Bird's eye view of temperaments by accuracy: Difference between revisions
improve & clarify conventions |
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 |
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Note count: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]]) | Note count: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]]) | ||
Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none | Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none if odd 9 is 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. | ||
It has two main extensions to prime 11, both similarly complex, discussed in [[meantone vs meanpop]], though the one called [[undecimal meantone]] is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament [[ | It has two main extensions to prime 11, both similarly complex, discussed in [[meantone vs meanpop]], though the one called [[undecimal meantone]] is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament [[#Didacus]], which can be seen as every other gen of undecimal meantone. An alternative extension that splits the generator is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. This leads to [[#Migration]] if you accept the septimal meantone mapping of 7 (which becomes double as complex), or [[mohaha]] if you interpret it as no-7's. | ||
=== [[Mothra]] === | === [[Mothra]] === | ||