22ed7/4
| ← 21ed7/4 | 22ed7/4 | 23ed7/4 → |
(semiconvergent)
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.
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
Compared to 27edo:
- Primes 11 and 13 are much more accurate
- Primes 2 and 7 are much less accurate
- Other primes have roughly similar accuracy
and
- The mapping of primes 2, 3, 5 and 7 is the same as 27edo
- The mapping of primes 11, 13, 17, 19 and 23 is different from 27edo
While 27edo is quite close to the zeta peak 106zpi, 22ed7/4 is quite close to the zeta peak 107zpi.
22ed7/4's mapping of prime 2 is very inconsistent. This might not necessarily be a bad thing, as it might allow for the perceived octave equivalence to fall on different scale degrees each octave or two. This could make each perceived octave sound different to the last, which may make the music more interesting to listen to in the hands of a skilled composer.
Possible JI subgroups for interpreting 22ed7/4 could be:
- 2.3.5.7/2.11.13
or
- 2.3.5.14.11.13
Though other interpretations could be possible as well.
| 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 (¢) | +3.9 | +9.0 | -21.6 | +0.0 | -16.1 | +18.3 | +13.6 | +13.7 | +18.1 | -18.0 | -6.1 |
| Relative (%) | +8.8 | +20.1 | -48.6 | +0.0 | -36.1 | +41.2 | +30.6 | +30.8 | +40.7 | -40.5 | -13.6 | |
| Steps (reduced) |
100 (19) |
103 (22) |
105 (24) |
108 (0) |
110 (2) |
113 (5) |
115 (7) |
117 (9) |
119 (11) |
120 (12) |
122 (14) | |