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Very low accuracy temperaments: Difference between revisions

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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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* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.
* [[: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 ===
=== Septimal antitonic ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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[[Badness]] (Sintel): 1.83
[[Badness]] (Sintel): 1.83


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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Badness]] (Sintel): 0.392
[[Badness]] (Sintel): 0.392
=== Septimal antonian ===
{{See also| Trienstonic clan }}
Subgroup: 2.3.5.7
Comma list: 10/9, 15/14
Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}
Optimal tunings:
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}
{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}
Badness (Sintel): 0.606
=== Antonym ===
Subgroup: 2.3.5.7
Comma list: 7/6, 10/9
Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}
Optimal tunings:
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}
{{Optimal ET sequence|legend=0| 2cd, 3 }}
Badness (Sintel): 0.568
=== Antony ===
Subgroup: 2.3.5.7
Comma list: 8/7, 10/9
Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}
Optimal tunings:
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}
{{Optimal ET sequence|legend=0| 1c, 3d }}
Badness (Sintel): 0.720


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


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


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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
== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. 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
[[Comma list]]: 16/15, 27/25
{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
: Mapping generators: ~6/5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}
{{Optimal ET sequence|legend=1| 5, 15ccd }}
[[Badness]] (Sintel): 1.22
== Sept ==
Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.
This temperament used to be known as ''mujannab''.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 54/49, 64/63
{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
: [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
* [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
: error map: {{val| 0.000 -16.241 +6.406 +59.746 }}
{{Optimal ET sequence|legend=1| 7, 14d }}
[[Badness]] (Sintel): 2.68
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 45/44, 54/49, 64/63
Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}
Optimal tunings:
* WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})
{{Optimal ET sequence|legend=0| 7, 14de }}
Badness (Sintel): 2.02
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 45/44, 52/49, 64/63
Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
Optimal tunings:
* WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})
{{Optimal ET sequence|legend=0| 7, 14de }}
Badness (Sintel): 1.77


== Geryon ==
== Geryon ==
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== Plutus ==
== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}
{{See also| Meantone family }}


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: error map: {{val| +65.405 -84.203 -94.789 }}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}
: error map: {{val| 0.000 -181.329 -227.565 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~3 = 1988.549{{c}}, ~3/2 = 719.679{{c}}
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: error map: {{val| +86.594 -78.085 -111.407 }}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.995{{c}}, ~5/3 = 691.468{{c}}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}
: error map: {{val| 0.000 -192.891 -256.384 }}


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== Codex ==
== Codex ==
{{See also| Bug family }}
{{See also| Bug family }}
 
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. 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]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid dicot.  
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. 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]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid beta-dicot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: error map: {{val| -2.821 -31.998 +18.621 +34.699 }}
: [[error map]]: {{val| -2.821 -31.998 +18.621 +34.699 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}
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== Oxygen ==
== Oxygen ==
{{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]].
{{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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}


{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}
{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*
 
<nowiki/>*[[Optimal patent val]]: [[8edo|8]]


[[Badness]] (Sintel): 1.52
[[Badness]] (Sintel): 1.52
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* WE: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: error map: {{val| +68.274 -20.760 -249.765 }}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}
: error map: {{val| 0.000 -87.900 -386.314 }}


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[[Badness]] (Sintel): 0.474
[[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]]
[[Category:Temperament collections]]
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