User:CompactStar/8edso: Difference between revisions
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'''8 equal divisions of the superoctave''' is a [[super-pitch]] tuning system that tetratively divides the superoctave into 8 equally spaced steps. | '''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]] 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]]. | 8edso is notable for containing a close approximation to [[3/1]] (as well as log<sub>2</sub>(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 log<sub>2</sub>(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 == | == Intervals == | ||
| Line 8: | Line 8: | ||
!Step | !Step | ||
!Linear value | !Linear value | ||
!Cents | |||
!Tetrative intervals | |||
!Common JI approximation | |||
|- | |||
|0 | |||
|1 | |||
|0.000 | |||
| | |||
|[[1/1]] exact | |||
|- | |- | ||
|1 | |1 | ||
|1.11149118 | |1.11149118 | ||
|182.996 | |||
| | |||
|[[10/9]] | |||
|- | |- | ||
|2 | |2 | ||
|1.22436140 | |1.22436140 | ||
|350.435 | |||
|log(log(5)) = [[5/1]] reduced | |||
|[[5/4]], [[11/9]] | |||
|- | |- | ||
|3 | |3 | ||
|1.33973255 | |1.33973255 | ||
|506.334 | |||
| | |||
|[[4/3]] | |||
|- | |- | ||
|4 | |4 | ||
|1.45878181 | |1.45878181 | ||
|653.717 | |||
| | |||
|[[16/11]], [[19/13]] | |||
|- | |- | ||
|5 | |5 | ||
|1.58278746 | |1.58278746 | ||
|794.961 | |||
|log(3) = [[3/1]] reduced | |||
|[[8/5]], [[11/7]], [[19/12]] | |||
|- | |- | ||
|6 | |6 | ||
|1.71318047 | |1.71318047 | ||
|932.013 | |||
| | |||
|[[12/7]] | |||
|- | |- | ||
|7 | |7 | ||
|1.85160598 | |1.85160598 | ||
|1066.533 | |||
| | |||
|[[11/6]], [[13/7]] | |||
|- | |- | ||
|8 | |8 | ||
|2 | |2 | ||
|1200.000 | |||
| | |||
|[[2/1]] exact | |||
|} | |} | ||
Latest revision as of 09:53, 17 October 2026
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 |