Xenharmonic Wiki

Diminished family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Unify to 3 decimal places. +error maps
Spell out 1\4, 1\8, etc. in decimal and note the badness metric being used
Line 11: Line 11:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 1\4, ~3/2 = 696.983 (~25/24 = 96.983)
* [[CTE]]: ~6/5 = 300.000, ~3/2 = 696.983 (~25/24 = 96.983)
: [[error map]]: {{val| 0.000 -4.972 +10.670 }}
: [[error map]]: {{val| 0.000 -4.972 +10.670 }}
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.507 (~25/24 = 99.507)
* [[POTE]]: ~6/5 = 300.000, ~3/2 = 699.507 (~25/24 = 99.507)
: error map: {{val| 0.000 -2.448 +13.193 }}
: error map: {{val| 0.000 -2.448 +13.193 }}


{{Optimal ET sequence|legend=1| 4, 8, 12 }}
{{Optimal ET sequence|legend=1| 4, 8, 12 }}


[[Badness]]: 0.047231
[[Badness]] (Smith): 0.047231


== Diminished ==
== Diminished ==
Line 34: Line 34:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 1\4, ~3/2 = 691.955 (~21/20 = 91.955)
* [[CTE]]: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
: [[error map]]: {{val| 0.000 -10.000 +5.641 +23.129 }}
: [[error map]]: {{val| 0.000 -10.000 +5.641 +23.129 }}
* [[POTE]]: ~6/5 = 1\4, ~3/2 = 699.523 (~21/20 = 99.523)
* [[POTE]]: ~6/5 = 300.000, ~3/2 = 699.523 (~21/20 = 99.523)
: error map: {{val| 0.000 -2.432 +13.210 +30.698 }}
: error map: {{val| 0.000 -2.432 +13.210 +30.698 }}


{{Optimal ET sequence|legend=1| 4, 8d, 12 }}
{{Optimal ET sequence|legend=1| 4, 8d, 12 }}


[[Badness]]: 0.022401
[[Badness]] (Smith): 0.022401


=== 11-limit ===
=== 11-limit ===
Line 50: Line 50:
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}
Mapping: {{mapping| 4 0 3 5 14 | 0 1 1 1 0 }}


{{Multival|legend=1|4 4 4 0 -3 -5 -14 -2 -14 -14}}
Wedgie: {{multival| 4 4 4 0 -3 -5 -14 -2 -14 -14 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~3/2 = 691.955 (~21/20 = 91.955)
* CTE: ~6/5 = 300.000, ~3/2 = 691.955 (~21/20 = 91.955)
* POTE: ~6/5 = 1\4, ~3/2 = 709.109 (~15/14 = 109.109)
* POTE: ~6/5 = 300.000, ~3/2 = 709.109 (~15/14 = 109.109)


Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12, 32cddee, 44cddeee }}
Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12, 32cddee, 44cddeee }}


Badness: 0.022132
Badness (Smith): 0.022132


==== 13-limit ====
==== 13-limit ====
Line 68: Line 68:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~3/2 = 691.954 (~21/20 = 91.954)
* CTE: ~6/5 = 300.000, ~3/2 = 691.954 (~21/20 = 91.954)
* POTE: ~6/5 = 1\4, ~3/2 = 713.773 (~15/14 = 113.773)
* POTE: ~6/5 = 300.000, ~3/2 = 713.773 (~15/14 = 113.773)


Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12f, 20cdef }}
Optimal ET sequence: {{Optimal ET sequence| 4, 8d, 12f, 20cdef }}


Badness: 0.019509
Badness (Smith): 0.019509


=== Demolished ===
=== Demolished ===
Line 83: Line 83:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~3/2 = 687.747 (~21/20 = 87.747)
* CTE: ~6/5 = 300.000, ~3/2 = 687.747 (~21/20 = 87.747)
* POTE: ~6/5 = 1\4, ~3/2 = 689.881 (~21/20 = 89.881)
* POTE: ~6/5 = 300.000, ~3/2 = 689.881 (~21/20 = 89.881)


Optimal ET sequence: {{Optimal ET sequence| 12, 28, 40de }}
Optimal ET sequence: {{Optimal ET sequence| 12, 28, 40de }}


Badness: 0.026574
Badness (Smith): 0.026574


=== Cohedim ===
=== Cohedim ===
Line 102: Line 102:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~11/7 = 793.423 (~12/11 = 106.577)
* CTE: ~6/5 = 300.000, ~11/7 = 793.423 (~12/11 = 106.577)
* POTE: ~6/5 = 1\4, ~11/7 = 798.421 (~12/11 = 101.679)
* POTE: ~6/5 = 300.000, ~11/7 = 798.421 (~12/11 = 101.679)


Optimal ET sequence: {{Optimal ET sequence| 8bce, 12 }}
Optimal ET sequence: {{Optimal ET sequence| 8bce, 12 }}


Badness: 0.054965
Badness (Smith): 0.054965


==== 13-limit ====
==== 13-limit ====
Line 117: Line 117:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~11/7 = 791.375 (~12/11 = 108.625)
* CTE: ~6/5 = 300.000, ~11/7 = 791.375 (~12/11 = 108.625)
* POTE: ~6/5 = 1\4, ~11/7 = 797.701 (~12/11 = 102.299)
* POTE: ~6/5 = 300.000, ~11/7 = 797.701 (~12/11 = 102.299)


Optimal ET sequence: {{Optimal ET sequence| 8bcef, 12f }}
Optimal ET sequence: {{Optimal ET sequence| 8bcef, 12f }}


Badness: 0.041707
Badness (Smith): 0.041707


== Hemidim ==
== Hemidim ==
Line 136: Line 136:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~6/5 = 1\4, ~7/4 = 949.540 (~36/35 = 49.540)
* [[CTE]]: ~6/5 = 300.000, ~7/4 = 949.540 (~36/35 = 49.540)
: [[error map]]: {{val| 0.000 -2.875 +12.766 -19.286 }}
: [[error map]]: {{val| 0.000 -2.875 +12.766 -19.286 }}
* [[POTE]]: ~6/5 = 1\4, ~7/4 = 947.445 (~36/35 = 47.445)
* [[POTE]]: ~6/5 = 300.000, ~7/4 = 947.445 (~36/35 = 47.445)
: error map: {{val| 0.000 -7.065 +8.576 -21.381 }}
: error map: {{val| 0.000 -7.065 +8.576 -21.381 }}


{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}
{{Optimal ET sequence|legend=1| 4, …, 20c, 24 }}


[[Badness]]: 0.086378
[[Badness]] (Smith): 0.086378


=== 11-limit ===
=== 11-limit ===
Line 153: Line 153:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~7/4 = 949.872 (~36/35 = 49.872)
* CTE: ~6/5 = 300.000, ~7/4 = 949.872 (~36/35 = 49.872)
* POTE: ~6/5 = 1\4, ~7/4 = 948.342 (~36/35 = 48.342)
* POTE: ~6/5 = 300.000, ~7/4 = 948.342 (~36/35 = 48.342)


Optimal ET sequence: {{Optimal ET sequence| 4e, 20ce, 24 }}
Optimal ET sequence: {{Optimal ET sequence| 4e, 20ce, 24 }}


Badness: 0.056576
Badness (Smith): 0.056576


=== 13-limit ===
=== 13-limit ===
Line 168: Line 168:


Optimal tunings:  
Optimal tunings:  
* CTE: ~6/5 = 1\4, ~7/4 = 949.366 (~36/35 = 49.366)
* CTE: ~6/5 = 300.000, ~7/4 = 949.366 (~36/35 = 49.366)
* POTE: ~6/5 = 1\4, ~7/4 = 947.775 (~36/35 = 47.775)
* POTE: ~6/5 = 300.000, ~7/4 = 947.775 (~36/35 = 47.775)


Optimal ET sequence: {{Optimal ET sequence| 4ef, 24 }}
Optimal ET sequence: {{Optimal ET sequence| 4ef, 24 }}


Badness: 0.039030
Badness (Smith): 0.039030


== Semidim ==
== Semidim ==
Line 187: Line 187:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~15/14 = 1\8, ~3/2 = 702.765 (~36/35 = 47.235)
* [[CTE]]: ~15/14 = 150.000, ~3/2 = 702.765 (~36/35 = 47.235)
* [[POTE]]: ~15/14 = 1\8, ~3/2 = 707.014 (~36/35 = 42.986)
* [[POTE]]: ~15/14 = 150.000, ~3/2 = 707.014 (~36/35 = 42.986)


{{Optimal ET sequence|legend=1| 8d, 24, 32c }}
{{Optimal ET sequence|legend=1| 8d, 24, 32c }}


[[Badness]]: 0.107523
[[Badness]] (Smith): 0.107523


=== 11-limit ===
=== 11-limit ===
Line 202: Line 202:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 1\8, ~3/2 = 702.662 (~36/35 = 47.338)
* CTE: ~12/11 = 150.000, ~3/2 = 702.662 (~36/35 = 47.338)
* POTE: ~12/11 = 1\8, ~3/2 = 706.645 (~36/35 = 43.355)
* POTE: ~12/11 = 150.000, ~3/2 = 706.645 (~36/35 = 43.355)


Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32c }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32c }}


Badness: 0.047598
Badness (Smith): 0.047598


=== 13-limit ===
=== 13-limit ===
Line 217: Line 217:


Optimal tunings:  
Optimal tunings:  
* CTE: ~12/11 = 1\8, ~3/2 = 701.952 (~36/35 = 48.048)
* CTE: ~12/11 = 150.000, ~3/2 = 701.952 (~36/35 = 48.048)
* POTE: ~12/11 = 1\8, ~3/2 = 707.376 (~36/35 = 42.624)
* POTE: ~12/11 = 150.000, ~3/2 = 707.376 (~36/35 = 42.624)


Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32cf }}
Optimal ET sequence: {{Optimal ET sequence| 8d, 24, 32cf }}


Badness: 0.030597
Badness (Smith): 0.030597


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Diminished_family"