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Very low accuracy temperaments: Difference between revisions

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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.  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


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


Mapping generators: ~3/2, ~5
: mapping generators: ~3/2, ~5


[[POTE generator]]: ~5/4 = 313.690 or ~6/5 = 286.310
[[Optimal tuning]]s:
* [[POTE]]: ~3/2 = 600.000, ~5/4 = 313.690 (~6/5 = 286.310)


{{Optimal ET sequence|legend=1| 2, 4 }}
{{Optimal ET sequence|legend=1| 2, 4 }}


[[Badness]]: 0.021647
[[Badness]]: 0.021647
; Music
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.


=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 9/8, 15/14
Comma list: 9/8, 15/14


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


[[POTE generator]]: ~5/4 = 301.997 or ~6/5 = 298.003
Optimal tunings:
* POTE: ~3/2 = 600.000, ~5/4 = 301.997 (~6/5 = 298.003)


{{Optimal ET sequence|legend=1| 2, 4 }}
Optimal ET sequence: {{optimal ET sequence| 2, 4 }}


[[Badness]]: 0.019373
Badness: 0.019373


==== 11-limit ====
==== 11-limit ====
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Comma list: 9/8, 12/11, 15/14
Comma list: 9/8, 12/11, 15/14


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


POTE generator: ~5/4 = 340.293 or ~6/5 = 259.707
Optimal tunings:
* POTE: ~3/2 = 600.000, ~5/4 = 340.293 (~6/5 = 259.707)


{{Optimal ET sequence|legend=1| 2, 4 }}
Optimal ET sequence: {{optimal ET sequence| 2, 4 }}


Badness: 0.021810
Badness: 0.021810
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Comma list: 9/8, 11/10, 15/14
Comma list: 9/8, 11/10, 15/14


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


POTE generator: ~5/4 = 322.080 or ~6/5 = 277.920
Optimal tunings:
* POTE: ~3/2 = 600.000, ~5/4 = 322.080 (~6/5 = 277.920)


{{Optimal ET sequence|legend=1| 2, 4e }}
Optimal ET sequence: {{optimal ET sequence| 2, 4e }}


Badness: 0.018993
Badness: 0.018993
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Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 9/8, 35/32
Comma list: 9/8, 35/32


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


[[POTE generator]]: ~5/4 = 315.962 or ~6/5 = 284.038
Optimal tunings:
* POTE: ~3/2 = 600.000, ~5/4 = 315.962 (~6/5 = 284.038)


{{Optimal ET sequence|legend=1| 2d, 4 }}
Optimal ET sequence: {{optimal ET sequence| 2d, 4 }}


[[Badness]]: 0.037537
Badness: 0.037537


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


Mapping: [{{val| 2 3 0 10 7 }}, {{val| 0 0 1 -1 0 }}]
Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}


POTE generator: ~5/4 = 318.173 or ~6/5 = 281.827
Optimal tunings:
* POTE: ~3/2 = 600.000, ~5/4 = 318.173 (~6/5 = 281.827)


{{Optimal ET sequence|legend=1| 2d, 4 }}
Optimal ET sequence: {{optimal ET sequence| 2d, 4 }}


Badness: 0.033786
Badness: 0.033786
=== Music ===
[[File:Antitonic(8).mp3]]
short composition by Wensik in POTE-tuned 5-limit antitonic using an 8-note ternary scale.


== Alteraugment ==
== Alteraugment ==
Like [[augmented]], but the generator provides 5/4 instead of 3/2.
Alteraugment is like [[augmented]], but the generator provides 5/4 instead of 3/2. [[User:VectorGraphics|Vector Graphics]] suggests the name kinsborough for this temperament.


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


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


[[POTE generator]]: ~10/9 = 59.935
[[Optimal tuning]]s:
* [[POTE]]: ~4/3 = 400.000, ~10/9 = 59.935


{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc, 18bbcc }}
{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc, 18bbcc }}
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[[Badness]]: 0.078042
[[Badness]]: 0.078042


Vector Graphics suggests the name kinsborough for this temperament.
== Yo (2c & 3)  ==
This temperament family is characterized by the [[color notation|yo 2nd]] [[10/9]] being tempering out.


== Yo (2c&3)  ==
[[Subgroup]]: 2.3.5
This temperament family is characterized by the [[Color notation|yo 2nd]] [[10/9]] being tempering out.
 
Subgroup: 2.3.5


[[Comma list]]: [[10/9]]
[[Comma list]]: [[10/9]]


[[Mapping]]: [{{val| 1 0 -1 }}, {{val| 0 1 2 }}]
{{Mapping|legend=1| 1 0 -1 | 0 1 2 }}


[[POTE generator]]: ~3/2 = 775.970
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~3/2 = 775.970


{{Optimal ET sequence|legend=1| 3, 11bc, 14bc, 17bb }}
{{Optimal ET sequence|legend=1| 3, 11bc, 14bc, 17bb }}
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Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 28/25
Comma list: 10/9, 28/25


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


[[POTE generator]]: ~3/2 = 848.221
Optimal tunings:
* POTE: ~2 = 1200.000, ~3/2 = 848.221


{{Optimal ET sequence|legend=1| 3, 7bc, 10bc }}
Optimal ET sequence: {{optimal ET sequence| 3, 7bc, 10bc }}


[[Badness]]: 0.053389
Badness: 0.053389


=== Phlegyas ===
=== Phlegyas ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 35/32
Comma list: 10/9, 35/32


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


[[POTE generator]]: ~3/2 = 743.135
Optimal tunings:
* POTE: ~2 = 1200.000, ~3/2 = 743.135


{{Optimal ET sequence|legend=1| 3, 5c, 8c, 13bcc }}
Optimal ET sequence: {{optimal ET sequence| 3, 5c, 8c, 13bcc }}


[[Badness]]: 0.051293
Badness: 0.051293


=== Charon ===
=== Charon ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 49/45
Comma list: 10/9, 49/45


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


[[POTE generator]]: ~3/2 = 782.604 (or ~15/14 = 182.604)
Optimal tunings:
* POTE: ~2 = 1200.000, ~3/2 = 782.604 (or ~15/14 = 182.604)


{{Optimal ET sequence|legend=1| 6, 14bc, 20bb }}
Optimal ET sequence: {{optimal ET sequence| 6, 14bc, 20bb }}


[[Badness]]: 0.056404
Badness: 0.056404


=== Nessus ===
=== Nessus ===
{{See also| Semaphoresmic clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10/9, 49/48
Comma list: 10/9, 49/48


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


[[POTE generator]]: ~7/6 = 213.117
Optimal tunings:
* POTE: ~2 = 1200.000, ~7/6 = 213.117


{{Optimal ET sequence|legend=1| 5c, 6, 11bc, 17bb }}
Optimal ET sequence: {{optimal ET sequence| 5c, 6, 11bc, 17bb }}


[[Badness]]: 0.059070
Badness: 0.059070


== Geryon ==
== Geryon ==
{{see also| Dicot family }}
{{See also| Dicot family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 8/7, 25/21


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


[[POTE generator]]: ~5/4 = 385.440
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~5/4 = 385.440


{{Optimal ET sequence|legend=1| 3d }}, {{val| 16 26 37 48 }}, {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}
{{Optimal ET sequence|legend=1| 3d }}, {{val| 16 26 37 48 }}, {{val| 19 31 44 57 }}, {{val| 22 36 51 66 }}
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== Malacoda ==
== Malacoda ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 35/32
[[Comma list]]: 15/14, 35/32


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


[[POTE generator]]: ~7/6 = 276.283
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~7/6 = 276.283


{{Optimal ET sequence|legend=1| 1, 4, 9c, 13bcc }}
{{Optimal ET sequence|legend=1| 1, 4, 9c, 13bcc }}
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== Ugolino ==
== Ugolino ==
{{see also| Bug family }}
{{See also| Bug family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 3 5 }}]
{{Mapping|legend=1| 1 0 0 -1 | 0 2 3 5 }}


[[POTE generator]]: ~6/5 = 278.360
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~6/5 = 278.360


{{Optimal ET sequence|legend=1| 4, 9d, 13bd }}
{{Optimal ET sequence|legend=1| 4, 9d, 13bd }}
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== Uncle ==
== Uncle ==
{{see also| Trienstonic clan#Uncle }}
{{See also| Trienstonic clan #Uncle }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 4096/3645
[[Comma list]]: 4096/3645


[[Mapping]]: [{{val| 1 1 6 }}, {{val| 0 1 -6 }}]
{{Mapping|legend=1| 1 1 6 | 0 1 -6 }}


Optimal tuning ([[CWE]]): ~2 = 1\1, ~3/2 = 731.732
[[Optimal tuning]]s:
* [[CWE]]): ~2 = 1200.000, ~3/2 = 731.732


Supporting ETs: [[5edo|5]], [[8edo|8c]], [[13edo|13]], [[18edo|18]], [[23edo|23bc]], [[28edo|28bc]], [[33edo|33bcc]], [[41edo|41bbc]], [[51edo|51bbccc]]
Supporting ETs: [[5edo|5]], [[8edo|8c]], [[13edo|13]], [[18edo|18]], [[23edo|23bc]], [[28edo|28bc]], [[33edo|33bcc]], [[41edo|41bbc]], [[51edo|51bbccc]]
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== Medusa ==
== Medusa ==
{{see also| Pelogic family }}
{{See also| Pelogic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 64/63
[[Comma list]]: 15/14, 64/63


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


[[POTE generator]]: ~3/2 = 685.632
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~3/2 = 685.632


{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
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Comma list: 15/14, 22/21, 33/32
Comma list: 15/14, 22/21, 33/32


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


POTE generator: ~3/2 = 685.809
Optimal tunings:
* POTE: ~2 = 1200.000, ~3/2 = 685.809


{{Optimal ET sequence|legend=1| 2, 5c, 7 }}
Optimal ET sequence: {{optimal ET sequence| 2, 5c, 7 }}


Badness: 0.026828
Badness: 0.026828


== Plutus ==
== Plutus ==
{{see also| Meantone family }}
{{See also| Meantone family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 81/80
[[Comma list]]: 15/14, 81/80


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


[[POTE generator]]: ~3/2 = 682.895
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000, ~3/2 = 682.895


{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}
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Comma list: 15/14, 22/21, 81/80
Comma list: 15/14, 22/21, 81/80


Mapping: [{{val| 1 0 -4 -5 -6 }}, {{val| 0 1 4 5 6 }}]
Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}


POTE generator: ~3/2 = 685.234
Optimal tunings:
* POTE: ~2 = 1200.000, ~3/2 = 685.234


{{Optimal ET sequence|legend=1| 2cde, 5de, 7 }}
Optimal ET sequence: {{optimal ET sequence| 2cde, 5de, 7 }}


Badness: 0.032521
Badness: 0.032521
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[[Comma list]]: [[45/32]]
[[Comma list]]: [[45/32]]


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


[[CTE|CTE generator]]: ~[[3/2]] = 509.892
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 509.892


[[Optimal ET sequence]]: None aside from extremely absurd alternative ETs
[[Optimal ET sequence]]: None aside from extremely absurd alternative ETs
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=== 7-limit ===
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 45/28, 8/7
Comma list: 8/7, 45/28


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


[[CTE|CTE generator]]: ~[[3/2]] = 509.892
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 509.892


[[Optimal ET sequence]]: None aside from extremely absurd alternative ETs
Optimal ET sequence: None aside from extremely absurd alternative ETs


[[Badness]]: 0.118764
Badness: 0.118764


== Devolution ==
== Devolution ==
[[Subgroup]]: 3.5.7
[[Subgroup]]: 3.5.7


[[Comma list]]: [[35/27]]
[[Comma list]]: [[35/27]]


[[Mapping]]: [{{val|1 1 2}}, {{val|0 1 -1}}]
{{Mapping|legend=1| 1 1 2 | 0 1 -1 }}


[[CTE|CTE generator]]: ~[[5/3]] = 701.863
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~5/3 = 701.863


{{Optimal ET sequence|legend=1| 8cdd }}, otherwise none aside from extremely absurd alternative ETs  
{{Optimal ET sequence|legend=1| 8cdd }}, otherwise none aside from extremely absurd alternative ETs  
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== Codex ==
== Codex ==
{{see also| Bug family }}
{{See also| Bug family }}
 
This temperament is named as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5s and 7s. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] in the same way decimal is of [[semaphore]].  
This temperament is named as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5s and 7s. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] in the same way decimal is of [[semaphore]].  


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[[Comma list]]: [[27/25]], [[54/49]]
[[Comma list]]: [[27/25]], [[54/49]]


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


[[CTE|CTE generator]]: ~[[7/5]] = 1\2, ~[[5/3]] = 935.289
[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 1\2, ~5/3 = 935.289


{{Optimal ET sequence|legend=1| 10cd, 14d }}
{{Optimal ET sequence|legend=1| 10cd, 14d }}
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