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Subgroup temperaments: Difference between revisions

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Badness (Sintel): 0.450
Badness (Sintel): 0.450
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].
[[Subgroup]]: 2.3.7.11/5.13
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


=== Darian calendar ===
=== Darian calendar ===
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 [[subgroup]]. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].


668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
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{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


=== Hypnosis ===
=== Hydrothermal ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.


[[Subgroup]]: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7/5


[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})


{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~2, ~13/9
: mapping generators: ~7/5, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}


{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
<nowiki/>* Wart for 7/5
 
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
 
[[Subgroup]]: 2.3.7/5
 
[[Comma list]]: [[50/49]]
 
{{Mapping|legend=3| 2 3 1 | 0 1 0 }}
 
[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
 
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


=== Argentic ===
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].  


[[Subgroup]]: 2.3.7/5
[[Subgroup]]: 2.3.7/5


[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})


{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


Badness (Sintel): 0.119
[[Badness]] (Sintel): 0.119


==== Edson ====
==== Edson ====
{{See also| Chromatic pairs #Edson }}
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}
[[Comma list]]: [[196/195]] ({{monzo| 2 -1 2 0 -1 }}), [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[364/363]] ({{monzo| 2 -1 1 -2 1 }})


{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3


{{Mapping|legend=5| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
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