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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 = 11}}


1789edo can be adapted for use with the 2.5.11.13.29.31 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.  


'''Table of selected intervals'''
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.
{| class="wikitable"
 
|+
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]].
!Step
 
!Name
On the patent val in the 7-limit, 1789edo supports {{nowrap|99 & 373}} temperament called [[maviloid]]. In addition, it also tempers out [[2401/2400]].
!JI Approximation or Monzo
 
=== Odd harmonics ===
{{Harmonics in equal|1789}}
 
=== Jacobin temperaments ===
{{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 {{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 & 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. 
 
=== Miscellany ===
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
|-
! Step
! Eliora's naming system
! JI approximation or other interpretations*
|-
| 0
| Unison
| 1/1
|-
| 25
| Oquatonic comma
| {{monzo| 65 -28 }}
|-
| 35
| Pythagorean comma
| [[531441/524288]]
|-
| 36
|
| 145/143
|-
| 61
| Lesser diesis
| [[128/125]]
|-
| 74
|
| 319/310
|-
| 122
|
| 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
|
|-
| 777
| Maviloid generator
| 875/648
|-
| 822
| Jacobin superfourth, Mongolian fourth
| [[11/8]]
|-
| 1032
| Secor fifth, Tricesimoprimal Miracle fifth
| (31/29)<sup>6</sup>
|-
| 1039
| Sextilimeans fifth
|
|-
| 1046
| Minor fifth
| [[3/2]]**
|-
|-
|0
| 1047
|Unison
| Major fifth
|1/1 exact
| [[3/2]]**
|-
|-
|25
| 1213
|28-thirds comma
| Classical minor sixth
|[65 -28]
| [[8/5]]
|-
|-
|61
| 1444
|Lesser diesis
| Harmonic seventh
|[[128/125]]
| [[7/4]]
|-
|-
|576
| 1535
|Major third
| 29th harmonic
|[[5/4]]
| [[29/16]]
|-
|-
|677
| 1579
|Jacobin naiadic
| 59th harmonic
|[[13/10]]
| [[59/32]]
|-
|-
|822
| 1707
|Jacobin superfourth
| 31st harmonic
|[[11/8]]
| [[31/16]]
|-
|-
|1789
| 1789
|Octave
| Octave
|2/1 exact
| 2/1
|}
|}
<nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable


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


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.
== 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
|}


Addition of 29 and 31 harmonic intervals may also be suitable to spice up an otherwise monotonous scale.
=== Rank-2 temperaments ===
{| 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
| 531441/524288
| [[Commatose]]
|-
| "
| 125\1789
| 83.85
| 16807/16000
| [[Sextilimeans]]
|-
| "
| 144\1789
| 96.59
| 200/189
| [[Hemiluna]] (1789bd)
|-
| "
| 377\1789
| 252.88
| 53094899/45875200
| [[Double bastille]]
|-
| "
| 567\1789
| 380.32
| 34328125/27557888
| [[Genojacobin]]
|-
| "
| 754\1789
| 505.76
| {{monzo| 104 0 57 0 -14 5 }}
| [[Pure bastille]]
|-
| "
| 777\1789
| 521.18
| 875/648
| [[Maviloid]]
|-
| "
| 778\1789
| 521.86
| 80275/59392
| [[Estates general]]
|-
| "
| 822\1789
| 551.37
| 11/8
| [[Onzonic]]
|-
| "
| 865\1789
| 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


== Scales ==
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=1zrnsGODQSg ''Etude la (R)evolution''] (2022)


* Jacobin[37]
[[Category:Jacobin]]
* Jacobin[74]
[[Category:Listen]]
* Jacobin[111]
* Jacobin[222]
* Decimal[


[[Category:Equal divisions of the octave]]
{{Todo| review | clarify }}
[[Category:Prime EDO]]

Latest revision as of 19:35, 30 August 2026

← 1788edo 1789edo 1790edo →
Prime factorization 1789 (prime)
Step size 0.670766 ¢ 
Fifth 1046\1789 (701.621 ¢)
Semitones (A1:m2) 166:137 (111.3 ¢ : 91.89 ¢)
Dual sharp fifth 1047\1789 (702.292 ¢)
Dual flat fifth 1046\1789 (701.621 ¢)
Dual major 2nd 304\1789 (203.913 ¢)
Consistency limit 3
Distinct consistency limit 3

1789 equal divisions of the octave (abbreviated 1789edo or 1789ed2), also called 1789-tone equal temperament (1789tet) or 1789 equal temperament (1789et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1789 equal parts of about 0.671 ¢ each. Each step represents a frequency ratio of 21/1789, or the 1789th root of 2.

Theory

1789edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise, it is excellent in approximating harmonics 5, 9, 11, 13 and 21, making it suitable for a 2.9.5.21.11.13 subgroup interpretation.

Perhaps the most notable fact about 1789edo is that it 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.

2.9.5.11.13 subgroup is also represented strongly in 1789edo, where it tunes a temperament called commatose, defined as a 460 & 1789, which uses the Pythagorean comma as a generator. In the full 13-limit, 1789bd val, ⟨1789 2836 4154 5023] is of additional interest as it is better tuned than the patent val, where it tunes hemiluna.

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

Odd harmonics

Approximation of odd harmonics in 1789edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.334 +0.047 -0.240 +0.003 +0.052 -0.058 -0.287 -0.316 +0.307 +0.097 +0.233
Relative (%) -49.8 +7.1 -35.8 +0.4 +7.7 -8.7 -42.7 -47.1 +45.8 +14.4 +34.8
Steps
(reduced) 		2835
(1046) 		4154
(576) 		5022
(1444) 		5671
(304) 		6189
(822) 		6620
(1253) 		6989
(1622) 		7312
(156) 		7600
(444) 		7858
(702) 		8093
(937)

Jacobin temperaments

Main article: 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.

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.

Miscellany

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

Selected intervals in 1789edo
Step Eliora's naming system JI approximation or other interpretations*
0 Unison 1/1
25 Oquatonic comma [65 -28⟩
35 Pythagorean comma 531441/524288
36 145/143
61 Lesser diesis 128/125
74 319/310
122 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) [-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
777 Maviloid generator 875/648
822 Jacobin superfourth, Mongolian fourth 11/8
1032 Secor fifth, Tricesimoprimal Miracle fifth (31/29)6
1039 Sextilimeans fifth
1046 Minor fifth 3/2**
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

* 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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-5671 1789⟩ [⟨1789 5671]] −0.00044 0.00044 0.06
2.9.5 [-70 36 -19⟩, [129 -7 -46⟩ [⟨1789 5671 4154]] −0.00710 0.00942 1.40
2.9.5.7 420175/419904, [34 2 -21 3⟩, [-55 15 2 1⟩ [⟨1789 5671 4154 5022]] +0.01606 0.04093 6.10
2.5.11.13 6656/6655, [43 -18  5 -5⟩, [-38 -32 10 21⟩ [⟨1789 4154 6189 6620]] −0.00490 0.01405 2.09
2.5.11.13.29 6656/6655, 371293/371200, [-18 -6 -1 3 5⟩, [34 -20 5 0 -1⟩ [⟨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 [⟨1789 4154 6189 6620 8691 8863]] −0.00363 0.01268 1.89

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 35\1789 23.48 531441/524288 Commatose
" 125\1789 83.85 16807/16000 Sextilimeans
" 144\1789 96.59 200/189 Hemiluna (1789bd)
" 377\1789 252.88 53094899/45875200 Double bastille
" 567\1789 380.32 34328125/27557888 Genojacobin
" 754\1789 505.76 [104 0 57 0 -14 5⟩ Pure bastille
" 777\1789 521.18 875/648 Maviloid
" 778\1789 521.86 80275/59392 Estates general
" 822\1789 551.37 11/8 Onzonic
" 865\1789 580.21 6875/4914 Eternal revolutionary (1789bd)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Eliora
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