28ed4/3: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''28ed4/3''' can be thought of as a [[2.3.7.11 subgroup]] analogue to [[20edf]] or [[Carlos Gamma]]. It very closely approximates the intervals of [[8/7]] (at 13 steps) and [[7/6]] (at 15 steps), along with [[11/8]] (at 31 steps). This tuning is close to every other step of [[135edo]] or to [[107edt]]. | '''28ed4/3''' can be thought of as a [[2.3.7.11 subgroup]] analogue to [[20edf]] or [[Carlos Gamma]]. It very closely approximates the intervals of [[8/7]] (at 13 steps) and [[7/6]] (at 15 steps), along with [[11/8]] (at 31 steps); these approximations are related to the temperament in this subgroup tempering out [[117649/117612]] and [[67110351/67108864]], which is the subgroup restriction of [[keenanose]]. This tuning is close to every other step of [[135edo]] or to [[107edt]]. | ||
==Intervals== | |||
These are the intervals up to a perfect fourth up. | |||
{| class="wikitable mw-collapsible" | |||
|+ Intervals of 28ed4/3 | |||
|- | |||
! Degrees | |||
! 2.3.7.11.97 subgroup approximation | |||
! Cents | |||
|- | |||
|1 | |||
|[[99/98]], 98/97, 97/96 | |||
|17.8 | |||
|- | |||
|2 | |||
|[[49/48]], 99/97 | |||
|35.6 | |||
|- | |||
|3 | |||
|[[33/32]] | |||
|53.4 | |||
|- | |||
|4 | |||
|1067/1024, 1617/1552 | |||
|71.1 | |||
|- | |||
|5 | |||
|539/512 | |||
|88.9 | |||
|- | |||
|6 | |||
|1089/1024 | |||
|106.7 | |||
|- | |||
|7 | |||
|8192/7623 | |||
|124.5 | |||
|- | |||
|8 | |||
|4096/3773 | |||
|142.3 | |||
|- | |||
|9 | |||
|3584/3267 | |||
|160.1 | |||
|- | |||
|10 | |||
|256/231 | |||
|177.9 | |||
|- | |||
|11 | |||
|384/343, 776/693 | |||
|195.7 | |||
|- | |||
|12 | |||
|112/99, 388/343, 768/679 | |||
|213.4 | |||
|- | |||
|13 | |||
|[[8/7]] | |||
|231.2 | |||
|- | |||
|14 | |||
|[[97/84]], 112/97, 343/297, 396/343 | |||
|249.0 | |||
|- | |||
|15 | |||
|[[7/6]] | |||
|266.8 | |||
|- | |||
|16 | |||
|[[33/28]] | |||
|284.6 | |||
|- | |||
|17 | |||
|231/194, 343/288 | |||
|302.4 | |||
|- | |||
|18 | |||
|[[77/64]] | |||
|320.2 | |||
|- | |||
|19 | |||
|1089/896 | |||
|338.0 | |||
|- | |||
|20 | |||
|3773/3072 | |||
|355.7 | |||
|- | |||
|21 | |||
|2541/2048 | |||
|373.5 | |||
|- | |||
|22 | |||
|4096/3267 | |||
|391.3 | |||
|- | |||
|23 | |||
|2048/1617 | |||
|409.1 | |||
|- | |||
|24 | |||
|3072/2401 | |||
|426.89571354395364193329708622509 | |||
|- | |||
|25 | |||
|[[128/99]] | |||
|444.68303494161837701385113148447 | |||
|- | |||
|26 | |||
|[[64/49]], 388/297 | |||
|462.47035633928311209440517674385 | |||
|- | |||
|27 | |||
|[[128/97]], 194/147, 392/297 | |||
|480.25767773694784717495922200323 | |||
|- | |||
|'''28''' | |||
|[[4/3]] | |||
|'''498.04499913461258225551326726261''' | |||
|} | |||
== Harmonics == | == Harmonics == | ||
Revision as of 01:46, 3 January 2025
| ← 27ed4/3 | 28ed4/3 | 29ed4/3 → |
(semiconvergent)
28ed4/3 can be thought of as a 2.3.7.11 subgroup analogue to 20edf or Carlos Gamma. It very closely approximates the intervals of 8/7 (at 13 steps) and 7/6 (at 15 steps), along with 11/8 (at 31 steps); these approximations are related to the temperament in this subgroup tempering out 117649/117612 and 67110351/67108864, which is the subgroup restriction of keenanose. This tuning is close to every other step of 135edo or to 107edt.
Intervals
These are the intervals up to a perfect fourth up.
| Degrees | 2.3.7.11.97 subgroup approximation | Cents |
|---|---|---|
| 1 | 99/98, 98/97, 97/96 | 17.8 |
| 2 | 49/48, 99/97 | 35.6 |
| 3 | 33/32 | 53.4 |
| 4 | 1067/1024, 1617/1552 | 71.1 |
| 5 | 539/512 | 88.9 |
| 6 | 1089/1024 | 106.7 |
| 7 | 8192/7623 | 124.5 |
| 8 | 4096/3773 | 142.3 |
| 9 | 3584/3267 | 160.1 |
| 10 | 256/231 | 177.9 |
| 11 | 384/343, 776/693 | 195.7 |
| 12 | 112/99, 388/343, 768/679 | 213.4 |
| 13 | 8/7 | 231.2 |
| 14 | 97/84, 112/97, 343/297, 396/343 | 249.0 |
| 15 | 7/6 | 266.8 |
| 16 | 33/28 | 284.6 |
| 17 | 231/194, 343/288 | 302.4 |
| 18 | 77/64 | 320.2 |
| 19 | 1089/896 | 338.0 |
| 20 | 3773/3072 | 355.7 |
| 21 | 2541/2048 | 373.5 |
| 22 | 4096/3267 | 391.3 |
| 23 | 2048/1617 | 409.1 |
| 24 | 3072/2401 | 426.89571354395364193329708622509 |
| 25 | 128/99 | 444.68303494161837701385113148447 |
| 26 | 64/49, 388/297 | 462.47035633928311209440517674385 |
| 27 | 128/97, 194/147, 392/297 | 480.25767773694784717495922200323 |
| 28 | 4/3 | 498.04499913461258225551326726261 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8.25 | +1.29 | +1.29 | +6.30 | -6.96 | -7.02 | -6.96 | +2.58 | -1.95 | -6.87 | +2.58 |
| Relative (%) | -46.4 | +7.2 | +7.2 | +35.4 | -39.1 | -39.5 | -39.1 | +14.5 | -11.0 | -38.6 | +14.5 | |
| Steps (reduced) |
67 (11) |
107 (23) |
135 (23) |
157 (17) |
174 (6) |
189 (21) |
202 (6) |
214 (18) |
224 (0) |
233 (9) |
242 (18) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.30 | +2.52 | +7.58 | +2.58 | +4.35 | -5.67 | +7.45 | +7.58 | -5.73 | +2.67 | -3.14 |
| Relative (%) | +35.4 | +14.1 | +42.6 | +14.5 | +24.4 | -31.9 | +41.9 | +42.6 | -32.2 | +15.0 | -17.7 | |
| Steps (reduced) |
250 (26) |
257 (5) |
264 (12) |
270 (18) |
276 (24) |
281 (1) |
287 (7) |
292 (12) |
296 (16) |
301 (21) |
305 (25) | |
|
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