Metallic MOS: Difference between revisions

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|}

== L:s sequences ==

== ''L'':''s'' sequences ==

Each scale has exactly two step sizes: large and small, or "L" and "s". We can refer to the ratio between these large and small steps as

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</math>

We'll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of L:s corresponding to these scales a generator's ''L:s sequence''.

We'll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of ''L'':''s'' corresponding to these scales a generator's ''L:s sequence''.

=== Golden case ===

The golden generator's L:s sequence is simple. Every L:s ratio is φ:

The golden generator's ''L'':''s'' sequence is simple. Every ''L'':''s'' ratio is φ:

<math>\qquad L{:}s = \phi

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=== Noble cases ===

A noble generator's L:s sequence is slightly more complex. Not every—but almost every—L:s is φ. Only the first few are not.

A noble generator's ''L'':''s'' sequence is slightly more complex. Not every—but almost every—''L'':''s'' is φ. Only the first few are not.

=== Beyond golden cases ===

Instead of every scale's L:s equaling the same value, as is the case for the golden mean, the silver mean's L:s sequence alternates between its isotopes that are greater than 1:

Instead of every scale's ''L'':''s'' equaling the same value, as is the case for the golden mean, the silver mean's ''L'':''s'' sequence alternates between its isotopes that are greater than 1:

<math>

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</math>

And the bronze mean's L:s sequence cycles through its isotopes that are greater than 1:

And the bronze mean's ''L'':''s'' sequence cycles through its isotopes that are greater than 1:

<math>

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</math>

Any n-metallic mean's L:s sequence will cycle through its isotopes that are greater than 1.

Any n-metallic mean's ''L'':''s'' sequence will cycle through its isotopes that are greater than 1.

=== Isotopic cases ===

Isotopic L:s sequences are just like those of their mean's, but offset.

Isotopic ''L'':''s'' sequences are just like those of their mean's, but offset.

For example, the silver mean's first isotope's generator's L:s sequence alternates between {{nowrap|L:s {{=}} δ''s''}} and {{nowrap|L:s {{=}} δ''s'' − 1}}, just like the silver generator's, however—unlike the silver generator's—it begins with {{nowrap|L:s {{=}} δ''s'' − 1}}.

For example, the silver mean's first isotope's generator's ''L'':''s'' sequence alternates between {{nowrap|''L'':''s'' {{=}} δ''s''}} and {{nowrap|''L'':''s'' {{=}} δ''s'' − 1}}, just like the silver generator's, however—unlike the silver generator's—it begins with {{nowrap|''L'':''s'' {{=}} δ''s'' − 1}}.

=== Aristocratic cases ===

Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We'll call the periodic part of an L:s sequence its ''L:s cycle''. So most of the L:s sequence will be the L:s cycle, with only the first few scales not being so.

Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We'll call the periodic part of an ''L'':''s'' sequence its ''L:s cycle''. So most of the ''L'':''s'' sequence will be the ''L'':''s'' cycle, with only the first few scales not being so.

== Interval patterns ==

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=== Noble cases ===

Noble scales at first do not—but eventually do—reach a point where they start exhibiting this interval pattern (paralleling how their L:s sequences only eventually exhibit {{nowrap|L:s {{=}} φ}}).

Noble scales at first do not—but eventually do—reach a point where they start exhibiting this interval pattern (paralleling how their ''L'':''s'' sequences only eventually exhibit {{nowrap|''L'':''s'' {{=}} φ}}).

Once we've iterated past the point that our scale exhibits {{nowrap|L:s {{=}} φ}}, some of the smaller intervals will begin to be related by φ, but its larger intervals will never be related by φ.

Once we've iterated past the point that our scale exhibits {{nowrap|''L'':''s'' {{=}} φ}}, some of the smaller intervals will begin to be related by φ, but its larger intervals will never be related by φ.

=== Beyond golden cases ===

The silver generator, as it did for its L:s sequence, alternates in quality between its two >1 isotopes for its intervals. For half of its scales,

The silver generator, as it did for its ''L'':''s'' sequence, alternates in quality between its two >1 isotopes for its intervals. For half of its scales,

<math>

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Bronze's scales cycle through three different interval patterns related to its respective <math>>1</math> isotopes.

This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and L:s sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''.

This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and ''L'':''s'' sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''.

=== Isotopic cases ===

As with L:s sequences, isotopic interval pattern sequences are identical to their metallic mean's, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle.

As with ''L'':''s'' sequences, isotopic interval pattern sequences are identical to their metallic mean's, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle.

=== Aristocratic cases ===

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= Mathematical explanations =

We'll now start going through mathematical explanations for the behavior we've observed about met-MOS generators, L:s sequences, and interval patterns.

We'll now start going through mathematical explanations for the behavior we've observed about met-MOS generators, ''L'':''s'' sequences, and interval patterns.

== Infinite scale sequences ==

Every metallic generator generates an infinitely long scale sequence.

This property is not unique to metallic generators, though—it is attributable to their being irrational numbers. A rational generator's scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because {{nowrap|L {{=}} s}} and {{nowrap|L:s {{=}} 1}}. For example, the generator

This property is not unique to metallic generators, though—it is attributable to their being irrational numbers. A rational generator's scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because {{nowrap|''L'' {{=}} s}} and {{nowrap|''L'':''s'' {{=}} 1}}. For example, the generator

<math>\qquad \frac{5}{12} = 0.41\overline{6}

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What is special about metallic generators' infinite scale sequences—as opposed to those of other irrational generators—is that no matter how infinitesimally small their steps become, they will maintain their metallic ratios to each other.

The ratio {{nowrap|L:s {{=}} φ}} is unique in that it is the only ratio in which the MOS is [[strictly proper]], and all of its descendent MOS's are also strictly proper.

The ratio {{nowrap|''L'':''s'' {{=}} φ}} is unique in that it is the only ratio in which the MOS is [[strictly proper]], and all of its descendent MOS's are also strictly proper.

== Weighted mediants ==

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Despite this exponential profusion of scales, however, as one travels deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which

# begins the periodic phase of the L:s sequence for the metal they're based on, and

# begins the periodic phase of the ''L'':''s'' sequence for the metal they're based on, and

# begins supporting the interval pattern for the metal they're based on.

The golden generator is essentially the noble generator for the interval {{sfrac|0|1}} to {{sfrac|1|1}}, which—being the root of the Stern–Brocot tree—is as golden as we can get: we see {{nowrap|L:s {{=}} φ}} and φ's distinctive interval pattern from the very start.

The golden generator is essentially the noble generator for the interval {{sfrac|0|1}} to {{sfrac|1|1}}, which—being the root of the Stern–Brocot tree—is as golden as we can get: we see {{nowrap|''L'':''s'' {{=}} φ}} and φ's distinctive interval pattern from the very start.

And the noble generator between {{sfrac|0|1}} and {{sfrac|1|3}}

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</math>

is very close to the root of the tree; it has initial L:s ratio of <math>\phi + 2</math>, then attains {{nowrap|L:s {{=}} φ}} after only one iteration. And it begins the golden interval pattern after just one iteration too.

is very close to the root of the tree; it has initial ''L'':''s'' ratio of <math>\phi + 2</math>, then attains {{nowrap|''L'':''s'' {{=}} φ}} after only one iteration. And it begins the golden interval pattern after just one iteration too.

On the other hand, the noble generator equal to 0.275267—while only a smidgen off from the other noble generator we just looked at—necessitates iterating ''six'' times before attaining {{nowrap|L:s {{=}} φ}}. This corresponds to it being the noble generator between <math>\frac {5}{18}</math> and <math>\frac {3}{11}</math>, an interval which lies five levels deeper in the Stern–Brocot tree than the interval from {{sfrac|0|1}} to {{sfrac|1|3}}.

On the other hand, the noble generator equal to 0.275267—while only a smidgen off from the other noble generator we just looked at—necessitates iterating ''six'' times before attaining {{nowrap|''L'':''s'' {{=}} φ}}. This corresponds to it being the noble generator between <math>\frac {5}{18}</math> and <math>\frac {3}{11}</math>, an interval which lies five levels deeper in the Stern–Brocot tree than the interval from {{sfrac|0|1}} to {{sfrac|1|3}}.

So if we want a golden scale, and we also happen to want a generator near 0.276393, then we're in luck. But if we want a golden generator that is close to 0.275267, we may be disappointed to hear that it is not "golden" enough for us.

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== Isotopic arithmetic progression ==

Now we'll explain why the L:s sequences for metallic means cycle through their isotopes.

Now we'll explain why the ''L'':''s'' sequences for metallic means cycle through their isotopes.

=== Why they decrease by 1 ===

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[[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¢ 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 ~~<math>~~s~~</math>~~-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 <math>L</math>, and <math>s</math>'s size is 1, then 1 should be subtracted from <math>L</math>.

<math>

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=== Why they cycle ===

That is true of scale iterations where ~~<math>~~L - s > s~~</math>~~. For the other type of scale iteration, where ~~<math>~~L - s < s~~</math>~~, 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>

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</math>

This alone would not suffice to explain how the L:s sequences lock into a cycle of isotopes. But here's where the magic of the metallic means comes into play. φ has the property that

This alone would not suffice to explain how the ''L'':''s'' sequences lock into a cycle of isotopes. But here's where the magic of the metallic means comes into play. φ has the property that

<math>\qquad \phi - 1 = \frac{1}{\phi}

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</math>

That's why the golden L:s sequence locks into {{nowrap|L:s {{=}} φ}} forever.

That's why the golden ''L'':''s'' sequence locks into {{nowrap|''L'':''s'' {{=}} φ}} forever.

A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value's reciprocal is equal to the silver mean.

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Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2's, 3's, or <math>n</math> if we base our noble on the <math>n</math>th metallic mean. For example, our earlier example 0.226541 is <math>[0; 4, \overline{2}]</math>

=== Application: L:s sequences===

=== Application: ''L'':''s'' sequences===

Continued fractions can compute L:s sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we'll look at the L:s sequence for <math>g = [0; 2, 2, \overline{1}] \approx 0.419821</math>.

Continued fractions can compute ''L'':''s'' sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we'll look at the ''L'':''s'' sequence for <math>g = [0; 2, 2, \overline{1}] \approx 0.419821</math>.

<math>

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</math>

We've stated that {{nowrap|L:s {{=}} φ}} for every golden scale, while L:s for noble scales eventually do, just not at first. Noble L:s sequences lock onto φ at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1's is a no-op).

We've stated that {{nowrap|''L'':''s'' {{=}} φ}} for every golden scale, while ''L'':''s'' for noble scales eventually do, just not at first. Noble ''L'':''s'' sequences lock onto φ at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1's is a no-op).

Thus it makes sense that logarithmic phi's L:s sequence remains fixed from the beginning, because with a continued fraction of <math>[0; 1]</math> we get the L:s sequence

Thus it makes sense that logarithmic phi's ''L'':''s'' sequence remains fixed from the beginning, because with a continued fraction of <math>[0; 1]</math> we get the ''L'':''s'' sequence

<math>

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</math>

And it makes sense that the silver mean's generator would alternate between two L:s ratios, because it will alternate between

And it makes sense that the silver mean's generator would alternate between two ''L'':''s'' ratios, because it will alternate between

<math>

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=== Application: finding generator ===

To compute the L:s sequence, we depleted terms of the generator's continued fraction. By doing the opposite—gradually building up the generator's continued fraction by incrementing terms—we can determine the path our generator takes through the Stern–Brocot tree.

To compute the ''L'':''s'' sequence, we depleted terms of the generator's continued fraction. By doing the opposite—gradually building up the generator's continued fraction by incrementing terms—we can determine the path our generator takes through the Stern–Brocot tree.

We'll use the example of the golden generator, with continued fraction <math>[0; 2, \overline{1}]</math>:

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=== Golden case ===

We know that the golden generator's {{nowrap|L:s {{=}} φ}}, but we can also say this about them:

We know that the golden generator's {{nowrap|''L'':''s'' {{=}} φ}}, but we can also say this about them:

<math>\qquad (L+s){:}L = \phi

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</math>

So, wherever we have a scale where {{nowrap|L:s {{=}} δ''s''}}, we'll also see the interval pattern

So, wherever we have a scale where {{nowrap|''L'':''s'' {{=}} δ''s''}}, we'll also see the interval pattern

<math>

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</math>

Every other scale the silver generator generates has an L:s other than δ''s'', namely, its isotope, {{nowrap|δ''s'' − 1}}. These scales have a different pattern:

Every other scale the silver generator generates has an ''L'':''s'' other than δ''s'', namely, its isotope, {{nowrap|δ''s'' − 1}}. These scales have a different pattern:

<math> L{:}s = (L+2s){:}(L+s) = \delta_s - 1

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</math>

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''s'' 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''s'' − 1}}, s and L's interval sequences are completely different.

Another way of looking at this is: for δ''s'' 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''s'' − 1}},''s'' and L's interval sequences are completely different.

Finally, for the bronze ratio,

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</math>

As expected, {{nowrap|L:s {{=}} (3L + 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 =

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= Glossary =

* '''Cycle, interval pattern:''' The periodic part at the end of an interval pattern sequence.

* '''Cycle, L:s:''' The periodic part at the end of an sequence, cycling through the isotopes of the given metallic mean.

* '''Cycle, ''L'':''s'':''' The periodic part at the end of an sequence, cycling through the isotopes of the given metallic 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}}.

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* '''Sequence, cardinality:''' For a given scale sequence, the corresponding sequence of their cardinalities.

* '''Sequence, interval pattern:''' For a given scale sequence, the corresponding sequence of their interval patterns.

* '''Sequence, L:s:''' For a given scale sequence, the corresponding sequence of their ratios.

* '''Sequence, ''L'':''s'':''' For a given scale sequence, the corresponding sequence of their ratios.

* '''Sequence, scale:''' The ordered sequence of scales a generator generates, each one containing the previous, strictly adding new pitches to it.

@@ Line 102: / Line 102: @@
 |}
-== L:s sequences ==
+== ''L'':''s'' sequences ==
 Each scale has exactly two step sizes: large and small, or "L" and "s". We can refer to the ratio between these large and small steps as
@@ Line 108: / Line 108: @@
 </math>
-We'll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of L:s corresponding to these scales a generator's ''L:s sequence''.
+We'll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of ''L'':''s'' corresponding to these scales a generator's ''L:s sequence''.
 === Golden case ===
-The golden generator's L:s sequence is simple. Every L:s ratio is &phi;:
+The golden generator's ''L'':''s'' sequence is simple. Every ''L'':''s'' ratio is &phi;:
 <math>\qquad L{:}s = \phi
@@ Line 117: / Line 117: @@
 === Noble cases ===
-A noble generator's L:s sequence is slightly more complex. Not every&mdash;but almost every&mdash;L:s is &phi;. Only the first few are not.
+A noble generator's ''L'':''s'' sequence is slightly more complex. Not every&mdash;but almost every&mdash;''L'':''s'' is &phi;. Only the first few are not.
 === Beyond golden cases ===
-Instead of every scale's L:s equaling the same value, as is the case for the golden mean, the silver mean's L:s sequence alternates between its isotopes that are greater than 1:
+Instead of every scale's ''L'':''s'' equaling the same value, as is the case for the golden mean, the silver mean's ''L'':''s'' sequence alternates between its isotopes that are greater than 1:
 <math>
@@ Line 131: / Line 131: @@
 </math>
-And the bronze mean's L:s sequence cycles through its isotopes that are greater than 1:
+And the bronze mean's ''L'':''s'' sequence cycles through its isotopes that are greater than 1:
 <math>
@@ Line 143: / Line 143: @@
 </math>
-Any n-metallic mean's L:s sequence will cycle through its isotopes that are greater than 1.
+Any n-metallic mean's ''L'':''s'' sequence will cycle through its isotopes that are greater than 1.
 === Isotopic cases ===
-Isotopic L:s sequences are just like those of their mean's, but offset.
+Isotopic ''L'':''s'' sequences are just like those of their mean's, but offset.
-For example, the silver mean's first isotope's generator's L:s sequence alternates between {{nowrap|L:s {{=}} &delta;<sub>''s''</sub>}} and {{nowrap|L:s {{=}} &delta;<sub>''s''</sub> &minus; 1}}, just like the silver generator's, however&mdash;unlike the silver generator's&mdash;it begins with {{nowrap|L:s {{=}} &delta;<sub>''s''</sub> &minus; 1}}.
+For example, the silver mean's first isotope's generator's ''L'':''s'' sequence alternates between {{nowrap|''L'':''s'' {{=}} &delta;<sub>''s''</sub>}} and {{nowrap|''L'':''s'' {{=}} &delta;<sub>''s''</sub> &minus; 1}}, just like the silver generator's, however&mdash;unlike the silver generator's&mdash;it begins with {{nowrap|''L'':''s'' {{=}} &delta;<sub>''s''</sub> &minus; 1}}.
 === Aristocratic cases ===
-Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We'll call the periodic part of an L:s sequence its ''L:s cycle''. So most of the L:s sequence will be the L:s cycle, with only the first few scales not being so.
+Again, aristocratic scales synthesize both the complexities of noble scales and beyond golden scales. We'll call the periodic part of an ''L'':''s'' sequence its ''L:s cycle''. So most of the ''L'':''s'' sequence will be the ''L'':''s'' cycle, with only the first few scales not being so.
 == Interval patterns ==
@@ Line 183: / Line 183: @@
 === Noble cases ===
-Noble scales at first do not&mdash;but eventually do&mdash;reach a point where they start exhibiting this interval pattern (paralleling how their L:s sequences only eventually exhibit {{nowrap|L:s {{=}} &phi;}}).
+Noble scales at first do not&mdash;but eventually do&mdash;reach a point where they start exhibiting this interval pattern (paralleling how their ''L'':''s'' sequences only eventually exhibit {{nowrap|''L'':''s'' {{=}} &phi;}}).
-Once we've iterated past the point that our scale exhibits {{nowrap|L:s {{=}} &phi;}}, some of the smaller intervals will begin to be related by &phi;, but its larger intervals will never be related by &phi;.
+Once we've iterated past the point that our scale exhibits {{nowrap|''L'':''s'' {{=}} &phi;}}, some of the smaller intervals will begin to be related by &phi;, but its larger intervals will never be related by &phi;.
 === Beyond golden cases ===
-The silver generator, as it did for its L:s sequence, alternates in quality between its two &gt;1 isotopes for its intervals. For half of its scales,
+The silver generator, as it did for its ''L'':''s'' sequence, alternates in quality between its two &gt;1 isotopes for its intervals. For half of its scales,
 <math>
@@ Line 226: / Line 226: @@
 Bronze's scales cycle through three different interval patterns related to its respective <math>>1</math> isotopes.
-This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and L:s sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''.
+This pattern continues for other metallic means. As another entry to our family of sequence terms (along with scale sequence and ''L'':''s'' sequence) we shall use the term ''interval pattern sequence'', and for the periodic part at the end, the ''interval pattern cycle''.
 === Isotopic cases ===
-As with L:s sequences, isotopic interval pattern sequences are identical to their metallic mean's, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle.
+As with ''L'':''s'' sequences, isotopic interval pattern sequences are identical to their metallic mean's, cycling through a set of interval patterns from the beginning, except starting at a different position in that cycle.
 === Aristocratic cases ===
@@ Line 235: / Line 235: @@
 = Mathematical explanations =
-We'll now start going through mathematical explanations for the behavior we've observed about met-MOS generators, L:s sequences, and interval patterns.
+We'll now start going through mathematical explanations for the behavior we've observed about met-MOS generators, ''L'':''s'' sequences, and interval patterns.
 == Infinite scale sequences ==
 Every metallic generator generates an infinitely long scale sequence.
-This property is not unique to metallic generators, though&mdash;it is attributable to their being irrational numbers. A rational generator's scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because {{nowrap|L {{=}} s}} and {{nowrap|L:s {{=}} 1}}. For example, the generator
+This property is not unique to metallic generators, though&mdash;it is attributable to their being irrational numbers. A rational generator's scale sequence eventually terminates, hitting bedrock when the period has been divided up into equal steps, i.e. where the notion of large steps and small steps no longer applies because {{nowrap|''L'' {{=}} s}} and {{nowrap|''L'':''s'' {{=}} 1}}. For example, the generator
 <math>\qquad \frac{5}{12} = 0.41\overline{6}
@@ Line 251: / Line 251: @@
 What is special about metallic generators' infinite scale sequences&mdash;as opposed to those of other irrational generators&mdash;is that no matter how infinitesimally small their steps become, they will maintain their metallic ratios to each other.
-The ratio {{nowrap|L:s {{=}} &phi;}} is unique in that it is the only ratio in which the MOS is [[strictly proper]], and all of its descendent MOS's are also strictly proper.
+The ratio {{nowrap|''L'':''s'' {{=}} &phi;}} is unique in that it is the only ratio in which the MOS is [[strictly proper]], and all of its descendent MOS's are also strictly proper.
 == Weighted mediants ==
@@ Line 374: / Line 374: @@
 Despite this exponential profusion of scales, however, as one travels deeper down the tree, the generators lose musical interest. They become less and less metallic. We can quantify their metallicity in terms of how many iterations of their scale sequence are required before they reach the scale which
-# begins the periodic phase of the L:s sequence for the metal they're based on, and
+# begins the periodic phase of the ''L'':''s'' sequence for the metal they're based on, and
 # begins supporting the interval pattern for the metal they're based on.
-The golden generator is essentially the noble generator for the interval {{sfrac|0|1}} to {{sfrac|1|1}}, which&mdash;being the root of the Stern&ndash;Brocot tree&mdash;is as golden as we can get: we see {{nowrap|L:s {{=}} &phi;}} and &phi;'s distinctive interval pattern from the very start.
+The golden generator is essentially the noble generator for the interval {{sfrac|0|1}} to {{sfrac|1|1}}, which&mdash;being the root of the Stern&ndash;Brocot tree&mdash;is as golden as we can get: we see {{nowrap|''L'':''s'' {{=}} &phi;}} and &phi;'s distinctive interval pattern from the very start.
 And the noble generator between {{sfrac|0|1}} and {{sfrac|1|3}}
@@ Line 383: / Line 383: @@
 </math>
-is very close to the root of the tree; it has initial L:s ratio of <math>\phi + 2</math>, then attains {{nowrap|L:s {{=}} &phi;}} after only one iteration. And it begins the golden interval pattern after just one iteration too.
+is very close to the root of the tree; it has initial ''L'':''s'' ratio of <math>\phi + 2</math>, then attains {{nowrap|''L'':''s'' {{=}} &phi;}} after only one iteration. And it begins the golden interval pattern after just one iteration too.
-On the other hand, the noble generator equal to 0.275267&mdash;while only a smidgen off from the other noble generator we just looked at&mdash;necessitates iterating ''six'' times before attaining {{nowrap|L:s {{=}} &phi;}}. This corresponds to it being the noble generator between <math>\frac {5}{18}</math> and <math>\frac {3}{11}</math>, an interval which lies five levels deeper in the Stern&ndash;Brocot tree than the interval from {{sfrac|0|1}} to {{sfrac|1|3}}.
+On the other hand, the noble generator equal to 0.275267&mdash;while only a smidgen off from the other noble generator we just looked at&mdash;necessitates iterating ''six'' times before attaining {{nowrap|''L'':''s'' {{=}} &phi;}}. This corresponds to it being the noble generator between <math>\frac {5}{18}</math> and <math>\frac {3}{11}</math>, an interval which lies five levels deeper in the Stern&ndash;Brocot tree than the interval from {{sfrac|0|1}} to {{sfrac|1|3}}.
 So if we want a golden scale, and we also happen to want a generator near 0.276393, then we're in luck. But if we want a golden generator that is close to 0.275267, we may be disappointed to hear that it is not "golden" enough for us.
@@ Line 418: / Line 418: @@
 == Isotopic arithmetic progression ==
-Now we'll explain why the L:s sequences for metallic means cycle through their isotopes.
+Now we'll explain why the ''L'':''s'' sequences for metallic means cycle through their isotopes.
 === Why they decrease by 1 ===
@@ Line 427: / Line 427: @@
 [[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¢ 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 <math>s</math>-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 <math>L</math>, and <math>s</math>'s size is 1, then 1 should be subtracted from <math>L</math>.
 <math>
@@ Line 440: / Line 440: @@
 === Why they cycle ===
-That is true of scale iterations where <math>L - s > s</math>. For the other type of scale iteration, where <math>L - s < s</math>, the result is simply reciprocated:
+That is true of scale iterations where {{nowrap|''L'' &minus; ''s'' &gt; s}}. For the other type of scale iteration, where {{nowrap|''L'' &minus; ''s'' &lt; s}}, the result is simply reciprocated:
 <math>
@@ Line 450: / Line 450: @@
 </math>
-This alone would not suffice to explain how the L:s sequences lock into a cycle of isotopes. But here's where the magic of the metallic means comes into play. &phi; has the property that
+This alone would not suffice to explain how the ''L'':''s'' sequences lock into a cycle of isotopes. But here's where the magic of the metallic means comes into play. &phi; has the property that
 <math>\qquad  \phi - 1 = \frac{1}{\phi}
@@ Line 466: / Line 466: @@
 </math>
-That's why the golden L:s sequence locks into {{nowrap|L:s {{=}} &phi;}} forever.
+That's why the golden ''L'':''s'' sequence locks into {{nowrap|''L'':''s'' {{=}} &phi;}} forever.
 A similar case is true for the silver mean, except we have to subtract 1 from it twice before the resulting value's reciprocal is equal to the silver mean.
@@ Line 512: / Line 512: @@
 Crossing nobles with beyond golden cases results in continued fractions which can start with anything but eventually settle on all 2's, 3's, or <math>n</math> if we base our noble on the <math>n</math>th metallic mean. For example, our earlier example 0.226541 is <math>[0; 4, \overline{2}]</math>
-=== Application: L:s sequences===
+=== Application: ''L'':''s'' sequences===
-Continued fractions can compute L:s sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we'll look at the L:s sequence for <math>g = [0; 2, 2, \overline{1}] \approx 0.419821</math>.
+Continued fractions can compute ''L'':''s'' sequences by repeatedly depleting the terms of the continued fraction for the generator. For example, we'll look at the ''L'':''s'' sequence for <math>g = [0; 2, 2, \overline{1}] \approx 0.419821</math>.
 <math>
@@ Line 527: / Line 527: @@
 </math>
-We've stated that {{nowrap|L:s {{=}} &phi;}} for every golden scale, while L:s for noble scales eventually do, just not at first. Noble L:s sequences lock onto &phi; at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1's is a no-op).
+We've stated that {{nowrap|''L'':''s'' {{=}} &phi;}} for every golden scale, while ''L'':''s'' for noble scales eventually do, just not at first. Noble ''L'':''s'' sequences lock onto &phi; at the point where depleting the continued fraction more no longer changes it (removing a 1 from the beginning of an infinite string of 1's is a no-op).
-Thus it makes sense that logarithmic phi's L:s sequence remains fixed from the beginning, because with a continued fraction of <math>[0; 1]</math> we get the L:s sequence
+Thus it makes sense that logarithmic phi's ''L'':''s'' sequence remains fixed from the beginning, because with a continued fraction of <math>[0; 1]</math> we get the ''L'':''s'' sequence
 <math>
@@ Line 540: / Line 540: @@
 </math>
-And it makes sense that the silver mean's generator would alternate between two L:s ratios, because it will alternate between
+And it makes sense that the silver mean's generator would alternate between two ''L'':''s'' ratios, because it will alternate between
 <math>
@@ Line 553: / Line 553: @@
 === Application: finding generator ===
-To compute the L:s sequence, we depleted terms of the generator's continued fraction. By doing the opposite&mdash;gradually building up the generator's continued fraction by incrementing terms&mdash;we can determine the path our generator takes through the Stern&ndash;Brocot tree.
+To compute the ''L'':''s'' sequence, we depleted terms of the generator's continued fraction. By doing the opposite&mdash;gradually building up the generator's continued fraction by incrementing terms&mdash;we can determine the path our generator takes through the Stern&ndash;Brocot tree.
 We'll use the example of the golden generator, with continued fraction <math>[0; 2, \overline{1}]</math>:
@@ Line 698: / Line 698: @@
 === Golden case ===
-We know that the golden generator's {{nowrap|L:s {{=}} &phi;}}, but we can also say this about them:
+We know that the golden generator's {{nowrap|''L'':''s'' {{=}} &phi;}}, but we can also say this about them:
 <math>\qquad (L+s){:}L = \phi
@@ Line 786: / Line 786: @@
 </math>
-So, wherever we have a scale where {{nowrap|L:s {{=}} &delta;<sub>''s''</sub>}}, we'll also see the interval pattern
+So, wherever we have a scale where {{nowrap|''L'':''s'' {{=}} &delta;<sub>''s''</sub>}}, we'll also see the interval pattern
 <math>
@@ Line 804: / Line 804: @@
 </math>
-Every other scale the silver generator generates has an L:s other than &delta;<sub>''s''</sub>, namely, its isotope, {{nowrap|&delta;<sub>''s''</sub> &minus; 1}}. These scales have a different pattern:
+Every other scale the silver generator generates has an ''L'':''s'' other than &delta;<sub>''s''</sub>, namely, its isotope, {{nowrap|&delta;<sub>''s''</sub> &minus; 1}}. These scales have a different pattern:
 <math> L{:}s = (L+2s){:}(L+s) = \delta_s - 1
@@ Line 887: / Line 887: @@
 </math>
-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 &delta;<sub>''s''</sub> and &phi;, 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|&delta;<sub>''s''</sub> &minus; 1}}, s and L's interval sequences are completely different.
+Another way of looking at this is: for &delta;<sub>''s''</sub> and &phi;, 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|&delta;<sub>''s''</sub> &minus; 1}},''s'' and L's interval sequences are completely different.
 Finally, for the bronze ratio,
@@ Line 896: / Line 896: @@
 </math>
-As expected, {{nowrap|L:s {{=}} (3L + 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 =
@@ Line 1,302: / Line 1,302: @@
 = Glossary =
 * '''Cycle, interval pattern:''' The periodic part at the end of an interval pattern sequence.
-* '''Cycle, L:s:''' The periodic part at the end of an  sequence, cycling through the isotopes of the given metallic mean.
+* '''Cycle, ''L'':''s'':''' The periodic part at the end of an  sequence, cycling through the isotopes of the given metallic mean.
 * '''Generator, aristocratic:''' A generator which is found using the weighted mediant formula on a non-period Stern&ndash;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}}] &#x2248; 0.232408}}.
@@ Line 1,320: / Line 1,320: @@
 * '''Sequence, cardinality:''' For a given scale sequence, the corresponding sequence of their cardinalities.
 * '''Sequence, interval pattern:''' For a given scale sequence, the corresponding sequence of their interval patterns.
-* '''Sequence, L:s:''' For a given scale sequence, the corresponding sequence of their  ratios.
+* '''Sequence, ''L'':''s'':''' For a given scale sequence, the corresponding sequence of their  ratios.
 * '''Sequence, scale:''' The ordered sequence of scales a generator generates, each one containing the previous, strictly adding new pitches to it.