197edo
197 equal divisions of the octave (abbreviated 197edo or 197ed2), also called 197-tone equal temperament (197tet) or 197 equal temperament (197et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 197 equal parts of about 6.09 ¢ each. Each step represents a frequency ratio of 21/197, or the 197th root of 2.
Theory
197edo gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and catakleismic, the temperament tempering out both 225/224 and 4375/4374. Among patent vals, it gives the best results for both. In fact, the 11-limit patent val ⟨197 312 457 553 682] has a comma basis {225/224, 441/440, 4375/4374, 65536/65219}, so taking 225/224 and 441/440 together (prodigy temperament) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.
If we use 197e, the ⟨197 312 457 553 681] val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit marvel, we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. The 197ef val, ⟨197 312 457 553 681 728], is an excellent tuning for the 13-limit version of catakleismic.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.45 | -2.56 | -0.30 | -2.89 | +3.00 | +0.08 | +2.09 | -1.40 | +0.96 | -1.75 | -0.86 |
| Relative (%) | -23.8 | -42.0 | -4.9 | -47.5 | +49.2 | +1.3 | +34.3 | -23.0 | +15.8 | -28.7 | -14.2 | |
| Steps (reduced) |
312 (115) |
457 (63) |
553 (159) |
624 (33) |
682 (91) |
729 (138) |
770 (179) |
805 (17) |
837 (49) |
865 (77) |
891 (103) | |
Subsets and supersets
197edo is the 45th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-312 197⟩ | [⟨197 312]] | +0.4566 | 0.4568 | 7.50 |
| 2.3.5 | 15625/15552, [-53 32 1⟩ | [⟨197 312 457]] | +0.6717 | 0.4813 | 7.90 |
| 2.3.5.7 | 225/224, 4375/4374, [-25 6 -3 8⟩ | [⟨197 312 457 553]] | +0.5302 | 0.4834 | 7.94 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 52\197 | 316.75 | 6/5 | Catakleismic |
| 1 | 53\197 | 322.84 | 3087/2560 | Seniority |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
Music
- Pump1 – in pump12 1