1789edo
| ← 1788edo | 1789edo | 1790edo → |
1789 equal divisions of the octave (abbreviated 1789edo or 1789ed2), also called 1789-tone equal temperament (1789tet) or 1789 equal temperament (1789et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1789 equal parts of about 0.671 ¢ each. Each step represents a frequency ratio of 21/1789, or the 1789th root of 2.
Theory
1789edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise, it is excellent in approximating harmonics 5, 9, 11, 13 and 21, making it suitable for a 2.9.5.21.11.13 subgroup interpretation.
For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. Perhaps the most notable fact about 1789edo is 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.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.334 | +0.047 | -0.240 | +0.003 | +0.052 | -0.058 | -0.287 | -0.316 | +0.307 | +0.097 | +0.233 |
| relative (%) | -50 | +7 | -36 | +0 | +8 | -9 | -43 | -47 | +46 | +14 | +35 | |
| Steps (reduced) |
2835 (1046) |
4154 (576) |
5022 (1444) |
5671 (304) |
6189 (822) |
6620 (1253) |
6989 (1622) |
7312 (156) |
7600 (444) |
7858 (702) |
8093 (937) | |
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.
Subsets and supersets
1789edo is the 278th prime edo. 3578edo, which doubles it, is consistent in the 21-odd-limit.
Table of selected intervals
| 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 | 0.06 |
| 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 | 6656/6655, [43 -18 5 -5⟩, [-38 -32 10 21⟩ | [⟨1789 4154 6189 6620]] | -0.00490 | 0.01405 | 2.09 |
| 2.5.11.13.29 | 6656/6655, 371293/371200, [-18 -6 -1 3 5⟩, [34 -20 5 0 -1⟩ | [⟨1789 4154 6189 6620 8691]] | -0.00591 | 0.01272 | 1.90 |
| 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* | Cents* | 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 |
| 754\1789 | 505.76 | [104 0 57 0 -14 5⟩ | Pure bastille |
| 777\1789 | 521.18 | 875/648 | Maviloid |
| 778\1789 | 521.86 | 80275/59392 | Estates general |
| 822\1789 | 551.37 | 11/8 | Onzonic |
| 865\1789 | 580.21 | 6875/4914 | Eternal revolutionary (1789bd) |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Etude la (R)evolution (2022)