1789edo - Revision history

Eliora at 05:59, 5 June 2026

2026-06-05T05:59:36Z

← Older revision		Revision as of 05:59, 5 June 2026
Line 6:		Line 6:

	For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempering out\|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.		For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempering out\|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.

			1789bd val, {{Val\|1789 '''2836''' 4154 '''5023'''}} is better tuned than the patent val, and it tempers out 67108864/66976875, 48828125/48771072, 96889010407/96855122250.

	=== Odd harmonics ===		=== Odd harmonics ===
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	1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.		1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.

	Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit~~. It is also worth noting that 1789bd val is better tuned than the patent val~~.		Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit.

	1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap\|460 & 1789}} temperament.		1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap\|460 & 1789}} temperament.

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"

2026-03-13T13:33:17Z

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 13:33, 13 March 2026
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	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	\|}		\|}
	<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

	== Music ==		== Music ==

ArrowHead294 at 22:52, 20 February 2025

2025-02-20T22:52:04Z

← Older revision		Revision as of 22:52, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|1789~~}}		{{ED intro}}

	== Theory ==		== Theory ==
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	\|-		\|-
	! Step		! Step
	! Eliora's ~~Naming System~~		! Eliora's naming system
	! JI ~~Approximation~~ or ~~Other Interpretations~~*		! JI approximation or other interpretations*
	\|-		\|-
	\| 0		\| 0
Line 129:		Line 129:
	\| 1046		\| 1046
	\| Minor fifth		\| Minor fifth
	\| [[3/2]]†		\| [[3/2]]**
	\|-		\|-
	\| 1047		\| 1047
	\| Major fifth		\| Major fifth
	\| [[3/2]]†		\| [[3/2]]**
	\|-		\|-
	\| 1213		\| 1213
Line 161:		Line 161:
	<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable		<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable

	<~~sup>&dagger;<~~/~~sup~~> 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val		<nowiki />** 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val

	== Regular temperament properties ==		== Regular temperament properties ==
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	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	\|}		\|}
	<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

FloraC: /* Regular temperament properties */ fix broken table

2025-01-17T16:49:55Z

Regular temperament properties: fix broken table

← Older revision		Revision as of 16:49, 17 January 2025
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	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br />8ve stretch (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning error		! colspan="2" \| Tuning error
	\|-		\|-
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperament		! Temperament
	\|-		\|-
			\| 1
	\| 35\1789		\| 35\1789
	\| 23.48		\| 23.48
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	\| [[Commatose]]		\| [[Commatose]]
	\|-		\|-
			\| "
	\| 125\1789		\| 125\1789
	\| 83.85		\| 83.85
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	\| [[Sextilimeans]]		\| [[Sextilimeans]]
	\|-		\|-
			\| "
	\| 144\1789		\| 144\1789
	\| 96.59		\| 96.59
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	\| [[Hemiluna]] (1789bd)		\| [[Hemiluna]] (1789bd)
	\|-		\|-
			\| "
	\| 172\1789		\| 172\1789
	\| 115.37		\| 115.37
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	\| [[Tricesimoprimal miracloid]]		\| [[Tricesimoprimal miracloid]]
	\|-		\|-
			\| "
	\| 377\1789		\| 377\1789
	\| 252.88		\| 252.88
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	\| [[Double bastille]]		\| [[Double bastille]]
	\|-		\|-
			\| "
	\| 576\1789		\| 576\1789
	\| 386.36		\| 386.36
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	\| [[French decimal]]		\| [[French decimal]]
	\|-		\|-
			\| "
	\| 754\1789		\| 754\1789
	\| 505.76		\| 505.76
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	\| [[Pure bastille]]		\| [[Pure bastille]]
	\|-		\|-
			\| "
	\| 777\1789		\| 777\1789
	\| 521.18		\| 521.18
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	\| [[Maviloid]]		\| [[Maviloid]]
	\|-		\|-
			\| "
	\| 778\1789		\| 778\1789
	\| 521.86		\| 521.86
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	\| [[Estates general]]		\| [[Estates general]]
	\|-		\|-
			\| "
	\| 822\1789		\| 822\1789
	\| 551.37		\| 551.37
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	\| [[Onzonic]]		\| [[Onzonic]]
	\|-		\|-
			\| "
	\| 865\1789		\| 865\1789
	\| 580.21		\| 580.21
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	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct		<nowiki/>* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

FloraC: /* Theory */ cleanup

2025-01-17T16:47:58Z

Theory: cleanup

← Older revision		Revision as of 16:47, 17 January 2025
Line 5:		Line 5:
	1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1\|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1\|5]], [[9/1\|9]], [[11/1\|11]], [[13/1\|13]] and [[21/1\|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.		1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1\|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1\|5]], [[9/1\|9]], [[11/1\|11]], [[13/1\|13]] and [[21/1\|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.

	For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.		For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempering out\|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 15:		Line 15:
	Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is {{nowrap\|37 & 1789}}, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.		Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is {{nowrap\|37 & 1789}}, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.

	1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as {{nowrap\|1789 &~~amp;~~ 3125}}. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.		1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as {{nowrap\|1789 & 3125}}. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.

	=== Other ===		=== Other ===
Line 22:		Line 22:
	Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.		Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.

	1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap\|460 &~~amp;~~ 1789}} temperament.		1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap\|460 & 1789}} temperament.

	Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a {{nowrap\|52 &~~amp;~~ 1789}} temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.		Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a {{nowrap\|52 & 1789}} temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.

	On the patent val in the 7-limit, 1789edo supports {{nowrap\|99 &~~amp;~~ 373}} temperament called maviloid. In addition, it also tempers out [[2401/2400]].		On the patent val in the 7-limit, 1789edo supports {{nowrap\|99 & 373}} temperament called maviloid. In addition, it also tempers out [[2401/2400]].

	=== Subsets and supersets ===		=== Subsets and supersets ===
	1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]].		1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]].

	== Table of selected intervals ==		== Table of selected intervals ==

ArrowHead294 at 15:32, 16 January 2025

2025-01-16T15:32:53Z

← Older revision		Revision as of 15:32, 16 January 2025
Line 5:		Line 5:
	1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1\|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1\|5]], [[9/1\|9]], [[11/1\|11]], [[13/1\|13]] and [[21/1\|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.		1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1\|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1\|5]], [[9/1\|9]], [[11/1\|11]], [[13/1\|13]] and [[21/1\|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.

	For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[~~tempering out\|~~tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.		For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 13:		Line 13:
	{{Main\| The Jacobins }}		{{Main\| The Jacobins }}

	Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is 37 & 1789, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.		Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is {{nowrap\|37 & 1789}}, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.

	1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as 1789 & 3125. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.		1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as {{nowrap\|1789 & 3125}}. This is referencing the fact that Estates General were called by Louis XVI on 5th May 1789, written as 05/05, and 3125 is 5 to the 5th power and also provides an optimal patent val for tempering out the jacobin comma, contuing the lore.

	=== Other ===		=== Other ===
	1789edo can be used for the finite "French decimal" ~~temperament—that~~ is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.		1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.

	Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] ~~temperaments—more~~ exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.		Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.

	1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.		1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a {{nowrap\|460 & 1789}} temperament.

	Since 1789edo has a very precise 31/29, it supports tricesimoprimal ~~miracloid—a~~ version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.		Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a {{nowrap\|52 & 1789}} temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.

	On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]].		On the patent val in the 7-limit, 1789edo supports {{nowrap\|99 & 373}} temperament called maviloid. In addition, it also tempers out [[2401/2400]].

	=== Subsets and supersets ===		=== Subsets and supersets ===
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	\| {{monzo\| -5671 1789 }}		\| {{monzo\| -5671 1789 }}
	\| {{mapping\| 1789 5671 }}		\| {{mapping\| 1789 5671 }}
	\| ~~−0~~.00044		\| −0.00044
	\| 0.00044		\| 0.00044
	\| 0.06		\| 0.06
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	\| {{monzo\| -70 36 -19 }}, {{monzo\| 129 -7 -46 }}		\| {{monzo\| -70 36 -19 }}, {{monzo\| 129 -7 -46 }}
	\| {{mapping\| 1789 5671 4154 }}		\| {{mapping\| 1789 5671 4154 }}
	\| ~~−0~~.00710		\| −0.00710
	\| 0.00942		\| 0.00942
	\| 1.40		\| 1.40
Line 199:		Line 199:
	\| 6656/6655, {{monzo\| 43 -18 5 -5 }}, {{monzo\| -38 -32 10 21 }}		\| 6656/6655, {{monzo\| 43 -18 5 -5 }}, {{monzo\| -38 -32 10 21 }}
	\| {{mapping\| 1789 4154 6189 6620}}		\| {{mapping\| 1789 4154 6189 6620}}
	\| ~~−0~~.00490		\| −0.00490
	\| 0.01405		\| 0.01405
	\| 2.09		\| 2.09
Line 206:		Line 206:
	\| 6656/6655, 371293/371200, {{monzo\| -18 -6 -1 3 5 }}, {{monzo\| 34 -20 5 0 -1 }}		\| 6656/6655, 371293/371200, {{monzo\| -18 -6 -1 3 5 }}, {{monzo\| 34 -20 5 0 -1 }}
	\| {{mapping\| 1789 4154 6189 6620 8691 }}		\| {{mapping\| 1789 4154 6189 6620 8691 }}
	\| ~~−0~~.00591		\| −0.00591
	\| 0.01272		\| 0.01272
	\| 1.90		\| 1.90
Line 213:		Line 213:
	\| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321		\| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
	\| {{mapping\| 1789 4154 6189 6620 8691 8863 }}		\| {{mapping\| 1789 4154 6189 6620 8691 8863 }}
	\| ~~−0~~.00363		\| −0.00363
	\| 0.01268		\| 0.01268
	\| 1.89		\| 1.89
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	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	\|}		\|}
	<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

	== Music ==		== Music ==

ArrowHead294 at 11:39, 16 November 2024

2024-11-16T11:39:35Z

← Older revision		Revision as of 11:39, 16 November 2024
Line 164:		Line 164:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.9		\| 2.9
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	\| 0.01268		\| 0.01268
	\| 1.89		\| 1.89
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{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*
			! Temperament
	\|-		\|-
	\| 35\1789		\| 35\1789
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	\| 6875/4914		\| 6875/4914
	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	~~{{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

	== Music ==		== Music ==

ArrowHead294 at 06:16, 16 November 2024

2024-11-16T06:16:09Z

← Older revision		Revision as of 06:16, 16 November 2024
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	=== Other ===		=== Other ===
	1789edo can be used for the finite "French decimal" temperament - that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.		1789edo can be used for the finite "French decimal" temperament—that is, where all the interval targets in just intonation are expressed as terminating decimals. For example, [[5/4]], [[25/16]], [[128/125]], [[32/25]], 625/512, etc.

	Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments - more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.		Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] temperaments—more exactly those which are divisors of 576, and that includes all from [[2ed5/4]] to [[9ed5/4]], skipping [[7ed5/4]]. One such scale which stands for [[4ed5/4]], is a tuning for the [[hemiluna]] temperament in the 1789bd val in the 13-limit. It is also worth noting that 1789bd val is better tuned than the patent val.

	1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.		1789edo has an essentially perfect [[9/8]], a very common interval. 1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator, which is excess of six 9/8s over the octave in this case. It is defined as a 460 & 1789 temperament.

	Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid - a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.		Since 1789edo has a very precise 31/29, it supports tricesimoprimal miracloid—a version of secor with 31/29 as the generator and a flat, meantone-esque fifth of about 692.23 cents. Using the maximal evenness method, we find a 52 & 1789 temperament. Best subgroup for it is 2.5.7.11.19.29.31, since both 52edo and 1789edo support it well, and the comma basis is 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688.

	On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]].		On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]].
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	== Table of selected intervals ==		== Table of selected intervals ==
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style=white-space:nowrap \| Selected intervals in 1789edo		\|+ style="font-size: 105%; white-space: nowrap;" \| Selected intervals in 1789edo
			\|-
	! Step		! Step
	! Eliora's Naming System		! Eliora's Naming System
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	\| 2/1		\| 2/1
	\|}		\|}
	<nowiki>*~~</nowiki> based~~ on the 2.5.11.13.29.31 subgroup where applicable		<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable

	† 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val		<sup>&dagger;</sup> 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.9		\| 2.9
	\| {{monzo\| -5671 1789 }}		\| {{monzo\| -5671 1789 }}
	\| {{mapping\| 1789 5671 }}		\| {{mapping\| 1789 5671 }}
	\| -0.00044		\| −0.00044
	\| 0.00044		\| 0.00044
	\| 0.06		\| 0.06
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	\| {{monzo\| -70 36 -19 }}, {{monzo\| 129 -7 -46 }}		\| {{monzo\| -70 36 -19 }}, {{monzo\| 129 -7 -46 }}
	\| {{mapping\| 1789 5671 4154 }}		\| {{mapping\| 1789 5671 4154 }}
	\| -0.00710		\| −0.00710
	\| 0.00942		\| 0.00942
	\| 1.40		\| 1.40
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	\| 0.04093		\| 0.04093
	\| 6.10		\| 6.10
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.5.11.13		\| 2.5.11.13
	~~\| style="border-top: double;"~~ \| 6656/6655, {{monzo\| 43 -18 5 -5 }}, {{monzo\| -38 -32 10 21 }}		\| 6656/6655, {{monzo\| 43 -18 5 -5 }}, {{monzo\| -38 -32 10 21 }}
	~~\| style="border-top: double;"~~ \| {{mapping\| 1789 4154 6189 6620}}		\| {{mapping\| 1789 4154 6189 6620}}
	\| ~~style="border-top: double~~;~~" \| -~~0.00490		\| −0.00490
	~~\| style="border-top: double;"~~ \| 0.01405		\| 0.01405
	~~\| style="border-top: double;"~~ \| 2.09		\| 2.09
	\|-		\|-
	\| 2.5.11.13.29		\| 2.5.11.13.29
	\| 6656/6655, 371293/371200, {{monzo\| -18 -6 -1 3 5 }}, {{monzo\| 34 -20 5 0 -1 }}		\| 6656/6655, 371293/371200, {{monzo\| -18 -6 -1 3 5 }}, {{monzo\| 34 -20 5 0 -1 }}
	\| {{mapping\| 1789 4154 6189 6620 8691 }}		\| {{mapping\| 1789 4154 6189 6620 8691 }}
	\| -0.00591		\| −0.00591
	\| 0.01272		\| 0.01272
	\| 1.90		\| 1.90
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	\| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321		\| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
	\| {{mapping\| 1789 4154 6189 6620 8691 8863 }}		\| {{mapping\| 1789 4154 6189 6620 8691 8863 }}
	\| -0.00363		\| −0.00363
	\| 0.01268		\| 0.01268
	\| 1.89		\| 1.89
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-4"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	\|-
	! Generator*
	! Cents*
	~~! Associated<br>Ratio~~
	~~! Temperament~~
	\|-		\|-
	\| 35\1789		\| 35\1789
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	\| 6875/4914		\| 6875/4914
	\| [[Eternal revolutionary]] (1789bd)		\| [[Eternal revolutionary]] (1789bd)
	\|}		{{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}}

	== Music ==		== Music ==

Eliora: /* Rank-2 temperaments */ new mapping

2023-11-09T01:01:09Z

Rank-2 temperaments: new mapping

← Older revision		Revision as of 01:01, 9 November 2023
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	\| 865\1789		\| 865\1789
	\| 580.21		\| 580.21
	\| ~~{{monzo\| 294 0 -46 0 7 -57 }}~~		\| 6875/4914
	\| [[Eternal revolutionary]]		\| [[Eternal revolutionary]] (1789bd)
	\|}		\|}
	<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct		<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

FloraC: Rework on theory (the no-3 subgroup can't contain both 7 and 21); misc. cleanup

2023-10-30T13:00:52Z

Rework on theory (the no-3 subgroup can't contain both 7 and 21); misc. cleanup

← Older revision		Revision as of 13:00, 30 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	1789edo ~~can be adapted for use with~~ the 2.5~~.11.13.29.31.47.59.61~~ [[~~subgroup~~]]. ~~Perhaps the most notable fact about 1789edo~~, is ~~the fact that it tempers out the jacobin comma (~~[[~~6656~~/~~6655~~]]), and it ~~is also consistent on the subgroup~~ 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.		1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1\|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1\|5]], [[9/1\|9]], [[11/1\|11]], [[13/1\|13]] and [[21/1\|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation.

	1789edo ~~is consistent in~~ the ~~no-threes~~ 13~~-odd-limit~~. ~~Since its double,~~ [[~~3578edo~~]]~~, is consistent in~~ the ~~21-odd-limit, it can be thought of as a~~ [[~~KN subgroups\|21789~~]] 2.9.5.7.11.13.~~225.289.361.21 subgroup temperament~~, on ~~which it shares mapping with 3578edo and tempers out the same commas~~.		For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is that it [[tempering out\|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent on the subgroup 2.5.11.13 of the comma, which is quite appropriate for edo's number. Although there are temperaments which are better suited for tempering this comma, 1789edo is unique in that its number is the hallmark year of the French Revolution, thus making the tempering of the jacobin comma on topic.

	=== Odd harmonics ===		=== Odd harmonics ===
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	=== Jacobin temperaments ===		=== Jacobin temperaments ===
	{{Main \| The Jacobins }}		{{Main\| The Jacobins }}

	Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is 37 & 1789, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.		Since 1789edo tempers out the jacobin comma and it is defined by stacking three 11/8s to reach 13/10, one can use that as a generator. The resulting temperament is 37 & 1789, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	1789edo is the 278th [[prime edo]].		1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]].

	== Table of selected intervals ==		== Table of selected intervals ==
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-4"		{\| class="wikitable center-all left-4"
			\|+Table of rank-2 temperaments by generator
			\|-
	! Generator*		! Generator*
	! Cents*		! Cents*