Logarithmic approximants: Difference between revisions

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==Definition==

The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n''''/d'' is

The quadratic approximant ''q'' of an interval ''J'' with frequency ratio ''r'' = ''n'<nowiki/>'''/d'''''''' is'''

<math>\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\

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The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

~~== ==~~

==Relative sizes of intervals between 3 frequencies in arithmetic progression==

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The ratio of the large tone, ''T'' = 9/8, to the small tone, ''t'' = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

''T''''/t'' = 203.910/182.404 = 1.11790,

''T'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''

√5/2 = 1.11803.

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If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (3120/3103) is the bimodular comma formed from 10/7 and 9/8

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...

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...

==Golden temperaments==

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As [[Gene_Ward_Smith|Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (‘''selenia''’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|//gammic// comma ]]|-29 -11 20> (4.769 cents) and the ''semisuper'' comma (AKA ''[[vishnuzma|vishnuzma]]'') |23 6 -14> (3.338 cents). In particular,

It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|//gammic// comma]] |-29 -11 20> (4.769 cents) and the ''semisuper'' comma (AKA ''[[vishnuzma|vishnuzma]]'') |23 6 -14> (3.338 cents). In particular,

###''selenia'' = 7 ''gammic'' – 10 ''semisuper''

So to prove that ''selenia'' is small we must show that ''gammic''''/semisuper'' ≈ 10/7.

So to prove that ''selenia'' is small we must show that ''gammic'<nowiki/>'''/semisuper'''''<nowiki/>''' ≈ 10/7.'''

''Gammic'' and ''semisuper'' are both bimodular commas:

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This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.

The tuning referred to here as argent temperament appears to have been discovered 'about 1950' by Erv Wilson, who named it [http://anaphoria.com/meruthree.pdf 2-zig/2-zag]'. It was later rediscovered independently by [[Graham_Breed|Graham Breed]] and Paul Hahn, who described it in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.

The tuning referred to here as argent temperament appears to have been discovered 'about 1950' by Erv Wilson, who named it [http://anaphoria.com/meruthree.pdf 2-zig/2-zag]'. It was later rediscovered independently by [[Graham_Breed|Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000.

Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result.

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 ==<span style="font-family: Arial,Helvetica,sans-serif;">Definition</span>==
-The quadratic approximant ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' of an interval ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;">''r'' = ''n''''/d''</span> is
+The quadratic approximant ''<span style="font-family: Georgia,serif; font-size: 110%;">q</span>'' of an interval ''<span style="font-family: Georgia,serif; font-size: 110%;">J</span>'' with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;">''r'' = ''n'<nowiki/>'''/d'''''</span>''' is'''
 <math>\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
@@ Line 362: / Line 362: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
-== ==
 ==<span style="font-family: Arial,Helvetica,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span>==
@@ Line 393: / Line 391: @@
 The ratio of the large tone, <span style="font-family: Georgia,serif; font-size: 110%;">''T'' = <u>9/8</u></span>, to the small tone, <span style="font-family: Georgia,serif; font-size: 110%;">''t'' = <u>10/9</u></span>, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.
-<span style="font-family: Georgia,serif; font-size: 110%;">''T''''/t'' = 203.910/182.404 = 1.11790,</span>
+<span style="font-family: Georgia,serif; font-size: 110%;">''T'<nowiki/>'''/t'''''<nowiki/>''' = 203.910/182.404 = 1.11790,'''</span>
 <span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span>
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 If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (<u>3120/3103</u>) is the bimodular comma formed from <u>10/7</u> and <u>9/8</u>
-By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem ] the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;">''r'' = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2''a''+''b''</span>, where''<span style="font-family: Georgia,serif; font-size: 110%;"> a</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>'' are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;">''a'' = 0</span>). The frequency ratio of the tempered perfect eleventh (<span style="font-family: Georgia,serif; font-size: 110%;"><u>8/3</u> = <u>2.6666...</u></span>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant ]or Hilbert number, <span style="font-family: Georgia,serif; font-size: 110%;">2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 2.665144</span>...
+By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;">''r'' = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2''a''+''b''</span>, where''<span style="font-family: Georgia,serif; font-size: 110%;"> a</span>'' and ''<span style="font-family: Georgia,serif; font-size: 110%;">b</span>'' are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;">''a'' = 0</span>). The frequency ratio of the tempered perfect eleventh (<span style="font-family: Georgia,serif; font-size: 110%;"><u>8/3</u> = <u>2.6666...</u></span>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, <span style="font-family: Georgia,serif; font-size: 110%;">2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 2.665144</span>...
 ==Golden temperaments==
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 <span style="color: #333333;">As [[Gene_Ward_Smith|Gene Ward Smith]] has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280&gt; (‘''selenia''’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants.
-It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|//gammic// comma ]]|-29 -11 20&gt; (4.769 cents) and the ''semisuper'' comma (AKA ''[[vishnuzma|vishnuzma]]'') |23 6 -14&gt; (3.338 cents). In particular,
+It can be shown, using a suitable [[Comma-based_lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic_node|//gammic// comma]] |-29 -11 20&gt; (4.769 cents) and the ''semisuper'' comma (AKA ''[[vishnuzma|vishnuzma]]'') |23 6 -14&gt; (3.338 cents). In particular,
 <span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>''selenia'' = 7 ''gammic'' – 10 ''semisuper''</span>
-<span style="color: #333333;">So to prove that ''selenia'' is small we must show that ''gammic''''/semisuper'' ≈ 10/7.</span>
+<span style="color: #333333;">So to prove that ''selenia'' is small we must show that ''gammic'<nowiki/>'''/semisuper'''''<nowiki/>''' ≈ 10/7.'''</span>
 <span style="color: #333333;">''Gammic'' and ''semisuper'' are both bimodular commas:</span>
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 This article is based on original research by [[Martin_Gough|Martin Gough]]. See [[:File:Bimod_Approx_2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.
-The tuning referred to here as argent temperament appears to have been discovered 'about 1950' by Erv Wilson, who named it [http://anaphoria.com/meruthree.pdf 2-zig/2-zag]'. It was later rediscovered independently by [[Graham_Breed|Graham Breed]] and Paul Hahn, who described it in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.
+The tuning referred to here as argent temperament appears to have been discovered 'about 1950' by Erv Wilson, who named it [http://anaphoria.com/meruthree.pdf 2-zig/2-zag]'. It was later rediscovered independently by [[Graham_Breed|Graham Breed]] and Paul Hahn, who described it in posts ([https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12592.html#12599 #12599], [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_12637.html#12670 #12670]) to the Yahoo tuning list on 10 and 12 August 2000.
 Thanks to [[Gene_Ward_Smith|Gene Ward Smith]] for the Gelfond-Schneider result.