Now I get almost the same ratios as just intonation. I can make format rat use these fractions by telling rats that it has only 6 columns to work with. But if I’m much more tolerant and allow a 2 percent error, I get short continued fractions. ![]() This would not be a satisfactory way to try to tune an instrument. The equal temperament ratios do not give small integers. These are unconventional continued fractions because they contain negative terms. The MATLAB expression rat(X,tol) creates a truncated continued fraction approximation of X with an accuracy of tol. The ratios in just intonation or equal temperament produce instruments with tunings that generate vibrations with these desired frequencies. Here is a snapshot showing the first nine modes and the resulting wave traveling along the string. Our Experiments with MATLAB program vibrating_string provides a dynamic view. This one-dimensional model is all we need here.) (Two- and three-dimensional models are much more complicated. Since 12 semitones comprise a factor of 2, a single semitone is a factor of $\sqrt, \ \, n = 1, 2, …$$Īnd the frequency is simply the integer $n$. ![]() In other words the notes have equal frequency ratios. In most Western music, an octave is divided into 12 semitones in a geometric progression. In the theory of music, an octave is an interval with frequencies that range over a factor of two.
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