![\"\"](\"http:\/\/172.104.26.128\/wp-content\/uploads\/2012\/08\/music2.png\" "\\"music2\\"")
<\/a><\/p>\nThe Fourier transform<\/a> is a mathematical process designed to separate a mathematical function into its component frequencies. It works on a wide range of functions, including polynomial functions (e.g. parabolas). If you applied the Fourier Transform to a polynomial, the result would be an infinite series of waves added together – a bit of a mind bender. Interestingly, you can go the other direction using the Taylor series, which represents a wave as an infinite series of summed polynomials.<\/p>\n In mathematical theory, these operations are done on an infinitely large plane with continuous lines, which is impractical in real engineering exercises. The Fast Fourier Transform<\/a> is a variation which operates on a finite list of points along a graph, like that in a WAV file. Since there are potentially infinite frequencies in a sound sample, the output is placed into buckets based on the loudness of each pitch (10-20hz, 20-30hz, 30-40hz, etc). The size and number of these buckets are controlled by the end application and sound quality.<\/p>\n There is an open source project to read wave files, which I discussed previously here<\/a>. This library will identify the sample rate of the file – this is how many loudness measurements are stored for each second of audio. 44,100 is a common rate.<\/p>\n The Apache Commons Math<\/a> library provides code to execute an FFT. This bit of Scala code calls it with sampled data that has been read from a file.<\/p>\n The results are actually an array of vectors. To find the loudness of a pitch, you must compute the length of the vector. For a vector (a, b), this is computed by √(a2<\/sup> + b2<\/sup>).<\/p>\n The position of each bucket combined with the sample rate, determines the range of frequencies this tone represents. To find which notes are included in output, I take the center of the range:<\/p>\n In this application, all octaves will be combined. This tests each notes in the lowest octave (A…G) and finds the closest match:<\/p>\n I generated WAV files for all combinations of five notes in an octave – for the current version, 80% of generated test audio files are identified correctly. See this post on using Prolog to generate all chord combinations<\/a>.<\/p>\n Working with audio is challenging, but rewarding. Files are hard to test without perfect pitch, and you must be remember to start with the speaker volume on low \ud83d\ude42<\/p>\nwavFile.readFrames(frameData, dataSize)<\/pre>\n
\nval buckets : Array[Complex] = \n fft.transform(frameData, TransformType.FORWARD);\n<\/pre>\n
\nvar index = 0\nfor (val c : Complex {\nval magnitude : Double =\nMath.sqrt(c.getImaginary() * c.getImaginary() +\nc.getReal() * c.getReal())\n\nindex = index + 1\n<\/pre>\nval frq = (.5 + index) * sampleRate \/ dataSize;<\/pre>\n
var numOctaves = Math.log(frq\/noteFrq)\/Math.log(2)\nvar normFrq = Math.abs(numOctaves - Math.floor(numOctaves));\nif (normFrq < distance)\n\u2026found closer match...<\/pre>\n