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'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] with a 1/4-[[octave]] [[period]] [[generator|generated]] by a [[~]][[3/2]] perfect fifth or more generally by anything that melodically qualifies as a fifth. As a 5-limit temperament it [[tempering out|tempers out]] the diminished comma, [[648/625]], and [[36/35]] and [[50/49]] in the [[7-limit]]. A far more accurate interpretation is as a 2.3.19 temperament that makes [[~]][[19/16]] = 1\4 = 300{{cent}}, though its [[MOS]] structure of [[4L 4s]] is very flexible, so one could use 3\4 minus 8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned. [[12edo]] is the simplest nontrivial tuning. Other possible tunings include [[16edo]] and [[28edo]], both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately [[28/19]]. [[28edo]] is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[strongly consistent circle]] of them.
{{Infobox regtemp
| Title = Diminished
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[648/625]] (2.3.5); <br>[[36/35]], [[50/49]] (2.3.5.7)
| Edo join 1 = 4 | Edo join 2 = 12
| Mapping = 4; 1 1 1
| Generators = 3/2
| Generators tuning = 696.0
| Optimization method = CWE
| MOS scales = [[4L 4s]], [[4L 8s]], [[12L 4s]]
| Pergen = (P8/4, P5)
| Color name = Quadguti
| Odd limit 1 = 5 | Mistuning 1 = 15.6 | Complexity 1 = 8
| Odd limit 2 = 7-limit 21 | Mistuning 2 = 33.1 | Complexity 2 = 12
}}
'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] that [[tempering out|tempers out]] the diminished comma, [[648/625]], in the [[5-limit]], and [[36/35]] and [[50/49]] in the [[7-limit]]. It has a 1/4-[[octave]] [[period]] and is [[generator|generated]] by a [[~]][[3/2]] perfect fifth. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned.  


See [[Dimipent family #Diminished]] for technical data on the 5-limit temperament.  
It can be extended to the 2.3.5.7.19-[[subgroup]] where the 1/4-octave period stands in for both ~6/5 and ~19/16 since this ~19/16 is more accurate, though its mos structure of 4L 4s is very flexible, so one could use 3\4 minus ~8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example.


[[12edo]] is an obvious tuning. Another possible tuning is [[16edo]] which has the interesting feature of being relatively good in the 2.7.19 subgroup so that the fifth is approximately [[28/19]], or [[28edo]], which uses something like {{nowrap| ~[[95/64]] {{=}} ~([[19/16]])⋅([[5/4]]) }} as its fifth, of which the latter is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[consistent circle]] of them.
See [[Diminished family #Diminished]] for technical data.
__TOC__
{{Clear}}
== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–9 are in '''bold'''.  
In the following table, odd harmonics 1–9 are in '''bold'''.  


{| class="wikitable center-1 right-2 right-4 right-6 right-8"
{| class="wikitable center-1 right-2 right-4 right-6 right-8"
|-
! rowspan="2" | #
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 0
Line 18: Line 39:
|-
|-
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
|-
|-
| 0
| 0
Line 37: Line 58:
|-
|-
| 1
| 1
| 92.0
| 96.0
| 15/14, 21/20, 25/24, 49/48
| 15/14, 21/20, 25/24, 49/48
| 392.0
| 396.0
| '''5/4''', 9/7
| '''5/4''', 9/7
| 692.0
| 696.0
| '''3/2'''
| '''3/2'''
| 992.0
| 996.0
| '''7/4''', 9/5
| '''7/4''', 9/5
|-
|-
| 2
| 2
| 183.9
| 191.9
| '''9/8'''
| '''9/8'''
| 483.9
| 491.9
| 21/16
| 21/16
| 783.9
| 791.9
| 45/28, 63/40
| 45/28, 63/40
| 1083.9
| 1091.9
| 15/8
| 15/8
|}
|}
<nowiki>*</nowiki> in 7-limit CTE tuning
<nowiki/>* In 7-limit CWE tuning, octave reduced


== Scales ==
== Scales ==
Line 62: Line 83:


== Tunings ==
== Tunings ==
=== Prime-optimized tunings ===
=== Norm-based tunings ===
* 5-limit
{| class="wikitable mw-collapsible mw-collapsed"
** [[CTE]]: ~6/5 = 1\4, ~3/2 = 696.9833
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
** [[CWE]]: ~6/5 = 1\4, ~3/2 = 698.2661
|-
* 7-limit
! rowspan="2" |
** [[CTE]]: ~6/5 = 1\4, ~3/2 = 691.9545
! colspan="3" | Euclidean
** [[CWE]]: ~6/5 = 1\4, ~3/2 = 695.9618
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 696.9833{{c}}
| CWE: ~3/2 = 698.2661{{c}}
| POTE: ~3/2 = 699.5072{{c}}
|}
{| 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: ~3/2 = 691.9545{{c}}
| CWE: ~3/2 = 695.9618{{c}}
| POTE: ~3/2 = 699.5235{{c}}
|}


=== Others ===
=== Others ===
* 5-limit [[DKW theory|DKW]]: ~6/5 = 1\4, ~3/2 = 690.289
* 5-limit [[DKW theory|DKW]]: ~6/5 = 300.000{{c}}, ~3/2 = 690.289{{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<br>(Unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
|-
|-
| 2\4
| [[4edo|2\4]]
|  
|  
| 600.000
| 600.000
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|-
|-
|  
|  
| 49/48
| [[49/48]]
| 635.697
| 635.697
|  
|  
|-
|-
|  
|  
| 7/4
| [[7/4]]
| 668.826
| 668.826
|  
|  
|-
|-
|  
|  
| 25/24
| [[25/24]]
| 670.672
| 670.672
| 1/2-comma
| 1/2-comma
|-
|-
| 9\16
| [[16edo|9\16]]
|  
|  
| 675.000
| 675.000
Line 107: Line 152:
|-
|-
|  
|  
| 21/20
| [[21/20]]
| 684.467
| 684.467
|  
|  
|-
|-
|  
|  
| 21/16
| [[21/16]]
| 685.390
| 685.390
|  
|  
|-
| [[28edo|16\28]]
|
| 685.714
|
|-
|-
|  
|  
| 5/4
| [[5/4]]
| 686.314
| 686.314
| 1/4-comma
| 1/4-comma
|-
| [[40edo|23\40]]
|
| 690.000
| 40d val
|-
| [[52edo|30\52]]
|
| 692.308
| 52dd val
|-
| [[64edo|37\64]]
|
| 693.750
| 64dd val
|-
|-
|  
|  
| 15/8
| [[15/8]]
| 694.134
| 694.134
| 1/8-comma
| 1/8-comma
|-
|-
| 7\12
| [[12edo|7\12]]
|  
|  
| 700.000
| 700.000
Line 132: Line 197:
|-
|-
|  
|  
| 3/2
| [[3/2]]
| 701.955
| 701.955
| Untempered
| Untempered
|-
| [[32edo|19\32]]
|
| 712.500
| 32cd val
|-
|-
|  
|  
| 9/5
| [[9/5]]
| 717.596
| 717.596
| -1/4-comma
| -1/4-comma
|-
|-
|  
|  
| 15/14
| [[15/14]]
| 719.443
| 719.443
|  
|  
|-
| [[20edo|12\20]]
|
| 720.000
| 20cd val
|-
|-
|  
|  
| 9/7
| [[9/7]]
| 735.084
| 735.084
|  
|  
|-
|-
| 5\8
| [[8edo|5\8]]
|  
|  
| 750.000
| 750.000
| 8d val, upper bound of 7-odd-limit diamond monotone
| 8d val, upper bound of 7-odd-limit diamond monotone
|}
|}
<nowiki>*</nowiki> besides the octave
<nowiki/>* Besides the octave


== See also ==
== See also ==
* [[Diminished]] (disambiguation page)
* [[Diminished]] (disambiguation page)


[[Category:Temperaments]]
[[Category:Diminished| ]] <!-- Main article -->
[[Category:Diminished| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Dimipent family]]
[[Category:Diminished family]]
[[Category:Jubilismic clan]]
[[Category:Jubilismic clan]]
[[Category:Mint temperaments]]
[[Category:Mint temperaments]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
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