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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).
Perhaps the most notable fact about 1789edo is that it [[tempering out|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent to distance 2 on the subgroup 2.5.11.13 of the comma, which is naming-wise appropriate for edo's number while also providing an extremely strong tuning. 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]].  
2.9.5.11.13 subgroup is also represented strongly in 1789edo, where it tunes a temperament called ''commatose'', defined as a {{nowrap|460 & 1789}}, which uses the Pythagorean comma as a generator. In the full 13-limit, 1789bd val, {{Val|1789 '''2836''' 4154 '''5023'''}} is of additional interest as it is better tuned than the patent val, where it tunes [[hemiluna]].  


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


== Tempered commas ==
=== Odd harmonics ===
{| class="wikitable"
{{Harmonics in equal|1789}}
|+Tempered commas in 1789edo
 
!Prime
=== Jacobin temperaments ===
Subgroup
{{Main| The Jacobins }}
!Val
 
!Ratio
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''.
!Monzo
 
<small>(zeroes skipped for clarity in subgroups)</small>
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. 
!Cents
 
!1789edo Steps
=== Miscellany ===
!Name
For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. [[45/32]] and [[51/32]] are also strongly approximated.
 
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 of these, hemiluna (4ed5/4), is mentioned above.
 
=== 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
== Table of selected intervals ==
| Classical minor sixth
{| class="wikitable collapsible mw-collapsed"
| [[8/5]]
|+
|-
Selected intervals in 1789 EDO
| 1444
!Step
| Harmonic seventh
!Name
| [[7/4]]
!JI Approximation, Monzo, or another interpretation
<small>(based on the 2.5.11.13.29.31 subgroup where applicable)</small>
|-
|-
|0
| 1535
|Unison
| 29th harmonic
|1/1 exact
| [[29/16]]
|-
|-
|25
| 1579
|28-thirds comma
| 59th harmonic
|[65 -28]
| [[59/32]]
|-
|-
|36
| 1707
|
| 31st harmonic
|145/143
| [[31/16]]
|-
|-
|61
| 1789
|Lesser diesis
| Octave
|[[128/125]]
| 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"
|-
|-
|74
! rowspan="2" | [[Subgroup]]
|
! rowspan="2" | [[Comma list]]
|319/310
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|122
! [[TE error|Absolute]] (¢)
|
! [[TE simple badness|Relative]] (%)
|65/62
|-
|-
|172
| 2.9
|Tricesimoprimal Miracle semitone
| {{monzo| -5671 1789 }}
|[[31/29]]
| {{mapping| 1789 5671 }}
| −0.00044
| 0.00044
| 0.06
|-
|-
|226
| 2.9.5
|
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
|440/403
| {{mapping| 1789 5671 4154 }}
| −0.00710
| 0.00942
| 1.40
|-
|-
|290
| 2.9.5.7
|Jacobin minor interval
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
|160/143, 649/580
| {{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
|-
|-
|338
| 2.5.11.13.29
|Minor sqrt(13/10)
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
|[[Square root of 13 over 10]] I,
| {{mapping| 1789 4154 6189 6620 8691 }}
| −0.00591
| 0.01272
| 1.90
|-
|-
|339
| 2.5.11.13.29.31
|Major sqrt(13/10)
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
|[[Square root of 13 over 10]] II, (11/8)^20
| {{mapping| 1789 4154 6189 6620 8691 8863 }}
| −0.00363
| 0.01268
| 1.89
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|387
! Periods<br>per 8ve
|Jacobin major interval
! Generator*
|754/649
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
|576
| 1
|Major third
| 35\1789
|[[5/4]]
| 23.48
| 531441/524288
| [[Commatose]]
|-
|-
|677
| "
|Jacobin naiadic
| 125\1789
|[[13/10]]
| 83.85
| 16807/16000
| [[Sextilimeans]]
|-
|-
|822
| "
|Jacobin superfourth, Mongolian fourth
| 144\1789
|[[11/8]]
| 96.59
| 200/189
| [[Hemiluna]] (1789bd)
|-
|-
|1032
| "
|Secor fifth, Tricesimoprimal Miracle fifth
| 377\1789
|(31/29)^6
| 252.88
| 53094899/45875200
| [[Double bastille]]
|-
|-
|1046
| "
|Minor fifth
| 567\1789
|<u>[[3/2]] I</u>
| 380.32
| 34328125/27557888
| [[Genojacobin]]
|-
|-
|1047
| "
|Major fifth
| 754\1789
|[[3/2]] II
| 505.76
| {{monzo| 104 0 57 0 -14 5 }}
| [[Pure bastille]]
|-
|-
|1535
| "
|29th harmonic
| 777\1789
|[[29/16]]
| 521.18
| 875/648
| [[Maviloid]]
|-
|-
|1579
| "
|59th harmonic
| 778\1789
|[[59/32]]
| 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 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 ==
; [[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"