Val: Difference between revisions
m →Motivation: small clarification and make the long sentence easier to read |
m →Using a val: unlink "backslash notation" as rn what it links to is so terrible that i dont know how to even fix it, and a proper and clear definition is immediately given anyways, hence why i didnt link it in the first place |
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* 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) | * 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 (so represented by a frequency ratio of 2<sup>12/26</sup> = 1.377…/1 = 12\26) | * 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 | 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> (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. | ||
Now let us compare to the closest approximations: | Now let us compare to the closest approximations: | ||