Diminished (temperament): Difference between revisions

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| en = Diminished (temperament)

| de = Verminderte Temperaturen

}}

{{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

| Generator = 3/2

| Generator 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.

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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.

[[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__

== Interval chain ==

In the following table, odd harmonics 1–9 are in '''bold'''.

@@ Line 2: / Line 2: @@
 | en = Diminished (temperament)
 | de = Verminderte Temperaturen
+}}
+{{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
+| Generator = 3/2
+| Generator 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.
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 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.
+[[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 ==
 In the following table, odd harmonics 1–9 are in '''bold'''.