Interval span: Difference between revisions
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The span of some comma can be thought of as a very simple metric to measure the usefulness of the temperament vanishing that comma. This is because the formula for the [[TE simple badness|simple badness]], which is the product of the best-case tuning error and the complexity of the temperament, happens to reduce to the span of the comma in the situation where there is only one comma tempered out. | The span of some comma can be thought of as a very simple metric to measure the usefulness of the temperament vanishing that comma. This is because the formula for the [[TE simple badness|simple badness]], which is the product of the best-case tuning error and the complexity of the temperament, happens to reduce to the span of the comma in the situation where there is only one comma tempered out. | ||
[[Interval size unit|Various units have been used for the span of an interval]]. Typically [[cents]] are used, which can be thought of as the logarithm taken to the base <math>2^{1/1200}</math>. However, in mathematical situations it is often useful to work with the log base 2, meaning units of octaves, or even simply in the natural logarithm of base e, meaning units of "[[Wikipedia:Neper|nepers]]". [[Martin Gough]] introduced the unit of a "dineper" in his work on [[Logarithmic approximants]]. For a units analysis of interval size, see [[Dave Keenan & Douglas Blumeyer's guide to RTT | [[Interval size unit|Various units have been used for the span of an interval]]. Typically [[cents]] are used, which can be thought of as the logarithm taken to the base <math>2^{1/1200}</math>. However, in mathematical situations it is often useful to work with the log base 2, meaning units of octaves, or even simply in the natural logarithm of base e, meaning units of "[[Wikipedia:Neper|nepers]]". [[Martin Gough]] introduced the unit of a "dineper" in his work on [[Logarithmic approximants]]. For a units analysis of interval size, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Size of JI interval]] and the following section for [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Size of tempered interval|the size of a tempered interval]]. | ||
The term span is also used in a different mathematical sense to refer to the linear span of a set of vectors, see [[Wikipedia:Linear span|linear span]]. | The term span is also used in a different mathematical sense to refer to the linear span of a set of vectors, see [[Wikipedia:Linear span|linear span]]. | ||
[[Category:Interval]] | [[Category:Interval]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||