1794edo: Difference between revisions

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~~1794 equal divisions of the octave creates steps of 0.668896 cents each.~~

== 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~~|13, 23, 26, 39, ~~46, 69, 78, 138, 299, 598, and 897}}~~.

~~The~~ best subgroup for 1794 is 11.17.19.29.31.47.

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).

Using the ~~1794d~~ val ~~assigns~~ the 7/4 ~~to its [[~~26edo~~]] counterpart~~, ~~and opens the support for the 26edo [[slendric pentad]]. It supports the low-complexity 152&202&1794 temperament eliminating the [25 26~~ -14 -~~12⟩ comma~~.

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 }}.

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

Nonetheless, 1794edo does offer some simpler interpretations.

In the 2.11.17 realm, 1794edo ~~shares~~ the ~~[-67~~, ..~~. 43, ...-20⟩ comma with EDOs like~~ {{~~EDOs~~|~~148, 231, and 296~~}}. In the 2.17.19 ~~subgroup~~, 1794edo tempers out the [~~277~~, ...-~~21,~~ ...-~~45⟩ and mirrors 12edo when its m2 and m3 are assumed to be~~ 17~~/16 and~~ 19~~/16~~.

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.

~~== History ==~~

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

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

=== 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|Comma List]]

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3

| {{monzo| -2843 1794 }}

| {{mapping| 1794 2843 }}

| +0.0892

| 0.0892

| 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

|}

@@ Line 1: / Line 1: @@
-equal divisions of the octave creates steps of 0.668896 cents each.
+{{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|13, 23, 26, 39, 46, 69, 78, 138, 299, 598, and 897}}.
-The best subgroup for 1794 is 11.17.19.29.31.47.
+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).
-Using the 1794d val assigns the 7/4 to its [[26edo]] counterpart, and opens the support for the 26edo [[slendric pentad]]. It supports the low-complexity 152&202&1794 temperament eliminating the [25 26 -14 -12⟩ comma.
+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 }}.
-Remarkably, using the patent val, 1794edo tempers out the schisma.
+Nonetheless, 1794edo does offer some simpler interpretations.
-In the 2.11.17 realm, 1794edo shares the [-67, ... 43, ...-20⟩ comma with EDOs like {{EDOs|148, 231, and 296}}. In the 2.17.19 subgroup, 1794edo tempers out the [277, ...-21, ...-45⟩ and mirrors 12edo when its m2 and m3 are assumed to be 17/16 and 19/16.
+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.
-== History ==
-{{Wikipedia|1794}}
+Remarkably, using the patent val, 1794edo tempers out the [[schisma]].
-The number 1794 is known for being the fatal year of the French Revolution.
+=== 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|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| -2843 1794 }}
+| {{mapping| 1794 2843 }}
+| +0.0892
+| 0.0892
+| 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
+|}
+{{Todo| review }}