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Ragismic microtemperaments: Difference between revisions

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{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  


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== Supermajor ==
== Supermajor ==
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7, leading to a wedgie of {{multival| 37 46 75 -13 15 45 }}. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.
The generator for supermajor temperament is a supermajor third, 9/7, tuned about 0.002 cents flat. 37 of these give (2<sup>15</sup>)/3, 46 give (2<sup>19</sup>)/5, and 75 give (2<sup>30</sup>)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}
{{Multival|legend=1| 37 46 75 -13 15 45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 435.082
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{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
{{Multival|legend=1| 19 19 57 -14 37 79 }}


: mapping generators: ~28/27, ~3
: mapping generators: ~28/27, ~3
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{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
{{Mapping|legend=1| 1 36 48 61 | 0 -55 -73 -93 }}
{{Multival|legend=1| 55 73 93 -12 -7 11 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 449.1270
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: mapping generators: ~1157625/1048576, ~27/20
: mapping generators: ~1157625/1048576, ~27/20
{{Multival|legend=1| 21 56 -77 40 -181 -336 }}


[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 1\7, ~27/20 = 519.716
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: mapping generators: ~46305/32768, ~27/20
: mapping generators: ~46305/32768, ~27/20
{{Multival|legend=1| 22 48 -38 25 -122 -223 }}


[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 1\2, ~6912/6125 = 208.899
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: mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7
{{Multival|legend=1| 23 40 1 10 -63 -110 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~8/7 = 230.336
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: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an obvious tuning.  
Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament. [[1106edo]] is an strong tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~7411887/5242880, ~1310720/1058841
: mapping generators: ~7411887/5242880, ~1310720/1058841
{{Multival|legend=1| 58 102 -2 27 -166 -291 }}


[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 1\2, ~8/7 = 231.104
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{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}
{{Multival|legend=1| 35 62 -3 17 -103 -181 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3087/2560 = 322.804
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{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
{{Mapping|legend=1| 1 2 10 -25 | 0 -2 -37 134 }}
{{Multival|legend=1| 2 37 -134 54 -218 -415 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~{{monzo| -27 11 3 1 }} = 249.0207
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[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
[[Mapping]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}
{{Multival|legend=1| 31 41 53 -7 -3 8 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 448.456
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: mapping generators: ~2, ~6/5
: mapping generators: ~2, ~6/5
{{Multival|legend=1| 32 33 92 -22 56 121 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.557
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{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
{{Multival|legend=1| 20 64 -116 55 -240 -449 }}


[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388  
[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 1\4, ~5103/4096 = 380.388  
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: mapping generators: ~15/14, ~6/5
: mapping generators: ~15/14, ~6/5
{{Multival|legend=1| 50 60 110 -21 34 87 }}


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~6/5 = 315.577
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[[Badness]]: 0.080637
[[Badness]]: 0.080637


Badness (Dirichlet): 2.041
Badness (Sintel): 2.041


=== 11-limit ===
=== 11-limit ===
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Badness: 0.024329
Badness: 0.024329


Badness (Dirichlet): 0.804
Badness (Sintel): 0.804


=== 13-limit ===
=== 13-limit ===
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Badness: 0.016810
Badness: 0.016810


Badness (Dirichlet): 0.695
Badness (Sintel): 0.695


=== no-17's 19-limit ===
=== no-17's 19-limit ===
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{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}
{{Optimal ET sequence|legend=1| 80, 190, 270, 730, 1000 }}


Badness (Dirichlet): 0.556
Badness (Sintel): 0.556


== Keenanose ==
== Keenanose ==
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: mapping generators: ~2278125/1605632, ~448/405
: mapping generators: ~2278125/1605632, ~448/405
{{Multival|legend=1| 26 16 118 -35 114 229 }}


[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
[[Optimal tuning]] ([[POTE]]): ~2278125/1605632 = 1\2, ~448/405 = 176.805
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: mapping generators: ~83349/81920, ~3
: mapping generators: ~83349/81920, ~3
{{Multival|legend=1| 46 92 -46 39 -202 -365 }}


[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
[[Optimal tuning]] ([[POTE]]): ~83349/81920 = 1\46, ~3/2 = 701.6074
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{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
{{Multival|legend=1| 24 32 40 -5 -4 3 }}


: mapping generators: ~49/45, ~7/5
: mapping generators: ~49/45, ~7/5
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{{Main| Parakleismic }}
{{Main| Parakleismic }}


In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension, with the 7-limit wedgie being {{multival| 13 14 35 -8 19 42 }} and adding 3136/3125 and 4375/4374, and the 11-limit wedgie {{multival| 13 14 35 -36 -8 19 -102 42 -132 -222 }} adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.
In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. However while 118 no longer has better than a cent of accuracy in the 7- or 11-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, and 11-limit adding 385/384. For the 7-limit [[99edo]] may be preferred, but in the 11-limit it is best to stick with 118.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}
{{Mapping|legend=1| 1 5 6 12 | 0 -13 -14 -35 }}


{{Multival|legend=1| 13 14 35 -8 19 42 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 315.181
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: mapping generators: ~2, ~5/3
: mapping generators: ~2, ~5/3
{{Multival|legend=1| 25 24 79 -20 55 116 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 316.060
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{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
{{Multival|legend=1| 30 49 14 8 -62 -105 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1728/1715 = 16.613
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{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
{{Multival|legend=1| 19 31 9 5 -39 -66 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 26.287
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic microtemperaments| ]] <!-- main article -->
[[Category:Ragismic| ]] <!-- key article -->
[[Category:Ragismic| ]] <!-- key article -->
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