1789edo: Difference between revisions

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

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 it's number is the hallmark year of the French Revolution, thus making the temperance of the Jacobin comma on topic.

1789edo is consistent in the no-threes 13-limit.

=== Jacobin temperament ===

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.

Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament with the comma set 6656/6655 and {{Monzo|-74, -51, 0, 52}}. The first is the well known jacobin comma, and it means that a stack of three 11/8 is equated with 13/10. Second comma represents a stack of 52 13/10s being equal to 8/5.

The monotonicity 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.

=== French decimal temperament ===

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 1525 & 1789 temperament.^{[which mapping?]}

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]].

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

! Eliora's Naming System

! JI Approximation or Other Interpretations*

| 28-thirds comma

|35

|Pythagorean comma

|[[531441/524288]]

|875/648

| [[3/2]]†

| [[3/2]]†

|1213

|Classical minor sixth

|[[8/5]]

<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" | Subgroup

| [{{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

== Rank-2 temperaments by generator ==

! Generator<br>(reduced)

! Cents<br>(reduced)

! Associated<br>ratio

! Temperaments

|172\1789

|115.37

|31/29

|Tricesimoprimal Miracloid

| French decimal

| Maviloid

|822\1789

|551.37

|11/8

|Jacobin

[[Category:Equal divisions of the octave]]

[[Category:Prime EDO]]

{{Todo| review }}