30edt: Difference between revisions
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| Line 9: | Line 9: | ||
| | Degrees | | | Degrees | ||
| | Cents | | | Cents | ||
|Hekts | |||
| | Approximate Ratios | | | Approximate Ratios | ||
| | Sigma scale name | | | Sigma scale name | ||
|- | |- | ||
| | 0 | | | 0 | ||
| | 0 | | colspan="2"| 0 | ||
| | <span style="color: #660000;">[[1/1|1/1]]</span> | | | <span style="color: #660000;">[[1/1|1/1]]</span> | ||
| | C | | | C | ||
| Line 19: | Line 20: | ||
| | 1 | | | 1 | ||
| | 63.3985 | | | 63.3985 | ||
|43.333 | |||
| | 28/27, 27/26 | | | 28/27, 27/26 | ||
| | C#/Dbb | | | C#/Dbb | ||
| Line 24: | Line 26: | ||
| | 2 | | | 2 | ||
| | 126.797 | | | 126.797 | ||
|86.667 | |||
| | [[14/13|14/13]], [[15/14|15/14]], [[16/15|16/15]], 29/27 | | | [[14/13|14/13]], [[15/14|15/14]], [[16/15|16/15]], 29/27 | ||
| | Cx/Db | | | Cx/Db | ||
| Line 29: | Line 32: | ||
| | 3 | | | 3 | ||
| | 190.1955 | | | 190.1955 | ||
|130 | |||
| | 10/9~9/8 | | | 10/9~9/8 | ||
| | D | | | D | ||
| Line 34: | Line 38: | ||
| | 4 | | | 4 | ||
| | 253.594 | | | 253.594 | ||
|173.333 | |||
| | [[15/13|15/13]] | | | [[15/13|15/13]] | ||
| | D#/Ebb | | | D#/Ebb | ||
| Line 39: | Line 44: | ||
| | 5 | | | 5 | ||
| | 316.9925 | | | 316.9925 | ||
|216.667 | |||
| | 6/5 | | | 6/5 | ||
| | Dx/Eb | | | Dx/Eb | ||
| Line 44: | Line 50: | ||
| | 6 | | | 6 | ||
| | 380.391 | | | 380.391 | ||
|260 | |||
| | <span style="color: #660000;">[[5/4|5/4]]</span> | | | <span style="color: #660000;">[[5/4|5/4]]</span> | ||
| | E | | | E | ||
| Line 49: | Line 56: | ||
| | 7 | | | 7 | ||
| | 443.7895 | | | 443.7895 | ||
|303.333 | |||
| | 9/7 | | | 9/7 | ||
| | E#/Fbb | | | E#/Fbb | ||
| Line 54: | Line 62: | ||
| | 8 | | | 8 | ||
| | 507.188 | | | 507.188 | ||
|346.667 | |||
| | [[4/3|4/3]] | | | [[4/3|4/3]] | ||
| | Ex/Fb | | | Ex/Fb | ||
| Line 59: | Line 68: | ||
| | 9 | | | 9 | ||
| | 570.5865 | | | 570.5865 | ||
|390 | |||
| | 7/5 | | | 7/5 | ||
| | F | | | F | ||
| Line 64: | Line 74: | ||
| | 10 | | | 10 | ||
| | 633.985 | | | 633.985 | ||
|433.333 | |||
| | [[13/9|13/9]] | | | [[13/9|13/9]] | ||
| | F#/Gb | | | F#/Gb | ||
| Line 69: | Line 80: | ||
| | 11 | | | 11 | ||
| | 697.3835 | | | 697.3835 | ||
|476.667 | |||
| | 3/2 | | | 3/2 | ||
| | G | | | G | ||
| Line 74: | Line 86: | ||
| | 12 | | | 12 | ||
| | 760.782 | | | 760.782 | ||
|520 | |||
| | <span style="color: #660000;">[[14/9|14/9]]</span> | | | <span style="color: #660000;">[[14/9|14/9]]</span> | ||
| | G#/Hbb | | | G#/Hbb | ||
| Line 79: | Line 92: | ||
| | 13 | | | 13 | ||
| | 824.1805 | | | 824.1805 | ||
|563.333 | |||
| | 8/5 | | | 8/5 | ||
| | Gx/Hb | | | Gx/Hb | ||
| Line 84: | Line 98: | ||
| | 14 | | | 14 | ||
| | 887.579 | | | 887.579 | ||
|606.667 | |||
| | [[5/3|5/3]] | | | [[5/3|5/3]] | ||
| | H | | | H | ||
| Line 89: | Line 104: | ||
| | 15 | | | 15 | ||
| | 950.9775 | | | 950.9775 | ||
|650 | |||
| | 19/11 | | | 19/11 | ||
| | H#/Jbb | | | H#/Jbb | ||
| Line 94: | Line 110: | ||
| | 16 | | | 16 | ||
| | 1014.376 | | | 1014.376 | ||
|693.333 | |||
| | [[9/5|9/5]] | | | [[9/5|9/5]] | ||
| | Hx/Jb | | | Hx/Jb | ||
| Line 99: | Line 116: | ||
| | 17 | | | 17 | ||
| | 1077.7745 | | | 1077.7745 | ||
|736.667 | |||
| | 13/7 | | | 13/7 | ||
| | J | | | J | ||
| Line 104: | Line 122: | ||
| | 18 | | | 18 | ||
| | 1141.173 | | | 1141.173 | ||
|780 | |||
| | <span style="color: #660000;">[[27/14|27/14]]</span> | | | <span style="color: #660000;">[[27/14|27/14]]</span> | ||
| | J#/Kbb | | | J#/Kbb | ||
| Line 109: | Line 128: | ||
| | 19 | | | 19 | ||
| | 1204.5715 | | | 1204.5715 | ||
|823.333 | |||
| | 2/1 | | | 2/1 | ||
| | Jx/Kb | | | Jx/Kb | ||
|- | |- | ||
| | 20 | | | 20 | ||
| | 1267. | | | 1267.97 | ||
|866.667 | |||
| | [[27/13|27/13]] | | | [[27/13|27/13]] | ||
| | K | | | K | ||
| Line 119: | Line 140: | ||
| | 21 | | | 21 | ||
| | 1331.3685 | | | 1331.3685 | ||
|910 | |||
| | 28/13 | | | 28/13 | ||
| | K#/Lb | | | K#/Lb | ||
| Line 124: | Line 146: | ||
| | 22 | | | 22 | ||
| | 1394.767 | | | 1394.767 | ||
|953.333 | |||
| | [[9/4|9/4]] ([[9/8|9/8]] plus an octave) | | | [[9/4|9/4]] ([[9/8|9/8]] plus an octave) | ||
| | L | | | L | ||
| Line 129: | Line 152: | ||
| | 23 | | | 23 | ||
| | 1458.1655 | | | 1458.1655 | ||
|996.667 | |||
| | 7/3 | | | 7/3 | ||
| | L#/Abb | | | L#/Abb | ||
| Line 134: | Line 158: | ||
| | 24 | | | 24 | ||
| | 1521.564 | | | 1521.564 | ||
|1040 | |||
| | [[12/5|12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave) | | | [[12/5|12/5]] (<span style="color: #660000;">[[6/5|6/5]]</span> plus an octave) | ||
| | Lx/Ab | | | Lx/Ab | ||
| Line 139: | Line 164: | ||
| | 25 | | | 25 | ||
| | 1584.9625 | | | 1584.9625 | ||
|1083.333 | |||
| | 5/2 | | | 5/2 | ||
| | A | | | A | ||
| Line 144: | Line 170: | ||
| | 26 | | | 26 | ||
| | 1648.361 | | | 1648.361 | ||
|1126.667 | |||
| | [[13/5|13/5]] ([[13/10|13/10]] plus an octave) | | | [[13/5|13/5]] ([[13/10|13/10]] plus an octave) | ||
| | A#/Bbb | | | A#/Bbb | ||
| Line 149: | Line 176: | ||
| | 27 | | | 27 | ||
| | 1711.7595 | | | 1711.7595 | ||
|1170 | |||
| | 8/3 | | | 8/3 | ||
| | Ax/Bb | | | Ax/Bb | ||
| Line 154: | Line 182: | ||
| | 28 | | | 28 | ||
| | 1775.158 | | | 1775.158 | ||
|1213.333 | |||
| | [[14/5|14/5]] ([[7/5|7/5]] plus an octave) | | | [[14/5|14/5]] ([[7/5|7/5]] plus an octave) | ||
| | B | | | B | ||
| Line 159: | Line 188: | ||
| | 29 | | | 29 | ||
| | 1838.5565 | | | 1838.5565 | ||
|1256.667 | |||
| | 26/9 | | | 26/9 | ||
| | B#/Cb | | | B#/Cb | ||
| Line 164: | Line 194: | ||
| | 30 | | | 30 | ||
| | 1901.955 | | | 1901.955 | ||
|1300 | |||
| | [[3/1|3/1]] | | | [[3/1|3/1]] | ||
| | C | | | C | ||
Revision as of 20:01, 15 April 2019
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.
Intervals of 30edt
| Degrees | Cents | Hekts | Approximate Ratios | Sigma scale name |
| 0 | 0 | 1/1 | C | |
| 1 | 63.3985 | 43.333 | 28/27, 27/26 | C#/Dbb |
| 2 | 126.797 | 86.667 | 14/13, 15/14, 16/15, 29/27 | Cx/Db |
| 3 | 190.1955 | 130 | 10/9~9/8 | D |
| 4 | 253.594 | 173.333 | 15/13 | D#/Ebb |
| 5 | 316.9925 | 216.667 | 6/5 | Dx/Eb |
| 6 | 380.391 | 260 | 5/4 | E |
| 7 | 443.7895 | 303.333 | 9/7 | E#/Fbb |
| 8 | 507.188 | 346.667 | 4/3 | Ex/Fb |
| 9 | 570.5865 | 390 | 7/5 | F |
| 10 | 633.985 | 433.333 | 13/9 | F#/Gb |
| 11 | 697.3835 | 476.667 | 3/2 | G |
| 12 | 760.782 | 520 | 14/9 | G#/Hbb |
| 13 | 824.1805 | 563.333 | 8/5 | Gx/Hb |
| 14 | 887.579 | 606.667 | 5/3 | H |
| 15 | 950.9775 | 650 | 19/11 | H#/Jbb |
| 16 | 1014.376 | 693.333 | 9/5 | Hx/Jb |
| 17 | 1077.7745 | 736.667 | 13/7 | J |
| 18 | 1141.173 | 780 | 27/14 | J#/Kbb |
| 19 | 1204.5715 | 823.333 | 2/1 | Jx/Kb |
| 20 | 1267.97 | 866.667 | 27/13 | K |
| 21 | 1331.3685 | 910 | 28/13 | K#/Lb |
| 22 | 1394.767 | 953.333 | 9/4 (9/8 plus an octave) | L |
| 23 | 1458.1655 | 996.667 | 7/3 | L#/Abb |
| 24 | 1521.564 | 1040 | 12/5 (6/5 plus an octave) | Lx/Ab |
| 25 | 1584.9625 | 1083.333 | 5/2 | A |
| 26 | 1648.361 | 1126.667 | 13/5 (13/10 plus an octave) | A#/Bbb |
| 27 | 1711.7595 | 1170 | 8/3 | Ax/Bb |
| 28 | 1775.158 | 1213.333 | 14/5 (7/5 plus an octave) | B |
| 29 | 1838.5565 | 1256.667 | 26/9 | 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.
Compositions in 30edt
- "Room Full Of Steam", Mason Green. In the key of "Eb subminor".