1794edo: Difference between revisions

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== Theory ==

~~{{Primes~~ in ~~edo~~|~~1794~~|~~columns=~~15}}

1794edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all about halfway between its steps. Otherwise it has decent approximations to [[9/1|9]], [[15/1|15]], [[17/1|17]], and [[21/1|21]], making it suitable for a 2.9.15.21.17 [[subgroup]] interpretation with an optional addition of either [[11/1|11]] or [[13/1|13]].

~~1794edo's divisors are {{EDOs|~~2, ~~3, 6, 13, 23, 26, 39, 46, 69, 78, 138, 299, 598~~, and ~~897 }}~~.

For higher harmonics, the best subgroup for 1794 is 2.11.17.19.29.31.47. Notably, it offers a 1794 & [[2016edo|2016]] temperament, and the years 1794 and 2016 are known for having rather grotesque historical events in proportion to their era (see below).

~~The best subgroup for 1794~~ is 2.~~11.17.19.29.31.47. Notably~~, ~~it offers~~ a ~~1794 & [[2016edo~~|~~2016]] temperament~~, ~~and the years 1794 and 2016 are known for having rather grotesque historical events in proportion to their era (see below)~~.

It is possible to interpret 1794edo as an intersection of 26edo and 69edo. Using the {{val| 1794 2834 4160 5037 }} val in the 7-limit, which takes 5-limit from 69edo and 7/4 from 26edo, produces a comma basis 81/80, {{monzo| -41 1 17 0 }}, {{monzo| 20 13 14 -26 }}.

Nonetheless, 1794edo does offer some simpler interpretations.

In the 7-limit in the 1794c val, {{val| 1794 2843 '''4165''' 5036 }}, it tempers out the [[horwell comma]] and the [[landscape comma]], supporting [[mutt]]. However, it is ''not'' better tuned than [[171edo]]. Using the 1794bd val, {{val| 1794 '''2844''' 4166 '''5037''' }}, it tempers out {{monzo| 21 -8 -6 2 }}, {{monzo| -7 -15 6 6 }}, {{monzo| -2 -3 15 -10 }}. This mapping of harmonic 7 is the same as [[26edo]]'s.

In the 7-limit in the 1794c val, {{val| 1794 2843 '''4165''' 5036 }}, it tempers out the [[horwell comma]] and the [[landscape comma]], supporting [[mutt]]. However, it is ''not'' better tuned than [[171edo]]. Using the 1794bd val, {{val| 1794 '''2844''' 4166 '''5037''' }}, it tempers out {{monzo| 21 -8 -6 2 }}, {{monzo| -7 -15 6 6 }}, {{monzo| -2 -3 15 -10 }}. This mapping of harmonic 7 is the same as [[26edo]]'s. In the 2.11.17 realm, 1794edo supports the 148 & 83 temperament, defined by tempering out the 2.11.17 {{monzo| -67 43 -20 }} comma. 1794edo tempers out the 2.17.19 {{monzo| 277 -21 -45 }} and the corresponding temperament is 12 & 891.

Remarkably, using the patent val, 1794edo tempers out the schisma.

Remarkably, using the patent val, 1794edo tempers out the [[schisma]].

~~In the 2.11.17 realm, 1794edo shares the~~ {{~~monzo~~| ~~-67 43 -20~~ }} ~~comma with EDOs like~~ {{~~EDOs~~| ~~148, 231, and 296~~ }}~~. In the 2.17.19 subgroup~~, 1794edo ~~tempers out the~~ {{~~monzo~~| ~~277 -21 -45 }}. Most notably~~, ~~shares this property with {{EDOs| 12~~, 24, 36, ~~48 }}~~, ~~as well as {{EDOs| 855~~, ~~867~~, ~~879 and 891 }}~~, ~~with 855 being a multiple of 171. This is distantly reminiscent of the technique when 12edo's 1 and 3-step intervals~~, ~~for example C# and D# counting from C~~, ~~are assumed to be 17/16~~ and ~~19/16~~.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 1794 factors into {{factorization|1794}}, 1794edo has subset edos {{EDOs| 2, 3, 6, 13, 23, 26, 39, 46, 69, 78, 138, 299, 598, and 897 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

~~! rowspan="2" |Subgroup~~

~~! rowspan="2" |[[Comma list]]~~

~~! rowspan="2" |[[Mapping]]~~

~~! rowspan="2" |Optimal~~

~~8ve stretch (¢)~~

~~! colspan="2" |Tuning error~~

|-

![[~~TE error~~|~~Absolute~~]] ~~(¢)~~

! rowspan="2" | [[Subgroup]]

![[~~TE simple badness|Relative~~]] (%)

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning Error

|-

|~~2.3~~

! [[TE error|Absolute]] (¢)

|~~{{monzo|-2843, 1794}}~~

! [[TE simple badness|Relative]] (%)

~~|{{Val|1794 2843}}~~

~~|0.0892~~

~~|5.97~~

|-

|2.11.17.19.29.31.47

| 2.3

|9251/9248, 347072/346921, 492043/492032, 128116736/128086823, 151329376/151270111, 378544627/378535936

| {{monzo| -2843 1794 }}

|{{~~Val~~|1794 6206 7333 7621 8715 8888 9965}}

| {{mapping| 1794 2843 }}

| -0.0012

| +0.0892

|0.0259

| 0.0892

|1.74

| 5.97

|-

| 2.11.17.19.29.31.47

| 9251/9248, 347072/346921, 492043/492032, 128116736/128086823, 151329376/151270111, 378544627/378535936

| {{mapping| 1794 6206 7333 7621 8715 8888 9965 }}

| −0.0012

| 0.0259

| 1.74

|}

~~== Trivia ==~~

~~{{Wikipedia| 1794 }}~~

~~The number 1794 is known for being the fatal year of the French Revolution.~~

~~[[Category:Equal divisions of the octave~~|~~####]] ~~

@@ Line 1: / Line 1: @@
-{{EDO intro|1794}}
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-{{Primes in edo|1794|columns=15}}
+edo is in[[consistent]] to the [[5-odd-limit]] and [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all about halfway between its steps. Otherwise it has decent approximations to [[9/1|9]], [[15/1|15]], [[17/1|17]], and [[21/1|21]], making it suitable for a 2.9.15.21.17 [[subgroup]] interpretation with an optional addition of either [[11/1|11]] or [[13/1|13]].
-edo's divisors are {{EDOs|2, 3, 6, 13, 23, 26, 39, 46, 69, 78, 138, 299, 598, and 897 }}.
+For higher harmonics, the best subgroup for 1794 is 2.11.17.19.29.31.47. Notably, it offers a 1794 & [[2016edo|2016]] temperament, and the years 1794 and 2016 are known for having rather grotesque historical events in proportion to their era (see below).
-The best subgroup for 1794 is 2.11.17.19.29.31.47. Notably, it offers a 1794 & [[2016edo|2016]] temperament, and the years 1794 and 2016 are known for having rather grotesque historical events in proportion to their era (see below).
+It is possible to interpret 1794edo as an intersection of 26edo and 69edo. Using the {{val| 1794 2834 4160 5037 }} val in the 7-limit, which takes 5-limit from 69edo and 7/4 from 26edo, produces a comma basis 81/80, {{monzo| -41 1 17 0 }}, {{monzo| 20 13 14 -26 }}.
 Nonetheless, 1794edo does offer some simpler interpretations.
-In the 7-limit in the 1794c val, {{val| 1794 2843 '''4165''' 5036 }}, it tempers out the [[horwell comma]] and the [[landscape comma]], supporting [[mutt]]. However, it is ''not'' better tuned than [[171edo]]. Using the 1794bd val, {{val| 1794 '''2844''' 4166 '''5037''' }}, it tempers out {{monzo| 21 -8 -6 2 }}, {{monzo| -7 -15 6 6 }}, {{monzo| -2 -3 15 -10 }}. This mapping of harmonic 7 is the same as [[26edo]]'s.
+In the 7-limit in the 1794c val, {{val| 1794 2843 '''4165''' 5036 }}, it tempers out the [[horwell comma]] and the [[landscape comma]], supporting [[mutt]]. However, it is ''not'' better tuned than [[171edo]]. Using the 1794bd val, {{val| 1794 '''2844''' 4166 '''5037''' }}, it tempers out {{monzo| 21 -8 -6 2 }}, {{monzo| -7 -15 6 6 }}, {{monzo| -2 -3 15 -10 }}. This mapping of harmonic 7 is the same as [[26edo]]'s. In the 2.11.17 realm, 1794edo supports the 148 & 83 temperament, defined by tempering out the 2.11.17 {{monzo| -67 43 -20 }} comma. 1794edo tempers out the 2.17.19 {{monzo| 277 -21 -45 }} and the corresponding temperament is 12 & 891.
-Remarkably, using the patent val, 1794edo tempers out the schisma.
+Remarkably, using the patent val, 1794edo tempers out the [[schisma]].
-In the 2.11.17 realm, 1794edo shares the {{monzo| -67 43 -20 }} comma with EDOs like {{EDOs| 148, 231, and 296 }}. In the 2.17.19 subgroup, 1794edo tempers out the {{monzo| 277 -21 -45 }}. Most notably, shares this property with {{EDOs| 12, 24, 36, 48 }}, as well as {{EDOs| 855, 867, 879 and 891 }}, with 855 being a multiple of 171. This is distantly reminiscent of the technique when 12edo's 1 and 3-step intervals, for example C# and D# counting from C, are assumed to be 17/16 and 19/16.
+=== Odd harmonics ===
+{{Harmonics in equal|1794}}
+=== Subsets and supersets ===
+Since 1794 factors into {{factorization|1794}}, 1794edo has subset edos {{EDOs| 2, 3, 6, 13, 23, 26, 39, 46, 69, 78, 138, 299, 598, and 897 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |Subgroup
-! rowspan="2" |[[Comma list]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal
-ve stretch (¢)
-! colspan="2" |Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! rowspan="2" | [[Subgroup]]
-![[TE simple badness|Relative]] (%)
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
 |-
-|2.3
+! [[TE error|Absolute]] (¢)
-|{{monzo|-2843, 1794}}
+! [[TE simple badness|Relative]] (%)
-|{{Val|1794 2843}}
-|0.0892
-|0.0892
-|5.97
 |-
-|2.11.17.19.29.31.47
+| 2.3
-|9251/9248, 347072/346921, 492043/492032, 128116736/128086823,  151329376/151270111, 378544627/378535936
+| {{monzo| -2843 1794 }}
-|{{Val|1794 6206 7333 7621 8715 8888 9965}}
+| {{mapping| 1794 2843 }}
-| -0.0012
+| +0.0892
-|0.0259
+| 0.0892
-|1.74
+| 5.97
+|-
+| 2.11.17.19.29.31.47
+| 9251/9248, 347072/346921, 492043/492032, 128116736/128086823,  151329376/151270111, 378544627/378535936
+| {{mapping| 1794 6206 7333 7621 8715 8888 9965 }}
+| −0.0012
+| 0.0259
+| 1.74
 |}
-== Trivia ==
-{{Wikipedia| 1794 }}
-The number 1794 is known for being the fatal year of the French Revolution.
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+{{Todo| review }}