Bird's eye view of temperaments by accuracy: Difference between revisions
Fixed heading formatting (h1 is best avoided except for the title) |
→11-limit focus: note about 16/15 violating bound |
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[[#Generator tunings|Generator tunings]]: 3\31, 4\41, 7\72 | [[#Generator tunings|Generator tunings]]: 3\31, 4\41, 7\72 | ||
Bound-violating intervals: 16/15 (none in 11-odd-limit) | |||
Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering out {{nowrap| [[243/242|S9/S11 {{=}} (12/8)/(11/9)<sup>2</sup> {{=}} (3/2)/(11/9)<sup>2</sup>]] }}, splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering out [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[#Meantone]] for 31edo and [[#Garibaldi]] for 41edo. | Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering out {{nowrap| [[243/242|S9/S11 {{=}} (12/8)/(11/9)<sup>2</sup> {{=}} (3/2)/(11/9)<sup>2</sup>]] }}, splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)<sup>3</sup> = (3/2)/(8/7)<sup>3</sup>]] and then splitting the [[~]][[8/7]] in two by tempering out [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[#Meantone]] for 31edo and [[#Garibaldi]] for 41edo. | ||