Hodge dual: Difference between revisions
m Sintel moved page Dual multivals and multimonzos to Hodge dual: I want to start properly documenting this |
Give a basic definition + wikipedia link |
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{{wikipedia|Hodge star operator}} | |||
In [[exterior algebra]] applied to [[regular temperament theory]], the '''Hodge dual''', or '''Hodge star''' is an operation that converts the [[Plücker coordinates]] of a temperament into the corresponding coordinates of the [[comma basis]], and vice versa. | |||
Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of bracket product (which is similar to the {{w|dot product}}) relating multivals and multimonzos, which we discuss first. | Given ''n'' basis elements (i.e. the number of primes in a prime limit) and a ''k''-multival '''W''' in this basis, there is a ''dual'' {{nowrap|(''n'' − ''k'')}}-multimonzo '''W'''°. Similarly, given a k-multimonzo '''M''', there is a dual {{nowrap|(''n'' − ''k'')}}-multival '''M'''º. The dual may be defined in terms of bracket product (which is similar to the {{w|dot product}}) relating multivals and multimonzos, which we discuss first. | ||