1789edo: Difference between revisions

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== Theory ==

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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 ===

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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 ===

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 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 ===

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

== Table of selected intervals ==

{| 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

! Eliora's ~~Naming System~~

! Eliora's naming system

! JI ~~Approximation~~ or ~~Other Interpretations~~*

! JI approximation or other interpretations*

|-

| 0

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| 1046

| Minor fifth

| [[3/2]]†

| [[3/2]]**

|-

| 1047

| Major fifth

| [[3/2]]†

| [[3/2]]**

|-

| 1213

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| 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

<nowiki />** 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| {{monzo| -5671 1789 }}

| {{mapping| 1789 5671 }}

| -0.00044

| −0.00044

| 0.00044

| 0.06

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| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}

| {{mapping| 1789 5671 4154 }}

| -0.00710

| −0.00710

| 0.00942

| 1.40

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| 0.04093

| 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

| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}

| {{mapping| 1789 4154 6189 6620 8691 }}

| -0.00591

| −0.00591

| 0.01272

| 1.90

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| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321

| {{mapping| 1789 4154 6189 6620 8691 8863 }}

| -0.00363

| −0.00363

| 0.01268

| 1.89

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-4"

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

|+Table of rank-2 temperaments by generator

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperament

|-

| 1

| 35\1789

| 23.48

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| [[Commatose]]

|-

| "

| 125\1789

| 83.85

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| [[Sextilimeans]]

|-

| "

| 144\1789

| 96.59

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| [[Hemiluna]] (1789bd)

|-

| "

| 172\1789

| 115.37

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| [[Tricesimoprimal miracloid]]

|-

| "

| 377\1789

| 252.88

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| [[Double bastille]]

|-

| "

| 576\1789

| 386.36

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| [[French decimal]]

|-

| "

| 754\1789

| 505.76

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| [[Pure bastille]]

|-

| "

| 777\1789

| 521.18

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| [[Maviloid]]

|-

| "

| 778\1789

| 521.86

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| [[Estates general]]

|-

| "

| 822\1789

| 551.37

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| [[Onzonic]]

|-

| "

| 865\1789

| 580.21

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| [[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 />* [[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 ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1789}}
+{{ED intro}}
 == Theory ==
@@ 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.
+bd 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 ===
@@ Line 13: / Line 15: @@
 {{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 &amp; 1789}}, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.
-edo 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.
+edo 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 ===
-edo 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.
+edo 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.
-edo 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.
+edo 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 ===
 edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]].
 == Table of selected intervals ==
 {| 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
-! Eliora's Naming System
+! Eliora's naming system
-! JI Approximation or Other Interpretations*
+! JI approximation or other interpretations*
 |-
 | 0
@@ Line 128: / Line 131: @@
 | 1046
 | Minor fifth
-| [[3/2]]†
+| [[3/2]]**
 |-
 | 1047
 | Major fifth
-| [[3/2]]†
+| [[3/2]]**
 |-
 | 1213
@@ Line 158: / Line 161: @@
 | 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
+<nowiki />** 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 176: / Line 180: @@
 | {{monzo| -5671 1789 }}
 | {{mapping| 1789 5671 }}
-| -0.00044
+| −0.00044
 | 0.00044
 | 0.06
@@ Line 183: / Line 187: @@
 | {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
 | {{mapping| 1789 5671 4154 }}
-| -0.00710
+| −0.00710
 | 0.00942
 | 1.40
@@ Line 193: / Line 197: @@
 | 0.04093
 | 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
 | 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
 | {{mapping| 1789 4154 6189 6620 8691 }}
-| -0.00591
+| −0.00591
 | 0.01272
 | 1.90
@@ Line 211: / Line 215: @@
 | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
 | {{mapping| 1789 4154 6189 6620 8691 8863 }}
-| -0.00363
+| −0.00363
 | 0.01268
 | 1.89
@@ Line 217: / Line 221: @@
 === Rank-2 temperaments ===
-{| class="wikitable center-all left-4"
+{| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperament
 |-
+| 1
 | 35\1789
 | 23.48
@@ Line 230: / Line 236: @@
 | [[Commatose]]
 |-
+| "
 | 125\1789
 | 83.85
@@ Line 235: / Line 242: @@
 | [[Sextilimeans]]
 |-
+| "
 | 144\1789
 | 96.59
@@ Line 240: / Line 248: @@
 | [[Hemiluna]] (1789bd)
 |-
+| "
 | 172\1789
 | 115.37
@@ Line 245: / Line 254: @@
 | [[Tricesimoprimal miracloid]]
 |-
+| "
 | 377\1789
 | 252.88
@@ Line 250: / Line 260: @@
 | [[Double bastille]]
 |-
+| "
 | 576\1789
 | 386.36
@@ Line 255: / Line 266: @@
 | [[French decimal]]
 |-
+| "
 | 754\1789
 | 505.76
@@ Line 260: / Line 272: @@
 | [[Pure bastille]]
 |-
+| "
 | 777\1789
 | 521.18
@@ Line 265: / Line 278: @@
 | [[Maviloid]]
 |-
+| "
 | 778\1789
 | 521.86
@@ Line 270: / Line 284: @@
 | [[Estates general]]
 |-
+| "
 | 822\1789
 | 551.37
@@ Line 275: / Line 290: @@
 | [[Onzonic]]
 |-
+| "
 | 865\1789
 | 580.21
@@ Line 280: / Line 296: @@
 | [[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 />* [[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 ==