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833 cents scale
The 833 cents scale is a musical tuning and scale proposed by Heinz Bohlen based on combination tones, an interval of 833.09 cents, and, coincidentally, the Fibonacci sequence.〔"(An 833 Cents Scale: An experiment on harmony )", ''Huygens-Fokker.org''.〕 The golden ratio is 1.6180339..., which as a musical interval is 833.09 cents (). In the 833 cents scale this interval is taken as an alternative to the octave as the interval of repetition,〔"(833 Cent Golden Scale (Bohlen) )", ''Xenharmonic Wiki''.〕 however the golden ratio is not regarded as an equivalent interval (notes 833.09 cents apart are not "the same" in the 833 cents scale the way notes 1200 cents apart are in traditional tunings).
==Derivation==
Starting with any interval, take the interval produced by the highest original tone and the closest combination tone. Then do the same for that interval. These intervals "converge to a value close to 833 cents. That means nothing else than that for instance for an interval of 144:89 (833.11 cents) both the summation and the difference tone appear...again 833 cents distance from this interval".〔
For example, 220 Hz and 220 Hz (unison) produce combination tones at 0 and 440 Hz. 440 Hz is an octave above 220 Hz. 220 Hz and 440 Hz produce combination tones at 220 Hz and 660 Hz. 660 Hz is a perfect fifth (3:2) above 440 Hz, and produce combination tones at 220 Hz and 1,100 Hz. 1,100 Hz is a major sixth (5:3) above 660 Hz, and produce combination tones at 440 Hz and 1,760 Hz. 1100 Hz and 1760 Hz are a minor sixth (8:5), and so on. "It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent."〔
Once the interval of 833.09 cents is determined, a stack of them is produced:
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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