1794edo: Difference between revisions

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== Theory ==

~~{{Harmonics~~ in ~~equal|1794}}~~

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 }}~~.

~~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).

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).

It's 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}}.

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: @@
 {{Infobox ET}}
-{{EDO intro|1794}}
+{{ED intro}}
 == Theory ==
-{{Harmonics in equal|1794}}
+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 }}.
-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).
+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).
-It's 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}}.
+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 }}