1803edo
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1803 equal divisions of the octave (abbreviated 1803edo or 1803ed2), also called 1803-tone equal temperament (1803tet) or 1803 equal temperament (1803et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1803 equal parts of about 0.666 ¢ each. Each step represents a frequency ratio of 21/1803, or the 1803rd root of 2.
Theory
1803edo is inconsistent in the 5-odd-limit and the errors of harmonics 3, 5, and 7 are all quite large. To start with, the 1803c val and the patent val may be considered. Using the patent val, it tempers out 420175/419904 in the 7-limit, and 2080/2079, 4096/4095 and 4225/4224 in the 13-limit. In the 2.19.23.29 subgroup, 1803edo tempers out 2476099/2475904, and supports the corresponding rank-3 temperament eliminating this comma.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.208 | -0.290 | +0.226 | -0.249 | -0.236 | +0.071 | -0.082 | +0.203 | -0.009 | -0.232 | +0.012 |
| Relative (%) | +31.3 | -43.6 | +33.9 | -37.5 | -35.5 | +10.7 | -12.4 | +30.4 | -1.3 | -34.8 | +1.8 | |
| Steps (reduced) |
2858 (1055) |
4186 (580) |
5062 (1456) |
5715 (306) |
6237 (828) |
6672 (1263) |
7044 (1635) |
7370 (158) |
7659 (447) |
7919 (707) |
8156 (944) | |
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-2 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-2 temperaments, the associated rank-2 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.
Since 320edo is consistent in the 19-odd-limit, hectosaros leap week temperament is defined for the 2.3.7.13.17.19 subgroup. The resulting comma basis is 10081799/10077696, 39337984/39328497, 10754912/10744731, and 480024727/480020256. In addition, the generator is mapped to 6144/3757, which rounds to 524\1803, and is therefore consistent.
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 that of Maqam Sikah.
Another way to subdivide such a scale beyond simple generator stacking are the sub-cycles described on the Solar Calendar Leap Rules page, and presented in the Ford Circles Of Leap Cycles spreadsheet provided on the same page. Dividing the 231 into 45-141-45, and 293 into 45-79-45-79-45 produces an uneven and a rather unique 29-tone scale. In addition, further dissecting 231 into 45-45-6-45-45-45 produces a 35-tone scale.
Hectosaros Leap Day
Hectosaros Leap Day is defined as 437 & 1803 and is generated by 590\1803 interval, which is a submajor third, and it sounds close to magic. This time, it once again produces the 3L 4s scale, but it is extremely hard, with step ratio almost 17 to 1. Further mos produced are sephiroid, which makes it sound like würschmidt, but it is still quite hard for it. The best subgroup for it is 2.3.7.13.17.23.29, where it has the comma basis 5888/5887, 31213/31212, 2359296/2358811, 39337984/39328497, 102109696/102001683, and the generator maps to 64/51.
The next softest MOS is the 55-tone scale in the 52L 3s form, which has step sizes of 33 and 29. It's notable that in real life, these step sizes correspond to the subcycles of 33 years or 29 years that distinguish the leap year excess from exactly once every 4 years. This is also the scale provided on the next level by the Ford Circles Of Leap Cycles spreadsheet. Due to rough but somewhat noticeable similarity of step sizes (4\1803 is around just-noticeable difference), it can function as a well temperament for 55edo.
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 an almost equipentatonic scale which results in very hard diatonic scale.
The next scale which has a suitable hardness is the 92-note 5L 87s scale, with step sizes of 30 and 19. It can be dissected into 372-353-372-353-353 (darkest mode). In real life, 19 years corresponds to the Metonic cycle, and 353 years corresponds to the Rectified Hebrew cycle (not to be confused with the eponymous temperament, which only makes sense in 353edo and other EDOs that support it).
Regular temperament properties
Rank-2 temperaments by generator
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 524\1803 | 348.752 | 6144/3757 | Hectosaros leap week |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
- 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
Links
- Wikipedia Contributors, Saros (astronomy).
- Solar Calendar Leap Rules