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The '''1789 equal divisions of the octave''' ('''1789edo'''), or the '''1789-tone equal temperament''' ('''1789tet'''), '''1789 equal temperament''' ('''1789et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1789 [[equal]] parts of about 671 [[cent|millicent]]s each. It is the 278th [[prime edo]].
{{Infobox ET}}
 
{{ED intro}}
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.


== Theory ==
== Theory ==
{{Harmonics in equal|1789|columns = 10}}
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.


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


=== Jacobin temperament ===
=== Odd harmonics ===
A "proper" jacobin scale 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.
{{Harmonics in equal|1789}}


Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament.
=== Jacobin temperaments ===
{{Main| The Jacobins }}


=== French decimal temperament ===
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''.  
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).


Using the maximal evenness method of finding rank two temperaments, we get a 1524 & 1789 temperament.
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 ===
=== Other ===
For its elaborate xenharmonicity, 1789edo has an essentially perfect [[9/8]], a very common interval. The associated comma is {{monzo| 5671 -1789 }} in 2.9. This is a direct consequence of 1789edo being a [[dual-fifth system]]. 1046th and 1047th steps are not associated with JI intervals by themselves, but satisfy 3<sup>+</sup> × 3<sup>-</sup> = 9, which represents 9/1, and therefore when octave reduced this leads to 9/8.  
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.
 
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 {{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]].  


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


On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]].
== Table of selected intervals ==
== Table of selected intervals ==
{| class="wikitable mw-collapsible mw-collapsed"
{| 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
! Step
! Eliora's Naming System
! Eliora's naming system
! JI Approximation or Other Interpretations*
! JI approximation or other interpretations*
|-
|-
| 0
| 0
Line 36: Line 44:
|-
|-
| 25
| 25
| 28-thirds comma
| Oquatonic comma
| {{monzo| 65 -28 }}
| {{monzo| 65 -28 }}
|-
| 35
| Pythagorean comma
| [[531441/524288]]
|-
|-
| 36
| 36
Line 54: Line 66:
|
|
| 65/62
| 65/62
|-
| 125
| Sextilimeans generator
| 16807/16000
|-
|-
| 172
| 172
Line 79: Line 95:
| 754/649
| 754/649
|-
|-
|523
| 523
|Breedsmic neutral third
| Breedsmic neutral third
|49/40, 60/49
| 49/40, 60/49
|-
|-
| 576
| 576
Line 90: Line 106:
| Jacobin naiadic
| Jacobin naiadic
| [[13/10]]
| [[13/10]]
|-
| 750
| Sextilimeans fourth
|
|-
| 777
| Maviloid generator
| 875/648
|-
|-
| 822
| 822
Line 98: Line 122:
| Secor fifth, Tricesimoprimal Miracle fifth
| Secor fifth, Tricesimoprimal Miracle fifth
| (31/29)<sup>6</sup>
| (31/29)<sup>6</sup>
|-
| 1039
| Sextilimeans fifth
|
|-
|-
| 1046
| 1046
| Minor fifth
| Minor fifth
| [[3/2]]†
| [[3/2]]**
|-
|-
| 1047
| 1047
| Major fifth
| Major fifth
| [[3/2]]†
| [[3/2]]**
|-
|-
|1444
| 1213
|Harmonic seventh
| Classical minor sixth
|[[7/4]]
| [[8/5]]
|-
| 1444
| Harmonic seventh
| [[7/4]]
|-
|-
| 1535
| 1535
Line 127: Line 159:
| 2/1
| 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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 145: Line 177:
| 2.9
| 2.9
| {{monzo| -5671 1789 }}
| {{monzo| -5671 1789 }}
| [{{val| 1789 5671 }}]
| {{mapping| 1789 5671 }}
| -0.000441
| −0.00044
| 0.000441
| 0.00044
| 0.066
| 0.06
|-
|-
|2.3<sup>-</sup>.5
| 2.9.5
|{{monzo|8 14 -13}}, {{monzo|-223, 129, 8}}
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
|[{{val|1789 2835 4154}}]
| {{mapping| 1789 5671 4154 }}
|
| −0.00710
|
| 0.00942
|
| 1.40
|-
|-
|2.3<sup>+</sup>.5
| 2.9.5.7
|{{monzo|38 -2 -15}}, {{monzo|32 -111 62}}
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
|[{{val|1789 2836 4154}}]
| {{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
|-
|-
|2.3<sup>-</sup>.5.7
| 2.5.11.13.29
|[[2401/2400]], [8, 14, -13, 0⟩, [-56, 26, 10, -3⟩
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
|[{{val|1789 2835 4154 5022}}]
| {{mapping| 1789 4154 6189 6620 8691 }}
|0.068957
| −0.00591
|0.090989
| 0.01272
|
| 1.90
|-
|2.3<sup>+</sup>.5.7
|[8, -20, 9, 1⟩, [16, -9, -8, 6⟩, [-22, -7, 7, 6⟩
|[{{val|1789 2836 4154 5022}}]
| -0.036850
|0.108915
|
|-
|-
| 2.5.11.13.29.31
| 2.5.11.13.29.31
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321
| [{{val|1789 4154 6189 6620 8691 8863}}]
| {{mapping| 1789 4154 6189 6620 8691 8863 }}
| -0.003
| −0.00363
| 0.013
| 0.01268
|  
| 1.89
|}
|}


== Rank two temperaments by generator ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
(reduced)
|-
!Cents
! Periods<br>per 8ve
(reduced)
! Generator*
! Associated
! Cents*
ratio
! Associated<br>ratio*
!Temperaments
! Temperament
|-
| 1
| 35\1789
| 23.48
| 531441/524288
| [[Commatose]]
|-
| "
| 125\1789
| 83.85
| 16807/16000
| [[Sextilimeans]]
|-
|-
|576\1789
| "
|386.36
| 144\1789
|5/4
| 96.59
|French decimal
| 200/189
| [[Hemiluna]] (1789bd)
|-
|-
|777\1789
| "
|521.18
| 172\1789
| -
| 115.37
|Maviloid
| 31/29
| [[Tricesimoprimal miracloid]]
|-
|-
|822\1789
| "
|551.37
| 377\1789
|11/8
| 252.88
|Jacobin
| 53094899/45875200
| [[Double bastille]]
|-
| "
| 576\1789
| 386.36
| 5/4
| [[French decimal]]
|-
| "
| 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)
|}
|}
[[Category:Equal divisions of the octave]]
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Prime EDO]]
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=1zrnsGODQSg ''Etude la (R)evolution''] (2022)
 
[[Category:Jacobin]]
[[Category:Listen]]


{{Todo| review }}
{{Todo| review | clarify }}
Retrieved from "https://en.xen.wiki/w/1789edo"