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Ragismic microtemperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (10/)
Switch to Sintel's badness, WE & CWE tunings (11/11)
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Badness (Sintel): 1.14
Badness (Sintel): 1.14
== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}
[[Badness]] (Sintel): 7.92


== Oviminor ==
== Oviminor ==
Line 1,609: Line 1,629:
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''


In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses (648/625)]] fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament (''counterkleismic'', named by analogy to [[catakleismic]] and parakleismic), tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma).
In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)<sup>6</sup>) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,615: Line 1,635:
[[Comma list]]: 4375/4374, 158203125/157351936
[[Comma list]]: 4375/4374, 158203125/157351936


{{Mapping|legend=1| 1 20 20 61 | 0 -25 -24 -79 }}
{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
: Mapping generators: ~2, ~5/3
: mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.060{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}


{{Optimal ET sequence|legend=1| 19, 205, 224, 243, 467 }}
{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}


[[Badness]] (Sintel): 2.292
[[Badness]] (Sintel): 2.29


=== 11-limit ===
=== 11-limit ===
Line 1,629: Line 1,653:
Comma list: 540/539, 4375/4374, 2097152/2096325
Comma list: 540/539, 4375/4374, 2097152/2096325


Mapping: {{mapping| 1 20 20 61 -40 | 0 -25 -24 -79 59 }}
Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.071{{c}}
Optimal tunings:
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}


{{Optimal ET sequence|legend=0| 19, 205, 224 }}
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


Badness (Sintel): 2.346
Badness (Sintel): 2.35


==== 13-limit ====
==== 13-limit ====
Line 1,642: Line 1,668:
Comma list: 540/539, 625/624, 729/728, 10985/10976
Comma list: 540/539, 625/624, 729/728, 10985/10976


Mapping: {{mapping| 1 20 20 61 -40 56 | 0 -25 -24 -79 59 -71 }}
Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.070{{c}}
Optimal tunings:
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}


{{Optimal ET sequence|legend=0| 19, 205, 224, 1587cde, 1811ccdef, 2035ccddeef, 2259ccddeef, 2483ccddeef, 2707ccddeef }}
{{Optimal ET sequence|legend=0| 19, 205, 224 }}


Badness (Sintel): 1.400
Badness (Sintel): 1.40


=== Counterlytic ===
=== Counterlytic ===
Line 1,655: Line 1,683:
Comma list: 1375/1372, 4375/4374, 496125/495616
Comma list: 1375/1372, 4375/4374, 496125/495616


Mapping: {{mapping| 1 20 20 61 125 | 0 -25 -24 -79 -165 }}
Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
Optimal tunings:
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}


{{Optimal ET sequence|legend=1| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}


Badness (Sintel): 2.162
Badness (Sintel): 2.16


==== 13-limit ====
==== 13-limit ====
Line 1,668: Line 1,698:
Comma list: 625/624, 729/728, 1375/1372, 10985/10976
Comma list: 625/624, 729/728, 1375/1372, 10985/10976


Mapping: {{mapping| 1 20 20 61 125 56 | 0 -25 -24 -79 -165 -71 }}
Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.065{{c}}
Optimal tunings:
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}


{{Optimal ET sequence|legend=0| 19e, 205e, 224 }}
{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}


Badness (Sintel): 1.231
Badness (Sintel): 1.23


== Quincy ==
== Quincy ==
Line 1,682: Line 1,714:


{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2, ~1728/1715


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~1728/1715 = 16.613{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}


{{Optimal ET sequence|legend=1| 72, 217, 289 }}
{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}


[[Badness]] (Sintel): 2.016
[[Badness]] (Sintel): 2.02


=== 11-limit ===
=== 11-limit ===
Line 1,696: Line 1,733:
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.613{{c}}
Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}


{{Optimal ET sequence|legend=0| 72, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 217, 289 }}


Badness (Sintel): 1.021
Badness (Sintel): 1.02


=== 13-limit ===
=== 13-limit ===
Line 1,709: Line 1,748:
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}
Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
Optimal tunings:
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}


{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
Line 1,722: Line 1,763:
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}
Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.602{{c}}
Optimal tunings:
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}


{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}
Line 1,735: Line 1,778:
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}
Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~100/99 = 16.594{{c}}
Optimal tunings:
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}


{{Optimal ET sequence|legend=0| 72, 145, 217 }}
{{Optimal ET sequence|legend=0| 72, 145, 217 }}
Line 1,742: Line 1,787:


== Sfourth ==
== Sfourth ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Sfourth]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,749: Line 1,794:


{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
: mapping generators: ~2, ~49/48


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/48 = 26.287{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}}
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }}


{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}


[[Badness]] (Sintel): 3.120
[[Badness]] (Sintel): 3.12


=== 11-limit ===
=== 11-limit ===
Line 1,763: Line 1,813:
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.286{{c}}
Optimal tunings:
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}


{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}
{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}


Badness (Sintel): 1.788
Badness (Sintel): 1.78


==== 13-limit ====
==== 13-limit ====
Line 1,776: Line 1,828:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.310{{c}}
Optimal tunings:
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}


{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def }}
{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}


Badness (Sintel): 1.366
Badness (Sintel): 1.37


=== Sfour ===
=== Sfour ===
Line 1,789: Line 1,843:
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.246{{c}}
Optimal tunings:
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}


{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness (Sintel): 2.531
Badness (Sintel): 2.53


==== 13-limit ====
==== 13-limit ====
Line 1,802: Line 1,858:
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}
Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/48 = 26.239{{c}}
Optimal tunings:
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}


{{Optimal ET sequence|legend=0| 45, 46, 91, 137d }}
{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}


Badness (Sintel): 2.144
Badness (Sintel): 2.14


== Trideci ==
== Trideci ==
: ''For the 5-limit version of this temperament, see [[13th-octave temperaments #Tridecatonic]].''
: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''


The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from "tridecim" (Latin for "[[wikipedia:13|thirteen]]").
The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,818: Line 1,876:


{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3


[[Optimal tuning]] ([[POTE]]): ~256/245 = 92.3077{{c}}, ~3/2 = 699.1410{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}


{{Optimal ET sequence|legend=1| 26, 65, 91, 156d, 247cdd }}
{{Optimal ET sequence|legend=1| 26, 65, 91 }}


[[Badness]] (Sintel): 4.671
[[Badness]] (Sintel): 4.67


=== 11-limit ===
=== 11-limit ===
Line 1,832: Line 1,895:
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}
Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}


Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.6179{{c}}
Optimal tunings:
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}


{{Optimal ET sequence|legend=0| 26, 65, 91, 156d, 247cdde }}
{{Optimal ET sequence|legend=0| 26, 65, 91 }}


Badness (Sintel): 2.796
Badness (Sintel): 2.80


=== 13-limit ===
=== 13-limit ===
Line 1,845: Line 1,910:
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}
Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}


Optimal tuning (POTE): ~22/21 = 92.3077{{c}}, ~3/2 = 699.2969{{c}}
Optimal tunings:
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}


{{Optimal ET sequence|legend=0| 26, 65f, 91f, 156dff }}
{{Optimal ET sequence|legend=0| 26, 65f, 91f }}
 
Badness (Sintel): 2.164
 
== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}
 
{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
 
{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}


[[Badness]] (Sintel): 7.916
Badness (Sintel): 2.16


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Ragismic_microtemperaments"