User:Sintel/The mathematics of temperaments part 2: Difference between revisions

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 <h2>Commas</h2>
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-	With 5-limit, we have reduced the number of dimensions to 3 so far, coming from infinitely many this is a pretty good result. For most practical purposes this is still too many though. A 3-dimensional keyboard seems hard to play if you can even build one.
+	With 5-limit, we have reduced the number of dimensions to 3, and coming from infinitely many this is a pretty good result. For most practical purposes this is still too many though. A 3-dimensional keyboard seems hard to play if you can even build one.
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-	Let's go to rank-2 take a look at the 3-limit (also known as pythagorean tuning), the space generated by <math> \left\{ 2,3 \right\} </math>. Even though the major third isn't in this space, it turns out we can get arbitrarily close. Going up 4 perfect fifths, and down 2 octaves, we get the pythagorean third: <math> 2^{-6} \cdot 3^4 = 81/64 </math>. Its size is about 407.8¢, which is fairly close to the just major third at 386.3¢. You can find better ones further out, but let's try to work with this one since <math> 81/64 </math> is a reasonable looking ratio (the next better one is <math> 8192/6561</math> ).
+	Let's go even further, to rank-2, and take a look at the 3-limit (also known as pythagorean tuning): the space generated by <math> \left\{ 2,3 \right\} </math>. Even though the major third isn't in this space, it turns out we can get arbitrarily close. Going up 4 perfect fifths, and down 2 octaves, we get the pythagorean third: <math> 2^{-6} \cdot 3^4 = 81/64 </math>. Its size is about 407.8¢, which is fairly close to the just major third at 386.3¢. You can find better ones further out, but let's try to work with this one since <math> 81/64 </math> is a reasonable looking ratio (the next better one is <math> 8192/6561</math> ).
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