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Diminished (temperament): Difference between revisions

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'''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.  
'''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.  
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 [[strongly consistent circle]] of them.
 
See [[Diminished family #Diminished]] for technical data.  


== Interval chain ==
== Interval chain ==
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{| 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
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|-
|-
! 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 39: Line 44:
| 92.0
| 92.0
| 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 ==
* [[Diminished12]] – in [[44edo]] tuning
* [[Diminished12]] – in [[44edo]] tuning
== Tunings ==
=== Prime-optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit prime-optimized tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! 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 prime-optimized 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 ===
* 5-limit [[DKW theory|DKW]]: ~6/5 = 300.000{{c}}, ~3/2 = 690.289{{c}}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 2\4
|
| 600.000
| Lower bound of 7-odd-limit diamond monotone
|-
|
| 49/48
| 635.697
|
|-
|
| 7/4
| 668.826
|
|-
|
| 25/24
| 670.672
| 1/2-comma
|-
| 9\16
|
| 675.000
|
|-
|
| 21/20
| 684.467
|
|-
|
| 21/16
| 685.390
|
|-
|
| 5/4
| 686.314
| 1/4-comma
|-
|
| 15/8
| 694.134
| 1/8-comma
|-
| 7\12
|
| 700.000
| 9-odd-limit diamond monotone (singleton)
|-
|
| 3/2
| 701.955
| Untempered
|-
|
| 9/5
| 717.596
| -1/4-comma
|-
|
| 15/14
| 719.443
|
|-
|
| 9/7
| 735.084
|
|-
| 5\8
|
| 750.000
| 8d val, upper bound of 7-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave


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


[[Category:Temperaments]]
[[Category:Diminished| ]] <!-- Main article -->
[[Category:Dimipent family]]
[[Category:Rank-2 temperaments]]
[[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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