Octagar temperaments: Difference between revisions

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Temperaments considered below are slendi and slithy. Discussed elsewhere are:  
Temperaments considered below are slendi and slithy. Discussed elsewhere are:  
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Ripple]]'' (+36/35) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
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* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
* [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Shibboleth family]] and [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.252¢, ~49/48 = 29.187¢
* [[WE]]: ~2 = 1200.252{{c}}, ~49/48 = 29.187{{c}}
* [[CWE]]: ~2 = 1200.000¢, ~49/48 = 29.172¢
* [[CWE]]: ~2 = 1200.000{{c}}, ~49/48 = 29.172{{c}}


{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
{{Optimal ET sequence|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.803¢, ~49/48 = 29.175¢
* WE: ~2 = 1199.803{{c}}, ~49/48 = 29.175{{c}}
* CWE: ~2 = 1200.000¢, ~49/48 = 29.188¢
* CWE: ~2 = 1200.000{{c}}, ~49/48 = 29.188{{c}}


{{Optimal ET sequence|legend=0| 41, 206e }}
{{Optimal ET sequence|legend=0| 41, 206e }}
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Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.091¢, ~40/39 = 29.103¢
* WE: ~2 = 1199.091{{c}}, ~40/39 = 29.103{{c}}
* CWE: ~2 = 1200.000¢, ~40/39 = 29.158¢
* CWE: ~2 = 1200.000{{c}}, ~40/39 = 29.158{{c}}


{{Optimal ET sequence|legend=0| 41, 124ef, 165ef }}
{{Optimal ET sequence|legend=0| 41, 124ef, 165ef }}
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{{Mapping|legend=1| 1 3 5 4 | 0 -19 -36 -16 }}
{{Mapping|legend=1| 1 3 5 4 | 0 -19 -36 -16 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~21/20 = 89.282
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 89.282{{c}}


{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}


[[Badness]]: 0.107482
[[Badness]] (Smith): 0.107482


=== 11-limit ===
=== 11-limit ===
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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 = 1200.000, ~21/20 = 89.291
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.291{{c}}


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


Badness: 0.045895
Badness (Smith): 0.045895


=== 13-limit ===
=== 13-limit ===
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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 = 1200.000, ~21/20 = 89.286
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.286{{c}}


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


Badness: 0.028492
Badness (Smith): 0.028492


=== 17-limit ===
=== 17-limit ===
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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 = 1200.000, ~21/20 = 89.286
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.286{{c}}


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


Badness: 0.019811
Badness (Smith): 0.019811


=== 19-limit ===
=== 19-limit ===
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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 = 1200.000, ~19/18 = 89.288
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~19/18 = 89.288{{c}}


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


Badness: 0.016408
Badness (Smith): 0.016408


=== 23-limit ===
=== 23-limit ===
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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 = 1200.000, ~19/18 = 89.297
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~19/18 = 89.297{{c}}


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


Badness: 0.014946
Badness (Smith): 0.014946


== Tridecatonic ==
== Tridecatonic ==
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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 = 92.3077, ~3/2 = 701.9605
[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077{{c}}, ~3/2 = 701.9605{{c}}


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


[[Badness]]: 0.139213
[[Badness]] (Smith): 0.139213


=== 11-limit ===
=== 11-limit ===
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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 = 92.3077, ~3/2 = 701.7028
Optimal tuning (POTE): ~21/20 = 92.3077{{c}}, ~3/2 = 701.7028{{c}}


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


Badness: 0.063257
Badness (Smith): 0.063257


=== 13-limit ===
=== 13-limit ===
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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 = 92.3077, ~3/2 = 701.3573
Optimal tuning (POTE): ~21/20 = 92.3077{{c}}, ~3/2 = 701.3573{{c}}


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


Badness: 0.037851
Badness (Smith): 0.037851


== Dhaivatic ==
== Dhaivatic ==
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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 = 1200.000, ~6/5 = 319.198
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 319.198{{c}}


{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}
{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297 }}


[[Badness]]: 0.168373
[[Badness]] (Smith): 0.168373


=== 11-limit ===
=== 11-limit ===
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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 = 1200.000, ~6/5 = 319.197
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 319.197{{c}}


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


Badness: 0.051216
Badness (Smith): 0.051216


=== 13-limit ===
=== 13-limit ===
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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 = 1200.000, ~6/5 = 319.165
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 319.165{{c}}


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


Badness: 0.037468
Badness (Smith): 0.037468
   
   
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Octagar temperaments| ]] <!-- main article -->
[[Category:Octagar| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 10:41, 13 March 2026

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]]

Optimal tunings:

  • WE: ~2 = 1200.252 ¢, ~49/48 = 29.187 ¢
  • CWE: ~2 = 1200.000 ¢, ~49/48 = 29.172 ¢

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

Badness (Sintel): 3.616

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 tunings:

  • WE: ~2 = 1199.803 ¢, ~49/48 = 29.175 ¢
  • CWE: ~2 = 1200.000 ¢, ~49/48 = 29.188 ¢

Optimal ET sequence: 41, 206e

Badness (Sintel): 2.090

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 -12]]

Optimal tunings:

  • WE: ~2 = 1199.091 ¢, ~40/39 = 29.103 ¢
  • CWE: ~2 = 1200.000 ¢, ~40/39 = 29.158 ¢

Optimal ET sequence: 41, 124ef, 165ef

Badness (Sintel): 1.803

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 3 5 4], ⟨0 -19 -36 -16]]

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

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

Badness (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 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 (Smith): 0.037468

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