Diminished (temperament): Difference between revisions

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| en = Diminished (temperament)
| en = Diminished (temperament)
| de = Verminderte Temperaturen
| 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
| 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.  
'''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.
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.  
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'''.  
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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;" | 5-limit prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
|-
! rowspan="2" |  
! rowspan="2" |  
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|}
|}
{| 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" |  
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! 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
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|-
|-
|  
|  
| 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
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|-
|-
|  
|  
| 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

Latest revision as of 03:56, 15 March 2026

Diminished
Subgroups 2.3.5, 2.3.5.7
Comma basis 648/625 (2.3.5);
36/35, 50/49 (2.3.5.7)
Reduced mapping ⟨4; 1 1 1]
ET join 4 & 12
Generators (CWE) ~3/2 = 696.0 ¢
MOS scales 4L 4s, 4L 8s, 12L 4s
Ploidacot tetraploid monocot
Pergen (P8/4, P5)
Color name Quadguti
Minimax error 5-odd-limit: 15.6 ¢;
7-limit 21-odd-limit: 33.1 ¢
Target scale size 5-odd-limit: 8 notes;
7-limit 21-odd-limit: 12 notes

Diminished is a rank-2 temperament that 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 generated by a ~3/2 perfect fifth. The main interest in this temperament is in its mos scales, 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 ¢ 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 ~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.

Interval chain

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

# Period 0 Period 1 Period 2 Period 3
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.0 1/1 300.0 6/5, 7/6 600.0 7/5, 10/7 900.0 5/3, 12/7
1 96.0 15/14, 21/20, 25/24, 49/48 396.0 5/4, 9/7 696.0 3/2 996.0 7/4, 9/5
2 191.9 9/8 491.9 21/16 791.9 45/28, 63/40 1091.9 15/8

* In 7-limit CWE tuning, octave reduced

Scales

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 696.9833 ¢ CWE: ~3/2 = 698.2661 ¢ POTE: ~3/2 = 699.5072 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 691.9545 ¢ CWE: ~3/2 = 695.9618 ¢ POTE: ~3/2 = 699.5235 ¢

Others

  • 5-limit DKW: ~6/5 = 300.000 ¢, ~3/2 = 690.289 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(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
16\28 685.714
5/4 686.314 1/4-comma
23\40 690.000 40d val
30\52 692.308 52dd val
37\64 693.750 64dd val
15/8 694.134 1/8-comma
7\12 700.000 9-odd-limit diamond monotone (singleton)
3/2 701.955 Untempered
19\32 712.500 32cd val
9/5 717.596 -1/4-comma
15/14 719.443
12\20 720.000 20cd val
9/7 735.084
5\8 750.000 8d val, upper bound of 7-odd-limit diamond monotone

* Besides the octave

See also

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