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Catalog of rank-4 temperaments: Difference between revisions

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A '''rank-4 temperament''' has a [[period]] and three additional independent [[generator]]s, for a total of four [[rank and codimension|dimensions]]. Typical examples include [[7-limit]] [[JI]], full 11-limit temperament with a one-dimensional [[comma basis]], and full 13-limit temperament with a two-dimensional comma basis.  
A '''rank-4 temperament''' has a [[period]] and three additional independent [[generator]]s, for a total of four [[rank and codimension|dimensions]]. Typical examples include [[7-limit]] [[JI]], full 11-limit temperament with a one-dimensional [[comma basis]], and full 13-limit temperament with a two-dimensional comma basis.  


Temperaments in this catalog are ordered by [[Tenney height]]; temperaments whose commas have bigger denominators are listed first.  
Temperaments in this catalog are ordered by increasing [[Tenney norm]] of the first comma.  


== Cake (45/44) ==
== Cake (45/44) ==
Line 1,208: Line 1,208:


== Microorganismic (67110351/67108864) ==
== Microorganismic (67110351/67108864) ==
It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]]. It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.
Microorganismic splits the [[septischisma]] in three parts and the [[pythrabian comma]] in five parts, each of these parts being an interval representing an [[olympia]] as well as an [[argyria]]. Notable equal temperaments not appearing in the [[optimal ET sequence]] include {{EDOs| 8269, 8539, and 16808 }}.
 
It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[67110351/67108864]]
[[Comma list]]: 67110351/67108864
 
{{Mapping|legend=1| 1 0 0 1 7 | 0 1 0 1 -2 | 0 0 1 0 0 | 0 0 0 3 -5 }}
: mapping generators: ~2, ~3, ~5, ~539/512


[[Mapping]]: {{Mapping|1 0 0 7 7|0 1 0 1 -2|0 0 1 0 0|0 0 0 3 -5}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.001014{{C}}, ~3/2 = 701.953889{{C}}, ~5/4 = 386.311685{{C}}, ~539/512 = 88.956151{{C}}
: [[error map]]: {{val| +0.0010 -0.0001 -0.0000 -0.0015 -0.0014 }}
* [[CWE]]: ~2 = 1200.000000{{C}}, ~3/2 = 701.953627{{C}}, ~5/4 = 386.311920{{C}}, ~539/512 = 88.955922{{C}}
: error map: {{val| 0.0000 -0.0014 -0.0018 -0.0045 -0.0048 }}


[[Optimal tuning|Optimal tunings]]:
{{Optimal ET sequence|legend=0| 41, 53, 68, 94, 161, 176, 202, 217, 255, 270, 566, 742, 836, 1106, 1578, 1848, 2684, 3737, 4007, 4843, 5585, 6421, 6691, 12276, 14960, 27236, 33927, 42196, 48887, 69432, 76123, 192594de, 194442de, … }}


* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 =  2786.311685{{C}}, ~539/512 = 88.956151{{C}}
[[Badness]] (Sintel): 0.387


: [[error map]]: ⟨+0.001014 −0.000098 0.000000 −0.001537 −0.001401]
=== Koch ===
[[Subgroup]]: 2.3.5.7.11.13


* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 =  1901.953627{{C}}, ~5 =  2786.311919{{C}}, ~539/512 = 88.955923{{C}}
[[Comma list]]: 1990656/1990625, 67110351/67108864


: error map: ⟨0.000000 −0.001374 −0.001795 −0.004512 −0.004808]
{{Mapping|legend=1| 1 0 0 7 7 11 | 0 1 0 1 -2 3 | 0 0 1 0 0 -5 | 0 0 0 3 -5 -6 }}


[[Optimal ET sequence]]: {{Edos|11c, 12e, 15, 26, 38, 41, 53, 68, 94, 161, 176, 202, 217, 255, 270, 566, 742, 836, 1106, 1578, 1848, 2684, 3737, 4007, 4843, 5585, 6421, 6691, 12276, 14960…}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.000612{{c}}, ~3/2 = 701.951925{{c}}, ~5/4 = 386.317179{{c}}, ~539/512 = 88.957048{{c}}
: [[error map]]: {{val| +0.0006 -0.0025 +0.0047 −0.0016 -0.0040 +0.0024 }}
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.951815{{c}}, ~5/4 = 386.317182{{c}}, ~539/512 = 88.956884{{c}}
: error map: {{val| 0.0000 -0.0032 +0.0035 -0.0034 -0.0060 +0.0006 }}


[[Badness]] (Sintel): 0.387
{{Optimal ET sequence|legend=0| 53, 121e, 149e, 202, 217, 270, 783f, 836, 1038, 1106, 1308, 1578, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808, 30932, 39201, 47740, 70133, 86941 }}


Notable temperaments not appearing in the optimal ET sequence are {{Edos|8269, 8539, 16808}}.
[[Badness]] (Sintel): 0.912


=== Leeuwenhoek ===
=== Leeuwenhoek ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: [[67110351/67108864]], [[123201/123200]]
[[Comma list]]: 123201/123200, 67110351/67108864


[[Mapping]]: {{Mapping|1 0 0 1 7 7|0 2 0 2 −4 −7|0 0 1 0 0 1|0 0 0 3 −5 −1}}
{{Mapping|legend=1| 1 0 0 1 7 7 | 0 2 0 2 -4 -7 | 0 0 1 0 0 1 | 0 0 0 3 -5 -1 }}
: mapping generators: ~2, ~207515/119808, ~5, ~539/512


[[Optimal tuning|Optimal tunings]]:
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.001182{{c}}, ~207515/119808 = 950.977093{{c}}, ~5/4 = 386.311871{{c}}, ~539/512 = 88.956523{{c}}
* [[WE]]: ~2 = 1200.001182{{C}}, ~207515/119808 = 950.977093{{C}}, ~5 = 2786.311871{{C}}, ~539/512 = 88.956523{{C}}
: [[error map]]: {{val| +0.0012 -0.0008 +0.0005 -0.0010 −0.0007 -0.0013 }}
 
* [[CWE]]: ~2 = 1200.000000{{c}}, ~207515/119808 = 950.976202{{c}}, ~5/4 = 386.312215{{c}}, ~539/512 = 88.956363{{c}}
: [[error map]]: {{Val|+0.001182 −0.000815 +0.000522 −0.000971 −0.000653 −0.001325}}
: error map: {{val| 0.0000 -0.0026 -0.0015 -0.0044 -0.0046 -0.0052 }}
 
* [[CWE]]: ~2 = 1200.000000{{C}}, ~207515/119808 = 950.976202{{C}}, ~5 = 2786.312215{{C}}, ~539/512 = 88.956363{{C}}
 
: error map: {{Val|0.000000 −0.002596 −0.001499 −0.004414 −0.004565 −0.005226}}


[[Optimal ET sequence]]: {{Edos|15f, 38, 53, 120c, 135f, 188cf, 188, 202f, 217, 270, 742, 1012, 1619, 1889, 2414, 2631, 2684, 4573, 4843, 5585, 8269, 8539, 13112, 16808…}}
{{Optimal ET sequence|legend=0| 53, 120c, 135f, 188(cf), 202f, 217, 270, 742, 1012, 1619, 1889, 2414, 2631, 2684, 4573, 4843, 5585, 8269, 8539, 13112, 16808, 18697, 21651, 26966, 27236, 35505, 57156, 62741, 92661, 155402def, … }}


[[Badness]] (Sintel): 0.349
[[Badness]] (Sintel): 0.349
Line 1,256: Line 1,267:
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: [[67110351/67108864]], [[5767168/5767125]]
[[Comma list]]: 5767168/5767125, 35153041/35152000


[[Mapping]]: {{Mapping|1 0 0 7 −3 3|0 1 0 4 −7 −4|0 0 1 0 0 −1|0 0 0 9 −15 −8}}
{{Mapping|legend=1| 1 0 0 7 -3 3 | 0 1 0 4 -7 -4 | 0 0 1 0 0 -1 | 0 0 0 9 -15 -8 }}
: mapping generators: ~2, ~3, ~5, ~455/256


[[Optimal tuning|Optimal tunings]]:
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.000392{{c}}, ~3/2 = 701.954644{{c}}, ~5/4 = 386.314673{{c}}, ~455/256 = 995.667469{{c}}
* [[WE]]: ~2 = 1200.000392{{C}}, ~3 = 1901.954644{{C}}, ~5 = 2786.314673{{C}}, ~512/455 = 204.332922{{C}}
: [[error map]]: {{val| +0.0004 +0.000035 +0.0017 −0.0028 −0.0047 +0.0044 }}
 
* [[CWE]]: ~2 = 1200.000000{{c}}, ~3/2 = 701.954490{{c}}, ~5/4 = 386.314598{{c}}, ~455/256 = 995.667102{{c}}
: [[error map]]: {{Val|+0.000392 +0.000035 +0.001743 −0.002847 −0.004657 +0.004426}}
: error map: {{val| 0.0000 -0.0005 +0.0009 −0.0040 −0.0059 +0.0030 }}
 
* [[CWE]]: ~2 = 1200.000000{{C}}, ~207515/119808 = 950.976202{{C}}, ~5 = 2786.312215{{C}}, ~539/512 = 88.956363{{C}}
 
: error map: {{Val|0.000000 −0.000511 +0.000884 −0.004026 −0.005906 +0.002962}}


[[Optimal ET sequence]]: {{Edos|12e, 37b, 41, 53, 94, 176, 217, 270, 581, 851, 1469, 1563, 1739, 1780, 1833, 2009, 2050, 2103, 2320, 2414, 2631, 2684, 5274, 5315, 5585, 8269, 8539, 14124, 16808…}}
{{Optimal ET sequence|legend=0| 41, 53, 94, 176, 217, 270, 581, 851, 1469, 1563, 1739, 1780, 1833, 2009, 2050, 2103, 2320, 2414, 2631, 2684, 5274, 5315, 5585, 8269, 8539, 14124, 16808, 25658, 33927, 34197, 42466, 93471e, 99056 }}


[[Badness]] (Sintel): 0.924
[[Badness]] (Sintel): 0.924
=== Koch ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: [[67110351/67108864]], [[1990656/1990625]]
[[Mapping]]: {{Mapping|1 0 0 7 7 11|0 1 0 1 -2 3|0 0 1 0 0 -5|0 0 0 3 -5 -6}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 =  2786.311685{{C}}, ~539/512 = 88.956151{{C}}
: [[error map]]: {{Val|+0.000612 −0.002463 +0.004690 −0.001614 −0.003969 +0.002382}}
* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 =  1901.953627{{C}}, ~5 =  2786.311919{{C}}, ~539/512 = 88.955923{{C}}
: error map: {{Val|0.000000 −0.003186 +0.003468 −0.003438 −0.005995 +0.000566}}
[[Optimal ET sequence]]: {{Edos|11c, 15, 38f, 53, 121e, 149e, 202, 217, 270, 783f, 836, 1038, 1106, 1308, 1578, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808…}}
[[Badness]] (Sintel): 0.912


== Quartismic (117440512/117406179) ==
== Quartismic (117440512/117406179) ==
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