1789edo: Difference between revisions

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 is consistent in the no-threes 13-odd-limit. Since its double, [[3578edo]], is consistent in the 21-odd-limit, it can be thought of as a [[K*N subgroups|2*1789]] 2.9.5.7.11.13.225.289.361.21 subgroup temperament, on which it shares mapping with 3578edo and tempers out the same commas.

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

=== Onzonic 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 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. Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma basis 6656/6655 and {{monzo| -119 -46 15 47 }} in the 2.5.11.13 subgroup. The first is the jacobin comma discussed earlier.

Name "onzonic" comes from the French word for eleven, ''onze''.  

1789edo 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-2 temperaments, we get a 1525 & 1789 temperament with comma basis 28824005/28792192, 200126927/200000000, 6106906624/6103515625 in the 2.5.7.11.13 subgroup.

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⁺ × 3^- = 9, which represents 9/1, and therefore when octave reduced this leads to 9/8.  

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]].

On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out [[2401/2400]]. The 1789bd val in the 13-limit is better tuned than the patent val. It provides a tuning for the [[hemiluna]] 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 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.

1789edo supports the 2.9.5.11.13 subgroup temperament called ''commatose'' which uses the Pythagorean comma as a generator. It is defined as a 460 & 1789 temperament, and its comma basis is 62748517/62726400, 479773125/479756288, and 30530193408/30517578125.

==== Estates General ====

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. It is unambiguous in the 2.5.11.13.19.23.29.31 subgroup.

|+ style=white-space:nowrap | Selected intervals in 1789edo

! Eliora's Naming System

! JI Approximation or Other Interpretations*

|875/648

| [[3/2]]†

| [[3/2]]†

<nowiki>*</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

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

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

! colspan="2" | Tuning Error

| [{{val| 1789 5671 }}]

| -0.00044

| 0.066

| [{{val| 1789 5671 4154 }}]

| -0.00710

| [{{val| 1789 5671 4154 5022 }}]

| [{{val| 1789 4154 6189 6620 8691 8863 }}]

| -0.00363

{| class="wikitable center-all left-4"

! Generator<br>(Reduced)

! Cents<br>(Reduced)

! Associated<br>Ratio

|[[Commatose]]

|144\1789

|96.59

|200/189

|[[Hemiluna]] (1789bd)

| Tricesimoprimal miracloid

| French decimal

|[[Maviloid]]

|[[Estates general]]

|822\1789

|551.37

|11/8

|[[Onzonic]]