Very low accuracy temperaments: Difference between revisions

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 {{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 ==
@@ Line 185: / Line 185: @@
 * CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}
-{{Optimal ET sequence|legend=0| 2c, 3d }}
+{{Optimal ET sequence|legend=0| 1c, 3d }}
 Badness (Sintel): 0.720
@@ Line 294: / Line 294: @@
 == 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
@@ Line 321: / Line 321: @@
 {{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}
-: Mapping generators: ~9/8, ~7
+: Mapping generators: ~6/5, ~7
 [[Optimal tuning]]s:
-* [[POTE]]: ~9/8 = 1\5, ~7/4 = 1017.903 (~21/20 = 57.903)
+* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
-* [[CTE]]: ~9/8 = 1\5, ~7/4 = 968.8259 (~63/64 = 8.8259)
+: [[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]]: 0.048312
+[[Badness]] (Sintel): 1.22
+== Sept ==
+{{See also| Whitewood family }}
+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 ==
@@ Line 444: / Line 499: @@
 == Plutus ==
+{{Distinguish| Pluto }}
 {{See also| Meantone family }}
@@ Line 714: / Line 770: @@
 Badness (Sintel): 0.681
+== Sixseven ==
+Sixseven tempers out the [[7/6|septimal minor third (7/6)]], equating the [[6/1|sixth]] and [[7/1|seventh harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.
+[[Subgroup]]: 2.3.7
+[[Comma list]]: 7/6
+{{Mapping|legend=1| 1 0 1 | 0 1 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1218.135{{c}}, ~3/2 = 734.187{{c}}
+: [[error map]]: {{val| +18.135 +50.367 -198.368 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.927{{c}}
+: error map: {{val| 0.000 +36.972 -229.899 }}
+{{Optimal ET sequence|legend=1| 1, 2d, 3, 5dd }}
+[[Badness]] (Sintel): 0.329
+=== 2.3.7.13 subgroup ===
+This temperament is extended to the 2.3.7.13 subgroup naturally, tempering out both [[13/12]] and [[14/13]], tridecimal neutral seconds.
+Subgroup: 2.3.7.13
+Comma list: 7/6, 13/12
+Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}
+Optimal tunings:
+* WE: ~2 = 1220.937{{c}}, ~3/2 = 734.003{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 739.780{{c}}
+{{Optimal ET sequence|legend=0| 1, 2d, 3, 5ddf }}
+Badness (Sintel): 0.422
+=== Heaven ===
+Heaven tempers out [[11/9]] and [[22/21]] in the 2.3.7.11 subgroup.
+Subgroup: 2.3.7.11
+Comma list: 7/6, 11/9
+Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}
+Optimal tunings:
+* WE: ~2 = 1210.810{{c}}, ~3/2 = 783.079{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 785.173{{c}}
+{{Optimal ET sequence|legend=0| 1e, 2de, 3 }}
+Badness (Sintel): 0.577
+==== 2.3.7.11.13 subgroup ====
+Subgroup: 2.3.7.11.13
+Comma list: 7/6, 11/9, 13/12
+Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}
+Optimal tunings:
+* WE: ~2 = 1210.595{{c}}, ~3/2 = 783.176{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 784.989{{c}}
+{{Optimal ET sequence|legend=0| 1e, 2de, 3 }}
+Badness (Sintel): 0.627
 [[Category:Temperament collections]]