20edt: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} It corresponds to 12.6186 [[edo]]. | |||
== Intervals == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | degree | ! | degree | ||
! | cents value | ! | cents value | ||
!hekts | |||
! | corresponding <br>JI intervals | ! | corresponding <br>JI intervals | ||
! | comments | ! | comments | ||
|- | |- | ||
! colspan="3" | 0 | |||
| | '''exact [[1/1]]''' | | | '''exact [[1/1]]''' | ||
| | | | | | ||
| Line 15: | Line 17: | ||
| | 1 | | | 1 | ||
| | 95.0978 | | | 95.0978 | ||
|65 | |||
| | | | | | ||
| | | | | | ||
| Line 20: | Line 23: | ||
| | 2 | | | 2 | ||
| | 190.1955 | | | 190.1955 | ||
|130 | |||
| | | | | | ||
| | | | | | ||
| Line 25: | Line 29: | ||
| | 3 | | | 3 | ||
| | 285.2933 | | | 285.2933 | ||
|195 | |||
| | 33/28, [[13/11]] | | | 33/28, [[13/11]] | ||
| | | | | | ||
|- | |- | ||
| | 4 | | | 4 | ||
| | 380. | | | 380.391 | ||
|260 | |||
| | [[5/4]] | | | [[5/4]] | ||
| | | | | | ||
| Line 35: | Line 41: | ||
| | 5 | | | 5 | ||
| | 475.4888 | | | 475.4888 | ||
|325 | |||
| | | | | | ||
| | | | | | ||
| Line 40: | Line 47: | ||
| | 6 | | | 6 | ||
| | 570.5865 | | | 570.5865 | ||
|390 | |||
| | 39/28 | | | 39/28 | ||
| | | | | | ||
| Line 45: | Line 53: | ||
| | 7 | | | 7 | ||
| | 665.6843 | | | 665.6843 | ||
|455 | |||
| | [[22/15]] | | | [[22/15]] | ||
| | | | | | ||
| Line 50: | Line 59: | ||
| | 8 | | | 8 | ||
| | 760.7820 | | | 760.7820 | ||
|520 | |||
| | | | | | ||
| | | | | | ||
| Line 55: | Line 65: | ||
| | 9 | | | 9 | ||
| | 855.8798 | | | 855.8798 | ||
|585 | |||
| | [[18/11]] | | | [[18/11]] | ||
| | | | | | ||
| Line 60: | Line 71: | ||
| | 10 | | | 10 | ||
| | 950.9775 | | | 950.9775 | ||
|650 | |||
| | 45/26, [[26/15]] | | | 45/26, [[26/15]] | ||
| | | | | | ||
| Line 65: | Line 77: | ||
| | 11 | | | 11 | ||
| | 1046.0753 | | | 1046.0753 | ||
|725 | |||
| | [[11/6]] | | | [[11/6]] | ||
| | | | | | ||
|- | |- | ||
| | 12 | | | 12 | ||
| | 1141. | | | 1141.173 | ||
|780 | |||
| | | | | | ||
| | | | | | ||
| Line 75: | Line 89: | ||
| | 13 | | | 13 | ||
| | 1236.2708 | | | 1236.2708 | ||
|845 | |||
| | [[45/44|45/22]] | | | [[45/44|45/22]] | ||
| | | | | | ||
| Line 80: | Line 95: | ||
| | 14 | | | 14 | ||
| | 1331.3685 | | | 1331.3685 | ||
|910 | |||
| | [[14/13|28/13]] | | | [[14/13|28/13]] | ||
| | | | | | ||
| Line 85: | Line 101: | ||
| | 15 | | | 15 | ||
| | 1426.4663 | | | 1426.4663 | ||
|975 | |||
| | | | | | ||
| | | | | | ||
|- | |- | ||
| | 16 | | | 16 | ||
| | 1521. | | | 1521.564 | ||
|1040 | |||
| | [[12/5]] | | | [[12/5]] | ||
| | | | | | ||
| Line 95: | Line 113: | ||
| | 17 | | | 17 | ||
| | 1616.6618 | | | 1616.6618 | ||
|1105 | |||
| | 33/13, [[14/11|28/11]] | | | 33/13, [[14/11|28/11]] | ||
| | | | | | ||
| Line 100: | Line 119: | ||
| | 18 | | | 18 | ||
| | 1711.7595 | | | 1711.7595 | ||
|1170 | |||
| | | | | | ||
| | | | | | ||
| Line 105: | Line 125: | ||
| | 19 | | | 19 | ||
| | 1806.8573 | | | 1806.8573 | ||
|1235 | |||
| | | | | | ||
| | | | | | ||
|- | |- | ||
| | 20 | | | 20 | ||
| | 1901. | | | 1901.955 | ||
|1300 | |||
| | '''exact [[3/1]]''' | | | '''exact [[3/1]]''' | ||
| | [[3/2|just perfect fifth]] plus an octave | | | [[3/2|just perfect fifth]] plus an octave | ||
|} | |} | ||
== Harmonics == | |||
{{Harmonics in equal | |||
| steps = 20 | |||
| num = 3 | |||
| denom = 1 | |||
| intervals = integer | |||
}} | |||
{{Harmonics in equal | |||
| steps = 20 | |||
| num = 3 | |||
| denom = 1 | |||
| start = 12 | |||
| collapsed = 1 | |||
| intervals = integer | |||
}} | |||
{{stub}} | |||
Latest revision as of 19:20, 1 August 2025
| ← 19edt | 20edt | 21edt → |
20 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 20edt or 20ed3), is a nonoctave tuning system that divides the interval of 3/1 into 20 equal parts of about 95.1 ¢ each. Each step represents a frequency ratio of 31/20, or the 20th root of 3. It corresponds to 12.6186 edo.
Intervals
| degree | cents value | hekts | corresponding JI intervals |
comments |
|---|---|---|---|---|
| 0 | exact 1/1 | |||
| 1 | 95.0978 | 65 | ||
| 2 | 190.1955 | 130 | ||
| 3 | 285.2933 | 195 | 33/28, 13/11 | |
| 4 | 380.391 | 260 | 5/4 | |
| 5 | 475.4888 | 325 | ||
| 6 | 570.5865 | 390 | 39/28 | |
| 7 | 665.6843 | 455 | 22/15 | |
| 8 | 760.7820 | 520 | ||
| 9 | 855.8798 | 585 | 18/11 | |
| 10 | 950.9775 | 650 | 45/26, 26/15 | |
| 11 | 1046.0753 | 725 | 11/6 | |
| 12 | 1141.173 | 780 | ||
| 13 | 1236.2708 | 845 | 45/22 | |
| 14 | 1331.3685 | 910 | 28/13 | |
| 15 | 1426.4663 | 975 | ||
| 16 | 1521.564 | 1040 | 12/5 | |
| 17 | 1616.6618 | 1105 | 33/13, 28/11 | |
| 18 | 1711.7595 | 1170 | ||
| 19 | 1806.8573 | 1235 | ||
| 20 | 1901.955 | 1300 | exact 3/1 | just perfect fifth plus an octave |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +36.3 | +0.0 | -22.6 | -28.5 | +36.3 | -40.4 | +13.7 | +0.0 | +7.8 | +33.0 | -22.6 |
| Relative (%) | +38.1 | +0.0 | -23.7 | -29.9 | +38.1 | -42.5 | +14.4 | +0.0 | +8.2 | +34.7 | -23.7 | |
| Steps (reduced) |
13 (13) |
20 (0) |
25 (5) |
29 (9) |
33 (13) |
35 (15) |
38 (18) |
40 (0) |
42 (2) |
44 (4) |
45 (5) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +29.1 | -4.1 | -28.5 | -45.1 | +40.1 | +36.3 | +37.8 | +44.1 | -40.4 | -25.8 | -7.7 |
| Relative (%) | +30.6 | -4.3 | -29.9 | -47.4 | +42.2 | +38.1 | +39.7 | +46.3 | -42.5 | -27.2 | -8.1 | |
| Steps (reduced) |
47 (7) |
48 (8) |
49 (9) |
50 (10) |
52 (12) |
53 (13) |
54 (14) |
55 (15) |
55 (15) |
56 (16) |
57 (17) | |
|
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