198edt: Difference between revisions
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== Theory == | |||
198edt is related to [[125edo]], but with the [[3/1|perfect twelfth]] rather than the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|stretched]] by about 0.729 cents. Unlike 125edo, which is only [[consistent]] to the [[integer limit|10-integer-limit]], 198edt is consistent to the 12-integer-limit. In particular, it significantly improves the approximated [[prime harmonic]]s [[5/1|5]], [[11/1|11]] and [[13/1|13]] over 125edo, though the [[7/1|7]], [[17/1|17]] and [[19/1|19]], which are sharp to start with, are tuned worse here. | |||
=== Harmonics === | |||
{{Harmonics in equal|198|3|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|198|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 198edt (continued)}} | |||
=== Subsets and supersets === | |||
Since 198 factors into primes as {{nowrap| 2 × 3<sup>2</sup> × 11 }}, 198edt contains subset edts {{EDs|equave=t| 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99 }}. | |||
== See also == | |||
* [[125edo]] – relative edo | |||
* [[323ed6]] – relative ed6 | |||
Latest revision as of 12:37, 15 April 2025
| ← 197edt | 198edt | 199edt → |
198 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 198edt or 198ed3), is a nonoctave tuning system that divides the interval of 3/1 into 198 equal parts of about 9.61 ¢ each. Each step represents a frequency ratio of 31/198, or the 198th root of 3.
Theory
198edt is related to 125edo, but with the perfect twelfth rather than the octave being just. The octave is stretched by about 0.729 cents. Unlike 125edo, which is only consistent to the 10-integer-limit, 198edt is consistent to the 12-integer-limit. In particular, it significantly improves the approximated prime harmonics 5, 11 and 13 over 125edo, though the 7, 17 and 19, which are sharp to start with, are tuned worse here.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.73 | +0.00 | +1.46 | -0.62 | +0.73 | +2.82 | +2.19 | +0.00 | +0.11 | -1.60 | +1.46 |
| Relative (%) | +7.6 | +0.0 | +15.2 | -6.5 | +7.6 | +29.4 | +22.8 | +0.0 | +1.1 | -16.6 | +15.2 | |
| Steps (reduced) |
125 (125) |
198 (0) |
250 (52) |
290 (92) |
323 (125) |
351 (153) |
375 (177) |
396 (0) |
415 (19) |
432 (36) |
448 (52) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.63 | +3.55 | -0.62 | +2.92 | +3.63 | +0.73 | +3.18 | +0.84 | +2.82 | -0.87 | -0.98 | +2.19 |
| Relative (%) | -27.4 | +37.0 | -6.5 | +30.4 | +37.7 | +7.6 | +33.2 | +8.7 | +29.4 | -9.0 | -10.2 | +22.8 | |
| Steps (reduced) |
462 (66) |
476 (80) |
488 (92) |
500 (104) |
511 (115) |
521 (125) |
531 (135) |
540 (144) |
549 (153) |
557 (161) |
565 (169) |
573 (177) | |
Subsets and supersets
Since 198 factors into primes as 2 × 32 × 11, 198edt contains subset edts 2, 3, 6, 9, 11, 18, 22, 33, 66 and 99.