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{{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''' 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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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.
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 ==
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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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! 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]]
|  
|  
| 15/11
| 76.596
| 47e val
|-
|
| [[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
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|-
|-
|  
|  
| 3/2
| [[3/2]]
| 77.995
| 77.995
| 5-odd-limit minimax
| 5-odd-limit minimax
|-
| [[123edo|8\123]]
|
| 78.049
| 123e val
|-
|-
|  
|  
| 11/7
| [[11/7]]
| 78.249
| 78.249
| 11-odd-limit minimax
| 11-odd-limit minimax
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|-
|-
|  
|  
| 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-limit 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 332: Line 382:
== References ==
== References ==


[[Category:Temperaments]]
[[Category:Valentine| ]] <!-- main article -->
[[Category:Valentine| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
[[Category:Gamelismic clan]]
[[Category:Gamelismic clan]]

Latest revision as of 00:21, 13 February 2026

Valentine
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 126/125, 1029/1024 (7-limit);
121/120, 126/125, 176/175 (11-limit)
Reduced mapping ⟨1; 9 5 -3 7]
ET join 15 & 16
Generators (CWE) ~22/21 = 77.9 ¢
MOS scales …, 1L 14s, 15L 1s, 15L 16s
Ploidacot enneacot
Pergen (P8, P5/9)
Minimax error 7-odd-limit: 4.60 ¢;
11-limit 21-odd-limit: 9.01 ¢
Target scale size 7-odd-limit: 15 notes;
11-limit 21-odd-limit: 31 notes

Valentine is a regular temperament that divides a tempered 3/2 into 9 equal generators 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 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.[1] 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 scales 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 scales for scales in this temperament.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 77.9 21/20, 22/21, 25/24
2 155.8 12/11, 11/10, 35/32
3 233.7 8/7
4 311.6 6/5
5 389.5 5/4
6 467.4 21/16
7 545.3 11/8, 15/11
8 623.2 10/7
9 701.1 3/2
10 779.0 11/7, 25/16
11 856.9 18/11
12 934.8 12/7
13 1012.7 9/5, 25/14
14 1090.6 15/8
15 1168.5 55/28, 63/32, 96/49,
108/55, 125/64
16 46.4 33/32, 36/35, 50/49
17 124.3 15/14, 27/25
18 202.2 9/8
19 280.1 33/28
20 358.0 27/22
21 435.9 9/7
22 513.8 27/20
23 591.7 45/32
24 669.6 72/49
25 747.5 54/35
26 825.4 45/28
27 903.3 27/16
28 981.2 99/56
29 1059.1 90/49
30 1137.0 27/14
31 14.9 81/80, 99/98

* In 11-limit CWE tuning

Chords

Main article: Chords of valentine

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~21/20 = 77.7625 ¢ CSEE: ~21/20 = 77.7211 ¢ POEE: ~21/20 = 77.7256 ¢
Tenney CTE: ~21/20 = 77.8776 ¢ CWE: ~21/20 = 77.8673 ¢ POTE: ~21/20 = 77.8638 ¢
Benedetti,
Wilson 		CBE: ~21/20 = 77.9062 ¢ CSBE: ~21/20 = 77.9075 ¢ POBE: ~21/20 = 77.9104 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Equilateral CEE: ~22/21 = 78.0604 ¢ CSEE: ~22/21 = 77.9698 ¢ POEE: ~22/21 = 77.8485 ¢
Tenney CTE: ~22/21 = 77.9633 ¢ CWE: ~22/21 = 77.9007 ¢ POTE: ~22/21 = 77.8813 ¢
Benedetti,
Wilson 		CBE: ~22/21 = 77.9388 ¢ CSBE: ~22/21 = 77.9028 ¢ POBE: ~22/21 = 77.9093 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
1\16 75.000 Lower bound of 7-odd-limit diamond monotone
12/11 75.319
3\47 76.596 47e val
15/11 76.707
8/7 77.058
10/7 77.186
5/4 77.263 Lower bound of 5-odd-limit diamond tradeoff
2\31 77.419 Lower bound of 9- and 11-odd-limit,
11-limit 15- and 21-odd-limit diamond monotone
18/11 77.508
15/14 77.614
9\139 77.698 139e val
[26 25 3 -23⟩ 77.707 7-odd-limit least squares
15/8 77.733
12/7 77.761 7-odd-limit minimax
7\108 77.778 108e val
9/7 77.861 9-odd-limit minimax
5\77 77.922
[23 -13 -1⟩ 77.965 5-odd-limit least squares, Dave Benson's optimized tuning for Alpha
3/2 77.995 5-odd-limit minimax
8\123 78.049 123e val
11/7 78.249 11-odd-limit minimax
3\46 78.261
9/5 78.277
21/16 78.463
4\61 78.689
11/8 78.760
6/5 78.910 Upper bound of 5-odd-limit diamond tradeoff
1\15 80.000 Upper bound of 7-, 9- and 11-odd-limit,
11-limit 15- and 21-odd-limit diamond monotone
22/21 80.537
11/10 82.502
21/20 84.867

* Besides the octave

References

  1. ↑ Wendy Carlos, Pitch article (1989) Accessed 2025. https://www.wendycarlos.com/resources/pitch.html
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