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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Badness (Sintel): 0.606 | 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 }} | ||