Very low accuracy temperaments: Difference between revisions

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Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large “commas” such as 9/8, 32/27, or 15/14, equating wildly different interval sizes. All temperaments listed temper out an interval of at least size 15/14.
Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large “commas” such as 9/8, 32/27, or 15/14, equating wildly different interval sizes. All temperaments listed temper out an interval of at least size 15/14.


== Antitonic ==
== Antitonic ==
This temperament is characterized by [[9/8]] being tempering out, and has been termed a "troll temperament". It is named on account of 4/3 and 3/2 both being represented by the 600 cent half octave, which, in terms of diatonic function, serves as an Antitonic. Surprisingly, it has extensions all the way up to the 11-limit, as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?limit=11&ets=2_4p&tuning=po x31eq].  The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.  
This temperament is characterized by [[9/8]] being tempering out, and has been termed a "troll temperament". It is named on account of 4/3 and 3/2 both being represented by the 600 cent half octave, which, in terms of diatonic function, serves as an antitonic. Surprisingly, it has extensions all the way up to the 11-limit, as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?limit=11&ets=2_4p&tuning=po x31eq].  The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.  


Comma list: [[9/8]]
Subgroup: 2.3.5


POTE generator: ~5/4 = 313.6898 or ~6/5 = 286.3102
[[Comma]]: [[9/8]]


Mapping: [{{val| 2 3 0 }}, {{val| 0 0 1 }}]
[[Mapping]]: [{{val| 2 3 0 }}, {{val| 0 0 1 }}]


Mapping generators: ~3/2, ~5
Mapping generators: ~3/2, ~5
[[POTE generator]]: ~5/4 = 313.690 or ~6/5 = 286.310


{{Val list|legend=1| 2, 4 }}
{{Val list|legend=1| 2, 4 }}


Badness: 0.0216
[[Badness]]: 0.021647


=== 7-limit ===
=== 7-limit ===
Comma list: 9/8, 15/14
Subgroup: 2.3.5.7


POTE generator: ~5/4 = 301.9970 or ~6/5 = 298.0030
[[Comma list]]: 9/8, 15/14


Mapping: [{{val| 2 3 0 1 }}, {{val| 0 0 1 1 }}]
[[Mapping]]: [{{val| 2 3 0 1 }}, {{val| 0 0 1 1 }}]


Mapping generators: ~3/2, ~5
[[POTE generator]]: ~5/4 = 301.997 or ~6/5 = 298.003


{{Val list|legend=1| 2, 4 }}
{{Val list|legend=1| 2, 4 }}


Badness: 0.0194
[[Badness]]: 0.019373


==== 11-limit ====
==== 11-limit ====
Comma list: 9/8, 12/11, 21/20
Subgroup: 2.3.5.7.11


POTE generator: ~5/4 = 340.2928 or ~6/5 = 259.7072
Comma list: 9/8, 12/11, 15/14


Mapping: [{{val| 2 3 0 1 7 }}, {{val| 0 0 1 1 0 }}]
Mapping: [{{val| 2 3 0 1 7 }}, {{val| 0 0 1 1 0 }}]


Mapping generators: ~3/2, ~5
POTE generator: ~5/4 = 340.293 or ~6/5 = 259.707


{{Val list|legend=1| 2, 4 }}
Optimal GPV sequence: {{Val list| 2, 4 }}


Badness: 0.0218
Badness: 0.021810


=== Antaeus ===
=== Antaeus ===
Comma list: 9/8, 35/32
Subgroup: 2.3.5.7


POTE generator: ~5/4 = 315.962 or 6/5 = 284.038
[[Comma list]]: 9/8, 35/32


Mapping: [{{val| 2 3 0 10 }}, {{val| 0 0 1 -1 }}]
[[Mapping]]: [{{val| 2 3 0 10 }}, {{val| 0 0 1 -1 }}]


{{Multival|legend=1| 0 2 -2 3 -3 -10 }}
[[POTE generator]]: ~5/4 = 315.962 or ~6/5 = 284.038


{{Val list|legend=1| 2d, 4 }}
{{Val list|legend=1| 2d, 4 }}


Badness: 0.0375
[[Badness]]: 0.037537
 
==== 11-limit ====
Subgroup: 2.3.5.7.11


==== 11-limit  ====
Comma list: 9/8, 12/11, 35/32
Comma list: 9/8, 12/11, 35/32


POTE generator:
Mapping: [{{val| 2 3 0 10 7 }}, {{val| 0 0 1 -1 0 }}]
 
POTE generator: ~5/4 = 318.173 or ~6/5 = 281.827
 
Optimal GPV sequence: {{Val list| 2d, 4 }}
 
Badness: 0.033786
 
== Alteraugment ==
Like [[augmented]], but the generator provides 5/4 instead of 3/2.
 
Subgroup: 2.3.5


Mapping:
[[Comma]]: [[32/27]]


Mapping generators:
[[Mapping]]: [{{val| 3 5 7 }}, {{val| 0 0 1 }}]


Vals:
[[POTE generator]]: ~10/9 = 59.935


Badness:
{{Val list|legend=1| 3, 12bcc, 15bbcc, 18bbcc }}


== Alteraugment  ==
[[Badness]]: 0.078042


Like [[augmented]], but the generator provides 5/4 instead of 3/2.
== Yellow (2c&3) ==
This temperament family is characterized by [[10/9]] being tempering out.
 
Subgroup: 2.3.5
 
[[Comma list]]: [[10/9]]
 
[[Mapping]]: [{{val| 1 0 -1 }}, {{val| 0 1 2 }}]
 
[[POTE generator]]: ~3/2 = 775.970
 
{{Val list|legend=1| 3, 11bc, 14bc, 17bb }}
 
[[Badness]]: 0.016712
 
=== Brutus ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 10/9, 28/25
 
[[Mapping]]: [{{val| 1 0 -1 -4 }}, {{val| 0 1 2 4 }}]
 
[[POTE generator]]: ~3/2 = 848.221
 
{{Val list|legend=1| 3, 7bc, 10bc }}
 
[[Badness]]: 0.053389
 
=== Phlegyas ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 10/9, 35/32
 
[[Mapping]]: [{{val| 1 0 -1 6 }}, {{val| 0 1 2 -2 }}]
 
[[POTE generator]]: ~3/2 = 743.135
 
{{Val list|legend=1| 3, 5c, 8c, 13bcc }}
 
[[Badness]]: 0.051293
 
=== Charon ===
Subgroup: 2.3.5.7
 
[[Comma list]]: 10/9, 49/45
 
[[Mapping]]: [{{val| 2 0 -2 -1 }}, {{val| 0 1 2 2 }}]
 
[[POTE generator]]: ~3/2 = 782.604 (or ~10/9 = 182.604)
 
{{Val list|legend=1| 6, 14bc, 20bb }}


Comma list: [[32/27]]
[[Badness]]: 0.056404


POTE generator: ~10/9 = 59.9349 cents
=== Nessus ===
Subgroup: 2.3.5.7


Mapping: [{{val| 3 5 7 }}, {{val| 0 0 1 }}]
[[Comma list]]: 10/9, 49/48


{{Val list|legend=1| 3, 12bcc }}
[[Mapping]]: [{{val| 1 0 -1 2 }}, {{val| 0 2 4 1 }}]


Badness: 0.0780
[[POTE generator]]: ~7/6 = 213.117


== Brutus  ==
{{Val list|legend=1| 5c, 6, 11bc, 17bb }}
Comma list: 10/9, 28/25


POTE generator: ~3/2 = 848.221
[[Badness]]: 0.059070


Mapping: [{{val| 1 0 -1 -4 }}, {{val| 0 1 2 4 }}]
== Geryon ==
{{see also| Dicot family }}


{{Multival|legend=1| 1 2 4 1 4 4 }}
Subgroup: 2.3.5.7


{{Val list|legend=1| 3, 10bc }}
[[Comma list]]: 8/7, 25/21


Badness: 0.0534
[[Mapping]]: [{{val| 1 1 2 3 }}, {{val| 0 2 1 0 }}]


== Geryon  ==
[[POTE generator]]: ~5/4 = 385.440
Comma list: 8/7, 25/21


POTE generator: ~5/4 = 385.440
{{Val list|legend=1| 3d }}, {{val| 16 26 37 48 }}, {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}


Mapping: [{{val| 1 1 2 3 }}, {{val| 0 2 1 0 }}]
[[Badness]]: 0.051009


{{Multival|legend=1| 2 1 0 -3 -6 -3 }}
== Malacoda ==
Subgroup: 2.3.5.7


[[Val]]s: {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}
[[Comma list]]: 15/14, 35/32


Badness: 0.0510
[[Mapping]]: [{{val| 1 0 3 2 }}, {{val| 0 2 -1 1 }}]


== Phlegyas  ==
[[POTE generator]]: ~7/6 = 276.283
Comma list: 10/9, 35/32


POTE generator: ~3/2 = 743.135
{{Val list|legend=1| 1, 4, 9c, 13bcc }}


Mapping: [{{val| 1 0 -1 6 }}, {{val| 0 1 2 -2 }}]
[[Badness]]: 0.037207


{{Multival|legend=1| 1 2 -2 1 -6 -10 }}
== Ugolino ==
{{see also| Bug family }}


{{Val list|legend=1| 3, 10c, 11bcd, 13c }}
Subgroup: 2.3.5.7


Badness: 0.0513
[[Comma list]]: 15/14, 27/25


== Malacoda  ==
[[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 3 5 }}]
Comma list: 15/14, 35/32


POTE generator: ~8/7 = 276.283
[[POTE generator]]: ~6/5 = 278.360


Mapping: [{{val| 1 0 3 2 }}, {{val| 0 2 -1 1 }}]
{{Val list|legend=1| 4, 9d, 13bd }}


{{Multival|legend=1| 2 -1 1 -6 -4 5 }}
[[Badness]]: 0.043758


{{Val list|legend=1| 1, 4, 13bc }}
== Medusa ==
{{see also| Pelogic family }}


Badness: 0.0372
Subgroup: 2.3.5.7


== Ugolino  ==
[[Comma list]]: 15/14, 64/63
Comma list: 15/14, 27/25


POTE generator: ~6/5 = 278.360
[[Mapping]]: [{{val| 1 0 7 6 }}, {{val| 0 1 -3 -2 }}]


Mapping: [{{val| 1 0 0 -1 }}, {{val| 0 2 3 5 }}]
[[POTE generator]]: ~3/2 = 685.632


{{Multival|legend=1| 2 3 5 0 2 3 }}
{{Val list|legend=1| 2, 5c, 7 }}


{{Val list|legend=1| 4, 13bd }}
[[Badness]]: 0.042712


Badness: 0.0438
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Charon  ==
Comma list: 15/14, 22/21, 33/32
Comma list: 10/9, 49/45


POTE generator: ~3/2 = 782.604
Mapping: [{{val| 1 0 7 6 5 }}, {{val| 0 1 -3 -2 -1 }}]


Mapping: [{{val| 2 0 -2 -1 }}, {{val| 0 1 2 2 }}]
POTE generator: ~3/2 = 685.809


{{Multival|legend=1| 2 4 4 2 1 -2 }}
Optimal GPV sequence: {{Val list| 2, 5c, 7 }}


{{Val list|legend=1| 6, 13bd }}
Badness: 0.026828


Badness: 0.0564
== Plutus ==
{{see also| Meantone family }}


== Nessus  ==
Subgroup: 2.3.5.7
Comma list: 10/9, 49/48


POTE generator: ~8/7 = 213.117
[[Comma list]]: 15/14, 81/80


Mapping: [{{val| 1 0 -1 2 }}, {{val| 0 2 4 1 }}]
[[Mapping]]: [{{val| 1 0 -4 -5 }}, {{val| 0 1 4 5 }}]


{{Multival|legend=1| 2 4 1 2 -4 -9 }}
[[POTE generator]]: ~3/2 = 682.895


{{Val list|legend=1| 6, 11bc, 17b }}
{{Val list|legend=1| 2cd, 5d, 7 }}


Badness: 0.0591
[[Badness]]: 0.045275


== Medusa  ==
=== 11-limit ===
Comma list: 15/14, 64/63
Subgroup: 2.3.5.7.11


POTE generator: ~3/2 = 685.632
Comma list: 15/14, 22/21, 81/80


Mapping: [{{val| 1 0 7 6 }}, {{val| 0 1 -3 -2 }}]
Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]


{{Multival|legend=1| 1 -3 -2 -7 -6 4 }}
POTE generator: ~3/2 = 685.234


{{Val list|legend=1| 2, 7, 12c, 14cd }}
Optimal GPV sequence: {{Val list| 2cde, 5de, 7 }}


Badness: 0.0427
Badness: 0.032521


[[Category:Temperament]]
[[Category:Temperament]]
[[Category:Temperament collection]]

Revision as of 05:41, 2 January 2022

Todo: intro, cleanup

what are the criteria for inclusion here?

Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large “commas” such as 9/8, 32/27, or 15/14, equating wildly different interval sizes. All temperaments listed temper out an interval of at least size 15/14.

Antitonic

This temperament is characterized by 9/8 being tempering out, and has been termed a "troll temperament". It is named on account of 4/3 and 3/2 both being represented by the 600 cent half octave, which, in terms of diatonic function, serves as an antitonic. Surprisingly, it has extensions all the way up to the 11-limit, as confirmed by the data from x31eq. The original 5-limit is basically the 3-limit music of 2edo with the addition of harmonic 5 represented by an independent generator.

Subgroup: 2.3.5

Comma: 9/8

Mapping: [⟨2 3 0], ⟨0 0 1]]

Mapping generators: ~3/2, ~5

POTE generator: ~5/4 = 313.690 or ~6/5 = 286.310

Template:Val list

Badness: 0.021647

7-limit

Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

Mapping: [⟨2 3 0 1], ⟨0 0 1 1]]

POTE generator: ~5/4 = 301.997 or ~6/5 = 298.003

Template:Val list

Badness: 0.019373

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 15/14

Mapping: [⟨2 3 0 1 7], ⟨0 0 1 1 0]]

POTE generator: ~5/4 = 340.293 or ~6/5 = 259.707

Optimal GPV sequence: Template:Val list

Badness: 0.021810

Antaeus

Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: [⟨2 3 0 10], ⟨0 0 1 -1]]

POTE generator: ~5/4 = 315.962 or ~6/5 = 284.038

Template:Val list

Badness: 0.037537

11-limit

Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 35/32

Mapping: [⟨2 3 0 10 7], ⟨0 0 1 -1 0]]

POTE generator: ~5/4 = 318.173 or ~6/5 = 281.827

Optimal GPV sequence: Template:Val list

Badness: 0.033786

Alteraugment

Like augmented, but the generator provides 5/4 instead of 3/2.

Subgroup: 2.3.5

Comma: 32/27

Mapping: [⟨3 5 7], ⟨0 0 1]]

POTE generator: ~10/9 = 59.935

Template:Val list

Badness: 0.078042

Yellow (2c&3)

This temperament family is characterized by 10/9 being tempering out.

Subgroup: 2.3.5

Comma list: 10/9

Mapping: [⟨1 0 -1], ⟨0 1 2]]

POTE generator: ~3/2 = 775.970

Template:Val list

Badness: 0.016712

Brutus

Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: [⟨1 0 -1 -4], ⟨0 1 2 4]]

POTE generator: ~3/2 = 848.221

Template:Val list

Badness: 0.053389

Phlegyas

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: [⟨1 0 -1 6], ⟨0 1 2 -2]]

POTE generator: ~3/2 = 743.135

Template:Val list

Badness: 0.051293

Charon

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]

POTE generator: ~3/2 = 782.604 (or ~10/9 = 182.604)

Template:Val list

Badness: 0.056404

Nessus

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

Mapping: [⟨1 0 -1 2], ⟨0 2 4 1]]

POTE generator: ~7/6 = 213.117

Template:Val list

Badness: 0.059070

Geryon

See also: Dicot family

Subgroup: 2.3.5.7

Comma list: 8/7, 25/21

Mapping: [⟨1 1 2 3], ⟨0 2 1 0]]

POTE generator: ~5/4 = 385.440

Template:Val list, ⟨16 26 37 48], ⟨19 31 44 57], ⟨22 36 51 66]

Badness: 0.051009

Malacoda

Subgroup: 2.3.5.7

Comma list: 15/14, 35/32

Mapping: [⟨1 0 3 2], ⟨0 2 -1 1]]

POTE generator: ~7/6 = 276.283

Template:Val list

Badness: 0.037207

Ugolino

See also: Bug family

Subgroup: 2.3.5.7

Comma list: 15/14, 27/25

Mapping: [⟨1 0 0 -1], ⟨0 2 3 5]]

POTE generator: ~6/5 = 278.360

Template:Val list

Badness: 0.043758

Medusa

See also: Pelogic family

Subgroup: 2.3.5.7

Comma list: 15/14, 64/63

Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]

POTE generator: ~3/2 = 685.632

Template:Val list

Badness: 0.042712

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 33/32

Mapping: [⟨1 0 7 6 5], ⟨0 1 -3 -2 -1]]

POTE generator: ~3/2 = 685.809

Optimal GPV sequence: Template:Val list

Badness: 0.026828

Plutus

See also: Meantone family

Subgroup: 2.3.5.7

Comma list: 15/14, 81/80

Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]

POTE generator: ~3/2 = 682.895

Template:Val list

Badness: 0.045275

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

Mapping: [⟨1 0 -4 -5 -6], ⟨0 1 4 5 6]]

POTE generator: ~3/2 = 685.234

Optimal GPV sequence: Template:Val list

Badness: 0.032521

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