Bird's eye view of temperaments by accuracy: Difference between revisions
m add generator tunings for srutal archagall |
m add generator tunings for septimal meantone and mothra |
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Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none if odd 9 is omitted) | Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none if odd 9 is omitted) | ||
[[#Generator tunings|Generator tunings]]: 18\31 | |||
Septimal meantone is an extension of [[#meantone]] that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS ([[12L 7s]]) for more freedom and availability. | Septimal meantone is an extension of [[#meantone]] that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS ([[12L 7s]]) for more freedom and availability. | ||
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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) | ||
[[#Generator tunings|Generator tunings]]: 5\26, 6\31 | |||
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 out 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 out [[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. | 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 out 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 out [[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. | ||