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== Temperaments == | == Temperaments == | ||
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. | ||
=== Quadrantonismic === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
[[Mapping]]: <br> | |||
{| class="right-all" | |||
|- | |||
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -1 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 1 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || -2 || 0 || -1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | |||
|} | |||
: 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, ~13/8 | |||
{{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 }} | |||
=== Quadrantonic === | |||
[[Subgroup]]: 2.3.5.7.11.13.17 | |||
[[Mapping]]: <br> | |||
{| class="right-all" | |||
|- | |||
| [⟨ || 1 || 0 || 0 || 0 || 0 || -1 || ], | |||
|- | |||
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 1 || 0 || 1 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ], | |||
|- | |||
| ⟨ || 0 || 0 || 0 || 0 || -2 || -1 || ]] | |||
|} | |||
: 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 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 }} | |||
== Etymology == | == Etymology == | ||