Val: Difference between revisions
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 |
→Using a val: restore changes |
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* 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 (so represented by a frequency ratio of 2<sup>4/26</sup> = 1.112…/1 | * 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) | ||
* 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 | * 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) | ||
* 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 | * 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) | ||
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. | |||
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. Using [[backslash notation]] to denote the number of steps in an edo, these are 4\26, 8\26, and 12\26, respectively, so that the underlying "logic" of the approximations being followed is 4\26 + 8\26 = 12\26. | |||
Now let us compare to the closest approximations: | Now let us compare to the closest approximations: | ||