Interval span: Difference between revisions

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'''~~Span~~''' is a technical term referring to the size of an interval, typically in some logarithmic unit such as cents, octaves, or something else. The [[JIP]] can give the span of any just interval, whereas within regular temperaments, this is typically given by a [[tuning map]].

The '''span''' of an interval is a technical term referring to the size of an interval, typically in some logarithmic unit such as cents, octaves, or something else. The [[JIP]] can give the span of any just interval, whereas within regular temperaments, this is typically given by a [[tuning map]].

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.

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]].

The term span is also used in a broader mathematical sense to refer to the linear span of a set of vectors, see [[Wikipedia:Linear span]].

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-'''Span''' is a technical term referring to the size of an interval, typically in some logarithmic unit such as cents, octaves, or something else. The [[JIP]] can give the span of any just interval, whereas within regular temperaments, this is typically given by a [[tuning map]].
+The '''span''' of an interval is a technical term referring to the size of an interval, typically in some logarithmic unit such as cents, octaves, or something else. The [[JIP]] can give the span of any just interval, whereas within regular temperaments, this is typically given by a [[tuning map]].
 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.
 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]].
+The term span is also used in a broader mathematical sense to refer to the linear span of a set of vectors, see [[Wikipedia:Linear span]].