Val: Difference between revisions
I don't think you should try explaining backslash notation when you can link. You should also declare it before using it. |
the frequency ratios help ground the abstract mathematics in music to a beginner, so should be included. i will re-add the other changes in the next edit though Tag: Undo |
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* 5/4 = 5/(2 × 2) | * 5/4 = 5/(2 × 2) | ||
* 45/32 = (3 × 3 × 5)/(2 × 2 × 2 × 2 × 2) | * 45/32 = (3 × 3 × 5)/(2 × 2 × 2 × 2 × 2) | ||
Now all we do is substitute each occurrence of each prime with adding (or subtracting if we are dividing) the corresponding number of ''steps'' for that prime given by our val: | Now all we do is substitute each occurrence of each prime with adding (or subtracting if we are dividing) the corresponding number of ''steps'' for that prime given by our val: | ||
* 9/8 is mapped to (41 + 41) - (26 + 26 + 26) = 82 - 78 = 4 steps | * 9/8 is mapped to (41 + 41) - (26 + 26 + 26) = 82 - 78 = 4 steps (so represented by a frequency ratio of 2<sup>4/26</sup> = 1.112…/1 = 4\26) | ||
* 5/4 is mapped to 60 - (26 + 26) = 60 - 52 = 8 steps | * 5/4 is mapped to 60 - (26 + 26) = 60 - 52 = 8 steps (so represented by a frequency ratio of 2<sup>8/26</sup> = 1.237…/1 = 8\26) | ||
* 45/32 is mapped to (41 + 41 + 60) - (26 + 26 + 26 + 26 + 26) = 142 - 130 = 12 steps | * 45/32 is mapped to (41 + 41 + 60) - (26 + 26 + 26 + 26 + 26) = 142 - 130 = 12 steps (so represented by a frequency ratio of 2<sup>12/26</sup> = 1.377…/1 = 12\26) | ||
That is a successful use of a val. The arithmetic works out nicely: the approximation of 5/4 times the approximation of 9/8 is the approximation of 45/32. What about that funny backslash notation? That's just a shorthand: ''k''\''N'' = 2<sup>''k''/''N''</sup> (that is, the ''N''th root of 2, to the ''k''th power). Note that it can also be used ambiguously, as 2<sup>4/26</sup> + 2<sup>8/26</sup> = 2<sup>12/26</sup> is clearly invalid (the correct statement is 2<sup>4/26</sup> × 2<sup>8/26</sup> = 2<sup>12/26</sup>) but 4\26 + 8\26 = 12\26 need not be. It is used a lot in the xen community so is provided here for familiarization. | |||
That is a successful use of a val. The arithmetic works out nicely: the approximation of 5/4 times the approximation of 9/8 is the approximation of 45/32. | |||
Now let us compare to the closest approximations: | Now let us compare to the closest approximations: | ||