22ed7/4
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22 equal divisions of 7/4 (abbreviated 22ed7/4) is a nonoctave tuning system that divides the interval of 7/4 into 22 equal parts of about 44 ¢ each. Each step represents a frequency ratio of (7/4)1/22, or the 22nd root of 7/4.
| ← 21ed7/4 | 22ed7/4 | 23ed7/4 → |
(semiconvergent)
It is a heavily octave-compressed version of 27edo.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 44 | |
| 2 | 88.1 | 18/17, 21/20, 22/21, 23/22 |
| 3 | 132.1 | 15/14, 25/23 |
| 4 | 176.2 | 10/9, 11/10 |
| 5 | 220.2 | 8/7, 17/15, 25/22 |
| 6 | 264.2 | 7/6 |
| 7 | 308.3 | 6/5, 25/21 |
| 8 | 352.3 | 11/9, 21/17 |
| 9 | 396.3 | 5/4 |
| 10 | 440.4 | 9/7, 22/17 |
| 11 | 484.4 | 4/3, 21/16 |
| 12 | 528.5 | 15/11, 23/17 |
| 13 | 572.5 | 7/5, 25/18 |
| 14 | 616.5 | 10/7, 17/12 |
| 15 | 660.6 | 19/13, 22/15, 25/17 |
| 16 | 704.6 | 3/2 |
| 17 | 748.6 | 17/11, 23/15 |
| 18 | 792.7 | 11/7 |
| 19 | 836.7 | |
| 20 | 880.8 | 5/3 |
| 21 | 924.8 | 12/7, 17/10 |
| 22 | 968.8 | 7/4 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -11.0 | -8.3 | -22.0 | -11.9 | -19.3 | -22.0 | +11.1 | -16.7 | +21.1 | -11.8 | +13.7 |
| Relative (%) | -24.9 | -18.9 | -49.9 | -27.1 | -43.9 | -49.9 | +25.2 | -37.9 | +47.9 | -26.8 | +31.2 | |
| Steps (reduced) |
27 (5) |
43 (21) |
54 (10) |
63 (19) |
70 (4) |
76 (10) |
82 (16) |
86 (20) |
91 (3) |
94 (6) |
98 (10) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.3 | +11.1 | -20.3 | +0.1 | -16.8 | +16.4 | +10.8 | +10.1 | +13.7 | +21.3 | -11.7 |
| Relative (%) | +16.5 | +25.2 | -46.1 | +0.2 | -38.1 | +37.2 | +24.6 | +23.0 | +31.2 | +48.3 | -26.5 | |
| Steps (reduced) |
101 (13) |
104 (16) |
106 (18) |
109 (21) |
111 (1) |
114 (4) |
116 (6) |
118 (8) |
120 (10) |
122 (12) |
123 (13) | |
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +9.2 | +0.0 | +13.7 | +9.2 | +9.0 | +0.0 | +18.3 | +13.7 | -18.0 | +9.2 |
| Relative (%) | +0.0 | +20.6 | +0.0 | +30.8 | +20.6 | +20.1 | +0.0 | +41.2 | +30.8 | -40.5 | +20.6 | |
| Steps (reduced) |
27 (0) |
43 (16) |
54 (0) |
63 (9) |
70 (16) |
76 (22) |
81 (0) |
86 (5) |
90 (9) |
93 (12) |
97 (16) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.3 | +11.1 | -20.3 | +0.1 | -16.8 | +16.4 | +10.8 | +10.1 | +13.7 | +21.3 | -11.7 |
| Relative (%) | +16.5 | +25.2 | -46.1 | +0.2 | -38.1 | +37.2 | +24.6 | +23.0 | +31.2 | +48.3 | -26.5 | |
| Steps (reduced) |
101 (13) |
104 (16) |
106 (18) |
109 (21) |
111 (1) |
114 (4) |
116 (6) |
118 (8) |
120 (10) |
122 (12) |
123 (13) | |