1156/1155: Difference between revisions
m Misc. cleanup |
m →Temperaments: grammar |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 19: | Line 19: | ||
== 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 | [[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 [[quadrantonismic chords]] are enabled. | ||
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. This tempers the [[harmonic series segment]] of quartertones, 32::36, to reduce it to three equidistant elements: 33/32, 34/33~35/34, 36/35. | |||
Alternatively, 1089/1088 ({{S|33}}) can be added to the comma list, which reduces the segment to two distinct elements: 33/32~34/33~35/34, 36/35; 1225/1224 ({{S|35}}) works similarly, resulting in 33/32, 34/33~35/34~36/35. Tempering out both 1089/1088 and 1225/1224 while observing 1156/1155 is another major option, resulting in 33/32~34/33, 35/34~36/35, and merging all these temperaments will lead to [[uniwiz]], a rank-3 temperament with a single quartertone representing all the differently sized quartertones in the 2.3.5.7.11.17-subgroup. | |||
=== Quadrantonic === | === Quadrantonic === | ||
| Line 27: | Line 31: | ||
{| class="right-all" | {| class="right-all" | ||
|- | |- | ||
| [⟨ || 1 || 0 || 0 || 0 || | | [⟨ || 1 || 0 || 0 || 0 || -4 || -3 || ], | ||
|- | |- | ||
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ], | | ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ], | ||
| Line 35: | Line 39: | ||
| ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ], | | ⟨ || 0 || 0 || 0 || 1 || 1 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 0 || | | ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]] | ||
|} | |} | ||
: mapping generators: ~2, ~3, ~5, ~7, ~ | : mapping generators: ~2, ~3, ~5, ~7, ~22/17 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.9696{{c}}, ~3/2 = 702.0236{{c}}, ~5/4 = 386.4564{{c}}, ~7/4 = 969.0065{{c}}, ~22/17 = 447.0219{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0185{{c}}, ~5/4 = 386.4333{{c}}, ~7/4 = 968.9917{{c}}, ~22/17 = 447.0568{{c}} | |||
{{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 }} | {{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 }} | ||
[[Badness]] (Sintel): 0.337 | |||
=== Quadrantonismic === | === Quadrantonismic === | ||
| Line 49: | Line 57: | ||
{| class="right-all" | {| class="right-all" | ||
|- | |- | ||
| [⟨ || 1 || 0 || 0 || 0 || | | [⟨ || 1 || 0 || 0 || 0 || -4 || 0 || -3 || ], | ||
|- | |- | ||
| ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ], | | ⟨ || 0 || 1 || 0 || 0 || 1 || 0 || 1 || ], | ||
| Line 57: | Line 65: | ||
| ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ], | | ⟨ || 0 || 0 || 0 || 1 || 1 || 0 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 0 || | | ⟨ || 0 || 0 || 0 || 0 || 2 || 0 || 1 || ], | ||
|- | |- | ||
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]] | ||
|} | |} | ||
: mapping generators: ~2, ~3, ~5, ~7, ~ | : mapping generators: ~2, ~3, ~5, ~7, ~22/17, ~13 | ||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.9696{{c}}, ~3/2 = 702.0236{{c}}, ~5/4 = 386.4564{{c}}, ~7/4 = 969.0065{{c}}, ~22/17 = 447.0219{{c}}, ~13/8 = 840.6188{{c}} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0185{{c}}, ~5/4 = 386.4333{{c}}, ~7/4 = 968.9917{{c}}, ~22/17 = 447.0568{{c}}, ~13/8 = 840.5850{{c}} | |||
{{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 }} | {{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 }} | ||
[[Badness]] (Sintel): 0.836 | |||
== Etymology == | == Etymology == | ||