Wizard: Difference between revisions
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'''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]]. | {{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]]. | |||
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]]. | |||
[[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. | See [[Marvel temperaments #Wizard]] for technical data. | ||
== Interval chain == | == Interval chain == | ||
In the following table, odd harmonics and | In the following table, odd harmonics 1–21 and their inverses are in '''bold'''. | ||
{| class="wikitable center-1 right-2 right-4" | {| class="wikitable center-1 right-2 right-4" | ||
| Line 89: | Line 106: | ||
| 7/5 | | 7/5 | ||
|} | |} | ||
<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning | <nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave reduced | ||
== Chords == | == Chords and harmony == | ||
{{ | {{See also| Chords of wizard }} | ||
== Scales == | == Scales == | ||
| Line 98: | 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|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | ! Generator (¢) | ||
! Comments | ! Comments | ||
| Line 262: | Line 310: | ||
|} | |} | ||
<nowiki/>* Besides the octave | <nowiki/>* Besides the octave | ||
== References == | |||
[[Category:Wizard| ]] <!-- main article --> | [[Category:Wizard| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category:Marvel temperaments]] | [[Category:Marvel temperaments]] | ||
[[Category:Stearnsmic clan]] | [[Category:Stearnsmic clan]] | ||