6ed5: Difference between revisions
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Expand stub page with a couple short paragraphs about harmonics and subgroups, and a manually curated interval table |
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{{ED intro}} | {{ED intro}} | ||
== | == Harmonics == | ||
6ed5 does not approximate any sensible [[subgroup]] of integer [[harmonic]]s. If one does try to interpret it using an integer subgroup, then it looks something like a 5.19.33 subgroup: which has severely limited use cases. | |||
6ed5 does however have some [[chord]]s and [[interval]]s that sound good for its size, despite its poor approximations of pure harmonics. These shine when it is interpreted using a [[Subgroup temperaments#Fractional subgroup temperaments|fractional subgroup]]. | |||
{{Harmonics in equal | {{Harmonics in equal | ||
| steps = 6 | | steps = 6 | ||
| Line 15: | Line 15: | ||
| num = 5 | | num = 5 | ||
| denom = 1 | | denom = 1 | ||
| collapsed = yes | |||
| start = 12 | | start = 12 | ||
| | | columns = 14 | ||
| title = (higher harmonics) | |||
}} | }} | ||
== Intervals == | |||
{| class="wikitable mw-collapsible" | |||
|+ | |||
!Step | |||
!Interval (¢) | |||
!JI approximated <br>(subgroup A) | |||
!JI approximated <br>(subgroup B) | |||
!JI approximated <br>(subgroup C) | |||
|- | |||
|1 | |||
|424.39 | |||
|9/7 | |||
|14/11 | |||
|23/18 | |||
|- | |||
|2 | |||
|928.77 | |||
|12/7 | |||
|12/7 | |||
|53/31 | |||
|- | |||
|3 | |||
|1393.16 | |||
|9/4 | |||
|29/13 | |||
|38/17 | |||
|- | |||
|4 | |||
|1857.54 | |||
|29/10 | |||
|35/12 | |||
|38/13 | |||
|- | |||
|5 | |||
|2321.93 | |||
|19/5 | |||
|23/6 | |||
|65/17 | |||
|- | |||
|6 | |||
|2786.31 | |||
|5/1 | |||
|5/1 | |||
|5/1 | |||
|} | |||
* Subgroup A = low complexity subgroup: 5/1.9/4.9/7.12/7.19/5.29/10 | |||
* Subgroup B = compromise subgroup: 5/1.12/7.14/11.23/6.29/13.35/12 | |||
* Subgroup C = low error subgroup: 5/1.23/18.38/13.38/17.53/31.65/17 | |||
Other interpretations are possible. | |||
{{todo|expand|intro}} | |||
Revision as of 05:00, 17 December 2024
| ← 5ed5 | 6ed5 | 7ed5 → |
6 equal divisions of the 5th harmonic (abbreviated 6ed5) is a nonoctave tuning system that divides the interval of 5/1 into 6 equal parts of about 464 ¢ each. Each step represents a frequency ratio of 51/6, or the 6th root of 5.
Harmonics
6ed5 does not approximate any sensible subgroup of integer harmonics. If one does try to interpret it using an integer subgroup, then it looks something like a 5.19.33 subgroup: which has severely limited use cases.
6ed5 does however have some chords and intervals that sound good for its size, despite its poor approximations of pure harmonics. These shine when it is interpreted using a fractional subgroup.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +193 | -44 | -78 | +0 | +149 | -118 | +115 | -89 | +193 | +28 | -122 |
| Relative (%) | +41.6 | -9.6 | -16.8 | +0.0 | +32.0 | -25.4 | +24.8 | -19.1 | +41.6 | +6.1 | -26.4 | |
| Steps (reduced) |
3 (3) |
4 (4) |
5 (5) |
6 (0) |
7 (1) |
7 (1) |
8 (2) |
8 (2) |
9 (3) |
9 (3) |
9 (3) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +203 | +75 | -44 | -156 | +203 | +104 | +11 | -78 | -163 | +221 | +144 | +71 | +0 | -68 |
| Relative (%) | +43.8 | +16.2 | -9.6 | -33.6 | +43.8 | +22.5 | +2.3 | -16.8 | -35.0 | +47.7 | +31.1 | +15.2 | +0.0 | -14.6 | |
| Steps (reduced) |
10 (4) |
10 (4) |
10 (4) |
10 (4) |
11 (5) |
11 (5) |
11 (5) |
11 (5) |
11 (5) |
12 (0) |
12 (0) |
12 (0) |
12 (0) |
12 (0) | |
Intervals
| Step | Interval (¢) | JI approximated (subgroup A) |
JI approximated (subgroup B) |
JI approximated (subgroup C) |
|---|---|---|---|---|
| 1 | 424.39 | 9/7 | 14/11 | 23/18 |
| 2 | 928.77 | 12/7 | 12/7 | 53/31 |
| 3 | 1393.16 | 9/4 | 29/13 | 38/17 |
| 4 | 1857.54 | 29/10 | 35/12 | 38/13 |
| 5 | 2321.93 | 19/5 | 23/6 | 65/17 |
| 6 | 2786.31 | 5/1 | 5/1 | 5/1 |
- Subgroup A = low complexity subgroup: 5/1.9/4.9/7.12/7.19/5.29/10
- Subgroup B = compromise subgroup: 5/1.12/7.14/11.23/6.29/13.35/12
- Subgroup C = low error subgroup: 5/1.23/18.38/13.38/17.53/31.65/17
Other interpretations are possible.