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Valentine: Difference between revisions

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{{Infobox Regtemp
{{Infobox regtemp
| Title = Valentine
| Title = Valentine
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[126/125]], [[1029/1024]] (7-limit); <br /> [[121/120]], [[126/125]], [[176/175]] (11-limit)
| Comma basis = [[126/125]], [[1029/1024]] (7-limit); <br>[[121/120]], [[126/125]], [[176/175]] (11-limit)
| Edo join 1 = 15 | Edo join 2 = 16
| Edo join 1 = 15 | Edo join 2 = 16
| Generator = 21/20 | Generator tuning = 77.9 | Optimization method = CTE
| MOS scales = ..., [[1L 14s]], [[15L 1s]], [[15L 16s]]
| Mapping = 1; 9 5 -3 7
| Mapping = 1; 9 5 -3 7
| Generators = 22/21 | Generators tuning = 77.9 | Optimization method = CWE
| MOS scales = …, [[1L 14s]], [[15L 1s]], [[15L 16s]]
| Pergen = (P8, P5/9)
| Pergen = (P8, P5/9)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 31
| Odd limit 1 = 7 | Mistuning 1 = 4.60 | Complexity 1 = 15
| Odd limit 2 = (11-limit) 21 | Mistuning 2 = ? | Complexity 2 = 46
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 9.01 | Complexity 2 = 31
}}
}}
'''Valentine''' is a [[regular temperament]] that divides a tempered [[3/2]] into 9 equal [[generator]]s which are small semitones close to 78 cents; a stack of 3 generators is interpreted as [[8/7]] and a stack of 5 generators is interpreted as [[5/4]]. The generator serves as both [[21/20]] and [[25/24]]. It is a member of the [[starling temperaments]], by [[tempering out]] [[126/125]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the generator as [[22/21]], tempering out [[121/120]], [[176/175]], [[385/384]], and [[441/440]].  
'''Valentine''' is a [[regular temperament]] that divides a tempered [[3/2]] into 9 equal [[generator]]s which are small semitones close to 78 cents; a stack of 3 generators is interpreted as [[8/7]] and a stack of 5 generators is interpreted as [[5/4]]. The generator serves as both [[21/20]] and [[25/24]]. It is a member of the [[starling temperaments]], by [[tempering out]] [[126/125]], and the [[gamelismic clan]], by tempering out [[1029/1024]]. It extends naturally to the [[11-limit]] by treating the generator as [[22/21]], tempering out [[121/120]], [[176/175]], [[385/384]], and [[441/440]].  


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== Tunings ==
== Tunings ==
=== Prime-optimized tunings ===
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~15/14 = 77.7625¢
| CEE: ~21/20 = 77.7625{{c}}
| CSEE: ~15/14 = 77.7211¢
| CSEE: ~21/20 = 77.7211{{c}}
| POEE: ~21/20 = 77.7256{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~15/14 = 77.8776¢
| CTE: ~21/20 = 77.8776{{c}}
| CWE: ~15/14 = 77.8673¢
| CWE: ~21/20 = 77.8673{{c}}
| POTE: ~21/20 = 77.8638{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~15/14 = 77.9062¢
| CBE: ~21/20 = 77.9062{{c}}
| CSBE: ~15/14 = 77.9075¢
| CSBE: ~21/20 = 77.9075{{c}}
| POBE: ~21/20 = 77.9104{{c}}
|}
|}


{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
! colspan="2" | Euclidean
! colspan="3" | Euclidean
|-
|-
! Unskewed
! Constrained
! Skewed
! Constrained & skewed
! Destretched
|-
|-
! Equilateral
! Equilateral
| CEE: ~15/14 = 78.0604¢
| CEE: ~22/21 = 78.0604{{c}}
| CSEE: ~15/14 = 77.9698¢
| CSEE: ~22/21 = 77.9698{{c}}
| POEE: ~22/21 = 77.8485{{c}}
|-
|-
! Tenney
! Tenney
| CTE: ~15/14 = 77.9633¢
| CTE: ~22/21 = 77.9633{{c}}
| CWE: ~15/14 = 77.9007¢
| CWE: ~22/21 = 77.9007{{c}}
| POTE: ~22/21 = 77.8813{{c}}
|-
|-
! Benedetti, <br>Wilson
! Benedetti, <br>Wilson
| CBE: ~15/14 = 77.9388¢
| CBE: ~22/21 = 77.9388{{c}}
| CSBE: ~15/14 = 77.9028¢
| CSBE: ~22/21 = 77.9028{{c}}
| POBE: ~22/21 = 77.9093{{c}}
|}
|}


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| [[5/4]]
| [[5/4]]
| 77.263
| 77.263
|  
| Lower bound of 5-odd-limit diamond tradeoff
|-
|-
| '''[[31edo|2\31]]'''
| '''[[31edo|2\31]]'''
|  
|  
| '''77.419'''
| '''77.419'''
| '''Lower bound of 9- and 11-odd-limit, <br>11-limit 15-, and 21-odd-limit diamond monotone'''
| '''Lower bound of 9- and 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
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| 77.614
| 77.614
|  
|  
|-
| [[139edo|9\139]]
|
| 77.698
| 139e val
|-
|
| {{monzo|26 25 3 -23}}
| 77.707
| 7-odd-limit least squares
|-
|-
|  
|  
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| [[6/5]]
| [[6/5]]
| 78.910
| 78.910
|  
| Upper bound of 5-odd-limit diamond tradeoff
|-
|-
| '''[[15edo|1\15]]'''
| '''[[15edo|1\15]]'''
|  
|  
| '''80.000'''
| '''80.000'''
| '''Upper bound of 7-, 9- and 11-odd-limit, <br>11-limit 15-, and 21-odd-limit diamond monotone'''
| '''Upper bound of 7-, 9- and 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
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== References ==
== References ==


[[Category:Temperaments]]
[[Category:Valentine| ]] <!-- main article -->
[[Category:Valentine| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]
Retrieved from "https://en.xen.wiki/w/Valentine"