2118edo

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← 2117edo2118edo2119edo →
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