197ed12: Difference between revisions
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use ED intro; improve theory section |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | == Theory == | ||
This tuning | This tuning serves as a stretched version of [[55edo]], improving the accuracy of primes [[3/1|3]], [[7/1|7]], and [[11/1|11]], though prime [[5/1|5]] becomes worse. The [[patent val]]s match through the [[11-limit]], with the [[13-limit]] corresponding to the 55f [[val]] (written with [[wart notation]]). It thus tempers out the same [[comma]]s as 55edo in the 11-limit, and the same commas as 55f in the [[13-limit]]. | ||
==Harmonics== | == Harmonics == | ||
{{Harmonics in equal|197|12|1|prec=2|columns= | {{Harmonics in equal|197|12|1|prec=2|columns=11}} | ||
{{Harmonics in equal|197|12|1|prec=2|columns=11|start=12}} | |||
Latest revision as of 18:02, 20 March 2026
| ← 196ed12 | 197ed12 | 198ed12 → |
197 equal divisions of the 12th harmonic (abbreviated 197ed12) is a nonoctave tuning system that divides the interval of 12/1 into 197 equal parts of about 21.8 ¢ each. Each step represents a frequency ratio of 121/197, or the 197th root of 12.
Theory
This tuning serves as a stretched version of 55edo, improving the accuracy of primes 3, 7, and 11, though prime 5 becomes worse. The patent vals match through the 11-limit, with the 13-limit corresponding to the 55f val (written with wart notation). It thus tempers out the same commas as 55edo in the 11-limit, and the same commas as 55f in the 13-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.05 | -2.11 | +2.11 | +8.87 | -1.05 | -5.88 | +3.16 | -4.21 | +9.92 | -2.22 | +0.00 |
| Relative (%) | +4.8 | -9.6 | +9.6 | +40.6 | -4.8 | -26.9 | +14.5 | -19.3 | +45.4 | -10.2 | +0.0 | |
| Steps (reduced) |
55 (55) |
87 (87) |
110 (110) |
128 (128) |
142 (142) |
154 (154) |
165 (165) |
174 (174) |
183 (183) |
190 (190) |
197 (0) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.55 | -4.82 | +6.76 | +4.21 | +8.44 | -3.16 | -9.41 | -10.87 | -7.98 | -1.17 | +9.22 |
| Relative (%) | -34.6 | -22.1 | +30.9 | +19.3 | +38.7 | -14.5 | -43.1 | -49.8 | -36.6 | -5.4 | +42.2 | |
| Steps (reduced) |
203 (6) |
209 (12) |
215 (18) |
220 (23) |
225 (28) |
229 (32) |
233 (36) |
237 (40) |
241 (44) |
245 (48) |
249 (52) | |