Maths vs. Music - Tuning and harmony

[Alex K]

And last but not least, a comparison of an A major triad in pure tuning with the exact ratios 5/4 for the third and 3/2 for the fifth, and in equal tuning with 2^4/12 and 2^7/12 for the third and fifth.

https://dl.dropboxusercontent.com/u/4685...arison.mp3

It plays the exact ratios for 2 seconds, then the equal temperament for 2 seconds, then repeat. Equal temperament really sounds wrong when contrasted directly with the exact ratios...

The effect is maybe more subtle, but still very noticeable when everything is taken one octave lower, i.e. based on 220 hz.

https://dl.dropboxusercontent.com/u/4685...rison2.mp3

Interesting, I didn't realize harmonic beating could be generated digitally.

I use the software mathematica to define the waveform as a mathematical function, e.g. for the equal temperament A major triad

sin(2 pi * 440 * 1 * t)
+sin(2 pi * 440 * 2^(4/12) * t)
+sin(2 pi * 440 * 2^(7/12) * t)

corresponding to the fundamental and the 4th and 7th semitone, and then take values of this function in steps of 1/44100th of a second, normalize it such that the maximum and minimum are near +/- 32000 and write those values into a file as 16 bit integer numbers. Voilà your CD quality wav file.

For comparison, the triad with pure third and fifth would be
sin(2 pi * 440 * 1 * t)
+sin(2 pi * 440 * 5/4 * t)
+sin(2 pi * 440 * 3/2 * t)

the "pure" ratios 5/4 and 3/2 are numerically close to, but not quite equal to the corresponding steps 2^4/12 and 2^7/12 in equal temperament.