Chord complexity: Difference between revisions

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When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.

If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap|log(''n'' ~~{{asterisk}}~~ ''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.

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 When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio ''n''/''d'', these qualities tend to decrease as ''n'' and ''d'' increase, unless ''n''/''d'' is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.
-If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap|log(''n'' {{asterisk}} ''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).
+If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).
 Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.