Octagar temperaments: Difference between revisions

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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).
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 discussed elsewhere are:  
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
Line 19: Line 19:
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]


== Slendi ==
Considered below are slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].  
Slendi may be described as the {{nowrap| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.
 
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4000/3969, 84035/82944
 
{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
: mapping generators: ~2, ~49/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
: [[error map]]: {{val| +0.252 +2.368 -2.796 -1.568 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
: error map: {{val| 0.000 +2.129 -3.116 -2.198 }}
 
{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}
 
[[Badness]] (Sintel): 3.62
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 245/242, 46656/46585
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 40c, 41 }}
 
Badness (Sintel): 2.09
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 144/143, 245/242, 343/338
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
 
Optimal tunings:
* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 41 }}
 
Badness (Sintel): 1.80


== Slithy ==
== Slithy ==
Line 215: Line 165:


Badness (Sintel): 1.56
Badness (Sintel): 1.56
== Slendi ==
Slendi may be described as the {{nowrap| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4000/3969, 84035/82944
{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
: mapping generators: ~2, ~49/48
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
: [[error map]]: {{val| +0.252 +2.368 -2.796 -1.568 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
: error map: {{val| 0.000 +2.129 -3.116 -2.198 }}
{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}
[[Badness]] (Sintel): 3.62
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/242, 46656/46585
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
Optimal tunings:
* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}
{{Optimal ET sequence|legend=0| 1ce, 40c, 41 }}
Badness (Sintel): 2.09
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 245/242, 343/338
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
Optimal tunings:
* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}
{{Optimal ET sequence|legend=0| 1ce, 41 }}
Badness (Sintel): 1.80


== Dhaivatic ==
== Dhaivatic ==

Revision as of 12:46, 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 discussed elsewhere are:

Considered below are slithy, tridecatonic, slendi and dhaivatic, in the order of increasing badness.

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 -16 -31 -12], ⟨0 19 36 16]]

mapping generators: ~2, ~40/21

Optimal tunings:

  • WE: ~2 = 1199.5464 ¢, ~40/21 = 1110.2981 ¢
error map: ⟨-0.454 +0.996 -1.521 +1.387]
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7026 ¢
error map: ⟨0.000 +1.394 -1.021 +2.415]

Optimal ET sequence: 27, 67c, 94, 121, 215, 336d

Badness (Sintel): 2.72

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -16 -31 -12 -53], ⟨0 19 36 16 61]]

Optimal tunings:

  • WE: ~2 = 1199.5884 ¢, ~40/21 = 1110.3276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.6969 ¢

Optimal ET sequence: 27e, 67ce, 94, 121, 215, 336de

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -16 -31 -12 -53 0], ⟨0 19 36 16 61 4]]

Optimal tunings:

  • WE: ~2 = 1199.5032 ¢, ~40/21 = 1110.2545 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7037 ¢

Optimal ET sequence: 27e, 67ce, 94, 121, 215, 336def

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -16 -31 -12 -53 0 -57], ⟨0 19 36 16 61 4 66]]

Optimal tunings:

  • WE: ~2 = 1199.5051 ¢, ~40/21 = 1110.2555 ¢
  • CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7033 ¢

Optimal ET sequence: 27eg, 67ceg, 94, 121, 215, 336defg

Badness (Sintel): 1.01

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 -16 -31 -12 -53 0 -57 -30], ⟨0 19 36 16 61 4 66 37]]

Optimal tunings:

  • WE: ~2 = 1199.6184 ¢, ~19/10 = 1110.3583 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/10 = 1110.7036 ¢

Optimal ET sequence: 27eg, 67cegh, 94, 121, 215

Badness (Sintel): 0.998

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 -16 -31 -12 -53 0 -57 -30 -76], ⟨0 19 36 16 61 4 66 37 87]]

Optimal tunings:

  • WE: ~2 = 1199.6651 ¢, ~19/10 = 1110.3928 ¢
  • CWE: ~2 = 1200.0000 ¢, ~19/10 = 1110.6965 ¢

Optimal ET sequence: 27egi, 94, 121i, 215

Badness (Sintel): 1.07

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 0 -11 16], ⟨0 1 2 1]]

mapping generators: ~21/20, ~3

Optimal tunings:

  • WE: ~21/20 = 92.2197 ¢, ~3/2 = 701.2915 ¢
error map: ⟨-1.144 -1.807 -0.435 +6.837]
  • CWE: ~21/20 = 92.3077 ¢, ~3/2 = 701.2352 ¢
error map: ⟨0.000 -0.720 +0.772 +9.332]

Optimal ET sequence: 26, 39d, 65d

Badness (Sintel): 3.52

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨13 0 -11 16 45], ⟨0 1 2 1 0]]

Optimal tunings:

  • WE: ~21/20 = 92.2333 ¢, ~3/2 = 701.1371 ¢
  • CWE: ~21/20 = 92.3077 ¢, ~3/2 = 701.2751 ¢

Optimal ET sequence: 26, 39d, 65d

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨13 0 -11 16 45 7], ⟨0 1 2 1 0 2]]

Optimal tunings:

  • WE: ~21/20 = 92.2248 ¢, ~3/2 = 700.7272 ¢
  • CWE: ~21/20 = 92.3077 ¢, ~3/2 = 700.8621 ¢

Optimal ET sequence: 26, 39df, 65d

Badness (Sintel): 1.56

Slendi

Slendi may be described as the 41 & 42 temperament. It has a generator of 49/48, seventeen of which give 4/3. Its ploidacot is omega-17-cot.

The name was likely derived from slendro diesis, one of the names for the interval 49/48.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]

mapping generators: ~2, ~49/48

Optimal tunings:

  • WE: ~2 = 1200.2515 ¢, ~49/48 = 29.1870 ¢
error map: ⟨+0.252 +2.368 -2.796 -1.568]
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1715 ¢
error map: ⟨0.000 +2.129 -3.116 -2.198]

Optimal ET sequence: 1c, 40c, 41, 206

Badness (Sintel): 3.62

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 = 1199.8027 ¢, ~49/48 = 29.1751 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1875 ¢

Optimal ET sequence: 1ce, 40c, 41

Badness (Sintel): 2.09

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 = 1199.0915 ¢, ~49/48 = 29.1033 ¢
  • CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1576 ¢

Optimal ET sequence: 1ce, 41

Badness (Sintel): 1.80

Dhaivatic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 2100875/2097152

Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]

mapping generators: ~2, ~6/5

Optimal tunings:

  • WE: ~2 = 1199.9902 ¢, ~6/5 = 319.1955 ¢
error map: ⟨-0.010 +1.190 -2.374 +0.769]
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1980 ¢
error map: ⟨0.000 +1.202 -2.354 +0.798]

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

Badness (Sintel): 4.26

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 tunings:

  • WE: ~2 = 1200.0983 ¢, ~6/5 = 319.2229 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1985 ¢

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

Badness (Sintel): 1.69

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 tunings:

  • WE: ~2 = 1200.0732 ¢, ~6/5 = 319.1840 ¢
  • CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1658 ¢

Optimal ET sequence: 15, 79, 94

Badness (Sintel): 1.55

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