Tenney–Euclidean temperament measures: Difference between revisions
→TE simple badness: add note about span |
Span -> interval span |
||
| Line 75: | Line 75: | ||
=== Reduction to the Span of a Comma === | === Reduction to the Span of a Comma === | ||
It is notable that if M is codimension-1, we may view it as representing [[The dual|the dual]] of a single comma. In this situation, the simple badness happens to reduce to the [[ | It is notable that if M is codimension-1, we may view it as representing [[The dual|the dual]] of a single comma. In this situation, the simple badness happens to reduce to the [[Interval span|span]] of the comma, up to a constant multiplicative factor, so that the span of any comma can itself be thought of as measuring the complexity relative to the error of the temperament vanishing that comma. | ||
This relationship also holds if TOP is used rather than TE, as the TOP damage associated with tempering some comma n/d is log(n/d)/(n*d), and if we multiply by the complexity n*d, we simply get log(n/d) as our result. | This relationship also holds if TOP is used rather than TE, as the TOP damage associated with tempering some comma n/d is log(n/d)/(n*d), and if we multiply by the complexity n*d, we simply get log(n/d) as our result. | ||