User:Francium/239edt
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Prime factorization
239 (prime)
Step size
7.95797 ¢
Octave
151\239edt (1201.65 ¢)
Consistency limit
4
Distinct consistency limit
4
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239 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 239edt or 239ed3), is a nonoctave tuning system that divides the interval of 3/1 into 239 equal parts of about 7.96 ¢ each. Each step represents a frequency ratio of 31/239, or the 239th root of 3.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 100 | 15/14, 16/15, 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 | D♯, E♭ |
| 2 | 200 | 9/8, 10/9, 17/15, 19/17, 21/19 | E |
| 3 | 300 | 6/5, 19/16, 20/17 | F |
| 4 | 400 | 5/4, 14/11, 19/15, 23/18 | F♯, G♭ |
| 5 | 500 | 4/3, 23/17 | G |
| 6 | 600 | 7/5, 10/7, 17/12, 23/16 | G♯, A♭ |
| 7 | 700 | 3/2 | A |
| 8 | 800 | 8/5, 11/7, 19/12 | A♯, B♭ |
| 9 | 900 | 5/3, 17/10 | B |
| 10 | 1000 | 9/5, 16/9, 23/13 | C |
| 11 | 1100 | 15/8, 17/9, 19/10, 21/11, 23/12 | C♯, D♭ |
| 12 | 1200 | 2/1 | D |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.65 | +0.00 | +3.31 | -1.02 | +1.65 | -2.60 | -3.00 | +0.00 | +0.63 | +2.74 | +3.31 |
| Relative (%) | +20.8 | +0.0 | +41.6 | -12.9 | +20.8 | -32.7 | -37.7 | +0.0 | +7.9 | +34.5 | +41.6 | |
| Steps (reduced) |
151 (151) |
239 (0) |
302 (63) |
350 (111) |
390 (151) |
423 (184) |
452 (213) |
478 (0) |
501 (23) |
522 (44) |
541 (63) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.02 | -0.95 | -1.02 | -1.34 | -2.85 | +1.65 | +3.55 | +2.28 | -2.60 | -3.56 | -0.94 |
| Relative (%) | +0.3 | -11.9 | -12.9 | -16.9 | -35.8 | +20.8 | +44.6 | +28.7 | -32.7 | -44.8 | -11.8 | |
| Steps (reduced) |
558 (80) |
574 (96) |
589 (111) |
603 (125) |
616 (138) |
629 (151) |
641 (163) |
652 (174) |
662 (184) |
672 (194) |
682 (204) | |