Xenharmonic Wiki

Wizard: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
cleanup+recat+todo
 
(23 intermediate revisions by 7 users not shown)
Line 1: Line 1:
{{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]].


== 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]].


{| class="wikitable"
[[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"
|-
|-
! Eigenmonzo
! rowspan="2" | #
! Major Third
! 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
| 9/7
| 382.458
| 1033.61
| 20/11
|-
| 3
| 650.42
| '''16/11'''
| 50.42
| 33/32, 36/35
|-
|-
| 11/10
| 4
| 382.502
| 867.22
| 28/17, 33/20
| 267.22
| 7/6
|-
|-
| 4/3
| 5
| 383.007
| 1084.03
| '''15/8''', 28/15
| 484.03
| 45/34
|-
|-
| 8/7
| 6
| 383.117
| 100.83
| '''17/16''', 18/17
| 700.83
| '''3/2'''
|-
|-
| 11/9
| 7
| 383.161
| 317.64
| 6/5
| 917.64
| 17/10
|-
|-
| 10/9
| 8
| 383.262 (9 and 11 limit minimax)
| 534.45
| 15/11
| 1134.45
| 27/14, 48/25
|-
|-
| 9
| 751.25
| 17/11
| 151.25
| 12/11
| 12/11
| 383.263
|-
|-
| 14/11
| 10
| 383.269
| 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
| 7/6
| 383.282
| 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
|
|-
|-
| 7/5
| 30\166
| 383.408 (7 limit minimax)
|
| 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
|
|-
|-
| 6/5
|  
| 383.480 (5 limit minimax)
| 9/7
| 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
== References ==


[[Category:Wizard| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Marvel temperaments]]
{{IoT}}
[[Category:Stearnsmic clan]]
 
{{todo|expand}}
Retrieved from "https://en.xen.wiki/w/Wizard"