1789edo

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:

1789

Theory

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
0	Unison	1/1 exact
25	28-thirds comma	[65 -28]
61	Lesser diesis	128/125
172	Tricesimoprimal Miracle semitone	31/29
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
576	Major third	5/4
677	Jacobin naiadic	13/10
822	Jacobin superfourth	11/8
1535	29th harmonic	29/16
1707	31st harmonic	31/16
1789	Octave	2/1 exact

Scales

• Jacobin[37]
• Jacobin[74]
• Jacobin[111]
• Jacobin[222]
• Decimal[265]
• Decimal[1524]