Here is a a:link to Terumi Narushima's version
Here is a a:link to Warren Burt's scale version
Notes on a tuning for Satie’s VEXATIONS
An excerpt of the Microfest perfornance
Here is the final tuning
I decided to take a conservative approach to the problem. I thought I might rely on some historical model of music like Vexations that involves repeats over a long period of time. The one that came to mind was Pibroch, a style of bagpipe playing. The bagpipe tuning is quite impressive. It avoids simple ratios of most consonances leaving any repose in the melody to still be propelled forward by ratios of mild acoustical dissonance. The tuning also has some proportional triads. These type of triads have difference tones that support notes in the chord or scale which provides an overall unity, yet can depending on its complexity, can suspend the music in the air for undetermined periods of time without really resolving yet still reinforcing the tones being used. This is used to great effect with melodies repeated for hours.
Vexations with its preponderance of diminished triads made me look for the simplest proportional triad of this shape. The simplest one I could find was one of E. Wilson’s recurrent sequences A+C=F which he labeled as Meru 8.
This eventially will converge on a chain of minor thirds 306.75991106 cents in size but there is a fair bit of oscillating back and forth that gives sometimes for some nice variation.
If you are unfamiliar with these types of scales look here
To seed this formula, I took the Lucas Series that Satie was fond of. This series is like the Fibonacci series but starts with 1 and 3 instead of 1 and 2 and adds them together and continues this process with the answer and the last number added [1+3=4, 3+4=7, 4+7=11, etc.]. Using the 1-3-4-7-11 to seed the sequence that is then treated as harmonics, the series was continued until it converged to within a cent, and enough to place the 21 different pitches in a consistent order one finds notated in the score. Much to my surprise the first place where I could find this started on the 43,184th harmonic which effortlessly unfolded like a snail shell up to the 73,676,000th harmonic (odd harmonics happen in between to prevent a simpler reduction).
Here is the sequence
12 You can see here that at the beginning of the pattern you can have a lower number occur. this is why you have to take it so high
43184 THE SCALE STARTS HERE=and we take it out to 43 places to close out our cycles of
This puts the fundamental at a little over 3.5 kilometers in length which tempted me to proceed up high up the harmonic series until I reach a fundamental distance equal to the time sounds travel during the length of the performance, but like I said, I decided to take a conservative approach for the moment.
Even with this sequence we can see in the diagram below that besides the 43 tone scale we ended up using where the 11 unit is equal to our minor third generator. we could have also used all those scales from the 6 units of a 23 up through a 27, 31, 35, and 39 tone scale with the numerator being the number of units steps the minor third. The 43 tone scale used ends up being about .216 away from equal.