# 105edo

 ← 104edo 105edo 106edo →
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