7edt: Difference between revisions
+subsets and supersets; cleanup; -redundant categories |
|||
| (15 intermediate revisions by 5 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
== | Since one step of 7edt approximates a [[7/6]] subminor third (4.84{{c}} sharp) quite nicely, three steps are almost exactly [[8/5]] (tempering out [[1728/1715]], the orwellisma), and four steps are very nearly [[15/8]] (tempering out [[2430/2401]], the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some [[7-limit]] harmony, along with some elements of the [[11-limit]], such as the [[11/8]] major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma. | ||
Due to the proximity of the step size with 7/6, 7edt supports [[orwell]] temperament. One step of 7edt is almost identical to 12\53, the [[53edo]] orwell generator, at about 271.698 cents. 7edt is also a good tuning for [[Electra]] temperament, with two steps of 7edt being a close approximation to [[15/11]]. | |||
{| class="wikitable" | === Harmonics === | ||
! | {{Harmonics in equal|7|3|1|columns=15}} | ||
=== Subsets and supersets === | |||
7edt is the 4th [[prime equal division|prime edt]], after [[5edt]] and before [[11edt]]. | |||
== Intervals == | |||
{| class="wikitable center-1 right-2 right-3" | |||
|- | |||
! # | |||
! Cents | ! Cents | ||
! | ! [[Hekt]]s | ||
! Approximate | ! Approximate ratios | ||
! [[Electra]] notation<br>({{nowrap|J {{=}} 1/1}}) | |||
|- | |- | ||
| | | 0 | ||
| 0 | |||
| 0 | |||
| [[1/1]] | | [[1/1]] | ||
| J | |||
|- | |- | ||
| 1 | | 1 | ||
| | | 272 | ||
| | | 186 | ||
| [[7/6]] | | [[7/6]] | ||
| K | |||
|- | |- | ||
| 2 | | 2 | ||
| 543 | | 543 | ||
|371 | | 371 | ||
| [[ | | [[11/8]], [[15/11]] | ||
| L | |||
|- | |- | ||
| 3 | | 3 | ||
| 815 | | 815 | ||
|557 | | 557 | ||
| [[8/5]] | | [[8/5]] | ||
| M | |||
|- | |- | ||
| 4 | | 4 | ||
| | | 1087 | ||
| | | 743 | ||
| [[15/8]] | | [[15/8]] | ||
| N | |||
|- | |- | ||
| 5 | | 5 | ||
| | | 1359 | ||
| | | 929 | ||
| [[11/5]] | | [[11/5]] | ||
| O | |||
|- | |- | ||
| 6 | | 6 | ||
| 1630 | | 1630 | ||
|1114 | | 1114 | ||
| [[18/7]] | | [[18/7]] | ||
| P | |||
|- | |- | ||
| 7 | | 7 | ||
| | | 1902 | ||
|1300 | | 1300 | ||
| [[3/1]] | | [[3/1]] | ||
| J | |||
|} | |} | ||
[[Category:Orwell]] | |||
[[Category:Subminor third]] | |||
[[Category: | |||
[[Category: | |||
Latest revision as of 13:40, 23 February 2025
| ← 6edt | 7edt | 8edt → |
7 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 7edt or 7ed3), is a nonoctave tuning system that divides the interval of 3/1 into 7 equal parts of about 272 ¢ each. Each step represents a frequency ratio of 31/7, or the 7th root of 3.
Theory
Since one step of 7edt approximates a 7/6 subminor third (4.84 ¢ sharp) quite nicely, three steps are almost exactly 8/5 (tempering out 1728/1715, the orwellisma), and four steps are very nearly 15/8 (tempering out 2430/2401, the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony, along with some elements of the 11-limit, such as the 11/8 major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.
Due to the proximity of the step size with 7/6, 7edt supports orwell temperament. One step of 7edt is almost identical to 12\53, the 53edo orwell generator, at about 271.698 cents. 7edt is also a good tuning for Electra temperament, with two steps of 7edt being a close approximation to 15/11.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -113 | +0 | +45 | -69 | -113 | -108 | -68 | +0 | +89 | -76 | +45 | -93 | +50 | -69 | +91 |
| Relative (%) | -41.7 | +0.0 | +16.7 | -25.5 | -41.7 | -39.9 | -25.0 | +0.0 | +32.9 | -27.9 | +16.7 | -34.3 | +18.5 | -25.5 | +33.4 | |
| Steps (reduced) |
4 (4) |
7 (0) |
9 (2) |
10 (3) |
11 (4) |
12 (5) |
13 (6) |
14 (0) |
15 (1) |
15 (1) |
16 (2) |
16 (2) |
17 (3) |
17 (3) |
18 (4) | |
Subsets and supersets
7edt is the 4th prime edt, after 5edt and before 11edt.
Intervals
| # | Cents | Hekts | Approximate ratios | Electra notation (J = 1/1) |
|---|---|---|---|---|
| 0 | 0 | 0 | 1/1 | J |
| 1 | 272 | 186 | 7/6 | K |
| 2 | 543 | 371 | 11/8, 15/11 | L |
| 3 | 815 | 557 | 8/5 | M |
| 4 | 1087 | 743 | 15/8 | N |
| 5 | 1359 | 929 | 11/5 | O |
| 6 | 1630 | 1114 | 18/7 | P |
| 7 | 1902 | 1300 | 3/1 | J |