1803edo: Difference between revisions

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~~'''1803edo''', or the '''100 saros tuning''', divides the octave into steps of 665.6 millicents each. 1803 factors as 3 x 601.~~

== Theory ==

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 ===

=== 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 {{nowrap|94076 ≡ 320 (mod 1803)}} years are leap, and using the maximal evenness method of finding rank-2 temperaments, the associated rank-2 temperament is {{nowrap|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 {{dash|231, 293, 231, 293, 231, 293, 231|med}}. In addition, if one were to rearrange the steps (or raise the 4th degree by 62\1803) so they instead produce {{dash|231, 293, ''293'', ''231'', 231, 293, 231|med}}, 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 {{dash|45, 141, 45|med}}, and 293 into {{dash|45, 79, 45, 79, 45|med}} produces an uneven and a rather unique 29-tone scale. In addition, further dissecting 231 into {{dash|45, 45, 6, 45, 45, 45|med}} produces a 35-tone scale.

==== Hectosaros Leap Day ====

Hectosaros Leap Day is defined as {{nowrap|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 of almost 17:1. 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]].

~~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~~.

==== Hectosaros Lunisolar ====

Hectosaros Lunisolar is defined as {{nowrap|664 & 1803}} and is generated by 1078\1803, which measure about 717 cents, putting 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.

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. Therefore, it creates 3 different calendric scales that can be used as leap cycles~~ - ~~437~~-~~note leap day cycle with a 590\1803~~ generator ~~(a major third), 320~~-~~note leap week cycle with~~ 524\1803 ~~generator (a neutral third)~~, and ~~the lunisolar cycle with the 1078\1803 generator (a slightly sharp fifth).~~

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 {{dash|372, 353, 372, 353, 353|med}} (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 [[Rectified Hebrew|eponymous temperament]], which only makes sense in [[353edo]] and other EDOs that support it).

== Regular temperament properties ==

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

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

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 1

| 524\1803

| 348.752

| 6144/3757

| [[Hectosaros leap week]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

~~As such, 100 & 223 & 1803 saros temperament can be defined. In the 2.7.13.19.23.29 subgroup, 4952198/4950967 can be the starting comma.~~

== 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

* SolarLeapDay[437]

* SolarLeapWeek[320]

* Lunisolar[664]

== Links ==

* Wikipedia Contributors, [[wikipedia:Saros (astronomy)|Saros (astronomy)]].

* [~~https~~://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]

* [http://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]

@@ Line 1: / Line 1: @@
-'''1803edo''', or the '''100 saros tuning''', divides the octave into steps of 665.6 millicents each. 1803 factors as 3 x 601.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-{{harmonics in equal|1803|columns=10}}
+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 ===
+{{Harmonics in equal|1803}}
+=== 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 {{nowrap|94076 ≡ 320 (mod 1803)}} years are leap, and using the maximal evenness method of finding rank-2 temperaments, the associated rank-2 temperament is {{nowrap|320 &amp; 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&nbsp;4s]], which is also described in the Solar Calendar Leap Rules page as {{dash|231, 293, 231, 293, 231, 293, 231|med}}. In addition, if one were to rearrange the steps (or raise the 4th degree by 62\1803) so they instead produce {{dash|231, 293, ''293'', ''231'', 231, 293, 231|med}}, 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 {{dash|45, 141, 45|med}}, and 293 into {{dash|45, 79, 45, 79, 45|med}} produces an uneven and a rather unique 29-tone scale. In addition, further dissecting 231 into {{dash|45, 45, 6, 45, 45, 45|med}} produces a 35-tone scale.
+==== Hectosaros Leap Day ====
+Hectosaros Leap Day is defined as {{nowrap|437 &amp; 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&nbsp;4s scale, but it is extremely hard, with step ratio of almost 17:1. 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&nbsp;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]].
-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.
+==== Hectosaros Lunisolar ====
+Hectosaros Lunisolar is defined as {{nowrap|664 &amp; 1803}} and is generated by 1078\1803, which measure about 717 cents, putting 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.
-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. Therefore, it creates 3 different calendric scales that can be used as leap cycles - 437-note leap day cycle with a 590\1803 generator (a major third), 320-note leap week cycle with 524\1803 generator (a neutral third), and the lunisolar cycle with the 1078\1803 generator (a slightly sharp fifth).
+The next scale which has a suitable hardness is the 92-note 5L&nbsp;87s scale, with step sizes of 30 and 19. It can be dissected into {{dash|372, 353, 372, 353, 353|med}} (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 [[Rectified Hebrew|eponymous temperament]], which only makes sense in [[353edo]] and other EDOs that support it).
+== Regular temperament properties ==
+=== Rank-2 temperaments by generator ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 524\1803
+| 348.752
+| 6144/3757
+| [[Hectosaros leap week]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-As such, 100 & 223 & 1803 saros temperament can be defined. In the 2.7.13.19.23.29 subgroup, 4952198/4950967 can be the starting comma.
 == 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
-* SolarLeapDay[437]
-* SolarLeapWeek[320]
-* Lunisolar[664]
 == Links ==
 * Wikipedia Contributors, [[wikipedia:Saros (astronomy)|Saros (astronomy)]].
-* [https://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]
+* [http://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]
+{{Todo| review }}