Xenharmonic Wiki

Catalog of rank-4 temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
Eufalesio (talk | contribs)
autopatrolled
857 edits
Olympic add-29 extensions
Eufalesio (talk | contribs)
autopatrolled
857 edits
Chose another olympic add-29 extension that is much better and likely to be canon, made links
Line 919: Line 919:
== Olympic (131072/130977) ==
== Olympic (131072/130977) ==
Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.  
Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.  
It also has a [[strong extension]] to include [[29/1|prime 29]] though with higher complexity.


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
Line 940: Line 942:
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 2080/2079, 4096/4095
[[Comma list]]: [[2080/2079]], [[4096/4095]]


{{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
{{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
Line 957: Line 959:


==== Subgroup: 2.3.5.7.11.13.29 ====
==== Subgroup: 2.3.5.7.11.13.29 ====
[[Comma list]]: 2080/2079, 4096/4095, 112000/111969
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5104/5103]]


{{Mapping|legend=1| 1 0 0 0 17 12 -22| 0 1 0 0 -5 -2 4| 0 0 1 0 0 -1 4| 0 0 0 1 -2 -1 4}}
{{Mapping|legend=1| 1 0 0 0 17 12 -21| 0 1 0 0 -5 -2 11| 0 0 1 0 0 -1 0| 0 0 0 1 -2 -1 3}}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.964{{c}}, ~3/2 = 702.027{{c}}, ~5/4 = 386.259{{c}}, ~7/4 = 968.908{{c}}
* [[WE]]: ~2 = 1199.960{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.247{{c}}, ~7/4 = 968.895{{c}}
: [[error map]]: {{val|-0.036, 0.072, -0.055, 0.083, 0.124, -0.177, -0.016}}
: [[error map]]: {{val|-0.040, 0.053, -0.067, 0.069, 0.164, -0.171, 0.048}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.087{{c}}, ~5/4 = 386.334{{c}}, ~7/4 = 969.007{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.070{{c}}, ~5/4 = 386.337{{c}}, ~7/4 = 969.013{{c}}
: error map: {{val|0.000, 0.132, 0.020, 0.182, 0.233, -0.042, 0.134}}
: error map: {{val|0.000, 0.115, 0.023, 0.187, 0.304, -0.018, 0.237}}


{{Optimal ET sequence|legend=1|TBA}}
{{Optimal ET sequence|legend=1|TBA}}


[[Badness]] (Sintel): 0.638
[[Badness]] (Sintel): 0.505


=== Metaolympic ===
=== Metaolympic ===
Metaolympic is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma. It is better without prime 17, but prime 17 can be added easily into the temperament by also tempering out 4200/4199, which also takes advantage of the split prime 7, at the cost of higher complexity.  
Metaolympic is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma. It is better without prime 17, but prime 17 can be added easily into the temperament by also tempering out 4200/4199, which also takes advantage of the split prime 7 at the cost of higher complexity.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 2080/2079, 4096/4095, 35000/34969
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[35000/34969]]


Mapping: {{mapping| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}
[[Mapping]]: {{mapping| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}
: mapping generators: ~2, ~3, ~5, ~187/100
: mapping generators: ~2, ~3, ~5, ~187/100


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9530{{c}}, ~3/2 = 702.0493{{c}}, ~5/4 = 386.2541{{c}}, ~187/100 = 1084.4393{{c}}
* [[WE]]: ~2 = 1199.9530{{c}}, ~3/2 = 702.0493{{c}}, ~5/4 = 386.2541{{c}}, ~187/100 = 1084.4393{{c}}
* [[error map]]: {{val|-0.047, 0.047, -0.154, 0.006, 0.208, -0.088, 0.184}}
* [[error map]]: {{val|-0.047, 0.047, -0.154, 0.006, 0.208, -0.088, 0.184}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}
* [[error map]]: {{val|0.000, 0.125, -0.057, 0.136, 0.359, 0.094, 0.363}}
* [[error map]]: {{val|0.000, 0.125, -0.057, 0.136, 0.359, 0.094, 0.363}}


{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}


Badness (Sintel): 1.59
[[Badness]] (Sintel): 1.59


==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
[[Subgroup]]: 2.3.5.7.11.13.17.19


Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199
[[Comma list]]: [[2080/2079]], [[3250/3249]], [[4096/4095]], [[4200/4199]]


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}
[[Mapping]]: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 1,006: Line 1,008:
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}


Badness (Sintel): 0.764
[[Badness]] (Sintel): 0.764


==== 2.3.5.7.11.13.17.19.29 subgroup ====
==== 2.3.5.7.11.13.17.19.29 subgroup ====
Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199, 112000/111969
[[Comma list]]: [[2080/2079]], [[3250/3249]], [[4096/4095]], [[4200/4199]], [[5104/5103]]
 
[[Mapping]]: {{mapping| 1 0 0 1 15 11 -13 6 -18| 0 1 0 0 -5 -2 5 -2 -11| 0 0 1 0 0 -1 2 1 0| 0 0 0 2 -4 -2 5 -1 6}}
 
Optimal tunings:
* [[WE]]: ~2 = 1199.951{{c}}, ~3/2 = 701.968{{c}}, ~5/4 = 386.215{{c}}, ~187/100 = 1084.473 {{c}} [[error map]]: {{val|-0.044, 0.056, -0.144, 0.008, 0.212, -0.080, 0.205, -0.068, -0.031}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.048{{c}}, ~5/4 = 386.316{{c}}, ~187/100 = 1084.517{{c}} [[error map]]: {{val|0.000, 0.124, -0.051, 0.134, 0.365, 0.090, 0.368, 0.111, 0.177}}
 
{{Optimal ET sequence|legend=0|TBA}}
 
[[Badness]] (Sintel): 1.290
 
==== 2.3.5.7.11.13.19 subgroup ====
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5776/5775]]
 
[[Mapping]]: {{mapping| 1 0 0 1 15 11 -13 6| 0 1 0 0 -5 -2 5 -2| 0 0 1 0 0 -1 2 1| 0 0 0 2 -4 -2 5 -1}}
 
Optimal tunings:
* [[WE]]: ~2 = 1199.964{{c}}, ~3/2 = 702.028{{c}}, ~5/4 = 386.250{{c}}, ~187/100 = 1084.468{{c}}  [[error map]]: {{val|-0.036, 0.073, -0.064, 0.073, 0.129, -0.166, -0.004}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.089{{c}}, ~5/4 = 386.326{{c}}, ~187/100 = 1084.501{{c}}  [[error map]]: {{val|0.000, 0.134, 0.012, 0.175, 0.236, -0.032, 0.135}}
 
{{Optimal ET sequence|legend=0|TBA}}
 
[[Badness]] (Sintel): 0.618
 
==== 2.3.5.7.11.13.19.29 subgroup ====
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5104/5103]], [[5776/5775]]


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18| 0 1 0 0 -5 -2 5 -2 4| 0 0 1 0 0 -1 2 1 4| 0 0 0 2 -4 -2 5 -1 8}}
[[Mapping]]: {{mapping| 1 0 0 1 15 11 -13 6 -18| 0 1 0 0 -5 -2 5 -2 -11| 0 0 1 0 0 -1 2 1 0| 0 0 0 2 -4 -2 5 -1 6}}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.951{{c}}, ~3/2 = 701.968{{c}}, ~5/4 = 386.215{{c}}, ~187/100 = 1084.473 {{c}} [[error map]]: {{val|-0.049, 0.013, -0.099, 0.070, 0.219, -0.162, 0.313, 0.000, -0.183}}
* [[WE]]: ~2 = 1199.959{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.245{{c}}, ~187/100 = 1084.468{{c}} [[error map]]: {{val|-0.041, 0.053, -0.069, 0.069, 0.163, -0.170, 0.005, 0.048}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.048{{c}}, ~5/4 = 386.316{{c}}, ~187/100 = 1084.517{{c}} [[error map]]: {{val|0.000, 0.093, 0.002, 0.208, 0.373, 0.026, 0.502, 0.189, 0.015}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.071{{c}}, ~5/4 = 386.333{{c}}, ~187/100 = 1084.506{{c}} [[error map]]: {{val|-0.000, 0.116, 0.019, 0.187, 0.302, -0.015, 0.171, 0.241}}


{{Optimal ET sequence|legend=0|TBA}}
{{Optimal ET sequence|legend=0|TBA}}


Badness (Sintel): 1.290
[[Badness]] (Sintel): 1.098


== Alphaxenic (131769/131072) ==
== Alphaxenic (131769/131072) ==
Retrieved from "https://en.xen.wiki/w/Catalog_of_rank-4_temperaments"