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{{Infobox ET}}
{{Infobox ET}}
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 0.671 [[cent]]s each. It is the 278th [[prime edo]].
{{EDO intro|1789}}


== Theory ==
== Theory ==
1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 [[subgroup]]. Perhaps the most notable fact about 1789edo, is the fact that it 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.
1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 [[subgroup]]. Perhaps the most notable fact about 1789edo, is the fact that it 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.


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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|1789|columns = 10}}
{{Harmonics in equal|1789}}


=== Jacobin temperaments ===
=== Jacobin temperaments ===
''Main article: [[The Jacobins]]''
{{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 37 & 1789, called onzonic. Name "onzonic" comes from the French word for eleven, ''onze''.  
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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 460 & 1789 temperament.  
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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 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]].  
=== Subsets and supersets ===
1789edo is the 278th [[prime edo]].


== Table of selected intervals ==
== Table of selected intervals ==
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| 65/62
| 65/62
|-
|-
|125
| 125
|Sextilimeans generator
| Sextilimeans generator
|16807/16000
| 16807/16000
|-
|-
| 172
| 172
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| [[13/10]]
| [[13/10]]
|-
|-
|750
| 750
|Sextilimeans fourth
| Sextilimeans fourth
|
|
|-
|-
| 777
| 777
| Maviloid generator
| Maviloid generator
|875/648
| 875/648
|-
|-
| 822
| 822
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| (31/29)<sup>6</sup>
| (31/29)<sup>6</sup>
|-
|-
|1039
| 1039
|Sextilimeans fifth
| Sextilimeans fifth
|
|
|-
|-
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| 2.9
| 2.9
| {{monzo| -5671 1789 }}
| {{monzo| -5671 1789 }}
| [{{val| 1789 5671 }}]
| {{mapping| 1789 5671 }}
| -0.00044
| -0.00044
| 0.00044
| 0.00044
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| 2.9.5
| 2.9.5
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }}
| [{{val| 1789 5671 4154 }}]
| {{mapping| 1789 5671 4154 }}
| -0.00710
| -0.00710
| 0.00942
| 0.00942
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| 2.9.5.7
| 2.9.5.7
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
| 420175/419904, {{monzo| 34 2 -21 3 }}, {{monzo| -55 15 2 1 }}
| [{{val| 1789 5671 4154 5022 }}]
| {{mapping| 1789 5671 4154 5022 }}
| +0.01606
| +0.01606
| 0.04093
| 0.04093
| 6.10
| 6.10
|-
|-
|style="border-top: double;" |2.5.11.13
| style="border-top: double;" | 2.5.11.13
|style="border-top: double;" |6656/6655, {{monzo|43 -18  5 -5}},  {{monzo|-38 -32 10 21}}
| style="border-top: double;" | 6656/6655, {{monzo| 43 -18  5 -5 }},  {{monzo| -38 -32 10 21 }}
|style="border-top: double;" |[{{val| 1789 4154 6189 6620}}]
| style="border-top: double;" | {{mapping| 1789 4154 6189 6620}}
|style="border-top: double;" | -0.00490
| style="border-top: double;" | -0.00490
|style="border-top: double;" |0.01405
| style="border-top: double;" | 0.01405
|style="border-top: double;" |2.09
| style="border-top: double;" | 2.09
|-
|-
|2.5.11.13.29
| 2.5.11.13.29
|6656/6655, 371293/371200, {{monzo|-18 -6 -1 3 5}}, {{monzo|34 -20 5 0 -1}}
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }}
|[{{val| 1789 4154 6189 6620 8691}}]
| {{mapping| 1789 4154 6189 6620 8691 }}
| -0.00591
| -0.00591
|0.01272
| 0.01272
|1.90
| 1.90
|-
|-
| 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 }}
| -0.00363
| -0.00363
| 0.01268
| 0.01268
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio
! Temperament
! Temperament
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| 23.48
| 23.48
| 531441/524288
| 531441/524288
|[[Commatose]]
| [[Commatose]]
|-
|-
|125\1789
| 125\1789
|83.85
| 83.85
|16807/16000
| 16807/16000
|[[Sextilimeans]]
| [[Sextilimeans]]
|-
|-
|144\1789
| 144\1789
|96.59
| 96.59
|200/189
| 200/189
|[[Hemiluna]] (1789bd)
| [[Hemiluna]] (1789bd)
|-
|-
| 172\1789
| 172\1789
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| [[Tricesimoprimal miracloid]]
| [[Tricesimoprimal miracloid]]
|-
|-
|377\1789
| 377\1789
|252.88
| 252.88
|53094899/45875200
| 53094899/45875200
|[[Double Bastille]]
| [[Double bastille]]
|-
|-
| 576\1789
| 576\1789
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| [[French decimal]]
| [[French decimal]]
|-
|-
|754\1789
| 754\1789
|505.76
| 505.76
|{{Monzo|104 0 57 0 -14 5}}
| {{monzo| 104 0 57 0 -14 5 }}
|[[Pure Bastille]]
| [[Pure bastille]]
|-
|-
| 777\1789
| 777\1789
| 521.18
| 521.18
| 875/648
| 875/648
|[[Maviloid]]
| [[Maviloid]]
|-
|-
| 778\1789
| 778\1789
| 521.86
| 521.86
| 80275/59392
| 80275/59392
|[[Estates general]]
| [[Estates general]]
|-
|-
|822\1789
| 822\1789
|551.37
| 551.37
|11/8
| 11/8
|[[Onzonic]]
| [[Onzonic]]
|-
|-
|865\1789
| 865\1789
|580.21
| 580.21
|{{Monzo|294 0 -46 0 7 -57}}
| {{monzo| 294 0 -46 0 7 -57 }}
|Eternal Revolutionary
| [[Eternal revolutionary]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|####]]


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


* [https://www.youtube.com/watch?v=1zrnsGODQSg Etude la (R)evolution, Op. 1, No. 2]by [[Eliora]]
<!-- 4-digit number -->
[[Category:Prime EDO]]
[[Category:Jacobin]]
[[Category:Jacobin]]
[[Category:Listen]]


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