1803edo

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Prime factorization 3 × 601
Step size 0.665557¢
Fifth 1055\1803 (702.163¢)
Semitones (A1:m2) 173:134 (115.1¢ : 89.18¢)
Consistency limit 3
Distinct consistency limit 3

1803 equal divisions of the octave (1803edo), or 1803-tone equal temperament (1803tet), 1803 equal temperament (1803et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 1803 equal parts of about 0.666 ¢ each.

Theory

Approximation of odd harmonics in 1803edo
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 -44 +34 -37 -36 +11 -12 +30 -1 -35
Steps
(reduced)
2858
(1055)
4186
(580)
5062
(1456)
5715
(306)
6237
(828)
6672
(1263)
7044
(1635)
7370
(158)
7659
(447)
7919
(707)

1803edo is consistent in the no-5, no-11 29-odd-limit.

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-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.

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.

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.

Regular temperament properties

Rank-2 temperaments by generator

Periods
per Octave
Generator
(Reduced)
Cents
(Reduced)
Associated
Ratio
Temperaments
1 524\1803 348.752 6144/3757 Hectosaros leap week

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