Bird's eye view of temperaments by accuracy: Difference between revisions
move "generator tunings" explanation to top of page, as it's important for accessibility and when clicked it might not be clear visually since the bottom of the page is reached, also add "note counts" explanation |
m missing size for septimal meantone & mild clarity increase |
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== 7-limit focus == | == 7-limit focus == | ||
=== [[Septimal meantone]] === | === [[Septimal meantone]] === | ||
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]], [[12L 7s]]) | ||
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) | ||
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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 [[#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 out [[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 out [[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. | 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 in half is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering out [[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 out [[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]] === | ||