Logarithmic approximants: 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:MartinGough|MartinGough]] and made on <tt>2015-10-24 05:58:43 UTC</tt>.<br>

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 06:09:20 UTC</tt>.<br>

: The original revision id was <tt>~~563714785~~</tt>.<br>

: The original revision id was <tt>563715177</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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Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

[[math]]

\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.

\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.

[[math]]

This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.

Another way to express the first of these relationships is

[[math]]

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},

\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},

[[math]]

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

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<!-- ws:start:WikiTextMathRule:40:

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

\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&lt;br/&gt;[[math]]

\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.</script><br />

--><script type="math/tex">\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.</script><br />

This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.<br />

Another way to express the first of these relationships is<br />

<!-- ws:start:WikiTextMathRule:41:

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

~~&lt;br /&gt;~~

\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},&lt;br/&gt;[[math]]

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},</script><br />

--><script type="math/tex">

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},</script><br />

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).<br />

<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:MartinGough|MartinGough]] and made on <tt>2015-10-24 05:58:43 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 06:09:20 UTC</tt>.<br>
-: The original revision id was <tt>563714785</tt>.<br>
+: The original revision id was <tt>563715177</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 381: / Line 381: @@
 Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:
 [[math]]
-\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.
+\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.
 [[math]]
 This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.
 Another way to express the first of these relationships is
 [[math]]
-\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},
+\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},
 [[math]]
 which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).
@@ Line 1,074: / Line 1,074: @@
 &lt;!-- ws:start:WikiTextMathRule:40:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&amp;lt;br/&amp;gt;[[math]]
+\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:40 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad \frac{q[10/7]}{q[7/5]}= \frac{ \tfrac{3} {2\sqrt{70}} } { \tfrac{2} {2\sqrt{35}} } = \tfrac{3}{2\sqrt{2}}.&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:40 --&gt;&lt;br /&gt;
 This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.&lt;br /&gt;
 Another way to express the first of these relationships is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:41:
 [[math]]&amp;lt;br/&amp;gt;
- &amp;lt;br /&amp;gt;
+\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},&amp;lt;br/&amp;gt;[[math]]
-\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&amp;lt;br/&amp;gt;[[math]]
+  --&gt;&lt;script type="math/tex"&gt;\qquad 3 (\tfrac{1}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:41 --&gt;&lt;br /&gt;
-  --&gt;&lt;script type="math/tex"&gt;
-\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:41 --&gt;&lt;br /&gt;
 which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).&lt;br /&gt;
 &lt;br /&gt;