Diminished (temperament): Difference between revisions

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| de = Verminderte Temperaturen

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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.

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. ~~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.

[[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 [[~~Dimipent~~ family#Diminished]] for technical data.

See [[Diminished family #Diminished]] for technical data.

== Interval chain ==

Line 16:

{| class="wikitable center-1 right-2 right-4 right-6 right-8"

|-

! rowspan="2" | &#~~35;~~

! rowspan="2" | #

! colspan="2" | Period 0

! colspan="2" | Period 1

Line 23:

|-

! Cents*

! Approx. ~~Ratios~~

! Approx. ratios

! Cents*

! Approx. ~~Ratios~~

! Approx. ratios

! Cents*

! Approx. ~~Ratios~~

! Approx. ratios

! Cents*

! Approx. ~~Ratios~~

! Approx. ratios

|-

| 0

Line 44:

| 92.0

| 15/14, 21/20, 25/24, 49/48

| ~~392~~.0

| 396.0

| '''5/4''', 9/7

| ~~692~~.0

| 696.0

| '''3/2'''

| ~~992~~.0

| 996.0

| '''7/4''', 9/5

|-

| 2

| ~~183~~.9

| 191.9

| '''9/8'''

| ~~483~~.9

| 491.9

| 21/16

| ~~783~~.9

| 791.9

| 45/28, 63/40

| ~~1083~~.9

| 1091.9

| 15/8

|}

<nowiki />* In 7-limit ~~CTE~~ tuning

<nowiki/>* In 7-limit CWE tuning, octave reduced

== Scales ==

* [[Diminished12]] ~~–~~ in [[44edo]] tuning

* [[Diminished12]] – in [[44edo]] tuning

== Tunings ==

=== Prime-optimized 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 prime-optimized 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 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 = ~~1\4~~, ~3/2 = 690.289

* 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~~

! Edo<br>generator

! [[Eigenmonzo|~~Eigenmonzo~~<br />(~~Unchanged-interval~~)]]*

! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*

! Generator (¢)

! Comments

Line 161:

Line 185:

| 8d val, upper bound of 7-odd-limit diamond monotone

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

== See also ==

* [[Diminished]] (disambiguation page)

~~[[Category:Temperaments]]~~

[[Category:Diminished| ]]

[[Category:~~Dimipent~~ family]]

[[Category:Rank-2 temperaments]]

[[Category:Diminished family]]

[[Category:Jubilismic clan]]

[[Category:Mint temperaments]]

[[Category:Starling temperaments]]

@@ Line 3: / Line 3: @@
 | de = Verminderte Temperaturen
 }}
-'''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.
 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. 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.
+[[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 [[Dimipent family#Diminished]] for technical data.
+See [[Diminished family #Diminished]] for technical data.
 == Interval chain ==
@@ Line 16: / Line 16: @@
 {| class="wikitable center-1 right-2 right-4 right-6 right-8"
 |-
-! rowspan="2" | &#35;
+! rowspan="2" | #
 ! colspan="2" | Period 0
 ! colspan="2" | Period 1
@@ Line 23: / Line 23: @@
 |-
 ! Cents*
-! Approx. Ratios
+! Approx. ratios
 ! Cents*
-! Approx. Ratios
+! Approx. ratios
 ! Cents*
-! Approx. Ratios
+! Approx. ratios
 ! Cents*
-! Approx. Ratios
+! Approx. ratios
 |-
 | 0
@@ Line 44: / Line 44: @@
 | 92.0
 | 15/14, 21/20, 25/24, 49/48
-| 392.0
+| 396.0
 | '''5/4''', 9/7
-| 692.0
+| 696.0
 | '''3/2'''
-| 992.0
+| 996.0
 | '''7/4''', 9/5
 |-
 | 2
-| 183.9
+| 191.9
 | '''9/8'''
-| 483.9
+| 491.9
 | 21/16
-| 783.9
+| 791.9
 | 45/28, 63/40
-| 1083.9
+| 1091.9
 | 15/8
 |}
-<nowiki />* In 7-limit CTE tuning
+<nowiki/>* In 7-limit CWE tuning, octave reduced
 == Scales ==
-* [[Diminished12]] &ndash; in [[44edo]] tuning
+* [[Diminished12]] – in [[44edo]] tuning
 == Tunings ==
 === Prime-optimized 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 prime-optimized 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 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 = 1\4, ~3/2 = 690.289
+* 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
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
+! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
 ! Generator (¢)
 ! Comments
@@ Line 161: / Line 185: @@
 | 8d val, upper bound of 7-odd-limit diamond monotone
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
 == See also ==
 * [[Diminished]] (disambiguation page)
-[[Category:Temperaments]]
 [[Category:Diminished| ]] <!-- Main article -->
-[[Category:Dimipent family]]
+[[Category:Rank-2 temperaments]]
+[[Category:Diminished family]]
 [[Category:Jubilismic clan]]
 [[Category:Mint temperaments]]
 [[Category:Starling temperaments]]