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:

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 06:13:51 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 12:53:21 UTC</tt>.

: The original revision id was <tt>~~563715357~~</tt>.

: The original revision id was <tt>563735661</tt>.

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

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

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######Figure 3. Geometrical representation of argent temperament

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:

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{ \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.

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

Another way to express the first of these relationships is

[[math]]

\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).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to __21/20__ (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to __15/14__ (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to __50/49__ (34.976 cents).

By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (//r// = 2√2//a//+//b//, where// a// and //b// are integers) are transcendental, with the exception of octave multiples (//a// = 0). The frequency ratio of the tempered perfect eleventh (__8/3__ = __2.6666...__) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, 2√2 = 2.665144...

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######Figure 3. Geometrical representation of argent temperament

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:

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:

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

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

--><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>

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.~~ ~~

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

Another way to express the first of these relationships is

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

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

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

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).

As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (r = 2√2a+b, where a and b are integers) are transcendental, with the exception of octave multiples (a = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow">Gelfond-Schneider constant </a>or Hilbert number, 2√2 = 2.665144...

@@ 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 06:13:51 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 12:53:21 UTC</tt>.<br>
-: The original revision id was <tt>563715357</tt>.<br>
+: The original revision id was <tt>563735661</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 379: / Line 379: @@
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament
-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:
+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{ \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.
+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
-Another way to express the first of these relationships is
 [[math]]
 \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).
+As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to __21/20__ (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to __15/14__ (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to __50/49__ (34.976 cents).
 By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2//a//+//b//&lt;/span&gt;, where//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//a// = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;__8/3__ = __2.6666...__&lt;/span&gt;) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 2.665144&lt;/span&gt;...
@@ Line 1,070: / Line 1,069: @@
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament&lt;br /&gt;
 &lt;br /&gt;
-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:&lt;br /&gt;
+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 &lt;u&gt;10/7&lt;/u&gt; and &lt;u&gt;7/5&lt;/u&gt;:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:40:
 [[math]]&amp;lt;br/&amp;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}}.&amp;lt;br/&amp;gt;[[math]]
   --&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;
+This means that in argent temperament the augmented fourth is very close to &lt;u&gt;10/7&lt;/u&gt; and the diminished fifth is very close to &lt;u&gt;7/5&lt;/u&gt;. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is&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;
@@ Line 1,082: / Line 1,080: @@
   --&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;
 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;
+As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to &lt;u&gt;21/20&lt;/u&gt; (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to &lt;u&gt;15/14&lt;/u&gt; (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to &lt;u&gt;50/49&lt;/u&gt; (34.976 cents).&lt;br /&gt;
 By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow"&gt;Gelfond-Schneider theorem &lt;/a&gt; the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;em&gt;a&lt;/em&gt;+&lt;em&gt;b&lt;/em&gt;&lt;/span&gt;, where&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;a&lt;/em&gt; = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;u&gt;8/3&lt;/u&gt; = &lt;u&gt;2.6666...&lt;/u&gt;&lt;/span&gt;) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow"&gt;Gelfond-Schneider constant &lt;/a&gt;or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 2.665144&lt;/span&gt;...&lt;br /&gt;
 &lt;br /&gt;