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:  
Line 27: Line 27:
{{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 }}
Line 40: Line 40:
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
Line 53: Line 53:
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
Line 68: Line 68:
{{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 }}
Line 81: Line 81:
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
Line 94: Line 94:
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
Line 107: Line 107:
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
Line 133: Line 133:
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
Line 142: Line 142:
{{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
Line 150: Line 150:
{{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 }}
Line 163: Line 163:
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
Line 176: Line 176:
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
Line 189: Line 189:
{{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 }}
Line 202: Line 202:
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

Revision as of 12:05, 26 April 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the octagar comma (monzo: [5 -4 3 -2⟩, ratio: 4000/3969).

Temperaments considered below are slendi and slithy. Discussed elsewhere are:

Slendi

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]

Wedgie: ⟨⟨ 17 28 8 5 -35 -60 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.181

Optimal ET sequence: 41, 206, 247bc, 288bcd, 329bcd

Badness: 0.142903

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 46656/46585

Mapping: [⟨1 2 3 3 4], ⟨0 -17 -28 -8 -22]]

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.180

Optimal ET sequence: 41, 206e

Badness: 0.063217

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 245/242, 343/338

Mapping: [⟨1 2 3 3 4 4], ⟨0 -17 -28 -8 -22 -1]]

Optimal tuning (POTE): ~2 = 1200.000, ~49/48 = 29.125

Optimal ET sequence: 41, 124ef, 165ef

Badness: 0.043644

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 3 5 4], ⟨0 -19 -36 -16]]

Wedgie: ⟨⟨ 19 36 16 13 -28 -64 ]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.282

Optimal ET sequence: 27, 94, 121, 215, 336d

Badness: 0.107482

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187, 4375/4356

Mapping: [⟨1 3 5 4 8], ⟨0 -19 -36 -16 -61]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.291

Optimal ET sequence: 94, 121, 215, 336de, 551bde

Badness: 0.045895

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539, 1575/1573

Mapping: [⟨1 3 5 4 8 4], ⟨0 -19 -36 -16 -61 -4]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286

Optimal ET sequence: 94, 121, 215, 336def

Badness: 0.028492

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 540/539, 595/594

Mapping: [⟨1 3 5 4 8 4 9], ⟨0 -19 -36 -16 -61 -4 -66]]

Optimal tuning (POTE): ~2 = 1200.000, ~21/20 = 89.286

Optimal ET sequence: 94, 121, 215, 336defg

Badness: 0.019811

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594

Mapping: [⟨1 3 5 4 8 4 9 7], ⟨0 -19 -36 -16 -61 -4 -66 -37]]

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.288

Optimal ET sequence: 94, 121, 215, 336defg

Badness: 0.016408

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483

Mapping: [⟨1 3 5 4 8 4 9 7 11], ⟨0 -19 -36 -16 -61 -4 -66 -37 -87]]

Optimal tuning (POTE): ~2 = 1200.000, ~19/18 = 89.297

Optimal ET sequence: 94, 121i, 215

Badness: 0.014946

Tridecatonic

See also: 13th-octave temperaments § Tridecatonic

The tridecatonic temperament (26 & 39) has a period of 1/13 octave. It tempers out the tridecalimma, [-11 26 -13⟩ in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26 & 39d, tempering out 4000/3969), trideci (26 & 39, tempering out 4375/4374), and bosonic (26 & 130, tempering out 321489/320000).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 8748/8575

Mapping: [⟨13 21 31 37], ⟨0 -1 -2 -1]]

Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.9605

Optimal ET sequence: 26, 39d, 65d

Badness: 0.139213

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 1944/1925, 2560/2541

Mapping: [⟨13 21 31 37 45], ⟨0 -1 -2 -1 0]]

Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.7028

Optimal ET sequence: 26, 39d, 65d

Badness: 0.063257

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 275/273, 1040/1029

Mapping: [⟨13 21 31 37 45 49], ⟨0 -1 -2 -1 0 -2]]

Optimal tuning (POTE): ~21/20 = 92.3077, ~3/2 = 701.3573

Optimal ET sequence: 26, 39df, 65d

Badness: 0.037851

Dhaivatic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 2100875/2097152

Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.198

Optimal ET sequence: 15, 64c, 79, 94, 203, 297

Badness: 0.168373

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 4000/3969

Mapping: [⟨1 -4 -3 6 0], ⟨0 21 20 -12 13]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.197

Optimal ET sequence: 15, 64c, 79, 94, 203, 297c

Badness: 0.051216

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 275/273, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -4 -3 6 0 -8], ⟨0 21 20 -12 13 44]]

Optimal tuning (POTE): ~2 = 1200.000, ~6/5 = 319.165

Optimal ET sequence: 15, 49bceff, 64cf, 79, 94

Badness: 0.037468

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