Chord complexity: Difference between revisions

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Now, we note that {{nowrap|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{c}}}}, rather than the typical units of cents or octaves—perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have

Now, we note that {{nowrap|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{cent}}}}, rather than the typical units of cents or octaves—perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have

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-Now, we note that {{nowrap|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{c}}}}, rather than the typical units of cents or octaves&mdash;perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have
+Now, we note that {{nowrap|log(''b''/''a'')}} can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref group="note">In fact, this can also be thought of as a representation of the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to {{nowrap|1200 log<sub>2</sub>''e'' {{=}} 1731.234{{cent}}}}, rather than the typical units of cents or octaves&mdash;perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have
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