1789edo

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Prime factorization 1789 (prime)
Step size 0.670766¢
Fifth 1046\1789 (701.621¢)
Semitones (A1:m2) 166:137 (111.3¢ : 91.89¢)
Sharp fifth 1047\1789 (702.292¢)
Flat fifth 1046\1789 (701.621¢)
Major 2nd 304\1789 (203.913¢)
Consistency limit 3
Distinct 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

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 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.

Odd harmonics

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)

Jacobin temperaments

Main article: 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.

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.

Other

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.

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.

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.

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

Table of selected intervals

Selected intervals in 1789edo
Step Eliora's Naming System JI Approximation or Other Interpretations*
0 Unison 1/1
25 Oquatonic comma [65 -28
35 Pythagorean comma 531441/524288
36 145/143
61 Lesser diesis 128/125
74 319/310
122 65/62
125 Sextilimeans generator 16807/16000
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
750 Sextilimeans fourth
777 Maviloid generator 875/648
822 Jacobin superfourth, Mongolian fourth 11/8
1032 Secor fifth, Tricesimoprimal Miracle fifth (31/29)6
1039 Sextilimeans fifth
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

Generator
(Reduced)
Cents
(Reduced)
Associated
Ratio
Temperament
35\1789 23.48 531441/524288 Commatose
125\1789 83.85 16807/16000 Sextilimeans
144\1789 96.59 200/189 Hemiluna (1789bd)
172\1789 115.37 31/29 Tricesimoprimal miracloid
377\1789 252.88 53094899/45875200 Double Bastille
576\1789 386.36 5/4 French decimal
777\1789 521.18 875/648 Maviloid
778\1789 521.86 80275/59392 Estates general
822\1789 551.37 11/8 Onzonic

Music