Octagar temperaments: Difference between revisions

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Review data for slendi. + intro
Switch to Sintel's badness, WE & CWE tunings
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[[Comma list]]: 4000/3969, 420175/419904
[[Comma list]]: 4000/3969, 420175/419904


{{Mapping|legend=1| 1 3 5 4 | 0 -19 -36 -16 }}
{{Mapping|legend=1| 1 -16 -31 -12 | 0 19 36 16 }}
: mapping generators: ~2, ~40/21


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/20 = 89.282{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5464{{c}}, ~40/21 = 1110.2981{{c}}
: [[error map]]: {{val| -0.454 +0.996 -1.521 +1.387 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7026{{c}}
: error map: {{val| 0.000 +1.394 -1.021 +2.415 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~40/21 = 1110.718{{c}} -->


{{Optimal ET sequence|legend=1| 27, 94, 121, 215, 336d }}
{{Optimal ET sequence|legend=1| 27, 67c, 94, 121, 215, 336d }}


[[Badness]] (Smith): 0.107482
[[Badness]] (Sintel): 2.72


=== 11-limit ===
=== 11-limit ===
Line 89: Line 95:
Comma list: 540/539, 2200/2187, 4375/4356
Comma list: 540/539, 2200/2187, 4375/4356


Mapping: {{mapping| 1 3 5 4 8 | 0 -19 -36 -16 -61 }}
Mapping: {{mapping| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.291{{c}}
Optimal tunings:
* WE: ~2 = 1199.5884{{c}}, ~40/21 = 1110.3276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.6969{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.709{{c}} -->


{{Optimal ET sequence|legend=0| 94, 121, 215, 336de, 551bde }}
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336de }}


Badness (Smith): 0.045895
Badness (Sintel): 1.52


=== 13-limit ===
=== 13-limit ===
Line 102: Line 111:
Comma list: 325/324, 352/351, 540/539, 1575/1573
Comma list: 325/324, 352/351, 540/539, 1575/1573


Mapping: {{mapping| 1 3 5 4 8 4 | 0 -19 -36 -16 -61 -4 }}
Mapping: {{mapping| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.286{{c}}
Optimal tunings:
* WE: ~2 = 1199.5032{{c}}, ~40/21 = 1110.2545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7037{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.7140{{c}} -->


{{Optimal ET sequence|legend=0| 94, 121, 215, 336def }}
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336def }}


Badness (Smith): 0.028492
Badness (Sintel): 1.18


=== 17-limit ===
=== 17-limit ===
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Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
Comma list: 325/324, 352/351, 375/374, 540/539, 595/594


Mapping: {{mapping| 1 3 5 4 8 4 9 | 0 -19 -36 -16 -61 -4 -66 }}
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/20 = 89.286{{c}}
Optimal tunings:
* WE: ~2 = 1199.5051{{c}}, ~40/21 = 1110.2555{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7033{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~40/21 = 1110.714{{c}} -->


{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
{{Optimal ET sequence|legend=0| 27eg, 67ceg, 94, 121, 215, 336defg }}


Badness (Smith): 0.019811
Badness (Sintel): 1.01


=== 19-limit ===
=== 19-limit ===
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Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594


Mapping: {{mapping| 1 3 5 4 8 4 9 7 | 0 -19 -36 -16 -61 -4 -66 -37 }}
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~19/18 = 89.288{{c}}
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~19/10 = 1110.3583{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.7036{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~19/10 = 1110.712{{c}} -->


{{Optimal ET sequence|legend=0| 94, 121, 215, 336defg }}
{{Optimal ET sequence|legend=0| 27eg, 67cegh, 94, 121, 215 }}


Badness (Smith): 0.016408
Badness (Sintel): 0.998


=== 23-limit ===
=== 23-limit ===
Line 141: Line 159:
Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483


Mapping: {{mapping| 1 3 5 4 8 4 9 7 11 | 0 -19 -36 -16 -61 -4 -66 -37 -87 }}
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~19/18 = 89.297{{c}}
Optimal tunings:
* WE: ~2 = 1199.6651{{c}}, ~19/10 = 1110.3928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.6965{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~19/10 = 1110.703{{c}} -->


{{Optimal ET sequence|legend=0| 94, 121i, 215 }}
{{Optimal ET sequence|legend=0| 27egi, 94, 121i, 215 }}


Badness (Smith): 0.014946
Badness (Sintel): 1.07


== Tridecatonic ==
== Tridecatonic ==
Line 158: Line 179:
[[Comma list]]: 4000/3969, 8748/8575
[[Comma list]]: 4000/3969, 8748/8575


{{Mapping|legend=1| 13 21 31 37 | 0 -1 -2 -1 }}
{{Mapping|legend=1| 13 0 -11 16 | 0 1 2 1 }}
: mapping generators: ~21/20, ~3


[[Optimal tuning]] ([[POTE]]): ~21/20 = 92.3077{{c}}, ~3/2 = 701.9605{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 92.2197{{c}}, ~3/2 = 701.2915{{c}}
: [[error map]]: {{val| -1.144 -1.807 -0.435 +6.837 }}
* [[CWE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2352{{c}}
: error map: {{val| 0.000 -0.720 +0.772 +9.332 }}
<!-- * [[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]] (Smith): 0.139213
[[Badness]] (Sintel): 3.52


=== 11-limit ===
=== 11-limit ===
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Comma list: 99/98, 1944/1925, 2560/2541
Comma list: 99/98, 1944/1925, 2560/2541


Mapping: {{mapping| 13 21 31 37 45 | 0 -1 -2 -1 0 }}
Mapping: {{mapping| 13 0 -11 16 45 | 0 1 2 1 0 }}


Optimal tuning (POTE): ~21/20 = 92.3077{{c}}, ~3/2 = 701.7028{{c}}
Optimal tunings:
* WE: ~21/20 = 92.2333{{c}}, ~3/2 = 701.1371{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2751{{c}}
<!-- * 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 (Smith): 0.063257
Badness (Sintel): 2.09


=== 13-limit ===
=== 13-limit ===
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Comma list: 99/98, 144/143, 275/273, 1040/1029
Comma list: 99/98, 144/143, 275/273, 1040/1029


Mapping: {{mapping| 13 21 31 37 45 49 | 0 -1 -2 -1 0 -2 }}
Mapping: {{mapping| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}


Optimal tuning (POTE): ~21/20 = 92.3077{{c}}, ~3/2 = 701.3573{{c}}
Optimal tunings:
* WE: ~21/20 = 92.2248{{c}}, ~3/2 = 700.7272{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 700.8621{{c}}
<!-- * 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 (Smith): 0.037851
Badness (Sintel): 1.56


== 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 }}
: mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 319.198{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9902{{c}}, ~6/5 = 319.1955{{c}}
: [[error map]]: {{val| -0.010 +1.190 -2.374 +0.769 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 319.1980{{c}}
: error map: {{val| 0.000 +1.202 -2.354 +0.798 }}
<!-- * [[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, 297c }}


[[Badness]] (Smith): 0.168373
[[Badness]] (Sintel): 4.26


=== 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{{c}}, ~6/5 = 319.197{{c}}
Optimal tunings:
* WE: ~2 = 1200.0983{{c}}, ~6/5 = 319.2229{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1985{{c}}
<!-- * 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 (Smith): 0.051216
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 225: Line 267:
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{{c}}, ~6/5 = 319.165{{c}}
Optimal tunings:
* WE: ~2 = 1200.0732{{c}}, ~6/5 = 319.1840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1658{{c}}
<!-- * 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, 79, 94 }}
 
Badness (Sintel): 1.55


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

Revision as of 12:39, 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

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

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

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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