Bird's eye view of temperaments by accuracy: Difference between revisions
m →Septimal meantone: justify septimal meantone's accuracy classification by clarifying that all bound-violating intervals are a consequence of odd 9. also note other extension to prime 11 |
m →Mothra: fix links per new convention and in preparation for new entries |
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Bound-violating intervals: [[9/8]], [[10/9]] (none in [[7-odd-limit]] or if 9 is omitted) | Bound-violating intervals: [[9/8]], [[10/9]] (none in [[7-odd-limit]] or if 9 is omitted) | ||
Mothra makes a near-just [[8/7]] equal to a third of a meantone fifth ([[~]][[3/2]]) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering 81/80 makes S6 = [[36/35]] (the distance between [[6/5]] and [[7/6]]) and S8 = [[64/63]] (the distance between [[8/7]] and [[9/8]]) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = [[49/48]] (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering [[1728/1715|S6/S7]], we have that 7/6 is a third of [[8/5]]. Combining it with [[ | Mothra makes a near-just [[8/7]] equal to a third of a meantone fifth ([[~]][[3/2]]) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering 81/80 makes S6 = [[36/35]] (the distance between [[6/5]] and [[7/6]]) and S8 = [[64/63]] (the distance between [[8/7]] and [[9/8]]) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = [[49/48]] (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering [[1728/1715|S6/S7]], we have that 7/6 is a third of [[8/5]]. Combining it with [[#Septimal meantone]] (among other things) results in [[31edo]], where it is quite accurate, while combining it with the less accurate [[#Flattone]] results in [[26edo]], where it is quite damaged. | ||
The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5. | The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5. | ||