2118edo: Difference between revisions

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{{novelty}}{{stub}}{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2118}}
{{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}}
Primes with less than 1 standard deviation in 2118edo are: 2, 3, 5, 7, 11, 19, 23, 29, 31, 43. Overall, it offers excellent double-13s 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.
=== 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 Hebrew|Rectified Hebrew]] temperament.


== 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]]
! 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
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|{{monzo|38 -2 15}}, {{monzo| -11 130 -84}}
! [[TE simple badness|Relative]] (%)
|[{{val|2118 3357 4918}}]
|-
| -0.0186
| 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 0}}
| 250047/250000, {{monzo| -1 -18 -3 13 }}, {{monzo| 38 -2 -15 }}
|[{{val|2118 3357 4918 5946}}]
| {{mapping| 2118 3357 4918 5946 }}
| -0.0150
| −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 }}
|[{{val|2118 3357 4918 5946 7927}}]
| {{mapping| 2118 3357 4918 5946 7927 }}
|<nowiki>-0.0096</nowiki>
| −0.0096
|0.0172
| 0.0172
|
|
|}
|}
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->

Latest revision as of 23:05, 20 February 2025

← 2117edo 2118edo 2119edo →
Prime factorization 2 × 3 × 353
Step size 0.566572 ¢ 
Fifth 1239\2118 (701.983 ¢) (→ 413\706)
Semitones (A1:m2) 201:159 (113.9 ¢ : 90.08 ¢)
Consistency limit 11
Distinct consistency limit 11

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

Approximation of prime harmonics in 2118edo
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
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