# 30edt

 ← 29edt 30edt 31edt →
Prime factorization 2 × 3 × 5
Step size 63.3985¢
Octave 19\30edt (1204.57¢)
Consistency limit 10
Distinct consistency limit 5

Division of the third harmonic into 30 equal parts (30edt) is related to 19 edo, but with the 3/1 rather than the 2/1 being just. The octave is about 4.5715 cents stretched and the step size is about 63.3985 cents. It is consistent to the 10-integer-limit.

Because 19edo has the 3rd, 5th, 7th, and 13th harmonics all flat (the latter two very flat), it benefits greatly from octave stretching. 30edt is one possible alternative; at the cost of sharpening the octave, it achieves much better matches to the odd harmonics; the 3 (tritave) is by definition just, the 5 slightly sharp, and the 7 and 13 slightly flat.

While the fifth is just, the fourth is noticeably sharper and less accurate than in 19edo, being close to that of 26edo.

## Harmonics

Approximation of harmonics in 30edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +4.57 +0.00 +9.14 +3.22 +4.57 -8.71 +13.71 +0.00 +7.79 -30.42 +9.14 -2.63 -4.13 +3.22 +18.29
Relative (%) +7.2 +0.0 +14.4 +5.1 +7.2 -13.7 +21.6 +0.0 +12.3 -48.0 +14.4 -4.2 -6.5 +5.1 +28.8
Steps
(reduced)
19
(19)
30
(0)
38
(8)
44
(14)
49
(19)
53
(23)
57
(27)
60
(0)
63
(3)
65
(5)
68
(8)
70
(10)
72
(12)
74
(14)
76
(16)

## Intervals of 30edt

Degrees Cents Hekts Approximate Ratios Lambda scale name Sigma scale name
0 0 0 1/1 C
1 63.3985 43.333 28/27, 27/26 C^/Dbv C#/Dbb
2 126.797 86.667 14/13, 15/14, 16/15, 29/27 Db Cx/Db
3 190.1955 130 10/9~9/8 C# D
4 253.594 173.333 15/13 C#^/Dv D#/Ebb
5 316.9925 216.667 6/5 D Dx/Eb
6 380.391 260 5/4 D^/Ev E
7 443.7895 303.333 9/7 E E#/Fbb
8 507.188 346.667 4/3 E^/Fbv Ex/Fb
9 570.5865 390 7/5 Fb F
10 633.985 433.333 13/9 E# F#/Gb
11 697.3835 476.667 3/2 E#^/Fv G
12 760.782 520 14/9 F G#/Hbb
13 824.1805 563.333 8/5 F^/Gv Gx/Hb
14 887.579 606.667 5/3 G H
15 950.9775 650 19/11 G^/Hbv H#/Jbb
16 1014.376 693.333 9/5 Hb Hx/Jb
17 1077.7745 736.667 13/7 G# J
18 1141.173 780 27/14 G#^/Hv J#/Kbb
19 1204.5715 823.333 2/1 H Jx/Kb
20 1267.97 866.667 27/13 H^/Jv K
21 1331.3685 910 28/13 J K#/Lb
22 1394.767 953.333 9/4 (9/8 plus an octave) J^/Av L
23 1458.1655 996.667 7/3 A L#/Abb
24 1521.564 1040 12/5 (6/5 plus an octave) A^/Bbv Lx/Ab
25 1584.9625 1083.333 5/2 Bb A
26 1648.361 1126.667 13/5 (13/10 plus an octave) A# A#/Bbb
27 1711.7595 1170 8/3 A#^/Bv Ax/Bb
28 1775.158 1213.333 14/5 (7/5 plus an octave) B
29 1838.5565 1256.667 26/9 B^/Cv B#/Cb
30 1901.955 1300 3/1 C

30edt contains all 19edo intervals within 3/1, all temepered progressively sharper. The accumulation of the .241 cent sharpening of the unit step relative to 19edo leads to the excellent 6edt approximations of 6/5 and 5/2. Non-redundantly with simpler edts, the 41 degree ~9/2 is only .6615 cents flatter than that in 6edo.

30edt also contains all the MOS contained in 15edt, being the double of this equal division. Being even, 30edt introduces MOS with an even number of periods per tritave such as a 6L 6s similar to Hexe Dodecatonic. This MOS has a period of 1/6 of the tritave and the generator is a single or double step. The major scale is sLsLsLsLsLsL, and the minor scale is LsLsLsLsLsLs. Being a "real" 3/2, the interval of 11 degrees generates an unfair Sigma scale of 8L 3s and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda MOS will generate a 4L 9s unfair Superlambda MOS which does not border on being atonal as the 17edt rendition does.

Mason Green
Ray Perlner