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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox regtemp
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Title = Wizard
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-03 14:48:36 UTC</tt>.<br>
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.17
: The original revision id was <tt>378901948</tt>.<br>
| 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)
: The revision comment was: <tt></tt><br>
| Edo join 1 = 22 | Edo join 2 = 72
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Mapping = 2; 6 -1 10 -3 6
<h4>Original Wikitext content:</h4>
| Generators = 17/15 | Generators tuning = 216.8 | Optimization method = CWE
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Marvel temperaments#Wizard]].
| 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]].  


=Spectrum of Wizard Tunings by Eigenmonzos=
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]].  
||~ Eigenmonzo ||~ Major Third ||
|| 9/7 || 382.458 ||
|| 11/10 || 382.502 ||
|| 4/3 || 383.007 ||
|| 8/7 || 383.117 ||
|| 11/9 || 383.161 ||
|| 10/9 || 383.262 (9 and 11 limit minimax) ||
|| 12/11 || 383.263 ||
|| 14/11 || 383.269 ||
|| 7/6 || 383.282 ||
|| 7/5 || 383.408 (7 limit minimax) ||
|| 6/5 || 383.480 (5 limit minimax) ||
|| 11/8 || 383.773 ||
|| 5/4 || 386.314 ||
</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;wizard&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Marvel%20temperaments#Wizard"&gt;Marvel temperaments&lt;/a&gt;.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Spectrum of Wizard Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Spectrum of Wizard Tunings by Eigenmonzos&lt;/h1&gt;


&lt;table class="wiki_table"&gt;
[[72edo]], [[94edo]], and especially [[166edo]] are among the good tuning options.  
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Major Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;382.458&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;382.502&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.007&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.117&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.161&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.262 (9 and 11 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.263&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.269&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.282&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.408 (7 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.480 (5 limit minimax)&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;383.773&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;386.314&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
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
|-
| 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, <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
|-
|
| 11/7
| 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, <br>and 2.3.5.7.11.17-subgroup 17- and 21-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave
 
== References ==
 
[[Category:Wizard| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Stearnsmic clan]]

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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