30edt
| ← 29edt | 30edt | 31edt → |
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
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.6 | +0.0 | +9.1 | +3.2 | +4.6 | -8.7 | +13.7 | +0.0 | +7.8 | -30.4 | +9.1 |
| Relative (%) | +7.2 | +0.0 | +14.4 | +5.1 | +7.2 | -13.7 | +21.6 | +0.0 | +12.3 | -48.0 | +14.4 | |
| Steps (reduced) |
19 (19) |
30 (0) |
38 (8) |
44 (14) |
49 (19) |
53 (23) |
57 (27) |
60 (0) |
63 (3) |
65 (5) |
68 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | -4.1 | +3.2 | +18.3 | -23.3 | +4.6 | -25.6 | +12.4 | -8.7 | -25.8 | +24.0 |
| Relative (%) | -4.2 | -6.5 | +5.1 | +28.8 | -36.7 | +7.2 | -40.4 | +19.5 | -13.7 | -40.8 | +37.9 | |
| Steps (reduced) |
70 (10) |
72 (12) |
74 (14) |
76 (16) |
77 (17) |
79 (19) |
80 (20) |
82 (22) |
83 (23) |
84 (24) |
86 (26) | |
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.
Music
- Room Full Of Steam (2018)