10edt

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← 9edt10edt11edt →
Prime factorization 2 × 5
Step size 190.196¢ 
Octave 6\10edt (1141.17¢) (→3\5edt)
Consistency limit 2
Distinct consistency limit 2

10 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 10edt or 10ed3), is a nonoctave tuning system that divides the interval of 3/1 into 10 equal parts of about 190 ¢ each. Each step represents a frequency ratio of 31/10, or the 10th root of 3.

Theory

10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the 5/4 inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as 7/5. We can use this to readily construct chords such as 4:5:7:12, although the 7/4, being 18 cents flat, does considerable damage to the consonance of this chord.

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.

One step of 10edt can serve as the generator for pocus temperament, a merge of sensamagic and 2.3.5.7.13 catakleismic, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

Harmonics

Approximation of harmonics in 10edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -58.8 +0.0 +72.5 +66.6 -58.8 +54.7 +13.7 +0.0 +7.8 +33.0 +72.5
Relative (%) -30.9 +0.0 +38.1 +35.0 -30.9 +28.8 +7.2 +0.0 +4.1 +17.3 +38.1
Steps
(reduced)
6
(6)
10
(0)
13
(3)
15
(5)
16
(6)
18
(8)
19
(9)
20
(0)
21
(1)
22
(2)
23
(3)
Approximation of prime harmonics in 10edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -58.8 +0.0 +66.6 +54.7 +33.0 -66.0 +40.1 +37.8 +87.4 +66.5 -49.0
Relative (%) -30.9 +0.0 +35.0 +28.8 +17.3 -34.7 +21.1 +19.9 +46.0 +35.0 -25.7
Steps
(reduced)
6
(6)
10
(0)
15
(5)
18
(8)
22
(2)
23
(3)
26
(6)
27
(7)
29
(9)
31
(1)
31
(1)

Interval table

Degrees Cents Hekts Approximate Ratios
0 1/1
1 190.196 130 10/9, 28/25
2 380.391 260 5/4
3 570.587 390 7/5
4 760.782 520 14/9
5 950.978 650 45/26, 26/15
6 1141.173 780 27/14
7 1331.369 910 15/7 (15/14 plus an octave)
8 1521.564 1040 12/5 (6/5 plus an octave)
9 1711.760 1170 27/10
10 1901.955 1300 3/1