Bird's eye view of temperaments by accuracy: Difference between revisions
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== 7-limit focus == | == 7-limit focus == | ||
=== [[Septimal meantone]] === | === [[Septimal meantone]] === | ||
Note count: 11 for { | Note count: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]]) | ||
Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] | Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none in [[7-odd-limit]] or if 9 and 25 are omitted) | ||
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. | ||
=== [[Mothra]] === | |||
Note counts: | |||
* 14 for {3, 5, 7, 9, 21, 35, 49} ([[5L 11s]]) | |||
* 18 to add {15} [[5L 16s]]) | |||
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 [[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. | |||
== 11-limit focus == | == 11-limit focus == | ||