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Ragismic microtemperaments: Difference between revisions

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The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}).  
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}).  


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 a 1/7-octave period (171.4286 ¢) and 1/21-apotome generator (5.4136 ¢).
Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|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{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).


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


[[Comma list]]: 4375/4374, 70368744177664/70338939985125
[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}


{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
: mapping generators: ~1157625/1048576, ~27/20
: mapping generators: ~1157625/1048576, ~27/20


[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 171.429{{c}}, ~27/20 = 519.716{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}


{{Optimal ET sequence|legend=1| 217, 224, 441, 1106, 1547 }}
{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}


[[Badness]] (Sintel): 0.737
[[Badness]] (Sintel): 0.737
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Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}
Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}


Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.704{{c}}
Optimal tunings:
* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}


{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771ee }}
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}


Badness (Sintel): 1.725
Badness (Sintel): 1.73


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}
Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}


Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.706{{c}}
Optimal tunings:
* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}


{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771eef }}
{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}


Badness (Sintel): 0.956
Badness (Sintel): 0.956


== Abigail ==
== Abigail ==
: ''For the 5-limit versino, see [[Miscellaneous 5-limit temperaments #Abigail]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''


Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things."</ref>
Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 4375/4374, 2147483648/2144153025
[[Comma list]]: 4375/4374, 2147483648/2144153025


{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}
{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
: Mapping generators: ~46305/32768, ~27/20
: mapping generators: ~46305/32768, ~1536/1225


[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 600.000{{c}}, ~6912/6125 = 208.899{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}


{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798 }}
{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}


[[Badness]] (Sintel): 0.936
[[Badness]] (Sintel): 0.936
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Comma list: 3025/3024, 4375/4374, 131072/130977
Comma list: 3025/3024, 4375/4374, 131072/130977


Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}
Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~1155/1024 = 208.901{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}


{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}
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Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095
Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095


Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}
Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}


Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~44/39 = 208.903{{c}}
Optimal tunings:
* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}


{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}
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