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Can anyone give a mathematical explanation of how to find n such that equally tempered divisions of the octave into n parts give rise to consonant intervals? I
by pervycreeper 13y ago
Can anyone give a mathematical explanation of how to find n such that equally tempered divisions of the octave into n parts give rise to consonant intervals? I have read that 53 works even better than 12, but I wonder if there are more (infinitely more, I would imagine) exponents, and how this sequence (the next number is the lowest n that is better than the previous one) can be found.
- analog31 13y agoI have played with this question in the past, and never found a better "formula" than a search along the following lines: For each n, find the closest interval to a perfect fifth (3/2 ratio), and write down the error in octaves. Graph error as a function of n, and you will see certain values of n with low error. On the same graph, do the 4/3 ratio, and so forth, for a small handful of small-numbered ratios. Somehow 19 springs to mind as having reasonably consonant intervals, but I don't remember for sure. 12 is the smallest, and is also an auspicious number in ancient cultures.
- malexw 13y agoI built a tool to compare the relationship of different n-note equal temperment tunings a few years ago[1]. The UI is a bit obtuse, but it compares the frequencies of the n-note system with the frequencies of the older just intonation system. In 19 Tone Equal Temperment, the Eb and A are within 1 cent of what they would be in a just intonation tuning, which is pretty cool. I find 22 TET interesting as well. [1] http://www.eng.uwaterloo.ca/~mawillia/music/ http://www.eng.uwaterloo.ca/~mawillia/music/
- ArkyBeagle 13y agoThe "most consonant" is Pythagorean for the simple fractional intervals, so you'd have to weight somehow against that. I'm using "minimum number of beats" as the definition of "consonant".