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Very low accuracy temperaments: Difference between revisions

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Delete antonym, antony, and geryon, from the precedent of deletion of {7/6}. Perhaps {9/8} should be the threshold for tempering out superparticulars
 
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{{Technical data page}}
{{Technical data page}}
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s. A lot of temperaments listed here don't have [[diamond monotone]] tunings because of their sheer inaccuracy. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.


== Antitonic ==
== Antitonic ==
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Comma list: 9/8, 15/14
Comma list: 9/8, 15/14


Mapping: {{mapping| 2 3 0 1 | 0 0 1 1 }}
{{Mapping|legend=0| 2 3 0 1 | 0 0 1 1 }}


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


Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}
{{Mapping|legend=0| 2 3 0 1 7 | 0 0 1 1 0 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 9/8, 11/10, 15/14
Comma list: 9/8, 11/10, 15/14


Mapping: {{mapping| 2 3 0 1 2 | 0 0 1 1 1 }}
{{Mapping|legend=0| 2 3 0 1 2 | 0 0 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 9/8, 35/32
Comma list: 9/8, 35/32


Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}
{{Mapping|legend=0| 2 3 0 10 | 0 0 1 -1 }}


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


Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}
{{Mapping|legend=0| 2 3 0 10 7 | 0 0 1 -1 0 }}


Optimal tunings:  
Optimal tunings:  
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{{Mapping|legend=1| 3 5 0 | 0 0 1 }}
{{Mapping|legend=1| 3 5 0 | 0 0 1 }}
: mapping generators: ~4/3, ~5
: mapping generators: ~4/3, ~5


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== Antonian ==
== Antonian ==
{{Main| Antonian }}
{{Main| Antonian }}
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].  


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Comma list: 10/9, 15/14
Comma list: 10/9, 15/14


Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}
{{Mapping|legend=0| 1 0 -1 -2 | 0 1 2 3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 10/9, 28/25
Comma list: 10/9, 28/25


Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}
{{Mapping|legend=0| 1 0 -1 -4 | 0 1 2 4 }}


Optimal tunings:  
Optimal tunings:  
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=== Phlegyas ===
=== Phlegyas ===
{{See also| Archytas clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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


Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}
{{Mapping|legend=0| 1 0 -1 6 | 0 1 2 -2 }}


Optimal tunings:  
Optimal tunings:  
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=== Charon ===
=== Charon ===
{{See also| Jubilismic clan }}
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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


Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
{{Mapping|legend=0| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3


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


Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
{{Mapping|legend=0| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


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Comma list: 10/9, 16/15, 22/21
Comma list: 10/9, 16/15, 22/21


Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}
{{Mapping|legend=0| 3 5 7 0 2 | 0 0 0 1 1 }}


Optimal tunings:  
Optimal tunings:  
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== Quad ==
== Quad ==
Quad is identical to 4et in the 5-limit, but has an independent generator for prime 7.  
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Badness]] (Sintel): 1.16
[[Badness]] (Sintel): 1.16


== Geryon ==
== Quint ==
{{See also| Dicot family }}
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 8/7, 25/21
[[Comma list]]: 16/15, 27/25


{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}


{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}
{{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }}


[[Badness]] (Sintel): 1.29
[[Badness]] (Sintel): 1.22


== Malacoda ==
== Malacoda ==
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{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}
{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}
: mapping generators: ~2, ~7/4
: mapping generators: ~2, ~7/4


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


Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}
{{Mapping|legend=0| 1 0 7 6 5 | 0 1 -3 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
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== Plutus ==
== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}
{{See also| Meantone family }}


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


Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}
{{Mapping|legend=0| 1 0 -4 -5 -6 | 0 1 4 5 6 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 1.08
Badness (Sintel): 1.08
== Involution ==
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.
[[Subgroup]]: 2.3.5
[[Comma list]]: [[45/32]]
{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}
{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
[[Badness]] (Sintel): 3.12
=== 7-limit ===
Subgroup: 2.3.5.7
Comma list: 8/7, 45/28
Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}
Optimal tunings:
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
Optimal ET sequence: none
Badness (Sintel): 3.01
== Devolution ==
Devolution was named by [[User:Akselai|Akselai]] in 2024.
[[Subgroup]]: 3.5.7
[[Comma list]]: [[35/27]]
{{Mapping|legend=1| 1 0 3 | 0 1 -1 }}
: mapping generators: ~3, ~5
[[Optimal tuning]]s:
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}
[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
[[Badness]] (Sintel): 0.751


== Codex ==
== Codex ==
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== Oxygen ==
== Oxygen ==
{{See also| Porcupine family }}
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*


<nowiki/>*[[Optimal patent val]]: [[8edo|8]]
<nowiki/>* [[optimal patent val]]: [[8edo|8]]


[[Badness]] (Sintel): 1.52
[[Badness]] (Sintel): 1.52
== Bixby ==
{{Main| Bixby }}
[[Subgroup]]: 2.3.5
[[Comma list]]: [[4/3]]
{{Mapping|legend=1| 1 2 0 | 0 0 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1020.058{{c}}, ~5/4 = 674.394{{c}}
: [[error map]]: {{val| -179.942 +138.161 -71.803 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 629.521{{c}}
: error map: {{val| 0.000 +498.045 +243.208 }}
{{Optimal ET sequence|legend=1| 1c, 2b, 3bbcc }}
[[Badness]] (Sintel): 0.424
== Archon ==
{{Main| Archon }}
[[Subgroup]]: 2.3.5
[[Comma list]]: [[5/4]]
{{Mapping|legend=1| 1 0 2 | 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}
{{Optimal ET sequence|legend=1| 2c }}
[[Badness]] (Sintel): 0.474
== Seesaw ==
{{Main| Seesaw }}
Seesaw tempers out the [[6/5|classic minor third (6/5)]], equating the [[5/1|fifth]] and [[6/1|sixth harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.
[[Subgroup]]: 2.3.5
[[Comma list]]: 6/5
{{Mapping|legend=1| 1 0 1 | 0 1 1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1155.569{{c}}, ~3/2 = 643.349{{c}}
: [[error map]]: {{val| -44.431 -103.037 +168.173 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 627.511{{c}}
: error map: {{val| 0.000 -74.444 +241.197 }}
{{Optimal ET sequence|legend=1| 2 }}
[[Badness]] (Sintel): 0.367
=== 2.3.5.11 subgroup ===
This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both [[11/10]] and [[12/11]], undecimal neutral seconds.
Subgroup: 2.3.5.11
Comma list: 6/5, 11/10
Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}
Optimal tunings:
* WE: ~2 = 1156.418{{c}}, ~3/2 = 643.202{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 627.023{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.499
=== Heavy windmill ===
Heavy windmill tempers out [[9/7]] and [[15/14]] in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 6/5, 9/7
Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}
Optimal tunings:
* WE: ~2 = 1161.600{{c}}, ~3/2 = 571.169{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 559.563{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.676
==== 11-limit ====
Subgroup: 2.3.5.7.11
Comma list: 6/5, 9/7, 11/10
Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}
Optimal tunings:
* WE: ~2 = 1166.584{{c}}, ~3/2 = 568.073{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 558.941{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.774
=== Light windmill ===
Light windmill tempers out [[8/7]] and [[21/20]] in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 6/5, 8/7
Mapping: {{mapping| 1 0 1 3 | 0 1 1 0 }}
Optimal tunings:
* WE: ~2 = 1134.018{{c}}, ~3/2 = 670.285{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 667.893{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.629
==== 11-limit ====
Subgroup: 2.3.5.7.11
Comma list: 6/5, 8/7, 11/10
Mapping: {{mapping| 1 0 1 3 2 | 0 1 1 0 1 }}
Optimal tunings:
* WE: ~2 = 1136.109{{c}}, ~3/2 = 672.403{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 668.374{{c}}
{{Optimal ET sequence|legend=0| 2 }}
Badness (Sintel): 0.681
[[Category:Temperament collections]]
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