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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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== Brahmagupta ==
== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]], {{monzo| 47 -7 -7 -7 }} = 140737488355328 / 140710042265625.  
{{todo | expand | inline=1 | comment=Explain the relationship to Sagittal's Athenian system.}}
 
Early in the design of the [[Sagittal]] notation system, Secor and Keenan found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament that has pure octaves and pure fifths, which can also be described as a 17-limit extension having 1/7th octave period (171.4286 ¢) and 1/21st apotome generator (5.4136 ¢).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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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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