Wizard
Wizard is a half-octave temperament generated by a wide whole tone of about 217 cents which is the semi-octave complement of ~5/4. Six generators minus a semi-octave represents 3/2, and ten generators minus a full octave represents 7/4, tempering out the commas 225/224 and 118098/117649.
Three generator steps may be identified with 16/11, and the generator itself is close in size to 17/15, which if used, would imply the semi-octave represents 17/12~24/17 and that 17/16 is obtained by stacking six generators octave reduced. As such, it is most naturally viewed as a temperament of the 2.3.5.7.11.17 subgroup, where it tempers out 225/224, 289/288, 385/384, and 561/560.
72edo, 94edo, and especially 166edo are among the good tuning options.
The name wizard appeared as early as 2003, presumably given by Gene Ward Smith[1].
See Marvel temperaments #Wizard for technical data.
Interval chain
In the following table, odd harmonics 1–21 and their inverses are in bold.
| # | Period 0 | Period 1 | ||
|---|---|---|---|---|
| Cents* | Approximate ratios | Cents* | Approximate ratios | |
| 0 | 0.00 | 1/1 | 600.00 | 17/12, 24/17 |
| 1 | 216.80 | 17/15 | 816.80 | 8/5 |
| 2 | 433.61 | 9/7 | 1033.61 | 20/11 |
| 3 | 650.42 | 16/11 | 50.42 | 33/32, 36/35 |
| 4 | 867.22 | 28/17, 33/20 | 267.22 | 7/6 |
| 5 | 1084.03 | 15/8, 28/15 | 484.03 | 45/34 |
| 6 | 100.83 | 17/16, 18/17 | 700.83 | 3/2 |
| 7 | 317.64 | 6/5 | 917.64 | 17/10 |
| 8 | 534.45 | 15/11 | 1134.45 | 27/14, 48/25 |
| 9 | 751.25 | 17/11 | 151.25 | 12/11 |
| 10 | 968.06 | 7/4 | 368.06 | 21/17 |
| 11 | 1184.86 | 119/60, 135/68, 168/85, 175/88, 240/121 |
584.86 | 7/5 |
* In 2.3.5.7.11.17-subgroup CWE tuning, octave reduced
Chords
Scales
Tunings
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 5/4 | 213.686 | ||
| 17/14 | 215.968 | ||
| 9\50 | 216.000 | Lower bound of 9- and 11-odd-limit, 11-limit 15-odd-limit, and 2.3.5.7.11.17-subgroup 17-odd-limit diamond monotone | |
| 15/14 | 216.111 | ||
| 11/8 | 216.227 | ||
| 22\122 | 216.393 | 122g val | |
| 17/9 | 216.492 | ||
| 5/3 | 216.520 | 5-odd-limit minimax | |
| 21/17 | 216.583 | ||
| 7/5 | 216.592 | 7-odd-limit minimax | |
| 13\72 | 216.667 | Lower bound of 2.3.5.7.11.17-subgroup 21-odd-limit diamond monotone | |
| 17/15 | 216.687 | ||
| 7/6 | 216.718 | ||
| 9/5 | 216.738 | 9- and 11-odd-limit minimax | |
| 11/7 | 216.731 | ||
| 21/20 | 216.733 | ||
| 11/6 | 216.737 | ||
| 11/9 | 216.839 | ||
| 30\166 | 216.867 | 166g val | |
| 7/4 | 216.883 | ||
| 21/16 | 216.924 | ||
| 17/10 | 216.949 | ||
| 3/2 | 216.993 | ||
| 17\94 | 217.021 | ||
| 17/11 | 217.071 | ||
| 15/11 | 217.119 | ||
| 17/16 | 217.493 | ||
| 11/10 | 217.498 | ||
| 9/7 | 217.542 | ||
| 15/8 | 217.654 | ||
| 4\22 | 218.182 | Upper bound of 9- and 11-odd-limit, 11-limit 15-odd-limit, and 2.3.5.7.11.17-subgroup 17- and 21-odd-limit diamond monotone |
* Besides the octave