1789edo

From Xenharmonic Wiki
Jump to navigation Jump to search

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.

English Wikipedia has an article on:

Theory

Approximation of prime intervals in 1789 EDO
Prime number 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
Error absolute (¢) +0.000 -0.334 +0.047 -0.240 +0.052 -0.058 -0.316 +0.307 +0.233 +0.048 -0.038 +0.193 +0.228 +0.277 -0.107 -0.167 -0.032 -0.060
relative (%) +0 -50 +7 -36 +8 -9 -47 +46 +35 +7 -6 +29 +34 +41 -16 -25 -5 -9
Steps (reduced) 1789 (0) 2835 (1046) 4154 (576) 5022 (1444) 6189 (822) 6620 (1253) 7312 (156) 7600 (444) 8093 (937) 8691 (1535) 8863 (1707) 9320 (375) 9585 (640) 9708 (763) 9937 (992) 10247 (1302) 10524 (1579) 10610 (1665)

1789edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup.

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).

In addition, 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.

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.

Tempered commas

Tempered commas in 1789edo
Prime

Subgroup

Val Ratio Monzo

(zeroes skipped for clarity in subgroups)

Cents 1789edo Steps Name
2.5 Patent (1251 digits/1251 digits) [4154,-1789⟩ 84.766 126.371 French decimalisma
7-limit Patent 2401/2400 [-5,-1,-2,4⟩ 0.721 1.075 Breedsma
2.5.11.13 Patent 6656/6655 [9,-1,-3,1⟩ 0.260 0.388 Jacobin comma
13-limit 1789bdeef 10648/10647 [3,-2,0,-1,3,-2⟩ 0.163 0.242 Harmonisma
2.5.11.13.31 Patent 387283/387200 [-7,-2,-2,1,3⟩ 0.553
2.5.11.13.31 Patent 2640704/2640625 [6,-6,3,-2,1⟩
2.11.13.29.31 Patent 3455881/3455756 [-2,2,4,-1,-3⟩
2.5.11.13.29.31 Patent 38132480000/38130225991 [11,4,-1,-2,-5,3⟩

Table of selected intervals

Selected intervals in 1789 EDO
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]