Limmic temperaments: Difference between revisions
m Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}" |
Community poll result: 7-limit {28/27, 49/48} is the canonical extension of 5-limit {256/243}, hence *septimal blackwood* and no need of a distinct name. |
||
| Line 1: | Line 1: | ||
{{interwiki | {{interwiki | ||
| en = Limmic temperaments | |||
| de = Blackwood-Limmisch | | de = Blackwood-Limmisch | ||
| es = | | es = | ||
| ja = | | ja = | ||
| Line 8: | Line 8: | ||
'''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it. | '''Limmic temperaments''' are [[temperament]]s that [[temper out]] the Pythagorean limma, [[256/243]]. As a consequence, [[3/2]] is always represented by 3\5, 720 [[cent]]s assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it. | ||
== | == Blackwood == | ||
Blackwood is the 5edo [[circle of fifths]] with an independent dimension for the harmonic 5. It can be described as the {{nowrap| 5 & 10 }} temperament. [[15edo]] is an obvious tuning. | |||
The only extension to the 7-limit that makes any sense is to map the [[7/4|harmonic seventh]] to 4\5, tempering out [[28/27]], [[49/48]], and [[64/63]]. This is known as ''blacksmith'' in earlier materials, including [[Graham Breed]]'s temperament finder. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 18: | Line 21: | ||
: mapping generators: ~9/8, ~5 | : mapping generators: ~9/8, ~5 | ||
[[Optimal tuning]] ([[POTE]]): ~9/8 = | [[Optimal tuning]] ([[POTE]]): ~9/8 = 240.000, ~5/4 = 399.594 | ||
{{Optimal ET sequence|legend=1| 5, 10, 15 }} | {{Optimal ET sequence|legend=1| 5, 10, 15 }} | ||
[[Badness]]: 0.063760 | [[Badness]] (Smith): 0.063760 | ||
[[File:BlackwoodMajor 15edo.mp3]] [[:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]] | [[File:BlackwoodMajor 15edo.mp3]] [[:File:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]] | ||
Blackwood major scale in 15edo | Blackwood major scale in 15edo | ||
=== 7-limit === | === 7-limit (blacksmith) === | ||
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of | [[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of septimal blackwood]] | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 39: | Line 42: | ||
{{Multival|legend=1| 0 5 0 8 0 -14 }} | {{Multival|legend=1| 0 5 0 8 0 -14 }} | ||
[[Optimal tuning]] ([[POTE]]): ~8/7 = | [[Optimal tuning]] ([[POTE]]): ~8/7 = 240.000, ~5/4 = 392.767 | ||
{{Optimal ET sequence|legend=1| 5, 10, 15, 40b, 55b }} | {{Optimal ET sequence|legend=1| 5, 10, 15, 40b, 55b }} | ||
[[Badness]]: 0.025640 | [[Badness]] (Smith): 0.025640 | ||
=== 11-limit === | === 11-limit === | ||
| Line 52: | Line 55: | ||
Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }} | Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }} | ||
Optimal tuning (POTE): ~8/7 = | Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 394.948 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5, 10, 15, 40be, 55be, 70bde, 85bcde }} | ||
Badness: 0.024641 | Badness (Smith): 0.024641 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 65: | Line 68: | ||
Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }} | Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }} | ||
Optimal tuning (POTE): ~8/7 = | Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 391.037 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5, 10, 15, 25e, 40bef }} | ||
Badness: 0.020498 | Badness (Smith): 0.020498 | ||
=== Farrier === | === Farrier === | ||
| Line 78: | Line 81: | ||
Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }} | Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }} | ||
Optimal tuning (POTE): ~8/7 = | Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 398.070 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5e, 10e, 15 }} | ||
Badness: 0.029200 | Badness (Smith): 0.029200 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 91: | Line 94: | ||
Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }} | Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }} | ||
Optimal tuning (POTE): ~8/7 = | Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 396.812 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5e, 10e, 15 }} | ||
Badness: 0.022325 | Badness (Smith): 0.022325 | ||
=== Ferrum === | === Ferrum === | ||
| Line 104: | Line 107: | ||
Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }} | Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }} | ||
Optimal tuning (POTE): ~8/7 = | Optimal tuning (POTE): ~8/7 = 240.000, ~5/4 = 374.763 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5e, 10 }} | ||
Badness: 0.030883 | Badness (Smith): 0.030883 | ||
== Blackweed == | == Blackweed == | ||
| Line 121: | Line 124: | ||
: sval mapping generators: ~9/8, ~11/7 | : sval mapping generators: ~9/8, ~11/7 | ||
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~11/7 = 786.2215 | [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~9/8 = 240.000, ~11/7 = 786.2215 | ||
{{Optimal ET sequence|legend=1| 15, 20, 35b }} | {{Optimal ET sequence|legend=1| 15, 20, 35b }} | ||