8ed6: Difference between revisions
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{{Mathematical interest}} | |||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
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== Intervals == | == Intervals == | ||
{| class="wikitable" | {| class="wikitable right-2" | ||
|- | |- | ||
! # | ! # | ||
| Line 16: | Line 17: | ||
|- | |- | ||
| 0 | | 0 | ||
| | | 0 | ||
| | | 1/1 | ||
|- | |- | ||
| 1 | | 1 | ||
| | | 388 | ||
| 5/4 | | 5/4 | ||
|- | |- | ||
| 2 | | 2 | ||
| 775 | | 775 | ||
| 25/16 | | 11/7, 25/16 | ||
|- | |- | ||
| 3 | | 3 | ||
| 1163 | | 1163 | ||
| | | | ||
|- | |- | ||
| 4 | | 4 | ||
| | | 1551 | ||
| 22/9 | | 22/9 | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 1939 | ||
| | | 49/16 | ||
|- | |- | ||
| 6 | | 6 | ||
| 2326 | | 2326 | ||
| | | | ||
|- | |- | ||
| 7 | | 7 | ||
| 2714 | | 2714 | ||
| 24/5 | | 24/5 | ||
|- | |- | ||
| 8 | | 8 | ||
| | | 3102 | ||
| | | 6/1 | ||
|} | |} | ||
Latest revision as of 22:19, 10 August 2025
|
This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 7ed6 | 8ed6 | 9ed6 → |
(semiconvergent)
(semiconvergent)
8 equal divisions of the 6th harmonic (abbreviated 8ed6) is a nonoctave tuning system that divides the interval of 6/1 into 8 equal parts of about 388 ¢ each. Each step represents a frequency ratio of 61/8, or the 8th root of 6.
Theory
8ed6 can be thought of as a subset (where the ~5/4 generator is stacked) of the 6/1-eigenmonzo tuning of würschmidt.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -37 | +37 | -74 | -72 | +0 | +121 | -110 | +74 | -109 | +114 | -37 |
| Relative (%) | -9.5 | +9.5 | -19.0 | -18.6 | +0.0 | +31.2 | -28.4 | +19.0 | -28.1 | +29.4 | -9.5 | |
| Steps (reduced) |
3 (3) |
5 (5) |
6 (6) |
7 (7) |
8 (0) |
9 (1) |
9 (1) |
10 (2) |
10 (2) |
11 (3) |
11 (3) | |
Intervals
| # | Cents | Approximate JI ratio(s) |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 388 | 5/4 |
| 2 | 775 | 11/7, 25/16 |
| 3 | 1163 | |
| 4 | 1551 | 22/9 |
| 5 | 1939 | 49/16 |
| 6 | 2326 | |
| 7 | 2714 | 24/5 |
| 8 | 3102 | 6/1 |