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Octagar temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the octagar comma, {{monzo| 5 -4 3 -2 }} = [[4000/3969]].
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).


Temperaments considered below are slendi and slithy. Discussed elsewhere are:  
Temperaments considered below are slendi and slithy. Discussed elsewhere are:  
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{{Multival|legend=1| 17 28 8 5 -35 -60 }}
{{Multival|legend=1| 17 28 8 5 -35 -60 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~49/48 = 29.181
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 29.181


{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
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Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~49/48 = 29.180
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.180


{{Optimal ET sequence|legend=1| 41, 206e }}
{{Optimal ET sequence|legend=0| 41, 206e }}


Badness: 0.063217
Badness: 0.063217
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Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -1 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -1 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~49/48 = 29.125
Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.125


{{Optimal ET sequence|legend=1| 41, 124ef, 165ef }}
{{Optimal ET sequence|legend=0| 41, 124ef, 165ef }}


Badness: 0.043644
Badness: 0.043644
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{{Multival|legend=1| 19 36 16 13 -28 -64 }}
{{Multival|legend=1| 19 36 16 13 -28 -64 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~21/20 = 89.282
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 89.282


{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
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Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}
Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~21/20 = 89.291
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.291


{{Optimal ET sequence|legend=1| 94, 121, 215, 336de, 551bde }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336de, 551bde }}


Badness: 0.045895
Badness: 0.045895
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Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}
Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~21/20 = 89.286
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286


{{Optimal ET sequence|legend=1| 94, 121, 215, 336def }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336def }}


Badness: 0.028492
Badness: 0.028492
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Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}
Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~21/20 = 89.286
Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286


{{Optimal ET sequence|legend=1| 94, 121, 215, 336defg }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}


Badness: 0.019811
Badness: 0.019811
Line 120: Line 120:
Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}
Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~19/18 = 89.288
Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288


{{Optimal ET sequence|legend=1| 94, 121, 215, 336defg }}
{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}


Badness: 0.016408
Badness: 0.016408
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Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}
Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~19/18 = 89.297
Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297


{{Optimal ET sequence|legend=1| 94, 121i, 215 }}
{{Optimal ET sequence|legend=0| 94, 121i, 215 }}


Badness: 0.014946
Badness: 0.014946
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{{See also| 13th-octave temperaments #Tridecatonic }}
{{See also| 13th-octave temperaments #Tridecatonic }}


The tridecatonic temperament (26 &amp; 39) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 &amp; 39d, tempering out 4000/3969), [[Ragismic microtemperaments #Trideci|trideci]] (26 &amp; 39, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] (26 &amp; 130, tempering out 321489/320000).
The tridecatonic temperament ({{nowrap| 26 & 39 }}) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic ({{nowrap| 26 & 39d }}, tempering out 4000/3969), [[ragismic microtemperaments #Trideci|trideci]] ({{nowrap| 26 & 39 }}, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] ({{nowrap| 26 & 130 }}, tempering out 321489/320000).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}
{{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}


[[Optimal tuning]] ([[POTE]]): ~21/20 = 1<span style="font-family:'Arial', sans-serif">\</span>13, ~3/2 = 701.9605
[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077, ~3/2 = 701.9605


{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
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Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}
Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}


Optimal tuning (POTE): ~21/20 = 1<span style="font-family:'Arial', sans-serif">\</span>13, ~3/2 = 701.7028
Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.7028


{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
{{Optimal ET sequence|legend=0| 26, 39d, 65d }}


Badness: 0.063257
Badness: 0.063257
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Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}
Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}


Optimal tuning (POTE): ~21/20 = 1<span style="font-family:'Arial', sans-serif">\</span>13, ~3/2 = 701.3573
Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573


{{Optimal ET sequence|legend=1| 26, 39df, 65d }}
{{Optimal ET sequence|legend=0| 26, 39df, 65d }}


Badness: 0.037851
Badness: 0.037851
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{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}
{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~6/5 = 319.198
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~6/5 = 319.198


{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
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Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}
Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~6/5 = 319.197
Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197


Optimal ET sequence: {{Optimal ET sequence| 15, 64c, 79, 94, 203, 297c }}
{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}


Badness: 0.051216
Badness: 0.051216
Line 215: Line 215:
Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}
Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}


Optimal tuning (POTE): ~2 = 1<span style="font-family:'Arial', sans-serif">\</span>1, ~6/5 = 319.165
Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165


Optimal ET sequence: {{Optimal ET sequence| 15, 49bceff, 64cf, 79, 94 }}
{{Optimal ET sequence|legend=0| 15, 49bceff, 64cf, 79, 94 }}


Badness: 0.037468
Badness: 0.037468
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