1803edo
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.208 | -0.290 | +0.226 | -0.249 | -0.236 | +0.071 | -0.082 | +0.203 | -0.009 | -0.232 |
| Relative (%) | +31.3 | -43.6 | +33.9 | -37.5 | -35.5 | +10.7 | -12.4 | +30.4 | -1.3 | -34.8 | |
| Steps (reduced) |
2858 (1055) |
4186 (580) |
5062 (1456) |
5715 (306) |
6237 (828) |
6672 (1263) |
7044 (1635) |
7370 (158) |
7659 (447) |
7919 (707) | |
In the 2.19.23.29 subgroup, 1803edo tempers out 2476099/2475904, and supports the corresponding rank 3 temperament eliminating this comma. In the 13 limit, 1803edo tempers out 2080/2079 and 4225/4224. In the 7-limit, it tempers out 420175/419904.
Relationship to the saros cycle
In real life, 1803 years is 100 times the saros cycle, designed to predict eclipses. In addition, it also makes for both the leap week and the leap day calendars that excellently approximate the March equinox - 22300 lunar months is almost exactly 658532 days or 94076 weeks. This can be used to produce a variety of different temperaments.
The simplest are the rank two temperaments produced by 1803 years being able to support a leap day, leap week, and a lunisolar calendar all in one.
Hectosaros Leap Week
Since 1803 years is equal to 94076 weeks, it produces a cycle where 94076 mod 1803 = 320 years are leap, and using the maximal evenness method of finding rank two temperaments, the associated rank two temperament is 320 & 1803, which if it had a name would be hectosaros leap week. The generator for such a temperament is 524\1803, a neutral third.
In the 19-limit, the intepretation with the smallest TE error is 320 & 1803g, tempering out 4225/4224, 5929/5928, 10830/10829, 11495/11492, 14161/14157, 67507/67500. Patent val approach is also possible, which results in the comma basis 4200/4199, 4225/4224, 14400/14399, 14875/14872, 104272/104247, 3414015/3411968. The generator in the patent val maps to 1224/1001.
A simple scale such a temperament it produces is 3L 4s, which is also described in the Solar Calendar Leap Rules page as 231 293 231 293 231 293 231. In addition, if one were to rearrange the steps (or raise the 4th degree by 62\1803) so they instead produce 231 293 293 231 231 293 231, the resulting scale is Maqam Sikah.
Hectosaros Leap Day
Hectosaros Leap Day is defined as 437 & 1803 and is generated by 590\1803 interval, which is a submajor third.
Hectosaros Lunisolar
Hectosaros Lunisolar is defined as 664 & 1803 and is generated by 1078\1803 interval measuring about 717-cents, which puts it in the far ultrapyth range, close to the sharp fifth of 5edo. A simple scale would be a very hard diatonic scale.
Scales
- HectosarosLeapDay[437]
- HectosarosLeapWeek[7], a MOS of type 3L 4s (mosh) - 231 293 231 293 231 293 231
- Hectosaros Maqam Sikah, a MODMOS of type 3L 4s (mosh) - 231 293 293 231 231 293 231
- HectosarosLeapWeek[320]
- HectosarosLunisolar[664]
Links
- Wikipedia Contributors, Saros (astronomy).
- Solar Calendar Leap Rules