1789edo: Difference between revisions
No edit summary |
|||
| Line 6: | Line 6: | ||
1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup. | 1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup. | ||
=== Jacobin temperament === | |||
The "proper" Jacobin temperament 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. This 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. | |||
Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament. | |||
=== 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 1524 & 1789 temperament. | |||
=== Other === | |||
For its elaborate xenharmonicity, 1789edo has an essentially perfect [[9/8]], a very common interval. The associated comma is [5671 -3578⟩. This is a direct consequence of 1789edo being a [[dual-fifth system]]. 1046th and 1047th steps aren't associated with JI intervals by themselves, but satisfy 3 × 3II = 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]]. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 33: | Line 40: | ||
|15.7 | |15.7 | ||
|- | |- | ||
|2. | |2.3II | ||
|{{monzo| -2836 1789 }} | |{{monzo| -2836 1789 }} | ||
|[{{val|1789 2836}}] | |[{{val|1789 2836}}] | ||
| Line 47: | Line 54: | ||
|1.9 | |1.9 | ||
|} | |} | ||
== Table of selected intervals == | == Table of selected intervals == | ||
{| class="wikitable collapsible mw-collapsed" | {| class="wikitable collapsible mw-collapsed" | ||
Revision as of 09:04, 20 March 2022
1789 EDO divides the octave into equal steps of 0.67 cents each. It is the 278th prime edo. Perhaps the most notable fact about 1789edo, is the fact that it tempers out the jacobin comma (6656/6655), 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.
Theory
| 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 (%) | -49.8 | +7.1 | -35.8 | +0.4 | +7.7 | -8.7 | -42.7 | -47.1 | +45.8 | +14.4 | |
| 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.
Jacobin temperament
The "proper" Jacobin temperament 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. This 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.
Using the maximal evenness method of finding rank two temperaments, we get a 37 & 1789 temperament.
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 1524 & 1789 temperament.
Other
For its elaborate xenharmonicity, 1789edo has an essentially perfect 9/8, a very common interval. The associated comma is [5671 -3578⟩. This is a direct consequence of 1789edo being a dual-fifth system. 1046th and 1047th steps aren't associated with JI intervals by themselves, but satisfy 3 × 3II = 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.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal
8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2835 1789⟩ | [⟨1789 2835]] | 0.105 | 0.105 | 15.7 |
| 2.3II | [-2836 1789⟩ | [⟨1789 2836]] | 0.106 | 0.106 | 15.8 |
| 2.5.11.13.29.31 | 6656/6655, 387283/387200, 2640704/2640625, 3455881/3455756, 594880000/594823321 | [⟨1789 4154 6189 6620 8691 8863]] | -0.003 | 0.013 | 1.9 |
Table of selected intervals
| Step | Name | JI Approximation, Monzo, or another interpretation
(based on the 2.5.11.13.29.31 subgroup where applicable) |
|---|---|---|
| 0 | Unison | 1/1 exact |
| 25 | 28-thirds comma | [65 -28] |
| 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) | Square root of 13 over 10 I, |
| 339 | Major sqrt(13/10) | Square root of 13 over 10 II, (11/8)^20 |
| 387 | Jacobin major interval | 754/649 |
| 576 | Major third | 5/4 |
| 677 | Jacobin naiadic | 13/10 |
| 822 | Jacobin superfourth, Mongolian fourth | 11/8 |
| 1032 | Secor fifth, Tricesimoprimal Miracle fifth | (31/29)^6 |
| 1046 | Minor fifth | 3/2 I |
| 1047 | Major fifth | 3/2 II |
| 1535 | 29th harmonic | 29/16 |
| 1579 | 59th harmonic | 59/32 |
| 1707 | 31st harmonic | 31/16 |
| 1789 | Octave | 2/1 exact |
Scales
- Jacobin[37]
- Jacobin[74]
- Jacobin[111]
- Jacobin[222]
- Decimal[265]
- Decimal[1524]