1794edo: Difference between revisions
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{{Infobox ET}} | |||
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== Theory == | == Theory == | ||
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]]. | |||
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 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 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]]. | |||
== | === 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 }} | |||
Latest revision as of 22:54, 20 February 2025
| ← 1793edo | 1794edo | 1795edo → |
1794 equal divisions of the octave (abbreviated 1794edo or 1794ed2), also called 1794-tone equal temperament (1794tet) or 1794 equal temperament (1794et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1794 equal parts of about 0.669 ¢ each. Each step represents a frequency ratio of 21/1794, or the 1794th root of 2.
Theory
1794edo is inconsistent to the 5-odd-limit and harmonics 3, 5, and 7 are all about halfway between its steps. Otherwise it has decent approximations to 9, 15, 17, and 21, making it suitable for a 2.9.15.21.17 subgroup interpretation with an optional addition of either 11 or 13.
For higher harmonics, the best subgroup for 1794 is 2.11.17.19.29.31.47. Notably, it offers a 1794 & 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 ⟨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, [-41 1 17 0⟩, [20 13 14 -26⟩.
Nonetheless, 1794edo does offer some simpler interpretations.
In the 7-limit in the 1794c 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, ⟨1794 2844 4166 5037], it tempers out [21 -8 -6 2⟩, [-7 -15 6 6⟩, [-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 [-67 43 -20⟩ comma. 1794edo tempers out the 2.17.19 [277 -21 -45⟩ and the corresponding temperament is 12 & 891.
Remarkably, using the patent val, 1794edo tempers out the schisma.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.283 | +0.308 | -0.264 | +0.103 | -0.147 | +0.275 | +0.026 | +0.061 | +0.146 | +0.122 | -0.181 |
| Relative (%) | -42.3 | +46.1 | -39.5 | +15.5 | -22.0 | +41.1 | +3.8 | +9.2 | +21.8 | +18.3 | -27.0 | |
| Steps (reduced) |
2843 (1049) |
4166 (578) |
5036 (1448) |
5687 (305) |
6206 (824) |
6639 (1257) |
7009 (1627) |
7333 (157) |
7621 (445) |
7880 (704) |
8115 (939) | |
Subsets and supersets
Since 1794 factors into 2 × 3 × 13 × 23, 1794edo has subset edos 2, 3, 6, 13, 23, 26, 39, 46, 69, 78, 138, 299, 598, and 897.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2843 1794⟩ | [⟨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 | [⟨1794 6206 7333 7621 8715 8888 9965]] | −0.0012 | 0.0259 | 1.74 |