses inversion to compress a 4-note GR stack into the range of a single GR interval.
The octave or pseudo-octave occurs at every tenth step of the scale. Perfect fifths appear sometimes too; if you know the scale well enough, you can use traditional harmony within its boundaries. Bohlen describes the variety of intervals that happen at different points in the scale, variable depending on which scale degree you're starting from.
36edo provides suitable tempered approximations of the intervals in the Golden Scale. It afford us the ability to compose in the GR scale and/or the traditional Western tonality.
Again, the best authority on the scale is Bohlen himself: http://www.huygens-fokker.org/bpsite/
But there is much still to be discovered with this tuning system. Combination tone harmony opens some interesting doors. The tuning is of course in need of its own theory, something I've been working on for the past year and a half. If you have any interest in the scale, please contact me at [email protected] - thanks!
-Dave G.
Music
- Gr Symphony A by David Guillot (Listen on SoundCloud)
- 833c by Bohlen scale improv by Yin Bell (Listen on SoundCloud)
External links
- An 833 Cents Scale - An experiment on harmony (Bohlen's original article)