1803edo: Difference between revisions

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== Theory ==

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~~.

1803edo is in[[consistent]] in the [[5-odd-limit]] and the errors of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|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 ===

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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]].

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 ====

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=== Rank-2 temperaments by generator ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|-

! Periods<br>per 8ve

! Generator*

@@ Line 3: / Line 3: @@
 == Theory ==
-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.
+edo is in[[consistent]] in the [[5-odd-limit]] and the errors of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|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 ===
@@ Line 25: / Line 25: @@
 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]].
+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 ====
@@ Line 35: / Line 35: @@
 === Rank-2 temperaments by generator ===
 {| class="wikitable center-all left-5"
+|+Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
 ! Generator*