1156/1155: Difference between revisions
→Temperaments: +data |
mNo edit summary |
||
| (3 intermediate revisions by one other user not shown) | |||
| Line 21: | Line 21: | ||
Tempering out this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone. | Tempering out this comma in the 17-limit results in the rank-6 '''quadrantonismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''quadrantonic temperament'''. In either case [[35/33]] is split into two equal parts, each representing 34/33~35/34, and enables [[quadrantonismic chords]]. If [[9801/9800]] is also added to the comma list, the quartertone above becomes literally a quarter of [[9/8]] and is tuned exactly middle of [[33/32]], the undecimal quartertone, and [[36/35]], the septimal quartertone. | ||
=== | === Quadrantonic === | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11.17 | ||
[[ | [[Sval]] [[mapping]]: <br> | ||
{| class="right-all" | {| class="right-all" | ||
|- | |- | ||
| [⟨ || 1 | | [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 1 || 0 || 0 || 1 | | ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 1 || 0 || 1 | | ⟨ || 0 || 0 || 1 || 0 || 1 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 1 || 1 | | ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 0 || -2 | | ⟨ || 0 || 0 || 0 || 0 || -2 || -1 || ]] | ||
|} | |} | ||
: mapping generators: ~2, ~3, ~5, ~7, ~34/11 | : sval mapping generators: ~2, ~3, ~5, ~7, ~34/11 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186 | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186 | ||
{{Optimal ET sequence|legend=1| 17cg, 19eg, 22 | {{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 27eg, 39dg, 43, 46, 65d, 68, 72, 118, 171, 183, 239, 282, 301, 311, 400, 422, 472, 494, 894, 1012g, 1205, 1388 }} | ||
=== | === Quadrantonismic === | ||
[[Subgroup]]: 2.3.5.7.11.13.17 | [[Subgroup]]: 2.3.5.7.11.13.17 | ||
| Line 52: | Line 50: | ||
{| class="right-all" | {| class="right-all" | ||
|- | |- | ||
| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ], | | [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -1 || ], | ||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ], | |||
|- | |- | ||
| ⟨ || 0 || 1 || 0 || | | ⟨ || 0 || 0 || 1 || 0 || 1 || 0 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 1 || | | ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || | | ⟨ || 0 || 0 || 0 || 0 || -2 || 0 || -1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 0 || | | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | ||
|} | |} | ||
: mapping generators: ~2, ~3, ~5, ~7, ~34/11 | : mapping generators: ~2, ~3, ~5, ~7, ~34/11, ~13 | ||
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186 | [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 701.9948, ~5/4 = 386.3991, ~7/4 = 968.9508, ~17/11 = 752.9186, ~13/8 | ||
{{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 27eg, | {{Optimal ET sequence|legend=1| 17cg, 19eg, 22, 26, 27eg, 29g, 39dfg, 43, 46, 65d, 68, 72, 111, 121, 140, 171, 183, 217, 282, 301, 311, 354, 400, 422, 494, 894, 1012g, 1133, 1205, 1506g, 1627e }} | ||
== Etymology == | == Etymology == | ||
| Line 77: | Line 77: | ||
[[Category:Quadrantonismic]] | [[Category:Quadrantonismic]] | ||
[[Category:Commas named for the intervals they stack]] | |||