1789edo: Difference between revisions

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 {{Wikipedia|1789}}
 ==Theory==
-{{primes in edo|1789|columns = 11}}
+{{primes in edo|1789|columns = 18}}
-edo can be adapted for use with the 2.5.11.13.29.31 subgroup.
+edo can be adapted for use with the 2.5.11.13.29.31.47.59.61 subgroup.
-'''Table of selected intervals'''
+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 ==
+{| class="wikitable"
+|+Tempered commas in 1789edo
+!Prime
+Subgroup
+!Val
+!Ratio
+!Monzo
+<small>(zeroes skipped for clarity in subgroups)</small>
+!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 ==
 {| class="wikitable collapsible mw-collapsed"
 |+
+Selected intervals in 1789 EDO
 !Step
 !Name
-!JI Approximation or Monzo
+!JI Approximation, Monzo, or another interpretation
 |-
 |0
@@ Line 42: / Line 127: @@
 |}
-===Temperaments===
+==Scales==
-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).
-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.
-== Scales ==
 * Jacobin[37]
 * Jacobin[74]