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Rework on theory (the no-3 subgroup can't contain both 7 and 21); misc. cleanup |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1|5]], [[9/1|9]], [[11/1|11]], [[13/1|13]] and [[21/1|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation. | 1789edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]] [[3/1|3]] is about halfway between its steps. Otherwise, it is excellent in approximating harmonics [[5/1|5]], [[9/1|9]], [[11/1|11]], [[13/1|13]] and [[21/1|21]], making it suitable for a 2.9.5.21.11.13 [[subgroup]] interpretation. | ||
Perhaps the most notable fact about 1789edo is that it [[tempering out|tempers out]] the jacobin comma ([[6656/6655]]), and it is also consistent to distance 2 on the subgroup 2.5.11.13 of the comma, which is naming-wise appropriate for edo's number while also providing an extremely strong tuning. 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. | |||
2.9.5.11.13 subgroup is also represented strongly in 1789edo, where it tunes a temperament called ''commatose'', defined as a {{nowrap|460 & 1789}}, which uses the Pythagorean comma as a generator. In the full 13-limit, 1789bd val, {{Val|1789 '''2836''' 4154 '''5023'''}} is of additional interest as it is better tuned than the patent val, where it tunes [[hemiluna]]. | |||
On the patent val in the 7-limit, 1789edo supports {{nowrap|99 & 373}} temperament called [[maviloid]]. In addition, it also tempers out [[2401/2400]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 13: | Line 17: | ||
{{Main| The Jacobins }} | {{Main| 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''. | 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 {{nowrap|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. | 1789edo supports the 2.5.11.13.19 subgroup temperament called ''estates general'' defined as {{nowrap|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. | ||
=== | === Miscellany === | ||
1789edo can be | For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. [[45/32]] and [[51/32]] are also strongly approximated. | ||
Since the 5/4 of 1789edo is on the 576th step, a highly divisible number, 1789edo can replicate a lot of [[ed5/4]] | 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 of these, hemiluna (4ed5/4), is mentioned above. | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]]. | 1789edo is the 278th [[prime edo]]. [[3578edo]], which doubles it, is consistent in the [[21-odd-limit]]. | ||
== Table of selected intervals == | == Table of selected intervals == | ||
{| class="wikitable mw-collapsible mw-collapsed" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ style=white-space:nowrap | Selected intervals in 1789edo | |+ style="font-size: 105%; white-space: nowrap;" | Selected intervals in 1789edo | ||
|- | |||
! Step | ! Step | ||
! Eliora's | ! Eliora's naming system | ||
! JI | ! JI approximation or other interpretations* | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 128: | Line 127: | ||
| 1046 | | 1046 | ||
| Minor fifth | | Minor fifth | ||
| [[3/2]] | | [[3/2]]** | ||
|- | |- | ||
| 1047 | | 1047 | ||
| Major fifth | | Major fifth | ||
| [[3/2]] | | [[3/2]]** | ||
|- | |- | ||
| 1213 | | 1213 | ||
| Line 158: | Line 157: | ||
| 2/1 | | 2/1 | ||
|} | |} | ||
<nowiki>* | <nowiki />* Based on the 2.5.11.13.29.31 subgroup where applicable | ||
<nowiki />** 1046\1789 as 3/2 is the patent val, 1047\1789 as 3/2 is the 1789b val | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 176: | Line 176: | ||
| {{monzo| -5671 1789 }} | | {{monzo| -5671 1789 }} | ||
| {{mapping| 1789 5671 }} | | {{mapping| 1789 5671 }} | ||
| | | −0.00044 | ||
| 0.00044 | | 0.00044 | ||
| 0.06 | | 0.06 | ||
| Line 183: | Line 183: | ||
| {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }} | | {{monzo| -70 36 -19 }}, {{monzo| 129 -7 -46 }} | ||
| {{mapping| 1789 5671 4154 }} | | {{mapping| 1789 5671 4154 }} | ||
| | | −0.00710 | ||
| 0.00942 | | 0.00942 | ||
| 1.40 | | 1.40 | ||
| Line 193: | Line 193: | ||
| 0.04093 | | 0.04093 | ||
| 6.10 | | 6.10 | ||
|- | |- style="border-top: double;" | ||
| 2.5.11.13 | |||
| 6656/6655, {{monzo| 43 -18 5 -5 }}, {{monzo| -38 -32 10 21 }} | |||
| {{mapping| 1789 4154 6189 6620}} | |||
| | | −0.00490 | ||
| 0.01405 | |||
| 2.09 | |||
|- | |- | ||
| 2.5.11.13.29 | | 2.5.11.13.29 | ||
| 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }} | | 6656/6655, 371293/371200, {{monzo| -18 -6 -1 3 5 }}, {{monzo| 34 -20 5 0 -1 }} | ||
| {{mapping| 1789 4154 6189 6620 8691 }} | | {{mapping| 1789 4154 6189 6620 8691 }} | ||
| | | −0.00591 | ||
| 0.01272 | | 0.01272 | ||
| 1.90 | | 1.90 | ||
| Line 211: | Line 211: | ||
| 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | ||
| {{mapping| 1789 4154 6189 6620 8691 8863 }} | | {{mapping| 1789 4154 6189 6620 8691 8863 }} | ||
| | | −0.00363 | ||
| 0.01268 | | 0.01268 | ||
| 1.89 | | 1.89 | ||
| Line 217: | Line 217: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left- | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br>per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br>ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
| 1 | |||
| 35\1789 | | 35\1789 | ||
| 23.48 | | 23.48 | ||
| Line 230: | Line 232: | ||
| [[Commatose]] | | [[Commatose]] | ||
|- | |- | ||
| " | |||
| 125\1789 | | 125\1789 | ||
| 83.85 | | 83.85 | ||
| Line 235: | Line 238: | ||
| [[Sextilimeans]] | | [[Sextilimeans]] | ||
|- | |- | ||
| " | |||
| 144\1789 | | 144\1789 | ||
| 96.59 | | 96.59 | ||
| Line 240: | Line 244: | ||
| [[Hemiluna]] (1789bd) | | [[Hemiluna]] (1789bd) | ||
|- | |- | ||
| | | " | ||
| 377\1789 | | 377\1789 | ||
| 252.88 | | 252.88 | ||
| Line 250: | Line 250: | ||
| [[Double bastille]] | | [[Double bastille]] | ||
|- | |- | ||
| | | " | ||
| | | 567\1789 | ||
| | | 380.32 | ||
| [[ | | 34328125/27557888 | ||
| [[Genojacobin]] | |||
|- | |- | ||
| " | |||
| 754\1789 | | 754\1789 | ||
| 505.76 | | 505.76 | ||
| Line 260: | Line 262: | ||
| [[Pure bastille]] | | [[Pure bastille]] | ||
|- | |- | ||
| " | |||
| 777\1789 | | 777\1789 | ||
| 521.18 | | 521.18 | ||
| Line 265: | Line 268: | ||
| [[Maviloid]] | | [[Maviloid]] | ||
|- | |- | ||
| " | |||
| 778\1789 | | 778\1789 | ||
| 521.86 | | 521.86 | ||
| Line 270: | Line 274: | ||
| [[Estates general]] | | [[Estates general]] | ||
|- | |- | ||
| " | |||
| 822\1789 | | 822\1789 | ||
| 551.37 | | 551.37 | ||
| Line 275: | Line 280: | ||
| [[Onzonic]] | | [[Onzonic]] | ||
|- | |- | ||
| " | |||
| 865\1789 | | 865\1789 | ||
| 580.21 | | 580.21 | ||
| | | 6875/4914 | ||
| [[Eternal revolutionary]] | | [[Eternal revolutionary]] (1789bd) | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||
Latest revision as of 19:35, 30 August 2026
| ← 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.
Perhaps the most notable fact about 1789edo is that it tempers out the jacobin comma (6656/6655), and it is also consistent to distance 2 on the subgroup 2.5.11.13 of the comma, which is naming-wise appropriate for edo's number while also providing an extremely strong tuning. 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.
2.9.5.11.13 subgroup is also represented strongly in 1789edo, where it tunes a temperament called commatose, defined as a 460 & 1789, which uses the Pythagorean comma as a generator. In the full 13-limit, 1789bd val, ⟨1789 2836 4154 5023] is of additional interest as it is better tuned than the patent val, where it tunes hemiluna.
On the patent val in the 7-limit, 1789edo supports 99 & 373 temperament called maviloid. In addition, it also tempers out 2401/2400.
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 (%) | -49.8 | +7.1 | -35.8 | +0.4 | +7.7 | -8.7 | -42.7 | -47.1 | +45.8 | +14.4 | +34.8 | |
| 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
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.
Miscellany
For higher harmonics, 1789edo can be adapted for use with the 2.9.5.21.11.13.29.31.47.59.61 subgroup. 45/32 and 51/32 are also strongly approximated.
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 of these, hemiluna (4ed5/4), is mentioned above.
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
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 35\1789 | 23.48 | 531441/524288 | Commatose |
| " | 125\1789 | 83.85 | 16807/16000 | Sextilimeans |
| " | 144\1789 | 96.59 | 200/189 | Hemiluna (1789bd) |
| " | 377\1789 | 252.88 | 53094899/45875200 | Double bastille |
| " | 567\1789 | 380.32 | 34328125/27557888 | Genojacobin |
| " | 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 distinct
Music
- Etude la (R)evolution (2022)