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
1............[A+
3
4 ............C=
7
11
5............. F]
10
15
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
21
20
22
36
32
43
56
54
79
88
97
135
142
176
223
239
311
365
415
534
604
726
899
1019
1260
1503
1745
2159
2522
3005
3662
4267
5164
6184
7272
8826
10451
12436
15010
17723
21262
25461
30159
36272
43184 THE SCALE STARTS HERE=and we take it out to 43 places to close out our cycles of
51421
61733
73343
87693
104917
124764
149426
178260
212457
254343
303024
361883
432603
515481
616226
735627
877364
1048829
1251108
1493590
1784456
2128472
2542419
3035564
3622062
4326875
5164036
6164481
7362439
8786098
10491356
12526475
14950579
17853795
21312573
30380270
36263152
43295730
51692843
61705087
73676000
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.
