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{{See also|Marvel temperaments #Wizard}}
{{Infobox regtemp
| Title = Wizard
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17
| Comma basis = [[225/224]], [[118098/117649]] (7-limit); <br>[[225/224]], [[385/384]], [[4000/3993]] (11-limit); <br>[[225/224]], [[289/288]], [[385/384]], [[561/560]]<br>(2.3.5.7.11.17)
| Edo join 1 = 22 | Edo join 2 = 72
| Mapping = 2; 6 -1 10 -3 6
| Generators = 17/15 | Generators tuning = 216.8 | Optimization method = CWE
| MOS scales = [[6L 4s]], [[6L 10s]], [[6L 16s]], [[22L 6s]]
| Odd limit 1 = 9 | Mistuning 1 = 3.05 | Complexity 1 = 28
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 3.05 | Complexity 2 = 50
}}
'''Wizard''' is a half-octave [[regular temperament|temperament]] [[generator|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 [[stearnsma|118098/117649]].


== Tuning spectrum ==
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 reduction|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]].


{| class="wikitable center-all"
[[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]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5279.html Yahoo! Tuning Group | ''Poptimal generators'']</ref>.
 
See [[Marvel temperaments #Wizard]] for technical data.
 
== Interval chain ==
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
| 600.00
| 17/12, 24/17
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
| 1
! major third<br>(¢)
| 216.80
! comments
| 17/15
| 816.80
| '''8/5'''
|-
|-
| 2
| 433.61
| 9/7
| 9/7
| 382.458
| 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, <br>175/88, 240/121
| 584.86
| 7/5
|}
<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave reduced
 
== Chords and harmony ==
{{See also| Chords of wizard }}
 
== Scales ==
* [[Wizard22]]
 
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~245/216 = 216.9187{{c}}
| CWE: ~245/216 = 216.7977{{c}}
| POTE: ~245/216 = 216.7438{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~25/22 = 216.9001{{c}}
| CWE: ~25/22 = 216.7961{{c}}
| POTE: ~25/22 = 216.7679{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(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, <br>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
|-
|-
| 11/10
|
| 382.502
| 11/7
| 216.731
|  
|  
|-
|-
| 4/3
|  
| 383.007
| 21/20
| 216.733
|  
|  
|-
|-
| 8/7
|  
| 383.117
| 11/6
| 216.737
|  
|  
|-
|-
|
| 11/9
| 11/9
| 383.161
| 216.839
|
|-
| 30\166
|
| 216.867
| 166g val
|-
|
| 7/4
| 216.883
|
|-
|
| 21/16
| 216.924
|
|-
|
| 17/10
| 216.949
|
|-
|
| 3/2
| 216.993
|  
|  
|-
|-
| 10/9
| 17\94
| 383.262
|  
| 9 and 11 limit minimax
| 217.021
|  
|-
|-
| 12/11
|  
| 383.263
| 17/11
| 217.071
|  
|  
|-
|-
| 14/11
|  
| 383.269
| 15/11
| 217.119
|  
|  
|-
|-
| 7/6
|  
| 383.282
| 17/16
| 217.493
|  
|  
|-
|-
| 7/5
|  
| 383.408
| 11/10
| 7 limit minimax
| 217.498
|  
|-
|-
| 6/5
|  
| 383.480
| 9/7
| 5 limit minimax
| 217.542
|  
|-
|-
| 11/8
|  
| 383.773
| 15/8
| 217.654
|  
|  
|-
|-
| 5/4
| 4\22
| 386.314
|  
|  
| 218.182
| Upper bound of 9- and 11-odd-limit, 11-limit 15-odd-limit, <br>and 2.3.5.7.11.17-subgroup 17- and 21-odd-limit diamond monotone
|}
|}
<nowiki/>* Besides the octave


[[Category:Temperaments]]
== References ==
 
[[Category:Wizard| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
 
[[Category:Stearnsmic clan]]
{{todo|expand}}

Latest revision as of 12:26, 26 March 2026

Wizard
Subgroups 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17
Comma basis 225/224, 118098/117649 (7-limit);
225/224, 385/384, 4000/3993 (11-limit);
225/224, 289/288, 385/384, 561/560
(2.3.5.7.11.17)
Reduced mapping ⟨2; 6 -1 10 -3 6]
ET join 22 & 72
Generators (CWE) ~17/15 = 216.8 ¢
MOS scales 6L 4s, 6L 10s, 6L 16s, 22L 6s
Ploidacot diploid alpha-hexacot
Minimax error 9-odd-limit: 3.05 ¢;
11-limit 21-odd-limit: 3.05 ¢
Target scale size 9-odd-limit: 28 notes;
11-limit 21-odd-limit: 50 notes

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 and harmony

See also: Chords of wizard

Scales

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~245/216 = 216.9187 ¢ CWE: ~245/216 = 216.7977 ¢ POTE: ~245/216 = 216.7438 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~25/22 = 216.9001 ¢ CWE: ~25/22 = 216.7961 ¢ POTE: ~25/22 = 216.7679 ¢

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

References

  1. ↑ Yahoo! Tuning Group | Poptimal generators
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