Opossum: Difference between revisions

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{{Infobox regtemp
| Title = Opossum
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[28/27]], [[126/125]] (7-limit);<br>[[28/27]], [[55/54]], [[77/75]] (11-limit)
| Edo join 1 = 8d | Edo join 2 = 15
| Mapping = 1; -3 -5 -9 -4
| Generators = 11/10
| Generators tuning = 160.5
| Optimization method = CWE
| MOS scales = [[1L 6s]], [[7L 1s]]
| Odd limit 1 = 7 | Mistuning 1 = 21.0 | Complexity 1 = 15
| Odd limit 2 = 11 | Mistuning 2 = 42.0 | Complexity 2 = 15
}}
'''Opossum''' is an alternative [[extension]] to [[porcupine]]. It is defined by [[tempering out]] [[28/27]] and [[126/125]].  
'''Opossum''' is an alternative [[extension]] to [[porcupine]]. It is defined by [[tempering out]] [[28/27]] and [[126/125]].  


See [[Porcupine family #Opossum]] for technical data.
See [[Porcupine family #Opossum]] for technical data.


== Tuning spectrum ==
== Interval chain ==
In the following table, odd harmonics 1–11 and their inverses are in '''bold'''.
 
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios*
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 160.0
| 10/9, 11/10, 12/11, 15/14
|-
| 2
| 320.0
| 6/5, 11/9
|-
| 3
| 480.0
| '''4/3''', 9/7
|-
| 4
| 640.0
| 10/7, '''16/11''', 22/15
|-
| 5
| 800.0
| '''8/5''', 11/7
|-
| 6
| 960.0
| 12/7, '''16/9'''
|-
| 7
| 1120.0
| 40/21, 48/25, 64/33
|-
| 8
| 80.0
| 16/15, 36/35
|-
| 9
| 240.0
| '''8/7'''
|}
<nowiki/>* In 15edo tuning, octave reduced
 
== Tunings ==
=== Norm-based tunings ===
{| 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
|-
! Tenney
| CTE: ~10/9 = 161.306{{c}}
| CWE: ~10/9 = 160.459{{c}}
| POTE: ~10/9 = 159.691{{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: ~11/10 = 161.365{{c}}
| CWE: ~11/10 = 160.464{{c}}
| POTE: ~11/10 = 159.807{{c}}
|}
 
=== Target tunings ===
{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo
! Generator
! Eigenmonzo
|-
| 5-odd-limit
| ~10/9 = 162.737{{c}}
| 5/4
| ~10/9 = 162.996{{c}}
| 262144/234375
|-
| 7-odd-limit
| ~10/9 = 159.019{{c}}
| 7/4
| ~10/9 = 158.732{{c}}
| {{Monzo| 0 -5 3 19 }}
|-
| 9-odd-limit
| ~10/9 = 159.019{{c}}
| 7/4
| ~10/9 = 159.481{{c}}
| {{Monzo| 0 3 2 22 }}
|-
| 11-odd-limit
| ~11/10 = 159.019{{c}}
| 7/4
| ~11/10 = 159.564{{c}}
| {{Monzo| -27 2 1 9 -1 }}
|-
| 13-odd-limit
| ~11/10 = 159.019{{c}}
| 7/4
| ~11/10 = 158.421{{c}}
| {{Monzo| 0 15 6 34 -1 -15 }}
|-
| 15-odd-limit
| ~11/10 = 159.019{{c}}
| 7/4
| ~11/10 = 159.377{{c}}
| {{Monzo| 0 32 23 35 -5 -21 }}
|}
 
=== 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
Line 34: Line 182:
|  
|  
| 150.000
| 150.000
|  
| 8d val, lower bound of 7-odd-limit diamond monotone
|-
|-
|  
|  
| 12/11
| 11/6
| 150.637
| 150.637
|  
|  
Line 47: Line 195:
|-
|-
|  
|  
| 14/13
| 13/7
| 153.100
| 153.100
|  
|  
Line 62: Line 210:
|-
|-
|  
|  
| 14/11
| 11/7
| 156.498
| 156.498
|  
|  
Line 69: Line 217:
|  
|  
| 156.522
| 156.522
|  
| 23bcf val
|-
|-
|  
|  
| 6/5
| 5/3
| 157.821
| 157.821
|  
|  
|-
|
| {{vector|0 15 6 34 -1 -15}}
| 158.421
| 13 limit least squares
|-
|-
| 5\38
| 5\38
|  
|  
| 157.895
| 157.895
|  
| 38bceff val
|-
|-
| 7\53
| 7\53
|  
|  
| 158.491
| 158.491
|  
| 53bcefff val
|-
|-
|  
|  
Line 97: Line 240:
|-
|-
|  
|  
| {{vector|0 -5 3 19}}
| 7/4
| 158.732
| 7 limit least squares
|-
|
| 8/7
| 159.019
| 159.019
| 7, 9, 11, 13 and 15 limit minimax
| 7-, 9-, 11-, 13- and 15-odd-limit minimax
|-
|-
|  
|  
| 18/13
| 13/9
| 159.154
| 159.154
|  
|  
|-
|
| {{vector|0 32 23 35 -5 -21}}
| 159.377
| 15 limit least squares
|-
|
| {{vector|0 3 2 22}}
| 159.481
| 9 limit least squares
|-
|
| 1815912315/1476395008
| 159.564
| 11 limit least squares
|-
|-
| 2\15
| 2\15
|  
|  
| 160.000
| 160.000
|  
| Upper bound of 7-odd-limit diamond monotone<br>9- and 11-odd-limit diamond monotone (singleton)
|-
|-
|  
|  
Line 139: Line 262:
| 5/4
| 5/4
| 162.737
| 162.737
| 5 limit minimax
| 5-odd-limit minimax
|-
|
| 262144/234375
| 162.996
| 5 limit least squares
|-
|-
|  
|  
| 16/15
| 15/8
| 163.966
| 163.966
|  
|  
Line 162: Line 280:
|-
|-
|  
|  
| 4/3
| 3/2
| 166.015
| 166.015
|  
|  
Line 172: Line 290:
|-
|-
|  
|  
| 16/13
| 13/8
| 179.736
| 179.736
|  
|  
|-
|-
|  
|  
| 10/9
| 9/5
| 182.404
| 182.404
|  
|  

Latest revision as of 09:12, 8 April 2026

Opossum
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 28/27, 126/125 (7-limit);
28/27, 55/54, 77/75 (11-limit)
Reduced mapping ⟨1; -3 -5 -9 -4]
ET join 8d & 15
Generators (CWE) ~11/10 = 160.5 ¢
MOS scales 1L 6s, 7L 1s
Ploidacot omega-tricot
Minimax error 7-odd-limit: 21.0 ¢;
11-odd-limit: 42.0 ¢
Target scale size 7-odd-limit: 15 notes;
11-odd-limit: 15 notes

Opossum is an alternative extension to porcupine. It is defined by tempering out 28/27 and 126/125.

See Porcupine family #Opossum for technical data.

Interval chain

In the following table, odd harmonics 1–11 and their inverses are in bold.

# Cents* Approximate ratios*
0 0.0 1/1
1 160.0 10/9, 11/10, 12/11, 15/14
2 320.0 6/5, 11/9
3 480.0 4/3, 9/7
4 640.0 10/7, 16/11, 22/15
5 800.0 8/5, 11/7
6 960.0 12/7, 16/9
7 1120.0 40/21, 48/25, 64/33
8 80.0 16/15, 36/35
9 240.0 8/7

* In 15edo tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~10/9 = 161.306 ¢ CWE: ~10/9 = 160.459 ¢ POTE: ~10/9 = 159.691 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~11/10 = 161.365 ¢ CWE: ~11/10 = 160.464 ¢ POTE: ~11/10 = 159.807 ¢

Target tunings

Target tunings
Target Minimax Least squares
Generator Eigenmonzo Generator Eigenmonzo
5-odd-limit ~10/9 = 162.737 ¢ 5/4 ~10/9 = 162.996 ¢ 262144/234375
7-odd-limit ~10/9 = 159.019 ¢ 7/4 ~10/9 = 158.732 ¢ [0 -5 3 19⟩
9-odd-limit ~10/9 = 159.019 ¢ 7/4 ~10/9 = 159.481 ¢ [0 3 2 22⟩
11-odd-limit ~11/10 = 159.019 ¢ 7/4 ~11/10 = 159.564 ¢ [-27 2 1 9 -1⟩
13-odd-limit ~11/10 = 159.019 ¢ 7/4 ~11/10 = 158.421 ¢ [0 15 6 34 -1 -15⟩
15-odd-limit ~11/10 = 159.019 ¢ 7/4 ~11/10 = 159.377 ¢ [0 32 23 35 -5 -21⟩

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
15/14 119.443
13/12 138.573
13/11 144.605
9/7 145.028
1\8 150.000 8d val, lower bound of 7-odd-limit diamond monotone
11/6 150.637
13/10 151.405
13/7 153.100
7/5 154.372
7/6 155.522
11/7 156.498
3\23 156.522 23bcf val
5/3 157.821
5\38 157.895 38bceff val
7\53 158.491 53bcefff val
15/13 158.710
7/4 159.019 7-, 9-, 11-, 13- and 15-odd-limit minimax
13/9 159.154
2\15 160.000 Upper bound of 7-odd-limit diamond monotone
9- and 11-odd-limit diamond monotone (singleton)
11/8 162.171
5/4 162.737 5-odd-limit minimax
15/8 163.966
11/10 165.004
15/11 165.762
3/2 166.015
11/9 173.704
13/8 179.736
9/5 182.404
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