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'''1789 EDO''' divides the octave into equal steps of 0.67 cents each. It is the 278th [[prime edo]]. Perhaps the most notable fact about 1789edo, is the fact that it tempers out the '''jacobin comma''' ([[6656/6655]]), 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 it's number is the hallmark year of the French Revolution, thus making the temperance of the Jacobin comma on topic.
{{Infobox ET}}
{{Wikipedia|1789}}
{{ED intro}}  
==Theory==
{{primes in edo|1789|columns = 18}}


1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup.  
== Theory ==
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.  


Since 1789edo contains the 2.5 subgroup, it can be used for the finite 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. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. This property of 1789edo is amplified by poor approximation of 3 and 7, allowing for cognitive separation of the intervals (or whatever is left of it at such small step size).
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.


In addition, 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]].
=== Odd harmonics ===
{{Harmonics in equal|1789}}


The "proper" Jacobin temperament in 1789edo, the maximum evenness scale that uses 822 as a generator, contains only 37 notes. The step sizes are 48 and 49, making them indistinguishable to human ear at this scale. This can be fixed by using divisors of 822 as a generaator, for example 137\1789 "6th root of 11/8" temperament having 222 notes. In addition, this can be re-interpreted by using 13/10 as a generator instead which produces a more vibrant 1205 out of 1789, and partitioning the resulting 13/5s in three around the octave. Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale.
=== Jacobin temperaments ===
{{Main| The Jacobins }}


== Tempered commas ==
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''.
{| class="wikitable"
 
|+Tempered commas in 1789edo
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. 
!Prime
 
Subgroup
=== Other ===
!Val
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.
!Ratio
 
!Monzo
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.
<small>(zeroes skipped for clarity in subgroups)</small>
 
!Cents
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 Steps
 
!Name
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 & 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="font-size: 105%; white-space: nowrap;" | Selected intervals in 1789edo
|-
|-
|2.5
! Step
|Patent
! Eliora's naming system
|(1251 digits/1251 digits)
! JI approximation or other interpretations*
|[4154,-1789⟩
|84.766
|126.371
|French decimalisma
|-
|-
|7-limit
| 0
|Patent
| Unison
|[[2401/2400]]
| 1/1
|[-5,-1,-2,4⟩
|0.721
|1.075
|Breedsma
|-
|-
|2.5.11.13
| 25
|Patent
| Oquatonic comma
|[[6656/6655]]
| {{monzo| 65 -28 }}
|[9,-1,-3,1⟩
|0.260
|0.388
|Jacobin comma
|-
|-
|13-limit
| 35
|1789bdeef
| Pythagorean comma
|[[10648/10647]]
| [[531441/524288]]
|[3,-2,0,-1,3,-2⟩
|0.163
|0.242
|Harmonisma
|-
|-
|2.5.11.13.31
| 36
|Patent
|387283/387200
|[-7,-2,-2,1,3⟩
|
|
|0.553
| 145/143
|-
| 61
| Lesser diesis
| [[128/125]]
|-
| 74
|
|
| 319/310
|-
|-
|2.5.11.13.31
| 122
|Patent
|2640704/2640625
|[6,-6,3,-2,1⟩
|
|
| 65/62
|-
| 125
| Sextilimeans generator
| 16807/16000
|-
| 172
| Tricesimoprimal Miracle semitone
| [[31/29]]
|-
| 226
|
|
| 440/403
|-
| 290
| Jacobin minor interval
| 160/143, 649/580
|-
| 338
| Minor sqrt(13/10)
|
|-
| 339
| Major sqrt(13/10)
| {{monzo| -69 0 0 0 20 }}
|-
| 387
| Jacobin major interval
| 754/649
|-
| 523
| Breedsmic neutral third
| 49/40, 60/49
|-
| 576
| Major third
| [[5/4]]
|-
| 677
| Jacobin naiadic
| [[13/10]]
|-
| 750
| Sextilimeans fourth
|
|
|-
|-
|2.11.13.29.31
| 777
|Patent
| Maviloid generator
|3455881/3455756
| 875/648
|[-2,2,4,-1,-3⟩
|-
|
| 822
|
| Jacobin superfourth, Mongolian fourth
| [[11/8]]
|-
| 1032
| Secor fifth, Tricesimoprimal Miracle fifth
| (31/29)<sup>6</sup>
|-
| 1039
| Sextilimeans fifth
|
|
|-
|-
|2.5.11.13.29.31
| 1046
|Patent
| Minor fifth
|38132480000/38130225991
| [[3/2]]**
|[11,4,-1,-2,-5,3⟩
|-
|
| 1047
|
| Major fifth
|
| [[3/2]]**
|-
| 1213
| Classical minor sixth
| [[8/5]]
|-
| 1444
| Harmonic seventh
| [[7/4]]
|-
| 1535
| 29th harmonic
| [[29/16]]
|-
| 1579
| 59th harmonic
| [[59/32]]
|-
| 1707
| 31st harmonic
| [[31/16]]
|-
| 1789
| Octave
| 2/1
|}
<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable
 
<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]]
! 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
| 0.06
|-
| 2.9.5
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
| {{mapping| 1789 5671 4154 }}
| −0.00710
| 0.00942
| 1.40
|-
| 2.9.5.7
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
| {{mapping| 1789 5671 4154 5022 }}
| +0.01606
| 0.04093
| 6.10
|- style="border-top: double;"
| 2.5.11.13
| 6656/6655, {{monzo| 43 -18  5 -5 }}, {{monzo| -38 -32 10 21 }}
| {{mapping| 1789 4154 6189 6620}}
| −0.00490
| 0.01405
| 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
| 0.01272
| 1.90
|-
| 2.5.11.13.29.31
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
| {{mapping| 1789 4154 6189 6620 8691 8863 }}
| −0.00363
| 0.01268
| 1.89
|}
|}


== Table of selected intervals ==
=== Rank-2 temperaments ===
{| class="wikitable collapsible mw-collapsed"
{| class="wikitable center-all left-5"
|+
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
Selected intervals in 1789 EDO
!Step
!Name
!JI Approximation, Monzo, or another interpretation
|-
|-
|0
! Periods<br>per 8ve
|Unison
! Generator*
|1/1 exact
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
|25
| 1
|28-thirds comma
| 35\1789
|[65 -28]
| 23.48
| 531441/524288
| [[Commatose]]
|-
|-
|61
| "
|Lesser diesis
| 125\1789
|[[128/125]]
| 83.85
| 16807/16000
| [[Sextilimeans]]
|-
|-
|172
| "
|Tricesimoprimal Miracle semitone
| 144\1789
|[[31/29]]
| 96.59
| 200/189
| [[Hemiluna]] (1789bd)
|-
|-
|338
| "
|Minor sqrt(13/10)
| 172\1789
|[[Square root of 13 over 10]] I,
| 115.37
| 31/29
| [[Tricesimoprimal miracloid]]
|-
|-
|339
| "
|Major sqrt(13/10)
| 377\1789
|[[Square root of 13 over 10]] II, (11/8)^20
| 252.88
| 53094899/45875200
| [[Double bastille]]
|-
|-
|576
| "
|Major third
| 576\1789
|[[5/4]]
| 386.36
| 5/4
| [[French decimal]]
|-
|-
|677
| "
|Jacobin naiadic
| 754\1789
|[[13/10]]
| 505.76
| {{monzo| 104 0 57 0 -14 5 }}
| [[Pure bastille]]
|-
|-
|822
| "
|Jacobin superfourth
| 777\1789
|[[11/8]]
| 521.18
| 875/648
| [[Maviloid]]
|-
|-
|1535
| "
|29th harmonic
| 778\1789
|[[29/16]]
| 521.86
| 80275/59392
| [[Estates general]]
|-
|-
|1707
| "
|31st harmonic
| 822\1789
|[[31/16]]
| 551.37
| 11/8
| [[Onzonic]]
|-
|-
|1789
| "
|Octave
| 865\1789
|2/1 exact
| 580.21
| 6875/4914
| [[Eternal revolutionary]] (1789bd)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=1zrnsGODQSg ''Etude la (R)evolution''] (2022)


==Scales==
[[Category:Jacobin]]
* Jacobin[37]
[[Category:Listen]]
* Jacobin[74]
* Jacobin[111]
* Jacobin[222]
* Decimal[265]
* Decimal[1524]


[[Category:Equal divisions of the octave]]
{{Todo| review | clarify }}
[[Category:Prime EDO]]
Retrieved from "https://en.xen.wiki/w/1789edo"