Cataharry family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}


The '''cataharry family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[cataharry comma]], 19683/19600.
The '''cataharry family''' of [[rank-3 temperament]]s [[tempering out|tempers out]] the [[cataharry comma]] ({{monzo|legend=1| -4 9 -2 -2 }}, [[ratio]]: 19683/19600).


== Cataharry ==
== Cataharry ==
Cataharry splits the [[3/1|3rd harmonic]] into two semitwelfths. Bending the semitwelfth up by a [[syntonic comma]] results in ~[[7/4]], and down results in ~[[12/7]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[19683/19600]]
[[Comma list]]: 19683/19600


{{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}
{{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}
: mapping generators: ~2, ~140/81, ~5


: mapping generators: ~2, ~140/81, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1072{{c}}, ~140/81 = 950.6740{{c}}, ~5/4 = 386.3883{{c}}
: [[error map]]: {{val| +0.107 -0.607 +0.289 +0.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6211{{c}}, ~5/4 = 386.4833{{c}}
: error map: {{val| 0.000 -0.713 +0.170 +0.280 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 19: Line 26:
{{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}
{{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}


[[Badness]]: 0.301 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.33


[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
[[Projection pair]]s: <code>3 19600/6561 7 1033052339200000000/150094635296999121</code> to 2.5.81/7


== Albus ==
== Albus ==
Line 29: Line 36:


{{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}
{{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}
: mapping generators: ~2, ~140/81, ~5


: mapping generators: ~2, ~140/81, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0427{{c}}, ~140/81 = 950.6653{{c}}, ~5/4 = 386.3435{{c}}
: [[error map]]: {{val| +0.043 -0.624 +0.115 +0.647 +0.242 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.6425{{c}}, ~5/4 = 386.3889{{c}}
: error map: {{val| 0.000 -0.670 +0.075 +0.568 +0.139 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}
{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}


[[Badness]]: 0.621 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.746


=== Madagascar ===
=== Madagascar ===
Line 53: Line 65:
: 15/13 length = 0.3019, 26/25 length = 0.9477
: 15/13 length = 0.3019, 26/25 length = 0.9477
: Angle (15/13, 26/25) = 81.6476
: Angle (15/13, 26/25) = 81.6476
Optimal tunings:
* WE: ~2 = 1199.9446{{c}}, ~26/15 = 950.7332{{c}}, ~5/4 = 387.0575{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.7647{{c}}, ~5/4 = 387.0080{{c}}


Minimax tuning:  
Minimax tuning:  
Line 59: Line 75:
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5


{{Optimal ET sequence|legend=1| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}
{{Optimal ET sequence|legend=0| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}


Badness: 0.560 × 10<sup>-3</sup>
Badness (Sintel): 0.524


Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11


Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3
Projection pairs: <code>3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675</code> to 2.5.13/3


Scales: [[madagascar19]]
Scales: [[madagascar19]]
Line 75: Line 91:


{{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}
{{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2224{{c}}, ~140/81 = 950.6271{{c}}, ~5/4 = 386.3424{{c}}
: [[error map]]: {{val| +0.222 -0.701 +0.474 -0.414 +0.559 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 950.5069{{c}}, ~5/4 = 386.5450{{c}}
: error map: {{val| 0.000 -0.941 +0.231 -0.809 +0.198 }}


{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}
{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}


[[Badness]]: 0.988 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.19


=== 13-limit ===
=== 13-limit ===
Line 87: Line 109:
Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}
Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}


{{Optimal ET sequence|legend=1| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}
Optimal tunings:
* WE: ~2 = 1200.1732{{c}}, ~26/15 = 950.6934{{c}}, ~5/4 = 387.0339{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.5975{{c}}, ~5/4 = 387.1756{{c}}


Badness: 0.786 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}
 
Badness (Sintel): 0.735


== Hagrid ==
== Hagrid ==
Line 97: Line 123:


{{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}
{{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}
: mapping generators: ~99/70, ~140/81, ~5


: mapping generators: ~99/70, ~140/81, ~5
[[Optimal tuning]]s:  
* [[WE]]: ~99/70 = 600.0301{{c}}, ~140/81 = 950.5810{{c}}, ~5/4 = 386.1803{{c}}
: [[error map]]: {{val| +0.060 -0.793 -0.013 -0.019 +1.557 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~140/81 = 950.5516{{c}}, ~5/4 = 386.2351{{c}}
: error map: {{val| 0.000 -0.852 -0.079 -0.096 +1.440 }}


{{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}
{{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}


[[Badness]]: 1.116 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.34


=== 13-limit ===
=== 13-limit ===
Line 111: Line 142:
Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}
Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}


: mapping generators: ~99/70, ~26/15, ~5
Optimal tunings:  
* WE: ~55/39 = 599.9896{{c}}, ~26/15 = 950.6467{{c}}, ~5/4 = 387.0194{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6571{{c}}, ~5/4 = 387.0028{{c}}


{{Optimal ET sequence|legend=1| 24, 34d, 58, 72, 130 }}
{{Optimal ET sequence|legend=0| 24, 34d, 58, 72, 130 }}


Badness: 0.876 × 10<sup>-3</sup>
Badness (Sintel): 0.819


== Draco ==
== Draco ==
Line 123: Line 156:


{{Mapping|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 0 0 1 -1 1 }}
{{Mapping|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 0 0 1 -1 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3870{{c}}, ~140/81 = 950.8204{{c}}, ~5/4 = 389.4825{{c}}
: [[error map]]: {{val| -0.613 -0.314 +1.943 +1.527 -2.000 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.2120{{c}}, ~5/4 = 389.3756{{c}}
: error map: {{val| 0.000 +0.469 +3.062 +2.706 -1.035 }}


{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}


[[Badness]]: 1.934 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.32


=== 13-limit ===
=== 13-limit ===
Line 135: Line 174:
Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}
Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}


{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cdf }}
Optimal tunings:
* WE: ~2 = 1199.3551{{c}}, ~26/15 = 950.7982{{c}}, ~5/4 = 389.3905{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.2095{{c}}, ~5/4 = 389.2432{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 24, 34d, 53, 58, 111, 280cdf }}


Badness: 0.999 × 10<sup>-3</sup>
Badness (Sintel): 0.934


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Cataharry| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 10:33, 7 July 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.

The cataharry family of rank-3 temperaments tempers out the cataharry comma (monzo: [-4 9 -2 -2⟩, ratio: 19683/19600).

Cataharry

Cataharry splits the 3rd harmonic into two semitwelfths. Bending the semitwelfth up by a syntonic comma results in ~7/4, and down results in ~12/7.

Subgroup: 2.3.5.7

Comma list: 19683/19600

Mapping: [⟨1 0 0 -2], ⟨0 2 0 9], ⟨0 0 1 -1]]

mapping generators: ~2, ~140/81, ~5

Optimal tunings:

  • WE: ~2 = 1200.1072 ¢, ~140/81 = 950.6740 ¢, ~5/4 = 386.3883 ¢
error map: ⟨+0.107 -0.607 +0.289 +0.423]
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.6211 ¢, ~5/4 = 386.4833 ¢
error map: ⟨0.000 -0.713 +0.170 +0.280]

Minimax tuning:

[[1 0 0 0⟩, [4/9 0 2/9 2/9⟩, [0 0 1 0⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7

Optimal ET sequence: 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587

Badness (Sintel): 1.33

Projection pairs: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7

Albus

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000

Mapping: [⟨1 0 0 -2 6], ⟨0 2 0 9 -12], ⟨0 0 1 -1 3]]

mapping generators: ~2, ~140/81, ~5

Optimal tunings:

  • WE: ~2 = 1200.0427 ¢, ~140/81 = 950.6653 ¢, ~5/4 = 386.3435 ¢
error map: ⟨+0.043 -0.624 +0.115 +0.647 +0.242]
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.6425 ¢, ~5/4 = 386.3889 ¢
error map: ⟨0.000 -0.670 +0.075 +0.568 +0.139]

Minimax tuning:

[[1 0 0 0 0⟩, [8/3 0 4/3 -2/3 -2/3⟩, [5 0 7/2 -2 -3/2⟩, [5 0 5/2 -1 -3/2⟩, [5 0 5/2 -2 -1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640

Badness (Sintel): 0.746

Madagascar

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675

Mapping: [⟨1 0 0 -2 6 -1], ⟨0 2 0 9 -12 3], ⟨0 0 1 -1 3 1]]

Mapping to lattice: [⟨0 -2 -3 -6 3 -6], ⟨0 0 -1 1 -3 -1]]

Lattice basis:

15/13 length = 0.3019, 26/25 length = 0.9477
Angle (15/13, 26/25) = 81.6476

Optimal tunings:

  • WE: ~2 = 1199.9446 ¢, ~26/15 = 950.7332 ¢, ~5/4 = 387.0575 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.7647 ¢, ~5/4 = 387.0080 ¢

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [8/3 0 4/3 -2/3 -2/3 0⟩, [5 0 7/2 -2 -3/2 0⟩, [5 0 5/2 -1 -3/2 0⟩, [5 0 5/2 -2 -1/2 0⟩, [8 0 11/2 -3 -5/2 0⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313

Badness (Sintel): 0.524

Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11

Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3

Scales: madagascar19

Snape

Subgroup: 2.3.5.7.11

Comma list: 441/440, 19683/19600

Mapping: [⟨1 0 0 -2 -7], ⟨0 2 0 9 22], ⟨0 0 1 -1 -3]]

Optimal tunings:

  • WE: ~2 = 1200.2224 ¢, ~140/81 = 950.6271 ¢, ~5/4 = 386.3424 ¢
error map: ⟨+0.222 -0.701 +0.474 -0.414 +0.559]
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.5069 ¢, ~5/4 = 386.5450 ¢
error map: ⟨0.000 -0.941 +0.231 -0.809 +0.198]

Optimal ET sequence: 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 676/675

Mapping: [⟨1 0 0 -2 -7 -1], ⟨0 2 0 9 22 3], ⟨0 0 1 -1 -3 1]]

Optimal tunings:

  • WE: ~2 = 1200.1732 ¢, ~26/15 = 950.6934 ¢, ~5/4 = 387.0339 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.5975 ¢, ~5/4 = 387.1756 ¢

Optimal ET sequence: 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409

Badness (Sintel): 0.735

Hagrid

Subgroup: 2.3.5.7.11

Comma list: 243/242, 9801/9800

Mapping: [⟨2 0 0 -4 -1], ⟨0 2 0 9 5], ⟨0 0 1 -1 0]]

mapping generators: ~99/70, ~140/81, ~5

Optimal tunings:

  • WE: ~99/70 = 600.0301 ¢, ~140/81 = 950.5810 ¢, ~5/4 = 386.1803 ¢
error map: ⟨+0.060 -0.793 -0.013 -0.019 +1.557]
  • CWE: ~99/70 = 600.0000 ¢, ~140/81 = 950.5516 ¢, ~5/4 = 386.2351 ¢
error map: ⟨0.000 -0.852 -0.079 -0.096 +1.440]

Optimal ET sequence: 24, 34d, 48, 58, 72, 130, 202

Badness (Sintel): 1.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363

Mapping: [⟨2 0 0 -4 -1 -2], ⟨0 2 0 9 5 3], ⟨0 0 1 -1 0 1]]

Optimal tunings:

  • WE: ~55/39 = 599.9896 ¢, ~26/15 = 950.6467 ¢, ~5/4 = 387.0194 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~26/15 = 950.6571 ¢, ~5/4 = 387.0028 ¢

Optimal ET sequence: 24, 34d, 58, 72, 130

Badness (Sintel): 0.819

Draco

Subgroup: 2.3.5.7.11

Comma list: 176/175, 19683/19600

Mapping: [⟨1 0 0 -2 -6], ⟨0 2 0 9 9], ⟨0 0 1 -1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3870 ¢, ~140/81 = 950.8204 ¢, ~5/4 = 389.4825 ¢
error map: ⟨-0.613 -0.314 +1.943 +1.527 -2.000]
  • CWE: ~2 = 1200.0000 ¢, ~140/81 = 951.2120 ¢, ~5/4 = 389.3756 ¢
error map: ⟨0.000 +0.469 +3.062 +2.706 -1.035]

Optimal ET sequence: 19e, 24, 34d, 53, 58, 111, 280cd

Badness (Sintel): 2.32

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 676/675

Mapping: [⟨1 0 0 -2 -6 -1], ⟨0 2 0 9 9 3], ⟨0 0 1 -1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3551 ¢, ~26/15 = 950.7982 ¢, ~5/4 = 389.3905 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.2095 ¢, ~5/4 = 389.2432 ¢

Optimal ET sequence: 19e, 24, 34d, 53, 58, 111, 280cdf

Badness (Sintel): 0.934

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