1789edo: Difference between revisions

{{ED intro}}  

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.  

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.

 

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]].  

 

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

{{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.

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.

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

|+ style="font-size: 105%; white-space: nowrap;" | Selected intervals in 1789edo

! Eliora's naming system

! JI approximation or other interpretations*

| [[3/2]]**

| [[3/2]]**

<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

! rowspan="2" | [[Comma list]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

| −0.00044

| −0.00710

|- 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

| −0.00591

| −0.00363

{| class="wikitable center-all left-5"

! Associated<br>ratio*

| "

| "

| 567\1789

| 380.32

| 34328125/27557888

| [[Genojacobin]]

| 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