Harmonic entropy: Difference between revisions
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Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit. | Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit. | ||
== References == | == References == | ||
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* [[Harmonic entropy of just intervals]] | * [[Harmonic entropy of just intervals]] | ||
* [[Low harmonic entropy linear temperaments]] | * [[Low harmonic entropy linear temperaments]] | ||
== Todo == | |||
{{todo|inline=1|complete section|text=Add 3HE, both for finite HE and for ''N'' → ∞.}} | |||
{{todo|inline=1|complete section|add table|text=Write-up of fast computation for infinite zeta-UHE, perhaps with a zeta table.}} | |||
{{todo|inline=1|add illustration|text=Addition of many more pictures.}} | |||
[[Category:Terms]] | [[Category:Terms]] | ||