2118edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
Primes approximated with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13's 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations. | |||
2118edo provides a 43-limit approximation of [[secor]] with [[46/43]] (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a {{nowrap|31 & 2118}} temperament, also in the 2.3.5.7.11.23.43 subgroup, and also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|2118}} | {{Harmonics in equal|2118}} | ||
=== Subsets and supersets === | |||
2118edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan #Rectified hebrew|rectified hebrew]] temperament. | |||
2118edo is 6 times the [[353edo]], meaning it can be used to play a compound of 6 chains of the [[Hemimean clan#Rectified | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| 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 | |||
|- | |- | ||
|2.3.5 | ! [[TE error|Absolute]] (¢) | ||
|{{monzo|38 -2 15}}, {{monzo| -11 130 -84}} | ! [[TE simple badness|Relative]] (%) | ||
| | |- | ||
| | | 2.3.5 | ||
|0.0156 | | {{monzo| 38 -2 15 }}, {{monzo| -11 130 -84 }} | ||
| {{mapping| 2118 3357 4918 }} | |||
| −0.0186 | |||
| 0.0156 | |||
| | | | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|250047/250000,{{monzo|-1 -18 -3 13}}, {{monzo|38 -2 -15 | | 250047/250000, {{monzo| -1 -18 -3 13 }}, {{monzo| 38 -2 -15 }} | ||
| | | {{mapping| 2118 3357 4918 5946 }} | ||
| | | −0.0150 | ||
|0.0148 | | 0.0148 | ||
| | | | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|9801/9800, 250047/250000, {{monzo|25 1 -4 0 -5}}, {{monzo|16 -7 -9 2 3}} | | 9801/9800, 250047/250000, {{monzo| 25 1 -4 0 -5 }}, {{monzo| 16 -7 -9 2 3 }} | ||
| | | {{mapping| 2118 3357 4918 5946 7927 }} | ||
| | | −0.0096 | ||
|0.0172 | | 0.0172 | ||
| | | | ||
|} | |} | ||
Latest revision as of 23:05, 20 February 2025
| ← 2117edo | 2118edo | 2119edo → |
2118 equal divisions of the octave (abbreviated 2118edo or 2118ed2), also called 2118-tone equal temperament (2118tet) or 2118 equal temperament (2118et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2118 equal parts of about 0.567 ¢ each. Each step represents a frequency ratio of 21/2118, or the 2118th root of 2.
Theory
Primes approximated with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13's 31-limit harmony, as both mappings of 13 (2118 and 2118f vals) have useful interpretations.
2118edo provides a 43-limit approximation of secor with 46/43 (206 steps), however this reduces to 103\1059, meaning that it is a compound of two circles of such secor. In addition, it offers a 205-step generator "meantone secor" which is described by a 31 & 2118 temperament, also in the 2.3.5.7.11.23.43 subgroup, and also offers a meantone fifth. The comma basis for the "meantone secor" temperament is 5376/5375, 9317/9315, 25921/25920, 151263/151250, and 10551296/10546875.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.028 | +0.089 | +0.013 | -0.043 | +0.266 | -0.140 | -0.063 | +0.054 | -0.115 | +0.007 |
| Relative (%) | +0.0 | +4.9 | +15.6 | +2.2 | -7.6 | +46.9 | -24.6 | -11.0 | +9.6 | -20.4 | +1.2 | |
| Steps (reduced) |
2118 (0) |
3357 (1239) |
4918 (682) |
5946 (1710) |
7327 (973) |
7838 (1484) |
8657 (185) |
8997 (525) |
9581 (1109) |
10289 (1817) |
10493 (2021) | |
Subsets and supersets
2118edo is 6 times the 353edo, meaning it can be used to play a compound of 6 chains of the rectified hebrew temperament.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [38 -2 15⟩, [-11 130 -84⟩ | [⟨2118 3357 4918]] | −0.0186 | 0.0156 | |
| 2.3.5.7 | 250047/250000, [-1 -18 -3 13⟩, [38 -2 -15⟩ | [⟨2118 3357 4918 5946]] | −0.0150 | 0.0148 | |
| 2.3.5.7.11 | 9801/9800, 250047/250000, [25 1 -4 0 -5⟩, [16 -7 -9 2 3⟩ | [⟨2118 3357 4918 5946 7927]] | −0.0096 | 0.0172 | |