Moving the bridge hack: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-10-~~14 16~~:04:10 UTC</tt>.<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-10-15 15:41:51 UTC</tt>.<br>

: The original revision id was <tt>~~372908500~~</tt>.<br>

: The original revision id was <tt>373260670</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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[[math]]

2^{-i/N} \text{ for } i = 1, 2, 3\dots

2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots

[[math]]

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[[math]]

2^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots

2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots

[[math]]

Line 27:

[[math]]

\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots

\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots

[[math]]

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<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&gt;

2^{-i/N} \text{ for } i = 1, 2, 3\dots&lt;br/&gt;[[math]]

2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots&lt;br/&gt;[[math]]

--><script type="math/tex">2^{-i/N} \text{ for } i = 1, 2, 3\dots</script><br />

--><script type="math/tex">2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots</script><br />

<br />

If the bridge is moved so that the new scale length is x, this adds (x-1) to all string lengths, so the new string lengths are simply<br />

Line 417:

<!-- ws:start:WikiTextMathRule:1:

[[math]]&lt;br/&gt;

2^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots&lt;br/&gt;[[math]]

2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots&lt;br/&gt;[[math]]

--><script type="math/tex">2^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots</script><br />

--><script type="math/tex">2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots</script><br />

<br />

The frequencies are inversely proportional to the string lengths. If we plug in i=0 to the above formula, we get x, so the frequency ratios relative to the open string are<br />

Line 424:

<!-- ws:start:WikiTextMathRule:2:

[[math]]&lt;br/&gt;

\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots&lt;br/&gt;[[math]]

\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots&lt;br/&gt;[[math]]

--><script type="math/tex">\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots</script><br />

--><script type="math/tex">\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots</script><br />

<br />

Converting those frequency ratios into cents in the usual way (taking the log to base 2 and multiplying by 1200) gives the new scale in cents.<br />

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-10-14 16:04:10 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-10-15 15:41:51 UTC</tt>.<br>
-: The original revision id was <tt>372908500</tt>.<br>
+: The original revision id was <tt>373260670</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 15: / Line 15: @@
 [[math]]
-^{-i/N} \text{ for } i = 1, 2, 3\dots
+^{-i/N} \text{ for } i = 0, 1, 2, 3\dots
 [[math]]
@@ Line 21: / Line 21: @@
 [[math]]
-^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots
+^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots
 [[math]]
@@ Line 27: / Line 27: @@
 [[math]]
-\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots
+\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots
 [[math]]
@@ Line 410: / Line 410: @@
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
-^{-i/N} \text{ for } i = 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
+^{-i/N} \text{ for } i = 0, 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;2^{-i/N} \text{ for } i = 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 If the bridge is moved so that the new scale length is x, this adds (x-1) to all string lengths, so the new string lengths are simply&lt;br /&gt;
@@ Line 417: / Line 417: @@
 &lt;!-- ws:start:WikiTextMathRule:1:
 [[math]]&amp;lt;br/&amp;gt;
-^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
+^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;2^{-i/N} + x - 1 \text{ for } i = 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:1 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 The frequencies are inversely proportional to the string lengths. If we plug in i=0 to the above formula, we get x, so the frequency ratios relative to the open string are&lt;br /&gt;
@@ Line 424: / Line 424: @@
 &lt;!-- ws:start:WikiTextMathRule:2:
 [[math]]&amp;lt;br/&amp;gt;
-\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
+\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Converting those frequency ratios into cents in the usual way (taking the log to base 2 and multiplying by 1200) gives the new scale in cents.&lt;br /&gt;