Diminished (temperament): Difference between revisions

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


== Interval chain ==
== Interval chain ==
Line 11: Line 15:


{| 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 23:
|-
|-
! 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 ==
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== Tunings ==
== Tunings ==
=== Prime-optimized 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 ===
=== 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
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| 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]]

Latest revision as of 12:15, 21 August 2025

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 strongly 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 92.0 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

Prime-optimized tunings

5-limit prime-optimized tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 696.9833 ¢ CWE: ~3/2 = 698.2661 ¢ POTE: ~3/2 = 699.5072 ¢
7-limit prime-optimized 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
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

* Besides the octave

See also

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