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Quartonic family: Difference between revisions

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== Quartonic ==
== Quartonic ==
The name ''quartonic'' means quarter-tone, which is the [[generator]] of this temperament.
The name ''quartonic'' refers to the [[quartertone]], since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. [[porcupine comma]]) is the [[generator]] of this temperament. The [[ploidacot]] for the temperament is omega-hendecacot, with eleven generator steps giving the [[4/3|perfect fourth]]. It is a member of the [[schismic–Mercator equivalence continuum]], with equivalence number ''n'' = 11/2.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
: mapping generators: ~2, ~250/243


[[Optimal tuning]] ([[CTE]]): 2 = 1200.0000{{c}}, ~250/243 = 45.2368{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9709{{c}}, ~250/243 = 45.2318{{c}}
: [[error map]]: {{val| -0.029 +0.437 -0.574 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 45.2349{{c}}
: error map: {{val| 0.000 +0.461 -0.541 }}


{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}


[[Badness]] (Smith): 0.117250
[[Badness]] (Sintel): 2.75


=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].  
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].  
* [[Hemimage comma|10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* [[10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[Landscape comma|250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[Dimcomp comma|390625/388962]] or [[canousma|4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[390625/388962]] or [[4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.
* [[16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.


== Septimal quartonic ==
== Septimal quartonic ==
Septimal quartonic tempers out the [[orwellisma]], the [[octagar comma]], as well as the [[ragisma]], and may be described as the {{nowrap| 26 & 27 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 45.2652{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1424{{c}}, ~36/35 = 45.1064{{c}}
: [[error map]]: {{val| -0.858 +0.160 -0.801 +3.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 45.1881{{c}}
: error map: {{val| 0.000 +0.976 +0.301 +5.234 }}


{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}


[[Badness]] (Smith): 0.042632
[[Badness]] (Sintel): 1.08


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}
Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1674{{c}}
Optimal tunings:
* WE: ~2 = 1198.8787{{c}}, ~36/35 = 44.9994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.0863{{c}}


{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}


Badness (Smith): 0.034031
Badness (Sintel): 1.13


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}
Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1632{{c}}
Optimal tunings:
* WE: ~2 = 1199.2991{{c}}, ~36/35 = 45.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.1049{{c}}


{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d, 186d }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d }}


Badness (Smith): 0.023875
Badness (Sintel): 0.987


=== Quarto ===
=== Quarto ===
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Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.4022{{c}}
Optimal tunings:
* WE: ~2 = 1198.3094{{c}}, ~36/35 = 45.0688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2329{{c}}


{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e, 80ee }}


Badness (Smith): 0.041786
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.3857{{c}}
Optimal tunings:
* WE: ~2 = 1198.9315{{c}}, ~36/35 = 45.1650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2640{{c}}


{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e }}


Badness (Smith): 0.027692
Badness (Sintel): 1.14


=== Quartz ===
=== Quartz ===
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Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3313{{c}}
Optimal tunings:
* WE: ~2 = 1200.5082{{c}}, ~33/32 = 45.4047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


{{Optimal ET sequence|legend=0| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness (Smith): 0.053285
Badness (Sintel): 1.76


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}
Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3168{{c}}
Optimal tunings:
* WE: ~2 = 1200.5899{{c}}, ~33/32 = 45.4096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


{{Optimal ET sequence|legend=0| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness (Smith): 0.028818
Badness (Sintel): 1.19


=== Biquartonic ===
=== Biquartonic ===
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Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
: mapping generators: ~99/70, ~36/35


Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~36/35 = 45.2678{{c}}
Optimal tunings:
* WE: ~99/70 = 599.5389{{c}}, ~36/35 = 45.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~36/35 = 45.1692{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 186de, 266dde }}


Badness (Smith): 0.060737
Badness (Sintel): 2.01


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}
Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}


Optimal tuning (CTE): ~55/39 = 600.0000{{c}}, ~40/39 = 45.2544{{c}}
Optimal tunings:
* WE: ~55/39 = 599.6870{{c}}, ~40/39 = 45.1293{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~40/39 = 45.1789{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness (Smith): 0.039891
Badness (Sintel): 1.65


==== 17-limit ====
==== 17-limit ====
Line 151: Line 179:
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}


Optimal tuning (CTE): ~17/12 = 600.0000{{c}}, ~34/33 = 45.2397{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7706{{c}}, ~40/39 = 45.1427{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~40/39 = 45.1790{{c}}


{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness (Smith): 0.028112
Badness (Sintel): 1.43


==== 19-limit ====
==== 19-limit ====
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Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}


Optimal tuning (CTE): ~17/12 = 600.000{{c}}, ~39/38 = 45.222{{c}}
Optimal tunings:
* WE: ~17/12 = 599.7848{{c}}, ~39/38 = 45.1288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~39/38 = 45.1634{{c}}


{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}


Badness (Smith): 0.0213
Badness (Sintel): 1.30


=== Yarm ===
=== Yarm ===
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Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0880{{c}}
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0617{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80, 239dd }}


Badness (Smith): 0.099950
Badness (Sintel): 3.30


==== 13-limit ====
==== 13-limit ====
Line 190: Line 224:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0842{{c}}
Optimal tunings:
* WE: ~2 = 1199.5798{{c}}, ~100/99 = 15.0551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0681{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness (Smith): 0.061645
Badness (Sintel): 2.55


==== 17-limit ====
==== 17-limit ====
Line 203: Line 239:
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0840{{c}}
Optimal tunings:
* WE: ~2 = 1199.6800{{c}}, ~100/99 = 15.0624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0705{{c}}


{{Optimal ET sequence|legend=0| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness (Smith): 0.046718
Badness (Sintel): 2.38


== Yarman I ==
== Yarman I ==
Retrieved from "https://en.xen.wiki/w/Quartonic_family"