Xenharmonic Wiki

Catalog of rank-4 temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,160 edits
"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 171: Line 171:
[[Comma list]]: [[385/384]]
[[Comma list]]: [[385/384]]


[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 1 }}, {{val| 0 0 1 0 -1 }}, {{val| 0 0 0 1 -1 }}]
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}


Mapping generators: ~2, ~3, ~5, ~7
: mapping generators: ~2, ~3, ~5, ~7


[[Transpose]]: [2 3 5 7 385/35]
[[Transpose]]: [2 3 5 7 385/35]
Line 180: Line 180:
* [[11-odd-limit]]
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 -1/3 2/3 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}, {{monzo| 7/3 1/3 -1/3 -1/3 2/3 }}]
: Eigenmonzos (unchanged-intervals): 2, 3, 7/5, 11/5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.7/5.11/5


{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}


[[Badness]]: 15.159 × 10<sup>-9</sup>
[[Badness]]: 15.2 × 10<sup>-9</sup>


=== Martwin ===
=== Martwin ===
Line 191: Line 191:
[[Comma list]]: 325/324, 385/384
[[Comma list]]: 325/324, 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]
{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}
 
Mapping generators: ~2, ~3, ~5, ~7


[[Transpose]]: [2 3 5 7 385/35 324/25]
[[Transpose]]: [2 3 5 7 385/35 324/25]
Line 203: Line 201:
* [[13-odd-limit|13-]] and [[15-odd-limit]]
* [[13-odd-limit|13-]] and [[15-odd-limit]]
: [{{val| 1 0 0 0 0 0 }}, {{val| 0 1 0 0 0 0 }}, {{val| 2/3 4/3 1/3 0 0 -1/3 }}, {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}, {{val| 19/6 -1/6 -1/6 -1/2 1/2 1/6 }}, {{val| 2/3 4/3 -2/3 0 0 2/3 }}]
: [{{val| 1 0 0 0 0 0 }}, {{val| 0 1 0 0 0 0 }}, {{val| 2/3 4/3 1/3 0 0 -1/3 }}, {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}, {{val| 19/6 -1/6 -1/6 -1/2 1/2 1/6 }}, {{val| 2/3 4/3 -2/3 0 0 2/3 }}]
: Eigenmonzos (unchanged-intervals): 2, 14/11, 13/10, 4/3
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3.11/7.13/5


{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}


[[Badness]]: 2.206 × 10<sup>-6</sup>
[[Badness]]: 2.21 × 10<sup>-6</sup>


=== Ancient ===
=== Ancient ===
Line 214: Line 212:
[[Comma list]]: 385/384, 625/624
[[Comma list]]: 385/384, 625/624


[[Mapping]]: [{{val| 1 0 0 0 7 -4 }}, {{val| 0 1 0 0 1 -1 }}, {{val| 0 0 1 0 -1 4 }}, {{val| 0 0 0 1 -1 0 }}]
{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}


[[Transpose]]: [2 3 5 7 385/35 625/48]
[[Transpose]]: [2 3 5 7 385/35 625/48]
Line 220: Line 218:
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}


[[Badness]]: 2.573 × 10<sup>-6</sup>
[[Badness]]: 2.57 × 10<sup>-6</sup>


=== Commas 351/350, 385/384 ===
=== Commas 351/350, 385/384 ===
Line 227: Line 225:
[[Comma list]]: 351/350, 385/384
[[Comma list]]: 351/350, 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 1 }}, {{val| 0 1 0 0 1 -3 }}, {{val| 0 0 1 0 -1 2 }}, {{val| 0 0 0 1 -1 1 }}]
{{Mapping|legend=1| 1 0 0 0 7 1 | 0 1 0 0 1 -3 | 0 0 1 0 -1 2 | 0 0 0 1 -1 1 }}


Mapping generators: ~2, ~3, ~5, ~7
{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}


{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
[[Badness]]: 2.98 × 10<sup>-6</sup>


=== Commas 352/351, 385/384 ===
=== Zaxa ===
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 352/351, 385/384
[[Comma list]]: 352/351, 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 -2 }}, {{val| 0 0 1 0 -1 -1 }}, {{val| 0 0 0 1 -1 -1 }}]
{{Mapping|legend=1| 1 0 0 0 7 12 |0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}


Mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829


{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
[[Badness]]: 3.35 × 10<sup>-6</sup>


=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Line 249: Line 249:
[[Comma list]]: 364/363, 385/384
[[Comma list]]: 364/363, 385/384


[[Mapping]]: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 1 3 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 -3 }}]
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 3 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 -3 }}


Mapping generators: ~2, ~3, ~5, ~7
{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}


{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
[[Badness]]: 3.32 × 10<sup>-6</sup>


=== Commas 385/384, 847/845 ===
=== Commas 385/384, 847/845 ===
Line 260: Line 260:
[[Comma list]]: 385/384, 847/845
[[Comma list]]: 385/384, 847/845


[[Mapping]]: [{{val| 1 0 0 0 7 7 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 1 1 -2 -2 }}, {{val| 0 0 0 2 -2 -1 }}]
{{Mapping|legend=1| 1 0 0 0 7 7 | 0 1 0 0 1 1 | 0 0 1 1 -2 -2 | 0 0 0 2 -2 -1 }}


Mapping generators: ~2, ~3, ~5, ~13/11
: mapping generators: ~2, ~3, ~5, ~13/11


{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
[[Badness]]: 4.15 × 10<sup>-6</sup>


== Swetismic (540/539) ==
== Swetismic (540/539) ==
Retrieved from "https://en.xen.wiki/w/Catalog_of_rank-4_temperaments"