User:CompactStar/8edso: Difference between revisions
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{{Novelty}} | |||
'''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 == | |||
{| class="wikitable" | |||
|+ | |||
!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 | |||
|} | |||
Latest revision as of 19:20, 26 March 2025
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This page presents a novelty topic.
It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks. |
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 |