1789edo

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1789edo
Prime factorization 1789 (prime)
Step size 0.67077
Fifth 1046\1789 (701.621¢)
Major 2nd 303\1789 (203.242¢)
Semitones (A1:m2) 166:137 (111.347¢:91.895¢)
Consistency limit 3

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 cents each. It is the 278th prime edo.

Theory

Approximation of odd harmonics in 1789edo
Harmonic 3 5 7 9 11 13 15 17 19 21
Error absolute (¢) -0.334 +0.047 -0.240 +0.003 +0.052 -0.058 -0.287 -0.316 +0.307 +0.097
relative (%) -50 +7 -36 +0 +8 -9 -43 -47 +46 +14
Steps
(reduced)
2835
(1046)
4154
(576)
5022
(1444)
5671
(304)
6189
(822)
6620
(1253)
6989
(1622)
7312
(156)
7600
(444)
7858
(702)

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 [-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?]

Other

For its elaborate xenharmonicity, 1789edo has an essentially perfect 9/8, a very common interval. The associated comma is [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.

Since it 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 evennes method, we find a 52 & 1789 temperament.

1789edo supports the 2.9.5.11.13 subgroup temperament which Eliora proposes be called commatose, and 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.

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.

Table of selected intervals

Selected intervals in 1789edo
Step Eliora's Naming System JI Approximation or Other Interpretations*
0 Unison 1/1
25 28-thirds comma [65 -28
35 Pythagorean comma 531441/524288
36 145/143
61 Lesser diesis 128/125
74 319/310
122 65/62
172 Tricesimoprimal Miracle semitone 31/29
226 440/403
290 Jacobin minor interval 160/143, 649/580
338 Minor sqrt(13/10)
339 Major sqrt(13/10) [-69 0 0 0 20
387 Jacobin major interval 754/649
523 Breedsmic neutral third 49/40, 60/49
576 Major third 5/4
677 Jacobin naiadic 13/10
777 Maviloid generator 875/648
822 Jacobin superfourth, Mongolian fourth 11/8
1032 Secor fifth, Tricesimoprimal Miracle fifth (31/29)6
1046 Minor fifth 3/2
1047 Major fifth 3/2
1213 Classical minor sixth 8/5
1444 Harmonic seventh 7/4
1535 29th harmonic 29/16
1579 59th harmonic 59/32
1707 31st harmonic 31/16
1789 Octave 2/1

* 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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.9 [-5671 1789 [1789 5671]] -0.00044 0.00044 0.066
2.9.5 [-70 36 -19, [129 -7 -46 [1789 5671 4154]] -0.00710 0.00942 1.40
2.9.5.7 420175/419904, [34 2 -21 3, [-55 15 2 1 [1789 5671 4154 5022]] +0.01606 0.04093 6.10
2.5.11.13.29.31 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 [1789 4154 6189 6620 8691 8863]] -0.00363 0.01268 1.89

Rank-2 temperaments by generator

Generator
(reduced)
Cents
(reduced)
Associated
ratio
Temperament
35\1789 23.48 531441/524288 Commatose
172\1789 115.37 31/29 Tricesimoprimal Miracloid
576\1789 386.36 5/4 French decimal
777\1789 521.18 875/648 Maviloid
778\1789 521.86 ... Estates General
822\1789 551.37 11/8 Jacobin