Logarithmic approximants - Revision history

Sintel: /* Sources and acknowledgements */ typo

2026-05-05T17:10:39Z

Sources and acknowledgements: typo

← Older revision		Revision as of 17:10, 5 May 2026
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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.		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.

	Argent tuning is based on the continued fraction convergents of <math>sqrt{2}</math>, which have been known since ancient times.		Argent tuning is based on the continued fraction convergents of <math>\sqrt{2}</math>, which have been known since ancient times.
	This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950".		This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950".
	The same construction was later arrived at independently by [[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 same construction was later arrived at independently by [[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.

Sintel: Clarify contributions

2026-05-05T17:04:12Z

Clarify contributions

← Older revision		Revision as of 17:04, 5 May 2026
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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.		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~~ 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.		Argent tuning is based on the continued fraction convergents of <math>sqrt{2}</math>, which have been known since ancient times.
			This application to tuning appears to have been first made by [[Erv Wilson]], who described it under the name '2-zig/2-zag' in a [http://anaphoria.com/meruthree.pdf note] dated December 1996, with a comment claiming it as his answer to [[Joseph Yasser]] "at about 1950".
			The same construction was later arrived at independently by [[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.		Thanks to [[Gene_Ward_Smith\|Gene Ward Smith]] for the Gelfond-Schneider result.

	[[Category:Essays]]		[[Category:Essays]]

Sintel: Cleanup formatting and fixing some errors.

2026-05-05T16:51:14Z

Cleanup formatting and fixing some errors.

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Overthink: /* Argent tuning */ fix math notation

2025-12-14T00:03:18Z

Argent tuning: fix math notation

← Older revision		Revision as of 00:03, 14 December 2025
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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 (5120/5103) is the bimodular comma formed from <u>10/7</u> and <u>9/8</u>		If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) 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 (''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 (<u>8/3</u> = <u>2.6666...</u>) 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 (<u>8/3</u> = <u>2.6666...</u>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2<sup>√2</sup> = 2.665144...

	==Golden temperaments==		==Golden temperaments==

Lériendil: Text replacement - "rgent temperament" to "rgent tuning"

2025-10-09T14:10:39Z

Text replacement - "rgent temperament" to "rgent tuning"

Sintel: /* Argent temperament */ Hemifamity: fix number

2025-05-26T02:03:44Z

Argent temperament: Hemifamity: fix number

← Older revision		Revision as of 02:03, 26 May 2025
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	As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to <u>21/20</u> (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to <u>15/14</u> (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to <u>50/49</u> (34.976 cents).		As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to <u>21/20</u> (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to <u>15/14</u> (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to <u>50/49</u> (34.976 cents).

	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>		If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) 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 (''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 (<u>8/3</u> = <u>2.6666...</u>) 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 (<u>8/3</u> = <u>2.6666...</u>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...

ArrowHead294: Remove unnecessary custom styling

2025-03-29T20:29:54Z

Remove unnecessary custom styling

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Sintel: -legacy, todo add intro

2025-03-29T18:07:25Z

-legacy, todo add intro

← Older revision		Revision as of 18:07, 29 March 2025
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	{{~~Legacy~~}}		{{todo\|intro\|inline=1}}
	='''<span style="font-size: 20px;">1. Introduction</span>'''=		='''<span style="font-size: 20px;">1. Introduction</span>'''=
	<span style="font-family: Arial,Helvetica,sans-serif;">A ''logarithmic approximant'' (or ''approximant'' for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:</span>		<span style="font-family: Arial,Helvetica,sans-serif;">A ''logarithmic approximant'' (or ''approximant'' for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:</span>

ArrowHead294 at 18:28, 13 March 2025

2025-03-13T18:28:36Z

← Older revision		Revision as of 18:28, 13 March 2025
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	Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.		Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.

	Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the [[~~Bohlen-Pierce\|equally tempered Bohlen-Pierce~~ scale]].		Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered [[Bohlen–Pierce scale]].

	Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET\|88 cent equal temperament]].		Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields [[88cET\|88 cent equal temperament]].

FloraC: +category

2025-02-18T10:50:35Z

+category

← Older revision		Revision as of 10:50, 18 February 2025
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	Thanks to [[Gene_Ward_Smith\|Gene Ward Smith]] for the Gelfond-Schneider result.		Thanks to [[Gene_Ward_Smith\|Gene Ward Smith]] for the Gelfond-Schneider result.

			[[Category:Essays]]

← Older revision		Revision as of 14:10, 9 October 2025
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	√5/2 = 1.11803.		√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.		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.

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	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).		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:		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).		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]]		[[File:Continued_fraction_jigsaw_41edo.png\|alt=Continued fraction jigsaw 41edo.png\|800x396px\|Continued fraction jigsaw 41edo.png]]
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	<span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo		<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]]		[[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>		<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>		<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.		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.		Thanks to [[Gene_Ward_Smith\|Gene Ward Smith]] for the Gelfond-Schneider result.

	[[Category:Essays]]		[[Category:Essays]]