I really think it's fascinating how mathematical music is, and how our brains have learned to perform a fairly sophisticated optimization without our sitting down and explicitly doing the math. It will be interesting, too, to see what other applications they may be able to find for this kind of algorithm.

http://www.technologyreview.com/blog/arxiv/27673/

[snip]

At first glance, tuning a musical instrument seems a straightforward task. However, any professional tuner will tell you that the reality is rather different.

Pluck a string and the sound it produces is the result of its fundamental frequency and its harmonics at frequencies that are whole number multiples of the fundamental. Clearly, harmonics have a simple linear relationship with the fundamental.

The problem arises because music consists of repeating patterns of notes based on octaves. Since the frequency of a note doubles from octave to octave, the frequencies grow exponentially as the octaves increase.

And therein lies the problem. The linear increase in the frequency of harmonics can never exactly match the exponential increase required when the notes are arranged in octaves. So there is always a compromise.

Western musical scales consists of notes that differ by a constant frequency ratio of 2^1/12, a system known as equal temperament. These notes are equidistant on a logarithmic scale but not on a linear scale.

In this system, notes that are an octave apart can all be in tune but other intervals, such as perfect fourths or fifths, are always slightly out of tune.

To get around this, a professional tuner 'stretches' the interval between some notes to correct these intervals. And that's where things get tricky.

The amount and type of stretching differs depending on the type of instrument (and even between instruments of the same type) and so cannot be calculated by electronic tuners, which have otherwise revolutionised tuning.

[snip]

 

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Interesting... will read more later.

We are a pretty musical family so I find this quite fascinating. Thank you!

 

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Well, at least they're only putting a tiny amount of people out of work (potentially).

 

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Either the author is a non-mathematician or he recognizes that most of his audience isn't, but he buried the lead... this employs a non-deterministic turing machine, which I find to be the most fascinating part. It says something about ourselves, that a non-deterministic algorithm employed subconsciously by the tuner would result in a more pleasant sound to the average and untrained listener.

 

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yeah, I admit I was surprised that they started the algorithm with a small random step, but I guess the end of the loop will see whether you walked in the right direction so you can converge on a minimum. but it could still be possible to accidentally wander away from an optimum solution if there is a small local minimum in the way. sorry if I'm putting it badly, it's been a long time since I took any math classes.

 

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