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IntroductionEdit

Goodman's Law states that the area below the curve[1] contained within the corresponding rectangle 1/n over the area above the curve in the corresponding rectangle is directly proportional to the frequency of the waves represented by the harmonic series. This value is also; moreover, inversely proportional to wavelength, as frequency is inversely proportional to wavelength.[2] It was discovered by Thomas.A.Goodman in 2013, during empirical research into the field of the Maths behind Music.



What is Goodman's Law?Edit

From a graph plotting the Harmonic series[3], it is evident that the area contained below the curve between n and n+1 [4], over the area contained within the rectangle 1/n is directly proportional to the frequency, and; moreover, inversely proportional to the wavelength of the waves represented by the graph. 


620px-Harmonic partials on strings.svg
602px-Integral Test.svg





Goodman's Law EquationEdit

The Mathematical Equation for Goodman's Law is :
Goodman's law equation






This can be simplified by replacing (ln(n+1)-ln(n)) with the lowercase Greek letter, Delta (d):


Goodman's Law Equation (Delta)






The Coefficient of Proportionality, kEdit

Glkof

The value of k in all circumstances is : 2.258891353 (to 9 d.p.), but is a non-recurring decimal, and it is; therefore, more accurate to use the equation for k in all calculations.[5]



Uses of Goodman's LawEdit

Goodman's Law can be used in order to acquire wavelength or frequency without the need for amplitude to be specified, so long as the wave is represented on the harmonic series (1/n), this also means that the amplitude of the wave can be worked out with just one variable (either wavelength or frequency).

ReferencesEdit

  1. Integral
  2. Wavelength and Frequency - Davidson, E. (2007)
  3. Harmonic Series (Mathematical)
  4. Integral
  5. Proportionality (mathematics)

See AlsoEdit

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