ELD: Difference between revisions

Line 10:

Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.

To find the steps for an n-ELDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in string length that you need to cover is not actually <math>p</math>, but only <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So, the formula for the length of step <math>k</math> of an n-ELDp is:

<math>

L(k) = 1 + (\frac kn)(p-1)

</math>

This way, when <math>k</math> is <math>0</math>, <math>L(k)</math> is simply <math>1</math>. And when <math>k</math> is <math>n</math>, <math>L(k)</math> is simply <math>1 + (p-1) = p</math>.

{| class="wikitable"

@@ Line 10: / Line 10: @@
 Note that because frequency is the inverse of length, if a frequency lower than the root pitch's frequency is asked for, the length will be greater than 1; at this point the physical analogy to a length of string breaks down somewhat, since it is not easy to imagine dynamically extending the length of a string to accommodate such pitches. However, it is not much of a stretch (pun intended) to tolerate lengths > 1, if the analogy is adapted to a switching from one string to another, and any string length imaginable is instantly available.
+To find the steps for an n-ELDp, begin by recognizing that while the ratio between your root pitch's string length and the length you would pluck to get the lowest pitch is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in string length that you need to cover is not actually <span><math>p</math></span>, but only <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the length of step <span><math>k</math></span> of an n-ELDp is:
+<math>
+L(k) = 1 + (\frac kn)(p-1)
+</math>
+This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>L(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>L(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
 {| class="wikitable"