7edt: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
Since one step of 7edt approximates a [[7/6]] subminor third (4.84 | 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]]. | 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]]. | ||
| Line 14: | Line 14: | ||
== Scale degrees of 7edt == | == Scale degrees of 7edt == | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |||
! Degrees | ! Degrees | ||
! Cents | ! Cents | ||
! [[Hekt]]s | ! [[Hekt]]s | ||
! Approximate Ratio | ! Approximate Ratio | ||
! [[Electra]] notation (J = 1/1) | ! [[Electra]] notation ({{nowrap|J {{=}} 1/1}}) | ||
|- | |- | ||
! colspan="3" | 0 | ! colspan="3" | 0 | ||
| Line 28: | Line 28: | ||
| 1 | | 1 | ||
| 271.708 | | 271.708 | ||
|185.714 | | 185.714 | ||
| [[7/6]] | | [[7/6]] | ||
| K | | K | ||
| Line 34: | Line 34: | ||
| 2 | | 2 | ||
| 543.416 | | 543.416 | ||
|371.429 | | 371.429 | ||
| [[15/11]], [[11/8]] | | [[15/11]], [[11/8]] | ||
| L | | L | ||
| Line 40: | Line 40: | ||
| 3 | | 3 | ||
| 815.124 | | 815.124 | ||
|557.143 | | 557.143 | ||
| [[8/5]] | | [[8/5]] | ||
| M | | M | ||
| Line 46: | Line 46: | ||
| 4 | | 4 | ||
| 1086.831 | | 1086.831 | ||
|742.857 | | 742.857 | ||
| [[15/8]] | | [[15/8]] | ||
| N | | N | ||
| Line 52: | Line 52: | ||
| 5 | | 5 | ||
| 1358.539 | | 1358.539 | ||
|928.571 | | 928.571 | ||
| [[11/5]] ([[11/10]] plus an octave) | | [[11/5]] ([[11/10]] plus an octave) | ||
| O | | O | ||
| Line 58: | Line 58: | ||
| 6 | | 6 | ||
| 1630.247 | | 1630.247 | ||
|1114.286 | | 1114.286 | ||
| [[18/7]] ([[9/7]] plus an octave) | | [[18/7]] ([[9/7]] plus an octave) | ||
| P | | P | ||
| Line 64: | Line 64: | ||
| 7 | | 7 | ||
| 1901.955 | | 1901.955 | ||
|1300 | | 1300 | ||
| [[3/1]] | | [[3/1]] | ||
| J | | J | ||
|} | |} | ||
[[category:Macrotonal]] | |||
[[category: | [[Category:Orwell]] | ||
[[Category: | [[Category:Subminor third]] | ||
[[Category: | |||
[[Category:Edt]] | [[Category:Edt]] | ||
Revision as of 16:19, 20 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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -113 | +0 | +45 | -69 | -113 | -108 | -68 | +0 | +89 | -76 | +45 |
| Relative (%) | -41.7 | +0.0 | +16.7 | -25.5 | -41.7 | -39.9 | -25.0 | +0.0 | +32.9 | -27.9 | +16.7 | |
| Steps (reduced) |
4 (4) |
7 (0) |
9 (2) |
10 (3) |
11 (4) |
12 (5) |
13 (6) |
14 (0) |
15 (1) |
15 (1) |
16 (2) | |
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -113 | +0 | -69 | -108 | -76 | -93 | -14 | +65 | +6 | -124 | +33 |
| Relative (%) | -41.7 | +0.0 | -25.5 | -39.9 | -27.9 | -34.3 | -5.2 | +23.9 | +2.2 | -45.5 | +12.0 | |
| Steps (reduced) |
4 (4) |
7 (0) |
10 (3) |
12 (5) |
15 (1) |
16 (2) |
18 (4) |
19 (5) |
20 (6) |
21 (0) |
22 (1) | |
Scale degrees of 7edt
| Degrees | Cents | Hekts | Approximate Ratio | Electra notation (J = 1/1) |
|---|---|---|---|---|
| 0 | 1/1 | J | ||
| 1 | 271.708 | 185.714 | 7/6 | K |
| 2 | 543.416 | 371.429 | 15/11, 11/8 | L |
| 3 | 815.124 | 557.143 | 8/5 | M |
| 4 | 1086.831 | 742.857 | 15/8 | N |
| 5 | 1358.539 | 928.571 | 11/5 (11/10 plus an octave) | O |
| 6 | 1630.247 | 1114.286 | 18/7 (9/7 plus an octave) | P |
| 7 | 1901.955 | 1300 | 3/1 | J |