1803edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:31:55Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:31, 13 March 2026
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	\| [[Hectosaros leap week]]		\| [[Hectosaros leap week]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==

ArrowHead294 at 22:59, 20 February 2025

2025-02-20T22:59:16Z

← Older revision		Revision as of 22:59, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1803~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 17:55, 15 January 2025

2025-01-15T17:55:56Z

ArrowHead294 at 12:07, 16 November 2024

2024-11-16T12:07:06Z

← Older revision		Revision as of 12:07, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	=== Rank-2 temperaments by generator ===		=== Rank-2 temperaments by generator ===
	{{rank-2 ~~begin}}~~		{\| 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		\| 1
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	\| 6144/3757		\| 6144/3757
	\| [[Hectosaros leap week]]		\| [[Hectosaros leap week]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Scales ==		== Scales ==

ArrowHead294 at 05:31, 16 November 2024

2024-11-16T05:31:03Z

← Older revision		Revision as of 05:31, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	=== Rank-2 temperaments by generator ===		=== Rank-2 temperaments by generator ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	\|-
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	~~! Associated<br>Ratio~~
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\| 6144/3757		\| 6144/3757
	\| [[Hectosaros leap week]]		\| [[Hectosaros leap week]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Scales ==		== Scales ==
	* HectosarosLeapWeek[7], a MOS of type [[3L 4s]] (mosh) - 231 293 231 293 231 293 231		* 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		* Hectosaros Maqam Sikah, a MODMOS of type [[3L 4s]] (mosh) – 231 293 293 231 231 293 231

	== Links ==		== Links ==

FloraC: Expand on theory and misc. cleanup

2023-10-30T12:44:08Z

Expand on theory and misc. cleanup

← Older revision		Revision as of 12:44, 30 October 2023
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	== Theory ==		== 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 ===		=== 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.		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 ====		==== Hectosaros Lunisolar ====
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	=== Rank-2 temperaments by generator ===		=== Rank-2 temperaments by generator ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
			\|+Table of rank-2 temperaments by generator
			\|-
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*

FloraC: Clarify the title row of the rank-2 temp table

2023-10-15T14:24:34Z

Clarify the title row of the rank-2 temp table

← Older revision		Revision as of 14:24, 15 October 2023
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	== Theory ==		== 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.

			=== Odd harmonics ===
	{{Harmonics in equal\|1803}}		{{Harmonics in equal\|1803}}

	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 ===		=== Relationship to the saros cycle ===
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	=== Rank-2 temperaments by generator ===		=== Rank-2 temperaments by generator ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	! Periods<br>per ~~Octave~~		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio
	! Temperaments		! Temperaments
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	\| [[Hectosaros leap week]]		\| [[Hectosaros leap week]]
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Scales ==		== Scales ==

FloraC: -redundant categories. This is just irrelevant to arabic music. If anything the mosh scale might be relevant

2023-08-29T08:48:16Z

-redundant categories. This is just irrelevant to arabic music. If anything the mosh scale might be relevant

← Older revision		Revision as of 08:48, 29 August 2023
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	== Theory ==		== Theory ==
	{{~~harmonics~~ in equal\|1803~~\|columns=10~~}}		{{Harmonics in equal\|1803}}

	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.		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.
Line 55:		Line 55:
	* [http://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]		* [http://individual.utoronto.ca/kalendis/leap/index.htm Solar Calendar Leap Rules]

	~~[[Category:Equal divisions of the octave\|####]] <!-- 4-digit number -->~~
	~~[[Category:Arabic]]~~
	~~[[Category:Maqam]]~~
	{{Todo\| review }}		{{Todo\| review }}

BudjarnLambeth at 06:57, 9 July 2023

2023-07-09T06:57:45Z

← Older revision		Revision as of 06:57, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|1803}}		{{EDO intro\|1803}}

BudjarnLambeth at 04:51, 9 July 2023

2023-07-09T04:51:46Z

← Older revision		Revision as of 04:51, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{Infobox ET}}
	{{EDO intro\|1803}}		{{EDO intro\|1803}}

← Older revision		Revision as of 17:55, 15 January 2025
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	=== Relationship to the saros cycle ===		=== 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.		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.		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 ====		==== 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 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.		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]].		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 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.		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 ====
	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.		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]].		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 ====
	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.		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.

	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 [[Rectified Hebrew\|eponymous temperament]], which only makes sense in [[353edo]] and other EDOs that support it).		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 ==		== Regular temperament properties ==
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	\| [[Hectosaros leap week]]		\| [[Hectosaros leap week]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Scales ==		== Scales ==
	* HectosarosLeapWeek[7], a MOS of type [[3L 4s]] (mosh) ~~–~~ 231 293 231 293 231 293 231		* 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		* Hectosaros Maqam Sikah, a MODMOS of type [[3L 4s]] (mosh) – 231 293 293 231 231 293 231

	== Links ==		== Links ==