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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]]
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