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See [[Gamelismic clan #Unidec]] for more information.
{{Infobox regtemp
| Title = Unidec
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[1029/1024]], [[4375/4374]] (7-limit); <br>[[385/384]], [[441/440]], [[4375/4374]] (11-limit)
| Edo join 1 = 26 | Edo join 2 = 46
| Mapping = 2; 6 11 -2 -3
| Generators = 14/11 | Generators tuning = 416.9 | Optimization method = CWE
| MOS scales = [[6L 2s]], [[6L 8s]], [[6L 14s]], [[20L 6s]]
| Odd limit 1 = 9 | Mistuning 1 = 2.41 | Complexity 1 = 46
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 2.95 | Complexity 2 = 46
}}
'''Unidec''' is a [[regular temperament|temperament]] [[generator|generated]] by a wide major third representing [[80/63]] which can be taken as [[14/11]] from the [[11-limit]] onwards. Two generators minus a [[semi-octave]] [[period]] make a [[~]][[8/7]], and three ~8/7's make a perfect fifth of [[3/2]], [[tempering out]] [[1029/1024]], the gamelisma, and [[4375/4374]], the ragisma, with the 11-limit [[extension]] adding [[385/384]] or [[441/440]] to the comma list.  


== Chords ==
Unidec has two viable 13-limit extensions: hendec and ekadash, discussed in [[Unidec extensions]].
{{main| Chords of unidec }}
 
See [[Gamelismic clan #Unidec]] for technical data.
 
== Interval chain ==
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
| 600.00
| 99/70, 140/99
|-
| 1
| 416.85
| 14/11
| 1016.80
| 9/5
|-
| 2
| 833.71
| 81/50, 160/99
| 233.71
| '''8/7'''
|-
| 3
| 50.56
| 33/32, 36/35
| 650.56
| '''16/11'''
|-
| 4
| 467.42
| '''21/16'''
| 1067.42
| 50/27
|-
| 5
| 884.27
| 5/3
| 284.27
| 33/28
|-
| 6
| 101.13
| 35/33
| 701.13
| '''3/2'''
|-
| 7
| 517.98
| 27/20
| 1117.98
| 21/11, 40/21
|-
| 8
| 934.83
| 12/7
| 334.83
| 40/33
|-
| 9
| 151.69
| 12/11
| 751.69
| 54/35
|-
| 10
| 568.54
| 25/18
| 1168.54
| 55/28, 63/32, 96/49, 108/55
|-
| 11
| 985.40
| 99/56
| 385.40
| '''5/4'''
|-
| 12
| 202.25
| '''9/8'''
| 802.25
| 35/22
|-
| 13
| 619.11
| 10/7
| 19.11
| 81/80, 99/98, 100/99
|}
<nowiki/>* In 11-limit CWE tuning, octave reduced
 
== Chords and harmony ==
{{See also| Chords of unidec }}


== Scales ==
== Scales ==
* [[Unidec26]]
* [[Unidec26]]


== Spectrum of unidec/hendec tunings by eigenmonzos ==
== Tunings ==
{| class="wikitable center-1 right-2"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
|-
! Eigenmonzo
! Tenney
! Minor Whole Tone
| CTE: ~80/63 = 416.9403{{c}}
| CWE: ~80/63 = 416.8688{{c}}
| POTE: ~80/63 = 416.8385{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~14/11 = 416.9262{{c}}
| CWE: ~14/11 = 416.8543{{c}}
| POTE: ~14/11 = 416.8350{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-3"
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]
! Generator (¢)
! Comments
! Comments
|-
|-
| 16/13
| 13/12
| 179.736
| 415.357
|  
|  
|-
|-
| 14/13
| 7/4
| 182.075
| 415.587
|  
|  
|-
|-
| 13/11
| 13/10
| 182.158
| 416.198
|  
|  
|-
|-
| 10/9
| 11/8
| 182.404
| 416.227
|  
|  
|-
|-
| 14/11
| 13/9
| 182.492
| 416.338
|  
|  
|-
|-
| 4/3
| 15/13
| 183.007
| 416.516
|  
|  
|-
|-
| 16/15
| 7/6
| 183.043
| 416.641
|  
|  
|-
|-
| 5/4
| 7/5
| 183.062
| 416.730
| 5-, 13- and 15-odd-limit minimax
| 7-odd-limit minimax
|-
|-
| 6/5
| 11/6
| 183.128
| 416.737
|  
|  
|-
|-
| 15/11
| 11/10
| 183.152
| 416.785
|  
|  
|-
|-
| 11/9
| 9/7
| 183.161
| 416.792
|  
| 9- and 11-odd-limit minimax
|-
|-
| 15/14
| 15/14
| 183.187
| 416.813
|  
|  
|-
|-
| 9/7
| 11/9
| 183.208
| 416.839
| 9- and 11-odd-limit minimax
|  
|-
|-
| 11/10
| 15/11
| 183.215
| 416.848
|  
|  
|-
|-
| 12/11
| 5/3
| 183.263
| 416.872
|  
|  
|-
|-
| 7/5
| 5/4
| 183.270
| 416.938
| 7-odd-limit minimax
| 5-, 13- and 15-odd-limit minimax
|-
|-
| 7/6
| 15/8
| 183.359
| 416.957
|  
|  
|-
|-
| 15/13
| 3/2
| 183.484
| 416.993
|  
|  
|-
|-
| 18/13
| 11/7
| 183.662
| 417.508
|  
|  
|-
|-
| 11/8
| 9/5
| 183.773
| 417.596
|  
|  
|-
|-
| 13/10
| 13/11
| 183.802
| 417.842
|  
|  
|-
|-
| 8/7
| 13/7
| 184.413
| 417.925
|  
|  
|-
|-
| 13/12
| 13/8
| 184.643
| 420.264
|  
|  
|}
|}


== Music ==
== Music ==
* [[Technical Notes for Newbeams #Hypnocloudsmack 2|Hypnocloudsmack 2]] by [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
* "[[Technical Notes for Newbeams #Hypnocloudsmack 2|Hypnocloudsmack 2]]" from ''Newbeams''


[[Category:Temperament]]
[[Category:Unidec| ]] <!-- main article -->
[[Category:Unidec| ]] <!-- main article -->
[[Category:Gamelismic]]
[[Category:Rank-2 temperaments]]
[[Category:Ragismic]]
[[Category:Gamelismic clan]]
 
[[Category:Ragismic microtemperaments]]
{{todo| add introduction }}

Latest revision as of 13:27, 26 March 2026

Unidec
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 1029/1024, 4375/4374 (7-limit);
385/384, 441/440, 4375/4374 (11-limit)
Reduced mapping ⟨2; 6 11 -2 -3]
ET join 26 & 46
Generators (CWE) ~14/11 = 416.9 ¢
MOS scales 6L 2s, 6L 8s, 6L 14s, 20L 6s
Ploidacot diploid gamma-hexacot
Minimax error 9-odd-limit: 2.41 ¢;
11-limit 21-odd-limit: 2.95 ¢
Target scale size 9-odd-limit: 46 notes;
11-limit 21-odd-limit: 46 notes

Unidec is a temperament generated by a wide major third representing 80/63 which can be taken as 14/11 from the 11-limit onwards. Two generators minus a semi-octave period make a ~8/7, and three ~8/7's make a perfect fifth of 3/2, tempering out 1029/1024, the gamelisma, and 4375/4374, the ragisma, with the 11-limit extension adding 385/384 or 441/440 to the comma list.

Unidec has two viable 13-limit extensions: hendec and ekadash, discussed in Unidec extensions.

See Gamelismic clan #Unidec for technical data.

Interval chain

In the following table, odd harmonics 1–21 and their inverses are in bold.

# Period 0 Period 1
Cents* Approximate ratios Cents* Approximate ratios
0 0.00 1/1 600.00 99/70, 140/99
1 416.85 14/11 1016.80 9/5
2 833.71 81/50, 160/99 233.71 8/7
3 50.56 33/32, 36/35 650.56 16/11
4 467.42 21/16 1067.42 50/27
5 884.27 5/3 284.27 33/28
6 101.13 35/33 701.13 3/2
7 517.98 27/20 1117.98 21/11, 40/21
8 934.83 12/7 334.83 40/33
9 151.69 12/11 751.69 54/35
10 568.54 25/18 1168.54 55/28, 63/32, 96/49, 108/55
11 985.40 99/56 385.40 5/4
12 202.25 9/8 802.25 35/22
13 619.11 10/7 19.11 81/80, 99/98, 100/99

* In 11-limit CWE tuning, octave reduced

Chords and harmony

See also: Chords of unidec

Scales

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~80/63 = 416.9403 ¢ CWE: ~80/63 = 416.8688 ¢ POTE: ~80/63 = 416.8385 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~14/11 = 416.9262 ¢ CWE: ~14/11 = 416.8543 ¢ POTE: ~14/11 = 416.8350 ¢

Tuning spectrum

Eigenmonzo
(unchanged interval) 		Generator (¢) Comments
13/12 415.357
7/4 415.587
13/10 416.198
11/8 416.227
13/9 416.338
15/13 416.516
7/6 416.641
7/5 416.730 7-odd-limit minimax
11/6 416.737
11/10 416.785
9/7 416.792 9- and 11-odd-limit minimax
15/14 416.813
11/9 416.839
15/11 416.848
5/3 416.872
5/4 416.938 5-, 13- and 15-odd-limit minimax
15/8 416.957
3/2 416.993
11/7 417.508
9/5 417.596
13/11 417.842
13/7 417.925
13/8 420.264

Music

Andrew Heathwaite
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