Xenharmonic Wiki

Valentine: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,668 edits
m Rewrite
Overthink (talk | contribs)
autopatrolled
2,773 edits
update scale sizes
 
(30 intermediate revisions by 7 users not shown)
Line 1: Line 1:
'''Valentine''' is a temperament that divides a tempered 3/2 into 9 equal generators which are small semitones; a stack of 3 of these generators is interpreted as 8/7 and a stack of 5 of them 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]].  
{{Infobox regtemp
| Title = Valentine
| 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)
| Edo join 1 = 15 | Edo join 2 = 16
| 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)
| Odd limit 1 = 7 | Mistuning 1 = 4.60 | Complexity 1 = 15
| 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 can be viewed as a counterpart of [[miracle]] in several ways. Miracle splits the generator of [[slendric]] in two while valentine splits it in three. Miracle is generated by [[15/14]][[~]][[16/15]], the classical diatonic semitone tempered together with the septimal major semitone, while valentine is generated by 21/20~25/24, the classical chromatic semitone tempered together with the septimal minor semitone. Miracle is known for its efficiency; the same is true of valentine. The [[11-odd-limit]] [[tonality diamond]] is covered by miracle with 22 generator steps, and by valentine with 21 generator steps.  
Valentine can be viewed as a counterpart of [[miracle]] in several ways. Miracle splits the generator of [[slendric]] in two while valentine splits it in three. Miracle is generated by [[15/14]][[~]][[16/15]], the classical diatonic semitone tempered together with the septimal major semitone, while valentine is generated by 21/20~25/24, the classical chromatic semitone tempered together with the septimal minor semitone. Miracle is known for its efficiency; the same is true of valentine. The [[11-odd-limit]] [[tonality diamond]] is covered by miracle with 22 generator steps, and by valentine with 21 generator steps.  


Valentine is very closely related to [[Carlos Alpha]], the rank-1 non-octave temperament of [[Wendy Carlos]], as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that she really intended Alpha to be the same thing as valentine, and that it is misdescribed as a rank-1 temperament. Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. [[Mos scale]]s of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.
Valentine is very closely related to [[Carlos Alpha]], the rank-1 non-octave temperament of [[Wendy Carlos]], as the generator chain of valentine is the same thing as Carlos Alpha. Indeed, the way Carlos uses Alpha in ''Beauty in the Beast'' suggests that valentine might be a better description of Alpha than the rank-1 temperament that we know as Alpha today.<ref><i>Wendy Carlos</i>, Pitch article (1989) Accessed 2025. https://www.wendycarlos.com/resources/pitch.html</ref> Carlos tells us that "[t]he melodic motions of Alpha are amazingly exotic and fresh, like you've never heard before", and since Alpha lives inside valentine this comment carries over and applies to it if you stick close melodically to generator steps, which is almost impossible not to do since the generator step is so small. [[Mos scale]]s of 15, 16, 31 and 46 notes are available to explore these exotic and fresh melodies, or the less exotic ones you might cook up otherwise.


See [[Valentine extensions]] for a discussion on [[13-limit]] extensions. See [[Gamelismic clan #Valentine]] for technical data.
See [[Valentine extensions]] for a discussion on [[13-limit]] extensions. See [[Gamelismic clan #Valentine]] for technical data. See [[Valentine scales]] for scales in this temperament.


== Interval chain ==
== Interval chain ==
Line 147: Line 159:


== Chords ==
== Chords ==
{{main| Chords of valentine }}
{{Main| Chords of valentine }}


== Tunings ==
== Tunings ==
=== Norm-based 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
|-
! Equilateral
| CEE: ~21/20 = 77.7625{{c}}
| CSEE: ~21/20 = 77.7211{{c}}
| POEE: ~21/20 = 77.7256{{c}}
|-
! Tenney
| CTE: ~21/20 = 77.8776{{c}}
| CWE: ~21/20 = 77.8673{{c}}
| POTE: ~21/20 = 77.8638{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~21/20 = 77.9062{{c}}
| CSBE: ~21/20 = 77.9075{{c}}
| POBE: ~21/20 = 77.9104{{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
|-
! Equilateral
| CEE: ~22/21 = 78.0604{{c}}
| CSEE: ~22/21 = 77.9698{{c}}
| POEE: ~22/21 = 77.8485{{c}}
|-
! Tenney
| CTE: ~22/21 = 77.9633{{c}}
| CWE: ~22/21 = 77.9007{{c}}
| POTE: ~22/21 = 77.8813{{c}}
|-
! Benedetti, <br>Wilson
| CBE: ~22/21 = 77.9388{{c}}
| CSBE: ~22/21 = 77.9028{{c}}
| POBE: ~22/21 = 77.9093{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Line 158: Line 223:
! Comments
! Comments
|-
|-
| [[16edo|1\16]]
| '''[[16edo|1\16]]'''
|  
|  
| 75.000
| '''75.000'''
| Lower bound of 7-odd-limit diamond monotone
| '''Lower bound of 7-odd-limit diamond monotone'''
|-
|-
|  
|  
| 11/6
| [[12/11]]
| 75.319
| 75.319
|  
|  
|-
| [[47edo|3\47]]
|
| 76.596
| 47e val
|-
|-
|  
|  
| 15/11
| [[15/11]]
| 76.707
| 76.707
|  
|  
|-
|-
|  
|  
| 7/4
| [[8/7]]
| 77.058
| 77.058
|  
|  
|-
|-
|  
|  
| 7/5
| [[10/7]]
| 77.186
| 77.186
|  
|  
|-
|-
|  
|  
| 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'''
|-
|-
|  
|  
| 11/9
| [[18/11]]
| 77.508
| 77.508
|  
|  
|-
|-
|  
|  
| 15/14
| [[15/14]]
| 77.614
| 77.614
|  
|  
|-
| [[139edo|9\139]]
|
| 77.698
| 139e val
|-
|
| {{monzo|26 25 3 -23}}
| 77.707
| 7-odd-limit least squares
|-
|-
|  
|  
| 15/8
| [[15/8]]
| 77.733
| 77.733
|  
|  
|-
|-
|  
|  
| 7/6
| [[12/7]]
| 77.761
| 77.761
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
| [[108edo|7\108]]
|
| 77.778
| 108e val
|-
|-
|  
|  
| 9/7
| [[9/7]]
| 77.861
| 77.861
| 9-odd-limit minimax
| 9-odd-limit minimax
Line 222: Line 307:
| 77.922
| 77.922
|  
|  
|-
|
| {{monzo|23 -13 -1}}
| 77.965
| 5-odd-limit least squares, Dave Benson's optimized tuning for Alpha
|-
|-
|  
|  
| 3/2
| [[3/2]]
| 77.995
| 77.995
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| [[123edo|8\123]]
|  
|  
| 11/7
| 78.049
| 123e val
|-
|
| [[11/7]]
| 78.249
| 78.249
| 11-odd-limit minimax
| 11-odd-limit minimax
Line 239: Line 334:
|-
|-
|  
|  
| 9/5
| [[9/5]]
| 78.277
| 78.277
|  
|  
|-
|-
|  
|  
| 21/16
| [[21/16]]
| 78.463
| 78.463
|  
|  
|-
|-
| [[61edo|4\61]]
|
| 78.689
|  
|  
| 11/8
|-
|
| [[11/8]]
| 78.760
| 78.760
|  
|  
|-
|-
|  
|  
| 5/3
| [[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-limti diamond monotone
| '''Upper bound of 7-, 9- and 11-odd-limit, <br>11-limit 15- and 21-odd-limit diamond monotone'''
|-
|-
|  
|  
| 21/11
| [[22/21]]
| 80.537
| 80.537
|  
|  
|-
|-
|  
|  
| 11/10
| [[11/10]]
| 82.502
| 82.502
|  
|  
|-
|-
|  
|  
| 21/20
| [[21/20]]
| 84.867
| 84.867
|  
|  
Line 280: Line 380:
<nowiki/>* Besides the octave
<nowiki/>* Besides the octave


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