Wizard: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Expand a bit
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{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]].  
'''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]].  


Line 97: Line 108:
<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave reduced
<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave reduced


== Chords ==
== Chords and harmony ==
{{Main| Chords of wizard }}
{{See also| Chords of wizard }}


== Scales ==
== Scales ==
Line 104: Line 115:


== Tunings ==
== 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 ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 269: Line 311:
<nowiki/>* Besides the octave
<nowiki/>* Besides the octave


== Notes ==
== References ==


[[Category:Wizard| ]] <!-- main article -->
[[Category:Wizard| ]] <!-- main article -->

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
Retrieved from "https://en.xen.wiki/index.php?title=Wizard&oldid=226921"