105edo

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← 104edo105edo106edo →
Prime factorization 3 × 5 × 7
Step size 11.4286¢ 
Fifth 61\105 (697.143¢)
Semitones (A1:m2) 7:10 (80¢ : 114.3¢)
Dual sharp fifth 62\105 (708.571¢)
Dual flat fifth 61\105 (697.143¢)
Dual major 2nd 18\105 (205.714¢) (→6\35)
Consistency limit 3
Distinct consistency limit 3

105 equal divisions of the octave (abbreviated 105edo or 105ed2), also called 105-tone equal temperament (105tet) or 105 equal temperament (105et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 105 equal parts of about 11.4 ¢ each. Each step represents a frequency ratio of 21/105, or the 105th root of 2.

Theory

105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone, such as grosstone and Huygens. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the optimal patent val, and provides a good tuning for the 13-limit extension, though 74edo is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the CTE generator for meantone.

Odd harmonics

Approximation of odd harmonics in 105edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -4.81 +2.26 +2.60 +1.80 -2.75 +5.19 -2.55 -2.10 -0.37 -2.21 +0.30
Relative (%) -42.1 +19.8 +22.8 +15.8 -24.0 +45.4 -22.4 -18.4 -3.2 -19.3 +2.6
Steps
(reduced)
166
(61)
244
(34)
295
(85)
333
(18)
363
(48)
389
(74)
410
(95)
429
(9)
446
(26)
461
(41)
475
(55)

Subsets and supersets

105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

Intervals

15-odd-limit interval mappings

The following tables show how 15-odd-limit intervals are represented in 105edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 105edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
9/5, 10/9 0.453 4.0
9/7, 14/9 0.798 7.0
13/12, 24/13 1.430 12.5
9/8, 16/9 1.804 15.8
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
13/9, 18/13 3.382 29.6
13/11, 22/13 3.495 30.6
15/13, 26/15 3.688 32.3
7/6, 12/7 4.014 35.1
5/3, 6/5 4.359 38.1
11/9, 18/11 4.551 39.8
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
15-odd-limit intervals in 105edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/11, 22/15 0.192 1.7
7/5, 10/7 0.345 3.0
11/6, 12/11 2.066 18.1
5/4, 8/5 2.258 19.8
15/8, 16/15 2.554 22.4
13/7, 14/13 2.584 22.6
7/4, 8/7 2.603 22.8
11/8, 16/11 2.747 24.0
13/10, 20/13 2.929 25.6
3/2, 4/3 4.812 42.1
11/10, 20/11 5.004 43.8
15/14, 28/15 5.157 45.1
13/8, 16/13 5.187 45.4
11/7, 14/11 5.349 46.8
11/9, 18/11 6.878 60.2
5/3, 6/5 7.070 61.9
7/6, 12/7 7.415 64.9
15/13, 26/15 7.741 67.7
13/11, 22/13 7.933 69.4
9/8, 16/9 9.624 84.2
13/12, 24/13 9.999 87.5
9/5, 10/9 11.882 104.0
9/7, 14/9 12.227 107.0
13/9, 18/13 14.811 129.6