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| == Quint == | | == 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. | | 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. |
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| [[Subgroup]]: 2.3.5.7 | | [[Subgroup]]: 2.3.5.7 |
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| {{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }} | | {{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }} |
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| : Mapping generators: ~6/5, ~7 | | : Mapping generators: ~6/5, ~7 |
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| [[Optimal tuning]]s: | | [[Optimal tuning]]s: |
| * [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong --> | | * [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}}) |
| : [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }} | | : [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }} |
| * [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135) | | * [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}}) |
| : error map: {{val| 0.000 +18.045 +93.686 +36.309 }} | | : error map: {{val| 0.000 +18.045 +93.686 +36.309 }} |
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| {{Optimal ET sequence|legend=1| 5, 15ccd }} | | {{Optimal ET sequence|legend=1| 5, 15ccd, 20cccd }} |
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| [[Badness]] (Sintel): 1.22 | | [[Badness]] (Sintel): 1.22 |
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| == Sept ==
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| {{See also| Whitewood family }}
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| 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.
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| This temperament used to be known as ''mujannab''.
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| [[Subgroup]]: 2.3.5.7
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| [[Comma list]]: 54/49, 64/63
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| {{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
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| [[Optimal tuning]]s:
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| * [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
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| : [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
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| * [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
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| : error map: {{val| 0.000 -16.241 +6.406 +59.746 }}
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| {{Optimal ET sequence|legend=1| 7, 14d }}
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| [[Badness]] (Sintel): 2.68
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| === 11-limit ===
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| Subgroup: 2.3.5.7.11
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| Comma list: 45/44, 54/49, 64/63
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| Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}
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| Optimal tunings:
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| * WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
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| * CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})
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| {{Optimal ET sequence|legend=0| 7, 14de }}
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| Badness (Sintel): 2.02
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| === 13-limit ===
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| Subgroup: 2.3.5.7.11.13
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| Comma list: 27/26, 45/44, 52/49, 64/63
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| Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
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| Optimal tunings:
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| * WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
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| * CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})
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| {{Optimal ET sequence|legend=0| 7, 14de }}
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| Badness (Sintel): 1.77
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| == Geryon == | | == Geryon == |
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| Badness (Sintel): 1.08 | | Badness (Sintel): 1.08 |
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| == Involution ==
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| Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.
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| [[Subgroup]]: 2.3.5
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| [[Comma list]]: [[45/32]]
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| {{Mapping|legend=1| 1 0 5 | 0 1 -2 }}
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| : mapping generators: ~2, ~3
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| [[Optimal tuning]]s:
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| * [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
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| : [[error map]]: {{val| +65.405 -84.203 -94.789 }}
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| * [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
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| : error map: {{val| 0.000 -181.329 -227.565 }}
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| {{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}
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| [[Badness]] (Sintel): 3.12
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| === 7-limit ===
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| Subgroup: 2.3.5.7
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| Comma list: 8/7, 45/28
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| Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}
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| Optimal tunings:
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| * WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
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| * CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}
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| Optimal ET sequence: none
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| Badness (Sintel): 3.01
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| == Devolution ==
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| Devolution was named by [[User:Akselai|Akselai]] in 2024.
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| [[Subgroup]]: 3.5.7
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| [[Comma list]]: [[35/27]]
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| {{Mapping|legend=1| 1 0 3 | 0 1 -1 }}
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| : mapping generators: ~3, ~5
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| [[Optimal tuning]]s:
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| * [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
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| : [[error map]]: {{val| +86.594 -78.085 -111.407 }}
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| * [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
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| : error map: {{val| 0.000 -192.891 -256.384 }}
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| [[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]
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| [[Badness]] (Sintel): 0.751
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| == 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]]. |
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| [[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 }}* |
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| <nowiki/>*[[Optimal patent val]]: [[8edo|8]] | | <nowiki/>* [[optimal patent val]]: [[8edo|8]] |
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| [[Badness]] (Sintel): 1.52 | | [[Badness]] (Sintel): 1.52 |