225/224: Difference between revisions

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The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between pairs of 7-limit ratios, for example as {{nowrap| ([[15/14]])/([[16/15]]) }} or {{nowrap| ([[45/32]])/([[7/5]]) }}. ~~Maybe most simply~~, it can be seen as the amount by which ~~{{nowrap|~~ [[15/8]] * [[15/8]] exceeds [[7/2]] }}, or ~~equivalently,~~ the amount by which ~~{{nowrap| (15/8)/(16/15) =~~ [[~~225~~/~~128~~]] ~~}} exceeds 7/4.~~

The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between pairs of 7-limit ratios, for example as {{nowrap| ([[15/14]])/([[16/15]]) }} or {{nowrap| ([[45/32]])/([[7/5]]) }}. Moreover, it can be seen as the amount by which [[225/128]], a stack of two [[15/8]]'s [[octave reduction|octave-reduced]], exceeds [[7/4]], or as the amount by which [[25/16]], a stack of two [[5/4]]'s, exceeds [[14/9]]. Hence, it is also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.

~~Another useful relation is as the difference between the~~ [[25/24]], ~~the classical chromatic semitone, and~~ [[28/27]]~~, the septimal third-tone~~. Hence, it is also the difference between [[32/25]] and [[9/7]], and between [[75/64]] and [[7/6]].

In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).

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 {{Wikipedia|Septimal kleisma}}
-The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between pairs of 7-limit ratios, for example as {{nowrap| ([[15/14]])/([[16/15]]) }} or {{nowrap| ([[45/32]])/([[7/5]]) }}. Maybe most simply, it can be seen as the amount by which {{nowrap| [[15/8]] * [[15/8]] exceeds [[7/2]] }}, or equivalently, the amount by which {{nowrap| (15/8)/(16/15) = [[225/128]] }} exceeds 7/4.
+The interval of '''225/224''', the '''marvel comma''', otherwise known as the '''septimal kleisma''', is a [[7-limit]] [[superparticular]] [[comma]]. It pops up as the difference between pairs of 7-limit ratios, for example as {{nowrap| ([[15/14]])/([[16/15]]) }} or {{nowrap| ([[45/32]])/([[7/5]]) }}. Moreover, it can be seen as the amount by which [[225/128]], a stack of two [[15/8]]'s [[octave reduction|octave-reduced]], exceeds [[7/4]], or as the amount by which [[25/16]], a stack of two [[5/4]]'s, exceeds [[14/9]]. Hence, it is also the difference between [[75/64]] and [[7/6]], and between [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone.
-Another useful relation is as the difference between the [[25/24]], the classical chromatic semitone, and [[28/27]], the septimal third-tone. Hence, it is also the difference between [[32/25]] and [[9/7]], and between [[75/64]] and [[7/6]].
 In terms of commas, it is the difference between [[81/80]] and [[126/125]] and is tempered out alongside these two commas in [[septimal meantone]]. In the 11-limit it factors neatly into ([[385/384]])([[540/539]]), and in the 13-limit, ([[351/350]])([[625/624]]) or ([[325/324]])([[729/728]]).