Logarithmic approximants: Difference between revisions

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√5/2 = 1.11803.

== Argent ~~temperament~~ ==

== Argent tuning ==

As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.

Line 407:

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent ~~temperament~~' is proposed instead.

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent tuning' is proposed instead.

Argent ~~temperament~~ has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

Argent tuning has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

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The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and ''minus'' the 41-tone comma).

Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent ~~temperament~~. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent tuning. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[File:Continued_fraction_jigsaw_41edo.png|alt=Continued fraction jigsaw 41edo.png|800x396px|Continued fraction jigsaw 41edo.png]]

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######Figure 2. Continued fraction jigsaw for 41edo

Figure 3 is a geometrical representation of argent ~~temperament~~ in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp#= Pythagorean apotome, p = Pythagorean comma.

Figure 3 is a geometrical representation of argent tuning in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp#= Pythagorean apotome, p = Pythagorean comma.

[[File:Silver_temperament_graphic.png|alt=Silver temperament graphic.png|800x587px|Silver temperament graphic.png]]

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

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

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 tuning 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. Another way to express the first of these relationships is

This means that in argent tuning 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}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},</math>

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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 ([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.

The tuning referred to here as argent tuning 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.

[[Category:Essays]]

@@ Line 396: / Line 396: @@
 √5/2 = 1.11803.
-== Argent temperament ==
+== Argent tuning ==
 As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.
@@ Line 407: / Line 407: @@
 This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).
-A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead.
+A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the [http://en.wikipedia.org/wiki/Silver_ratio silver ratio] (sometimes called the silver mean), ''δ''√2 + 1 = 2.4142. On this basis, and by analogy with [[Golden_Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent tuning' is proposed instead.
-Argent temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.
+Argent tuning has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.
 The continued fraction expansion of the silver ratio has a particularly simple form:
@@ Line 509: / Line 509: @@
 The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and ''minus'' the 41-tone comma).
-Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.
+Figure 2 is a ''continued fraction jigsaw'' showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to argent tuning. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.
 [[File:Continued_fraction_jigsaw_41edo.png|alt=Continued fraction jigsaw 41edo.png|800x396px|Continued fraction jigsaw 41edo.png]]
@@ Line 515: / Line 515: @@
 <span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo
-Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp<span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.
+Figure 3 is a geometrical representation of argent tuning in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mppp<span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.
 [[File:Silver_temperament_graphic.png|alt=Silver temperament graphic.png|800x587px|Silver temperament graphic.png]]
-<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent temperament
+<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent tuning
-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 <u>10/7</u> and <u>7/5</u>:
+Argent tuning 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 <u>10/7</u> and <u>7/5</u>:
 <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 <u>10/7</u> and the diminished fifth is very close to <u>7/5</u>. The discrepancy in each case is just 0.175 cents. Another way to express the first of these relationships is
+This means that in argent tuning the augmented fourth is very close to <u>10/7</u> and the diminished fifth is very close to <u>7/5</u>. 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}{2\sqrt{6}} – \tfrac{1}{4\sqrt{3}}) ≈ \tfrac{3}{2\sqrt{70}},</math>
@@ Line 761: / Line 761: @@
 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 ([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.
+The tuning referred to here as argent tuning 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.
 [[Category:Essays]]