30edt: Difference between revisions
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{{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=1|Approximation of harmonics in 30edt (continued)}} | {{Harmonics in equal|30|3|1|intervals=integer|columns=12|start=12|collapsed=1|Approximation of harmonics in 30edt (continued)}} | ||
== Intervals | == Intervals == | ||
{| class="wikitable center-all right-2 right-3 left-4" | {| class="wikitable center-all right-2 right-3 left-4" | ||
|- | |- | ||
Line 34: | Line 34: | ||
|- | |- | ||
| 1 | | 1 | ||
| 63. | | 63.4 | ||
| 43. | | 43.3 | ||
| 27/26, 28/27 | | [[27/26]], [[28/27]] | ||
| C^/Dbv | | C^/Dbv | ||
| C#/Dbb | | C#/Dbb | ||
|- | |- | ||
| 2 | | 2 | ||
| 126. | | 126.8 | ||
| 86. | | 86.7 | ||
| [[14/13]], [[15/14]], [[16/15]], [[29/27]] | | [[14/13]], [[15/14]], [[16/15]], [[29/27]] | ||
| Db | | Db | ||
Line 48: | Line 48: | ||
|- | |- | ||
| 3 | | 3 | ||
| 190. | | 190.2 | ||
| 130 | | 130.0 | ||
| 9/8, 10/9 | | [[9/8]], [[10/9]] | ||
| C# | | C# | ||
| D | | D | ||
|- | |- | ||
| 4 | | 4 | ||
| 253. | | 253.6 | ||
| 173. | | 173.3 | ||
| [[15/13]] | | [[15/13]] | ||
| C#^/Dv | | C#^/Dv | ||
Line 62: | Line 62: | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 317.0 | ||
| 216. | | 216.7 | ||
| 6/5 | | [[6/5]] | ||
| D | | D | ||
| Dx/Eb | | Dx/Eb | ||
|- | |- | ||
| 6 | | 6 | ||
| 380. | | 380.4 | ||
| 260 | | 260.0 | ||
| [[5/4]] | | [[5/4]] | ||
| D^/Ev | | D^/Ev | ||
Line 76: | Line 76: | ||
|- | |- | ||
| 7 | | 7 | ||
| 443. | | 443.8 | ||
| 303. | | 303.3 | ||
| 9/7 | | [[9/7]] | ||
| E | | E | ||
| E#/Fbb | | E#/Fbb | ||
|- | |- | ||
| 8 | | 8 | ||
| 507. | | 507.2 | ||
| 346. | | 346.7 | ||
| [[4/3]] | | [[4/3]] | ||
| E^/Fbv | | E^/Fbv | ||
Line 90: | Line 90: | ||
|- | |- | ||
| 9 | | 9 | ||
| 570. | | 570.6 | ||
| 390 | | 390.0 | ||
| 7/5 | | [[7/5]] | ||
| Fb | | Fb | ||
| F | | F | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 634.0 | ||
| 433. | | 433.3 | ||
| [[13/9]] | | [[13/9]] | ||
| E# | | E# | ||
Line 104: | Line 104: | ||
|- | |- | ||
| 11 | | 11 | ||
| 697. | | 697.4 | ||
| 476. | | 476.7 | ||
| 3/2 | | [[3/2]] | ||
| E#^/Fv | | E#^/Fv | ||
| G | | G | ||
|- | |- | ||
| 12 | | 12 | ||
| 760. | | 760.8 | ||
| 520 | | 520.0 | ||
| [[14/9]] | | [[14/9]] | ||
| F | | F | ||
Line 118: | Line 118: | ||
|- | |- | ||
| 13 | | 13 | ||
| 824. | | 824.2 | ||
| 563. | | 563.3 | ||
| 8/5 | | [[8/5]] | ||
| F^/Gv | | F^/Gv | ||
| Gx/Hb | | Gx/Hb | ||
|- | |- | ||
| 14 | | 14 | ||
| 887. | | 887.6 | ||
| 606. | | 606.7 | ||
| [[5/3]] | | [[5/3]] | ||
| G | | G | ||
Line 132: | Line 132: | ||
|- | |- | ||
| 15 | | 15 | ||
| | | 951.0 | ||
| 650 | | 650.0 | ||
| 19/11 | | [[19/11]] | ||
| G^/Hbv | | G^/Hbv | ||
| H#/Jbb | | H#/Jbb | ||
|- | |- | ||
| 16 | | 16 | ||
| 1014. | | 1014.4 | ||
| 693. | | 693.3 | ||
| [[9/5]] | | [[9/5]] | ||
| Hb | | Hb | ||
Line 146: | Line 146: | ||
|- | |- | ||
| 17 | | 17 | ||
| 1077. | | 1077.8 | ||
| 736. | | 736.7 | ||
| 13/7 | | [[13/7]] | ||
| G# | | G# | ||
| J | | J | ||
|- | |- | ||
| 18 | | 18 | ||
| 1141. | | 1141.2 | ||
| 780 | | 780.0 | ||
| [[27/14]] | | [[27/14]] | ||
| G#^/Hv | | G#^/Hv | ||
Line 160: | Line 160: | ||
|- | |- | ||
| 19 | | 19 | ||
| 1204. | | 1204.6 | ||
| 823. | | 823.3 | ||
| 2/1 | | [[2/1]] | ||
| H | | H | ||
| Jx/Kb | | Jx/Kb | ||
|- | |- | ||
| 20 | | 20 | ||
| | | 1268.0 | ||
| 866. | | 866.7 | ||
| [[27/13]] | | [[27/13]] | ||
| H^/Jv | | H^/Jv | ||
Line 174: | Line 174: | ||
|- | |- | ||
| 21 | | 21 | ||
| 1331. | | 1331.4 | ||
| 910 | | 910.0 | ||
| 28/13 | | [[28/13]] | ||
| J | | J | ||
| K#/Lb | | K#/Lb | ||
|- | |- | ||
| 22 | | 22 | ||
| 1394. | | 1394.8 | ||
| 953. | | 953.3 | ||
| [[9/4]] | | [[9/4]] | ||
| J^/Av | | J^/Av | ||
Line 188: | Line 188: | ||
|- | |- | ||
| 23 | | 23 | ||
| 1458. | | 1458.2 | ||
| 996. | | 996.7 | ||
| 7/3 | | [[7/3]] | ||
| A | | A | ||
| L#/Abb | | L#/Abb | ||
|- | |- | ||
| 24 | | 24 | ||
| 1521. | | 1521.6 | ||
| 1040 | | 1040.0 | ||
| [[12/5]] | | [[12/5]] | ||
| A^/Bbv | | A^/Bbv | ||
Line 202: | Line 202: | ||
|- | |- | ||
| 25 | | 25 | ||
| | | 1585.0 | ||
| 1083. | | 1083.3 | ||
| 5/2 | | [[5/2]] | ||
| Bb | | Bb | ||
| A | | A | ||
|- | |- | ||
| 26 | | 26 | ||
| 1648. | | 1648.4 | ||
| 1126. | | 1126.7 | ||
| [[13/5]] | | [[13/5]] | ||
| A# | | A# | ||
Line 216: | Line 216: | ||
|- | |- | ||
| 27 | | 27 | ||
| 1711. | | 1711.8 | ||
| 1170 | | 1170.0 | ||
| 8/3 | | [[8/3]] | ||
| A#^/Bv | | A#^/Bv | ||
| Ax/Bb | | Ax/Bb | ||
|- | |- | ||
| 28 | | 28 | ||
| 1775. | | 1775.2 | ||
| 1213. | | 1213.3 | ||
| [[14/5]] | | [[14/5]] | ||
| B | | B | ||
|- | |- | ||
| 29 | | 29 | ||
| 1838. | | 1838.6 | ||
| 1256. | | 1256.7 | ||
| 26/9 | | [[26/9]] | ||
| B^/Cv | | B^/Cv | ||
| B#/Cb | | B#/Cb | ||
|- | |- | ||
| 30 | | 30 | ||
| | | 1902.0 | ||
| 1300 | | 1300.0 | ||
| [[3/1]] | | [[3/1]] | ||
| colspan="2" | C | | colspan="2" | C |
Revision as of 11:50, 23 January 2025
← 29edt | 30edt | 31edt → |
30 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 30edt or 30ed3), is a nonoctave tuning system that divides the interval of 3/1 into 30 equal parts of about 63.4 ¢ each. Each step represents a frequency ratio of 31/30, or the 30th root of 3.
Theory
30edt is related to 19edo, but with the 3/1 rather than the 2/1 being just, which results in octaves being is stretched by about 4.5715 ¢ and the step size is about. 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.
30edt is a Phoenix tuning and exhibits all the benefits of such tunings.
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 | 24 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | -2.6 | -4.1 | +3.2 | +18.3 | -23.3 | +4.6 | -25.6 | +12.4 | -8.7 | -25.8 | +24.0 | +13.7 |
Relative (%) | -4.2 | -6.5 | +5.1 | +28.8 | -36.7 | +7.2 | -40.4 | +19.5 | -13.7 | -40.8 | +37.9 | +21.6 | |
Steps (reduced) |
70 (10) |
72 (12) |
74 (14) |
76 (16) |
77 (17) |
79 (19) |
80 (20) |
82 (22) |
83 (23) |
84 (24) |
86 (26) |
87 (27) |
Intervals
# | Cents | Hekts | Approximate ratios | Scale name | |
---|---|---|---|---|---|
Lambda | Sigma | ||||
0 | 0 | 0 | 1/1 | C | |
1 | 63.4 | 43.3 | 27/26, 28/27 | C^/Dbv | C#/Dbb |
2 | 126.8 | 86.7 | 14/13, 15/14, 16/15, 29/27 | Db | Cx/Db |
3 | 190.2 | 130.0 | 9/8, 10/9 | C# | D |
4 | 253.6 | 173.3 | 15/13 | C#^/Dv | D#/Ebb |
5 | 317.0 | 216.7 | 6/5 | D | Dx/Eb |
6 | 380.4 | 260.0 | 5/4 | D^/Ev | E |
7 | 443.8 | 303.3 | 9/7 | E | E#/Fbb |
8 | 507.2 | 346.7 | 4/3 | E^/Fbv | Ex/Fb |
9 | 570.6 | 390.0 | 7/5 | Fb | F |
10 | 634.0 | 433.3 | 13/9 | E# | F#/Gb |
11 | 697.4 | 476.7 | 3/2 | E#^/Fv | G |
12 | 760.8 | 520.0 | 14/9 | F | G#/Hbb |
13 | 824.2 | 563.3 | 8/5 | F^/Gv | Gx/Hb |
14 | 887.6 | 606.7 | 5/3 | G | H |
15 | 951.0 | 650.0 | 19/11 | G^/Hbv | H#/Jbb |
16 | 1014.4 | 693.3 | 9/5 | Hb | Hx/Jb |
17 | 1077.8 | 736.7 | 13/7 | G# | J |
18 | 1141.2 | 780.0 | 27/14 | G#^/Hv | J#/Kbb |
19 | 1204.6 | 823.3 | 2/1 | H | Jx/Kb |
20 | 1268.0 | 866.7 | 27/13 | H^/Jv | K |
21 | 1331.4 | 910.0 | 28/13 | J | K#/Lb |
22 | 1394.8 | 953.3 | 9/4 | J^/Av | L |
23 | 1458.2 | 996.7 | 7/3 | A | L#/Abb |
24 | 1521.6 | 1040.0 | 12/5 | A^/Bbv | Lx/Ab |
25 | 1585.0 | 1083.3 | 5/2 | Bb | A |
26 | 1648.4 | 1126.7 | 13/5 | A# | A#/Bbb |
27 | 1711.8 | 1170.0 | 8/3 | A#^/Bv | Ax/Bb |
28 | 1775.2 | 1213.3 | 14/5 | B | |
29 | 1838.6 | 1256.7 | 26/9 | B^/Cv | B#/Cb |
30 | 1902.0 | 1300.0 | 3/1 | C |
30edt contains all 19edo intervals within 3/1, all tempered progressively sharper. The accumulation of the 0.241 ¢ 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 ¢ 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 Template:Sl 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 Template:Sl and the major scale is LLLsLLLsLLs. The sharp 9/7 of 7 degrees, in addition to generating a Lambda mos will generate a Template:Sl unfair "Superlambda" mos which does not border on being atonal as the 17edt rendition does.
Music
- Room Full Of Steam (2018)