User:CompactStar/8edso: Difference between revisions
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| Line 8: | Line 8: | ||
!Step | !Step | ||
!Linear value | !Linear value | ||
!Cents | |||
!Tetrative intervals | !Tetrative intervals | ||
!Common JI approximation | !Common JI approximation | ||
| Line 13: | Line 14: | ||
|0 | |0 | ||
|1 | |1 | ||
|0.000 | |||
| | | | ||
|[[1/1]] exact | |[[1/1]] exact | ||
| Line 18: | Line 20: | ||
|1 | |1 | ||
|1.11149118 | |1.11149118 | ||
|182.996 | |||
| | | | ||
|[[10/9]] | |[[10/9]] | ||
| Line 23: | Line 26: | ||
|2 | |2 | ||
|1.22436140 | |1.22436140 | ||
|350.435 | |||
|log(log(5)) = [[5/1]] reduced | |log(log(5)) = [[5/1]] reduced | ||
|[[5/4]], [[11/9]] | |[[5/4]], [[11/9]] | ||
| Line 28: | Line 32: | ||
|3 | |3 | ||
|1.33973255 | |1.33973255 | ||
|506.334 | |||
| | | | ||
|[[4/3]] | |[[4/3]] | ||
| Line 33: | Line 38: | ||
|4 | |4 | ||
|1.45878181 | |1.45878181 | ||
|653.717 | |||
| | | | ||
|[[16/11]], [[19/13]] | |[[16/11]], [[19/13]] | ||
| Line 38: | Line 44: | ||
|5 | |5 | ||
|1.58278746 | |1.58278746 | ||
|794.961 | |||
|log(3) = [[3/1]] reduced | |log(3) = [[3/1]] reduced | ||
|[[8/5]], [[11/7]], [[19/12]] | |[[8/5]], [[11/7]], [[19/12]] | ||
| Line 43: | Line 50: | ||
|6 | |6 | ||
|1.71318047 | |1.71318047 | ||
|932.013 | |||
| | | | ||
|[[12/7]] | |[[12/7]] | ||
| Line 48: | Line 56: | ||
|7 | |7 | ||
|1.85160598 | |1.85160598 | ||
|1066.533 | |||
| | | | ||
|[[11/6]], [[13/7]] | |[[11/6]], [[13/7]] | ||
| Line 53: | Line 62: | ||
|8 | |8 | ||
|2 | |2 | ||
|1200.000 | |||
| | | | ||
|[[2/1]] exact | |[[2/1]] exact | ||
Revision as of 22:15, 1 July 2023
8 equal divisions of the superoctave is a super-pitch tuning system that tetratively divides the superoctave into 8 equally spaced steps.
8edso is notable for containing a close approximation to 3/1 (as well as log2(3)) according to the analytic extension of the tetration developed by Kneser, hereby producing a strong approximation to the super-pitch equivalent of the Pythagorean tuning. Furthermore, it provides good representation of 5/1 as well as log2(5), thus being a strong 2.3.5 super-subgroup tuning and therefore being the best candidate for the super-pitch equivalent of 12edo.
Intervals
| Step | Linear value | Cents | Tetrative intervals | Common JI approximation |
|---|---|---|---|---|
| 0 | 1 | 0.000 | 1/1 exact | |
| 1 | 1.11149118 | 182.996 | 10/9 | |
| 2 | 1.22436140 | 350.435 | log(log(5)) = 5/1 reduced | 5/4, 11/9 |
| 3 | 1.33973255 | 506.334 | 4/3 | |
| 4 | 1.45878181 | 653.717 | 16/11, 19/13 | |
| 5 | 1.58278746 | 794.961 | log(3) = 3/1 reduced | 8/5, 11/7, 19/12 |
| 6 | 1.71318047 | 932.013 | 12/7 | |
| 7 | 1.85160598 | 1066.533 | 11/6, 13/7 | |
| 8 | 2 | 1200.000 | 2/1 exact |