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

(One intermediate revision by the same user not shown)

Line 8:

<h2>Commas</h2>

<p>

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.

</p>

<p>

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> ).

</p>

<p>

Line 93:

</p>

<p>

[[User:Sintel/The mathematics of temperaments part 2|Continued in part 3 (WIP)]]

[[User:Sintel/The mathematics of temperaments part 3|Continued in part 3 (WIP)]]

</p>

</div>

@@ Line 8: / Line 8: @@
 <h2>Commas</h2>
 <p>
-	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.
 </p>
 <p>
-	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> ).
 </p>
 <p>
@@ Line 93: / Line 93: @@
 </p>
 <p>
-	[[User:Sintel/The mathematics of temperaments part 2|Continued in part 3 (WIP)]]
+	[[User:Sintel/The mathematics of temperaments part 3|Continued in part 3 (WIP)]]
 </p>
 </div>