Metallic MOS - Revision history

ArrowHead294 at 05:23, 26 January 2026

2026-01-26T05:23:38Z

← Older revision		Revision as of 05:23, 26 January 2026
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	But phi is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.		But phi is only the first of an infinite sequence of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.

	MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.~~09¢~~), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.		MOS concepts are logarithmic, not acoustic. In other words, we are not dealing with frequency ratios here. Frequency ratios related to metallic means, such as "[[acoustic phi]]" (approximately 833.09{{c}}), have interesting properties too—creating recursive combination tones, for example—but these musical applications of metallic means will not be discussed here.

	The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We'll be conforming here with the convention of choosing the period as the interval to lock to 1.		The met-MOS concepts discussed here may be used to define generators as fractions of an octave, as is most common. But these concepts are more abstract than that, and may be used to define generators as fractions of ''any'' period. As such, they only depend on the ratio between the generator and the period, and so for convenience we can lock one of these two values to 1 and only vary the value of the other. We'll be conforming here with the convention of choosing the period as the interval to lock to 1.
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	As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.		As for why we pick the lower half of the period rather than the upper half, this is somewhat arbitrary, but it seems objectively simpler to keep our lower bound at 0.

	Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.~~64¢~~ when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.		Sure, depending on the context, the generator complement greater than 0.5 may be the one we want to describe our scale in terms of. For example, we may be thinking of the generator as the perfect fifth instead of the perfect fourth. Or we may want to use 0.618034 instead of its complement 0.381966 (we've been using the latter and calling it the golden generator, but some readers may be more familiar with the former, known as "[[logarithmic phi]]", which is 741.64{{c}} when the period is an octave). But for purposes of cataloging we prefer the smaller, or ''reduced'' of the two complements.

	And this is a subtle point, but it's another reason to prefer leaning intervals parentward. We have a potential problem: we don't want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval {{sfrac\|0\|1}} to {{sfrac\|1\|1}}. We include this interval because it occupies space between {{sfrac\|0\|1}} and {{sfrac\|1\|2}}—so has potential to find useful generators—but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval {{sfrac\|0\|1}} to {{sfrac\|1\|1}} is {{sfrac\|1\|2}}, or exactly 0.5. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward {{sfrac\|0\|1}}. Since of these two ratios {{sfrac\|0\|1}} and {{sfrac\|1\|1}}, the parent ratio is {{sfrac\|0\|1}}, weighting parentward is the solution.		And this is a subtle point, but it's another reason to prefer leaning intervals parentward. We have a potential problem: we don't want to find generators <math>> 0.5</math>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval {{sfrac\|0\|1}} to {{sfrac\|1\|1}}. We include this interval because it occupies space between {{sfrac\|0\|1}} and {{sfrac\|1\|2}}—so has potential to find useful generators—but we have to be careful with it to avoid finding generators <math>> 0.5</math>. The method for this is simple. First, note that the unweighted mediant in the interval {{sfrac\|0\|1}} to {{sfrac\|1\|1}} is {{sfrac\|1\|2}}, or exactly 0.5. So if we want to avoid generators <math>> 0.5</math>, all we must do is make sure to weight more toward {{sfrac\|0\|1}}. Since of these two ratios {{sfrac\|0\|1}} and {{sfrac\|1\|1}}, the parent ratio is {{sfrac\|0\|1}}, weighting parentward is the solution.
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	[[File:MOS iteration rules for L and s.png\|452x452px]]		[[File:MOS iteration rules for L and s.png\|452x452px]]

	We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are ~~100¢~~ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.		We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100{{c}} or 4516.8{{c}}, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.

	So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ''L'', and ''s''{{`s}} size is 1, then 1 should be subtracted from ''L''.		So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ''L'', and ''s''{{`s}} size is 1, then 1 should be subtracted from ''L''.
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	</math>		</math>

	[[File:Silver horogram.png\|alt=horogram for g = 0.292893, 7 iterations\|horogram for g = 0.292893, 7 iterations\|none\|frame]]		[[File:Silver horogram.png\|alt=horogram for ''g'' = 0.292893, 7 iterations\|horogram for {{nowrap\|''g'' {{=}} 0.292893}}, 7 iterations\|none\|frame]]

	We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>''s''</sub>, we'll look at the right and left sides separately, as we did with the golden:		We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ<sub>''s''</sub>, we'll look at the right and left sides separately, as we did with the golden:
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	</math>		</math>

	As expected, {{nowrap\|''L'':''s'' {{=}} (3''L'' + s):L}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.		As expected, {{nowrap\|''L'':''s'' {{=}} (3''L'' + s):''L''}} is only true of every ''third'' scale the bronze generator generates. The remaining interval relationships are left as an exercise for the reader.

	= Gallery of generators =		= Gallery of generators =
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	== Non-Octave Golden ==		== Non-Octave Golden ==
	Golden lemba is a Multi-MOS, fitting two periods into an octave (~~600¢~~ period, 229.~~179¢~~ generator).		Golden lemba is a Multi-MOS, fitting two periods into an octave (600{{c}} period, 229.179{{c}} generator).

	Golden triforce similarly fits three periods into an octave (~~400¢~~ period, 152.~~786¢~~ generator).		Golden triforce similarly fits three periods into an octave (400{{c}} period, 152.786{{c}} generator).

	== Argent Temperament ==		== Argent Temperament ==
	[[File:Argent horogram.png\|alt=horogram for g = 0.414214, 7 iterations\|right\|388x388px]]		[[File:Argent horogram.png\|alt=horogram for {{nowrap\|''g'' {{=}} 0.414214}}, 7 iterations\|right\|388x388px]]

	Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention.		Scales based on the bronze mean and metallic means beyond it have not been extensively explored. However, the silver mean has gotten some attention.

ArrowHead294: /* Why they cycle */

2026-01-26T03:48:58Z

Why they cycle

← Older revision		Revision as of 03:48, 26 January 2026
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	=== Why they cycle ===		=== Why they cycle ===
	That is true of scale iterations where {{nowrap\|''L'' − ''s'' > s}}. For the other type of scale iteration, where {{nowrap\|''L'' − ''s'' < s}}, the result is simply reciprocated:		That is true of scale iterations where {{nowrap\|''L'' − ''s'' > ''s''}}. For the other type of scale iteration, where {{nowrap\|''L'' − ''s'' < ''s''}}, the result is simply reciprocated:

	<math>		<math>

ArrowHead294 at 03:48, 26 January 2026

2026-01-26T03:48:40Z

← Older revision		Revision as of 03:48, 26 January 2026
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	Find any ''s'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.		Find any ''s'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.

	Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.		Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and ''s''{{`s}} that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.

	=== Beyond golden cases ===		=== Beyond golden cases ===

ArrowHead294: /* Generator introduction counts */

2026-01-26T03:47:25Z

Generator introduction counts

← Older revision		Revision as of 03:47, 26 January 2026
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	! 4th Metallic mean or isotope		! 4th Metallic mean or isotope
	\|-		\|-
	~~\| '''~~1~~'''~~		! 1
	\| 1		\| 1
	\| 1		\| 1
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	\| 1		\| 1
	\|-		\|-
	~~\| '''~~2~~'''~~		! 2
	\| 1*		\| 1*
	\| 2		\| 2
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	\| 2		\| 2
	\|-		\|-
	~~\| '''~~3~~'''~~		! 3
	\| 2		\| 2
	\| 3*		\| 3*
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	\| 4		\| 4
	\|-		\|-
	~~\| '''~~4~~'''~~		! 4
	\| 4		\| 4
	\| 6		\| 6
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	\| 8		\| 8
	\|-		\|-
	~~\| '''~~5~~'''~~		! 5
	\| 8		\| 8
	\| 12		\| 12
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	\| 15*		\| 15*
	\|-		\|-
	~~\| '''~~6~~'''~~		! 6
	\| 16		\| 16
	\| 24		\| 24
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	\| 30		\| 30
	\|-		\|-
	~~\| '''~~7~~'''~~		! 7
	\| 32		\| 32
	\| 48		\| 48
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	\| 32		\| 32
	\|-		\|-
	~~\| '''~~6~~'''~~		! 6
	\| 16		\| 16
	\| 32		\| 32
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	\| 64		\| 64
	\|-		\|-
	~~\| '''~~7~~'''~~		! 7
	\| 32		\| 32
	\| 64		\| 64

ArrowHead294: /* Split segment ratios */

2026-01-26T03:46:20Z

Split segment ratios

← Older revision		Revision as of 03:46, 26 January 2026
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	Which we can see is the weight multiplied by the bounding ratios' denominators:		Which we can see is the weight multiplied by the bounding ratios' denominators:

	<math> 1.078674 = \phi ~~·~~ \frac 23		<math> 1.078674 = \phi \cdot \frac{2}{3}
	</math>		</math>

ArrowHead294 at 02:28, 26 January 2026

2026-01-26T02:28:59Z

← Older revision		Revision as of 02:28, 26 January 2026
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	We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.		We are reasoning about MOS concepts in the abstract here. These truths about large and small steps are true whether they are 100¢ or 4516.8¢, and all we really care about are their ratios. So if we treat our small steps' size as 1 then we can treat our large steps' size as equal to the ''L'':''s'' ratio.

	So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ~~<math>~~L~~</math>~~, and ~~<math>~~s~~</math>~~'s size is 1, then 1 should be subtracted from ~~<math>~~L~~</math>~~.		So the ''L'':''s'' ratio decreases by 1 because if an ''s''-sized chunk has been sliced off ''L'', and ''s''{{`s}} size is 1, then 1 should be subtracted from ''L''.

	<math>		<math>
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	</math>		</math>

	But we're only getting started. This situation has recursive potential. We can now substitute <math>L + s</math> in for ~~<math>~~L~~</math>~~ as long as we also substitute in ''L'' for ''s'', and we'll still get a ratio that equals φ:		But we're only getting started. This situation has recursive potential. We can now substitute <math>L + s</math> in for ''L'' as long as we also substitute in ''L'' for ''s'', and we'll still get a ratio that equals φ:

	<math>		<math>
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	</math>		</math>

	Now find any ''L'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ~~<math>~~L~~</math>~~ will be replaced with an ''L'' and an ''s''. After two iterations, the original ''L'' interval is now represented by two ''L''{{'s}} and an ''s''. And so forth.		Now find any ''L'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''L'' will be replaced with an ''L'' and an ''s''. After two iterations, the original ''L'' interval is now represented by two ''L''{{'s}} and an ''s''. And so forth.

	The same will hold for the right side of the interval pattern, for ''s'':		The same will hold for the right side of the interval pattern, for ''s'':
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	</math>		</math>

	Find any ~~<math>~~s~~</math>~~ in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.		Find any ''s'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.

	Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier L's and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.		Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier ''L''{{'s}} and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.

	=== Beyond golden cases ===		=== Beyond golden cases ===
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	</math>		</math>

	We'll repeat the technique we used for the golden case: find any ~~<math>~~L~~</math>~~ in the horogram and observe how it gets split up as we iterate through the scale sequence. However, the complexity that silver introduces is that we don't look to the next iteration to see the next entry in the interval pattern; we have to skip an iteration. So if we just look at all the odd rings, ring 1, 3, 5, 7, etc. then we'll see the pattern. The same is true of ''s''.		We'll repeat the technique we used for the golden case: find any ''L'' in the horogram and observe how it gets split up as we iterate through the scale sequence. However, the complexity that silver introduces is that we don't look to the next iteration to see the next entry in the interval pattern; we have to skip an iteration. So if we just look at all the odd rings, ring 1, 3, 5, 7, etc. then we'll see the pattern. The same is true of ''s''.

	And if we want to understand the interval pattern for {{nowrap\|δ<sub>''s''</sub> − 1}}, we'll look at the right and left sides separately:		And if we want to understand the interval pattern for {{nowrap\|δ<sub>''s''</sub> − 1}}, we'll look at the right and left sides separately:
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	The reason the other cases exhibited such overlapping is that the small step size of the next ratio in the equivalence pattern became an ''L'', which is the same as the ''L'' size of the preceding ratio. However, for the silver mean's first isotope here, no such link exists, since ''s'' is substituted not for ''L'', but {{nowrap\|''L'' + ''s''}}.		The reason the other cases exhibited such overlapping is that the small step size of the next ratio in the equivalence pattern became an ''L'', which is the same as the ''L'' size of the preceding ratio. However, for the silver mean's first isotope here, no such link exists, since ''s'' is substituted not for ''L'', but {{nowrap\|''L'' + ''s''}}.

	Another way of looking at this is: for δ<sub>''s''</sub> and φ, it was the case that both ~~<math>~~s~~</math>~~ and ~~<math>~~L~~</math>~~'s interval sequences were the same, just offset from each other by a step. Whereas for {{nowrap\|δ<sub>''s''</sub> − 1}},''s'' and L's interval sequences are completely different.		Another way of looking at this is: for δ<sub>''s''</sub> and φ, it was the case that both ''s'' and ''L''{{'s}} interval sequences were the same, just offset from each other by a step. Whereas for {{nowrap\|δ<sub>''s''</sub> − 1}}, ''s'' and ''L''{{'s}} interval sequences are completely different.

	Finally, for the bronze ratio,		Finally, for the bronze ratio,

ArrowHead294 at 02:22, 26 January 2026

2026-01-26T02:22:02Z

Show changes

ArrowHead294 at 02:17, 26 January 2026

2026-01-26T02:17:21Z

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

2025-10-09T14:11:12Z

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

← Older revision		Revision as of 14:11, 9 October 2025
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	== Imaginary ==		== Imaginary ==
	If Argent ~~temperament~~ splits the period into segments in the ratio of the silver ratio's isotope, what if we split the period into segments in the ratio of the silver mean itself? That gives us a generator of approximately 0.292893, which has been used by Billy Stiltner, who calls it Imaginary.		If Argent tuning splits the period into segments in the ratio of the silver ratio's isotope, what if we split the period into segments in the ratio of the silver mean itself? That gives us a generator of approximately 0.292893, which has been used by Billy Stiltner, who calls it Imaginary.

	Twice 0.292893 is equal to 0.585786, which is the complement of Argent Temperament's generator, 0.414214. Despite this similarity, these two generate tremendously different scales (compare with the silver horogram shown earlier).		Twice 0.292893 is equal to 0.585786, which is the complement of Argent Temperament's generator, 0.414214. Despite this similarity, these two generate tremendously different scales (compare with the silver horogram shown earlier).

ArrowHead294 at 18:50, 24 December 2024

2024-12-24T18:50:39Z

← Older revision		Revision as of 18:50, 24 December 2024
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{align*}}~~
	~~{\end{align*}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L &: s \\		L &: s \\
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{align*}}~~
	~~{\end{align*}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L &: s \\		L &: s \\
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{align*}}~~
	~~{\end{align*}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L &: s \\		L &: s \\
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{aligned}}~~
	~~{\end{aligned}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L &: s \\		L &: s \\
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{align*}}~~
	~~{\end{align*}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L &: s \\		L &: s \\
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	<math>		<math>
	~~\newenvironment{rcases}~~
	~~{\left.\begin{align*}}~~
	~~{\end{align*}\right\rbrace}~~

	\begin{rcases}		\begin{rcases}
	L&{:}s \\		L&{:}s \\

← Older revision		Revision as of 02:17, 26 January 2026
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	</math>		</math>

	Now find any ''L'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <math>L</math> will be replaced with an ''L'' and an ''s''. After two iterations, the original ''L'' interval is now represented by two ''L''{{'}}s and an ''s''. And so forth.		Now find any ''L'' in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <math>L</math> will be replaced with an ''L'' and an ''s''. After two iterations, the original ''L'' interval is now represented by two ''L''{{'s}} and an ''s''. And so forth.

	The same will hold for the right side of the interval pattern, for ''s'':		The same will hold for the right side of the interval pattern, for ''s'':
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	Find any <math>s</math> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.		Find any <math>s</math> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, ''s'' will be replaced with ''L''. After two iterations, the original ''s'' interval is now represented by an ''L'' and an ''s''. And so forth.

	Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier L's and s's that remain—only now spanning many ''L''{{'}}s and ''s''{{'}}s each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.		Every MOS scale contains every scale earlier in its scale sequence. In other words, any interval that existed in an earlier scale will remain in all later scales. These earlier L's and s's that remain—only now spanning many ''L''{{'s}} and ''s''{{`s}} each—are precisely the larger intervals in the scale that also exhibit the φ ratio to each other.

	=== Beyond golden cases ===		=== Beyond golden cases ===
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	* '''Generator, aristocratic:''' A generator which is found using the weighted mediant formula on a non-period Stern–Brocot tree interval with a beyond golden mean.		* '''Generator, aristocratic:''' A generator which is found using the weighted mediant formula on a non-period Stern–Brocot tree interval with a beyond golden mean.
	* '''Generator, bronze:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to {{nowrap\|[0; 4, {{overline\|3}}] ≈ 0.232408}}.		* '''Generator, bronze:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to {{nowrap\|[0; 4, {{overline\|3}}] ≈ 0.232408}}.
	* '''Generator, complement:''' A generator ''g''{{'}}s complement generator is equal to {{nowrap\|1 − ''g''}}.		* '''Generator, complement:''' A generator ''g''{{'s}} complement generator is equal to {{nowrap\|1 − ''g''}}.
	* '''Generator, golden:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to {{nowrap\|[0; 4, {{overline\|3}}] ≈ 0.381966}}.		* '''Generator, golden:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to {{nowrap\|[0; 4, {{overline\|3}}] ≈ 0.381966}}.
	* '''Generator, isotopic:''' A generator found using the weighted mediant formula on the period interval with any isotope of a beyond golden mean.		* '''Generator, isotopic:''' A generator found using the weighted mediant formula on the period interval with any isotope of a beyond golden mean.