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 | 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. | ||
=== | === Septimal antitonic === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
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[[Badness]] (Sintel): 1.83 | [[Badness]] (Sintel): 1.83 | ||
== | == Antonian == | ||
This temperament family is characterized by the [[color notation|yo 2nd]] [[10/9]] being | {{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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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}} | * [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}} | ||
: [[error map]]: {{val| +32.295 -53.513 -13.650 -5. | : [[error map]]: {{val| +32.295 -53.513 -13.650 -5.150 }} | ||
* [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}} | * [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}} | ||
: error map: {{val| 0.000 -101.955 -86.314 -71.236 }} | : error map: {{val| 0.000 -101.955 -86.314 -71.236 }} | ||
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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 | |||
== 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 | * [[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. | * [[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 | |||
[[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 }} | {{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 | ||