1789edo: Difference between revisions

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== Theory ==

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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. 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 ===

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

== Table of selected intervals ==

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|-

! 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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<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 ==

{~~{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

| {{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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| 6656/6655, {{monzo| 43 -18 5 -5 }}, {{monzo| -38 -32 10 21 }}

| {{mapping| 1789 4154 6189 6620}}

| ~~−0~~.00490

| −0.00490

| 0.01405

| 2.09

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| 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

~~{{comma basis end}~~}

|}

=== 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

|-

| 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

| 6875/4914

| [[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 distinct

== Music ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1789}}
+{{ED intro}}
 == Theory ==
@@ Line 13: / Line 13: @@
 {{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&mdash;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&mdash;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.
-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&mdash;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 ==
@@ Line 36: / Line 36: @@
 |-
 ! Step
-! Eliora's Naming System
+! Eliora's naming system
-! JI Approximation or Other Interpretations*
+! JI approximation or other interpretations*
 |-
 | 0
@@ Line 129: / Line 129: @@
 | 1046
 | Minor fifth
-| [[3/2]]†
+| [[3/2]]**
 |-
 | 1047
 | Major fifth
-| [[3/2]]†
+| [[3/2]]**
 |-
 | 1213
@@ Line 161: / Line 161: @@
 <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 ==
-{{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
 | {{monzo| -5671 1789 }}
 | {{mapping| 1789 5671 }}
-| &minus;0.00044
+| −0.00044
 | 0.00044
 | 0.06
@@ Line 176: / Line 185: @@
 | {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
 | {{mapping| 1789 5671 4154 }}
-| &minus;0.00710
+| −0.00710
 | 0.00942
 | 1.40
@@ Line 190: / Line 199: @@
 | 6656/6655, {{monzo| 43 -18  5 -5 }},  {{monzo| -38 -32 10 21 }}
 | {{mapping| 1789 4154 6189 6620}}
-| &minus;0.00490
+| −0.00490
 | 0.01405
 | 2.09
@@ Line 197: / Line 206: @@
 | 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
 | {{mapping| 1789 4154 6189 6620 8691 }}
-| &minus;0.00591
+| −0.00591
 | 0.01272
 | 1.90
@@ Line 204: / Line 213: @@
 | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
 | {{mapping| 1789 4154 6189 6620 8691 8863 }}
-| &minus;0.00363
+| −0.00363
 | 0.01268
 | 1.89
-{{comma basis end}}
+|}
 === 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
+|-
+| 1
 | 35\1789
 | 23.48
@@ Line 217: / Line 234: @@
 | [[Commatose]]
 |-
+| "
 | 125\1789
 | 83.85
@@ Line 222: / Line 240: @@
 | [[Sextilimeans]]
 |-
+| "
 | 144\1789
 | 96.59
@@ Line 227: / Line 246: @@
 | [[Hemiluna]] (1789bd)
 |-
+| "
 | 172\1789
 | 115.37
@@ Line 232: / Line 252: @@
 | [[Tricesimoprimal miracloid]]
 |-
+| "
 | 377\1789
 | 252.88
@@ Line 237: / Line 258: @@
 | [[Double bastille]]
 |-
+| "
 | 576\1789
 | 386.36
@@ Line 242: / Line 264: @@
 | [[French decimal]]
 |-
+| "
 | 754\1789
 | 505.76
@@ Line 247: / Line 270: @@
 | [[Pure bastille]]
 |-
+| "
 | 777\1789
 | 521.18
@@ Line 252: / Line 276: @@
 | [[Maviloid]]
 |-
+| "
 | 778\1789
 | 521.86
@@ Line 257: / Line 282: @@
 | [[Estates general]]
 |-
+| "
 | 822\1789
 | 551.37
@@ Line 262: / Line 288: @@
 | [[Onzonic]]
 |-
+| "
 | 865\1789
 | 580.21
 | 6875/4914
 | [[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 distinct
 == Music ==