Metallic MOS: Difference between revisions

The simplest ''metallic generator'' splits the period into two segments which are related by <span><math>φ</math></span>. Wilson named this generator “Fibonacci”, after the famous recurrence relation associated with <span><math>φ</math></span>, but for consistency here we’ll be calling this the ''golden generator''.

Another way to think about the period is the interval from <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span> (these are not frequency ratios, but just another way of writing 0 and 1, the motivation for which will become clear soon). We can find slightly more complex metallic generators by choosing an interval other than the entire period to split into two segments related by <span><math>φ</math></span>. For example, we could pick <span><math>\frac 13</math></span> to <span><math>\frac 12</math></span>, giving us approximately <span><math>0.419821</math></span>:

Disclaimer: while these two segments are indeed related by <span><math>φ</math></span>, it is not simply by their lengths, as it may appear in the diagram. For now, let it suffice to say that extensions of the golden mean such as this are known as [http://mathworld.wolfram.com/NobleNumber.html noble numbers]. As for why the order of the segments is flipped here — well, that too will be explained later.

The ''silver generator'' splits the entire period into two segments related by the silver mean, and is approximately equal to <span><math>0.292894</math></span>:

we find the ''bronze generator'', approximately <span><math>0.232408</math></span>:

Values of the [[wikipedia:Arithmetic_progression|arithmetic progression]] from the metallic mean downwards by <span><math>1</math></span> also impart metallic effects when used to split the period to find a generator. The simplest example uses the silver mean minus one,

finding a generator which is approximately <span><math>0.414214</math></span>:

We’ll be returning to these values regularly, so for convenience, we’ll refer to them as ''isotopes'' of their respective metallic mean, e.g. <span><math>δ_s - 1</math></span> is the first isotope of the silver mean (and we’ll make no claim as to the scientific appropriateness of this analogy).

Only isotopes greater than <span><math>1</math></span> find new generators; more on this later.

Isotopes theoretically could be formed by adding <span><math>1</math></span> repeatedly to each mean, instead of subtracting, but these also do not find new generators, and for simplicity we’ll not be considering these to be isotopes at all for our purposes here.

We can find even more metallic generators by extending the concept of noble numbers to metallic means beyond the golden, as well as their isotopes. For example, we could choose the silver mean, and <span><math>\frac 01</math></span> to <span><math>\frac 12</math></span> as our interval, finding approximately <span><math>0.226541</math></span>:

==<span><math>L{:}s</math></span> sequences==

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

We’ll call the ordered set of scales a generator generates its ''scale sequence'', and the ordered set of <span><math>L{:}s</math></span> corresponding to these scales a generator’s ''<span><math>L{:}s</math></span> sequence''.

The golden generator’s <span><math>L{:}s</math></span> sequence is simple. Every <span><math>L{:}s</math></span> ratio is <span><math>φ</math></span>:

A noble generator’s <span><math>L{:}s</math></span> sequence is slightly more complex. Not every — but almost every — <span><math>L{:}s</math></span> is <span><math>φ</math></span>. Only the first few are not.

Instead of every scale’s <span><math>L{:}s</math></span> equaling the same value, as is the case for the golden mean, the silver mean’s <span><math>L{:}s</math></span> sequence alternates between its isotopes that are greater than 1:

And the bronze mean’s <span><math>L{:}s</math></span> sequence cycles through its isotopes that are greater than 1:

Any n-metallic mean’s <span><math>L{:}s</math></span> sequence will cycle through its isotopes that are greater than 1.

Isotopic <span><math>L{:}s</math></span> sequences are just like those of their mean’s, but offset.

For example, the silver mean’s first isotope’s generator’s <span><math>L{:}s</math></span> sequence alternates between <span><math>L{:}s = δ_s</math></span> and <span><math>L{:}s = δ_s - 1</math></span>, just like the silver generator’s, however — unlike the silver generator’s — it begins with <span><math>L{:}s = δ_s - 1</math></span>.

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

But that’s not all. Due to the mathemagic of <span><math>φ</math></span>, we also get a recursive interval relationship pattern:

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

Once we’ve iterated past the point that our scale exhibits <span><math>L{:}s = φ</math></span>, some of the smaller intervals will begin to be related by <span><math>φ</math></span>, but its larger intervals will never be related by <span><math>φ</math></span>.

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

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

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

As with <span><math>L{:}s</math></span> 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.

We’ll now start going through mathematical explanations for the behavior we’ve observed about met-MOS generators, <span><math>L{:}s</math></span> sequences, and interval patterns.

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 <span><math>L = s</math></span> and <span><math>L{:}s = 1</math></span>. For example, the generator

generates scales with cardinality 2, 3, 5, 7, but when it reaches cardinality 12, it has generated 12edo. This occurs exactly at the moment when the generator has been repeated until it has returned exactly to from where it started (because <span><math>\frac{5}{12} · 12 = 5</math></span>, which is a multiple of the period, <span><math>1</math></span>); if we repeated the generator any more, we’d just go over the ground we already trod.

The ratio L:s = <span><math>φ</math></span> 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.

In our very first case — that of the golden generator — “related by” ''could'' be defined as simply “having a ratio of”. We’ll check the segment lengths to confirm this. One of the two segments is, of course, equal to the golden generator, approximately <span><math>0.381966</math></span>. The other is equal to the remainder of the golden generator with the period:

And as we can see, the ratio between those two lengths is <span><math>φ</math></span>:

But even then it’s not quite that simple, because it’s not a ''simple'' mediant between <span><math>\frac{a_1}{a_2}</math></span> and <span><math>\frac{b_1}{b_2}</math></span>, which would look like this:

Rather, it’s a [https://www.mathpages.com/home/kmath055/kmath055.htm ''weighted'' mediant], where the weight is <span><math>\frac{w_1}{w_2}</math></span>, which looks like this:

In particular, it is the mediant which is weighted by the desired metallic mean <span><math>μ</math></span>:

In some materials, the weighted mediant between two ratios using <span><math>φ</math></span> as the weight is referred to as the value "phi-way" between these two ratios. This analogy with the word "halfway" is misleading, because while the value halfway between two ratios splits the length into two segments which each are half of the total length, with the weighted mediant, the value splits the length into two segments neither of which is <span><math>φ</math></span> of the total length. In fact, that would not be possible, because while half is less than one, <span><math>φ</math></span> is greater than one, and part of a length can never be greater than the whole length. An alternative phrasing could be: the value <span><math>φ</math></span>-weighted from <span><math>\frac{a_1}{a_2}</math></span> to <span><math>\frac{b_1}{b_2}</math></span>; besides eschewing the misleading association with "halfway", another advantage of this nomenclature is its ability to specify, via the from/to, which direction the mediant is weighted. Another potential name for this construct is the "phidiant" (or "<span><math>φ</math></span>-diant") from <span><math>\frac{a_1}{a_2}</math></span> to <span><math>\frac{b_1}{b_2}</math></span>, which is especially nice if you a person who pronounces <span><math>φ</math></span> like /fiː/ (the same as "fee").

We can think of a mediant like a bizarro average of two ratios: however we may choose to weight one, it will always lie somewhere between the two ratios. That is why we can call these two ratios its bounds. This fact is easy enough to intuit: as the weight tends toward zero, the effects of <span><math>a_1</math></span> and <span><math>a_2</math></span> drops off to nothing, and as it tends toward infinity, their effects begin to utterly overwhelm <span><math>b_1</math></span> and <span><math>b_2</math></span>.

To confirm that the weighted mediant formula gives the same result for the golden generator as we were using before, we can plug in <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span> for our bounds:

And finally we can show how we got approximately <span><math>0.419821</math></span> as the value for the noble generator between <span><math>\frac 13</math></span> and <span><math>\frac 12</math></span>:

As we can see, Stern-Brocot tree actually covers ratios greater than 1, but for met-MOS purposes, we only need to consider the left half of it (in fact, we only need to consider the left quarter of it, since generators greater than <span><math>\frac 12</math></span> are complements of those less than <span><math>\frac 12</math></span> and generate the same scales; but more on that later).

# it’s much smaller, including only a single ratio, <span><math>\frac 01</math></span>;

Another benefit of finding by the child ratio is that every ratio has exactly two parent ratios, while its count of child ratios is variable (consider how many child ratios <span><math>\frac 01</math></span> has).

Each interval spans two levels of the tree, because a parent ratio will always be one level less than its child ratio. When classifying intervals by level, then, we should classify them by the child ratio. For example, we should consider the interval <span><math>\frac 17</math></span> to <span><math>\frac 16</math></span> a seventh-level interval, because it would not be available until we included the seventh level.

By the way, here is an easy way to identify which level of the tree a ratio is on: we can scan along the level to the left until we find the unit fraction which appears in the initial position, closest to <span><math>\frac 01</math></span>; if our level starts with <span><math>\frac 1n</math></span>, then the ratio is in the <span><math>n</math></span>th level.

because ⁠— of its two bounding ratios, <span><math>\frac 12</math></span> and <span><math>\frac 13</math></span> ⁠— the parent is <span><math>\frac 12</math></span>.

In Wilson’s work, he only used childward lean; had he included parentward lean for his noble generators, he would have found every single generator in one earlier level of the tree than he had previously. For example, he names the golden generator, leaning toward <span><math>\frac 12</math></span>, which of the two ratios is the child:

whereas we would name it, leaning toward <span><math>\frac 01</math></span>, which of our two ratios is the parent:

# begins the periodic phase of the <span><math>L{:}s</math></span> sequence for the metal they’re based on, and  

The golden generator is essentially the noble generator for the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>, which — being the root of the Stern-Brocot tree — is as golden as we can get: we see <span><math>L{:}s = φ</math></span> and <span><math>φ</math></span>'s distinctive interval pattern from the very start.

And the noble generator between <span><math>\frac 01</math></span> and <span><math>\frac 13</math></span>

is very close to the root of the tree; it has initial <span><math>L{:}s</math></span> ratio of <span><math>φ + 2</math></span>, then attains <span><math>L{:}s = φ</math></span> 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 <span><math>0.275267</math></span> — while only a smidgen off from the other noble generator we just looked at — necessitates iterating ''six'' times before attaining <span><math>L{:}s = φ</math></span>. This corresponds to it being the noble generator between <span><math>\frac {5}{18}</math></span> and <span><math>\frac {3}{11}</math></span>, an interval which lies five levels deeper in the Stern-Brocot tree than the interval from <span><math>\frac 01</math></span> to <span><math>\frac 13</math></span>.

So if we want a golden scale, and we also happen to want a generator near <span><math>0.276393</math></span>, then we’re in luck. But if we want a golden generator that is close to <span><math>0.275267</math></span>, we may be disappointed to hear that it is not “golden” enough for us.

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 <span><math>0</math></span>.

Sure, depending on the context, the generator complement greater than <span><math>0.5</math></span> 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 <span><math>0.618034</math></span> instead of its complement <span><math>0.381966</math></span> (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.

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 <span><math>> 0.5</math></span>. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span>. We include this interval because it occupies space between <span><math>\frac 01</math></span> and <span><math>\frac 12</math></span> — so has potential to find useful generators — but we have to be careful with it to avoid finding generators <span><math>> 0.5</math></span>. The method for this is simple. First, note that the unweighted mediant in the interval <span><math>\frac 01</math></span> to <span><math>\frac 11</math></span> is <span><math>\frac 12</math></span>, or exactly <span><math>0.5</math></span>. So if we want to avoid generators <span><math>> 0.5</math></span>, all we must do is make sure to weight more toward <span><math>\frac 01</math></span>. Since of these two ratios <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span>, the parent ratio is <span><math>\frac 01</math></span>, weighting parentward is the solution.

Now we’ll explain why the <span><math>L{:}s</math></span> sequences for metallic means cycle through their isotopes.

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 <span><math>1</math></span> then we can treat our large steps’ size as equal to the <span><math>L{:}s</math></span> ratio.

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

That is true of scale iterations where <span><math>L - s > s</math></span>. For the other type of scale iteration, where <span><math>L - s < s</math></span>, the result is simply reciprocated:

This alone would not suffice to explain how the <span><math>L{:}s</math></span> sequences lock into a cycle of isotopes. But here’s where the magic of the metallic means comes into play. <span><math>φ</math></span> has the property that

So, in the case of <span><math>φ</math></span>:

That’s why the golden <span><math>L{:}s</math></span> sequence locks into <span><math>L{:}s = φ</math></span> forever.

For example, the golden mean does have an isotope, <span><math>≈ 0.618034</math></span>, however, because the golden mean minus one is the same as the inverse of the golden mean,

The metallic means are all irrational numbers. Therefore their continued fractions are infinite, with a periodic pattern at the end. An advantage of continued fractions over the decimal system is that we can easily determine whether a number is rational by whether a periodic pattern at the end is required. Decimals, on the other hand, sometimes require periodic patterns at the end even when the number is rational, such as <span><math>\frac 13</math></span> which is <span><math>0.\overline{3}</math></span>, or <span><math>\frac 17</math></span> which is <span><math>0.\overline{142857}</math></span>; as continued fractions these two values are, respectively, <span><math>[0; 3]</math></span> and <span><math>[0; 7]</math></span>.

The golden mean has the continued fraction <span><math>[1; \overline{1}]</math></span>. The larger a term in a continued fraction, the closer the approximation of the value at that point; by this conception of irrationality, the golden mean is sometimes said to be the most irrational number possible, eluding close approximation by any ratio as much as possible at every turn.

The silver mean follows it closely with continued fraction <span><math>[2; \overline{2}]</math></span>, and the bronze mean with <span><math>[3; \overline{3}]</math></span>.

A handy way to quickly find the reciprocal of a number is to prepend its continued fraction with a 0. So, we can find the golden mean’s isotope <span><math>0.618034</math></span> is <span><math>[0; \overline{1}]</math></span>.

Actually any isotope’s continued fraction is found by simply depleting the initial term. For example, the silver mean’s isotopes are <span><math>[1; \overline{2}]</math></span> and <span><math>[0; \overline{2}]</math></span>. The initial term of a continued fraction, the one to the left of the semicolon, carries the same information as the digit of a decimal just to the left of the decimal point, i.e. any number starting with <span><math>[0;]</math></span> is between 0 and 1.

Noble generators start with other numbers but then settle on all 1’s; for example, our earlier example of <span><math>0.419821</math></span> is <span><math>[0; 2, 2, \overline{1}]</math></span>.

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 <span><math>n</math></span> if we base our noble on the <span><math>n</math></span>th metallic mean. For example, our earlier example <span><math>0.226541</math></span> is <span><math>[0; 4, \overline{2}]</math></span>

=== Application: <span><math>L{:}s</math></span> sequences===

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

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

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

And it makes sense that the silver mean’s generator would alternate between two <span><math>L{:}s</math></span> ratios, because it will alternate between

To compute the <span><math>L{:}s</math></span> 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 <span><math>[0; 2, \overline{1}]</math></span>:

If we look at the path that the generator <span><math>≈ 0.381966</math></span> takes through the scale tree — which intervals it crosses between as it goes — we’ll see that they are precisely the intervals bounded by these ratios, in this order.

Perhaps an even better way to approach the problem is to consider the terms of the continued fraction as map directions through the scale tree: each term represents a count of steps one should go in one direction, left or right, before switching directions. We start at <span><math>\frac11</math></span>, going left twice to <span><math>\frac12</math></span> and then to <span><math>\frac13</math></span>. Then we go right once to <span><math>\frac25</math></span>. Then we go left once to <span><math>\frac38</math></span>, and right once to <span><math>\frac{5}{13}</math></span>, and we can continue onwards.

The sum of the terms of any continued fraction in the Stern-Brocot tree is equal to its level in the tree. For example, the fifth level of the tree consists of <span><math>\frac 15</math></span>, <span><math>\frac 27</math></span>, <span><math>\frac 38</math></span>, and <span><math>\frac 37</math></span>:

A number of names and symbols have historically been used to denote metallic means. But many of them are ambiguous, or outright conflict with each other, and unfortunately none of them are optimal for met-MOS purposes. We’ve gotten by alright so far using traditional names and symbols in this discussion, but for the master charts we’re going to need to break from tradition in order to most clearly convey the patterns therein. So, we here propose a new notation using the Greek letter <span><math>μ</math></span>, read “mu” (<span><math>μ</math></span> because “m” figures so prominently in this domain: “m” for metallic, mean, or moment).

This <span><math>μ</math></span> notation is a direct mapping of the continued fraction for the metallic mean or isotope, as can clearly be seen in the following chart.

We know that the golden generator’s <span><math>L{:}s = φ</math></span>, but we can also say this about them:

In other words, any interval in the scale which spans exactly one large and one small step is <span><math>φ</math></span> times the size of one large step.

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

Horograms depict the scale sequences of MOS generators. To understand how the horogram illustrates the interval pattern, too, first consider just the left side of the interval pattern, for <span><math>L</math></span>:

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

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

Find any <span><math>s</math></span> in the horogram and observe how it gets split up as we iterate through the scale sequence. In the next iteration, <span><math>s</math></span> will be replaced with <span><math>L</math></span>. After two iterations, the original <span><math>s</math></span> interval is now represented by an <span><math>L</math></span> and an <span><math>s</math></span>. 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 <span><math>L</math></span>’s and <span><math>s</math></span>’s that remain ⁠— only now spanning many <span><math>L</math></span>’s and <span><math>s</math></span>’s each — are precisely the larger intervals in the scale that also exhibit the <span><math>φ</math></span> ratio to each other.

If the golden mean is the value for which <span><math>a{:}b = (a+b){:}a</math></span>, then the silver mean is the value for which

So, wherever we have a scale where <span><math>L{:}s = δ_s</math></span>, we’ll also see the interval pattern

Every other scale the silver generator generates has an <span><math>L{:}s</math></span> other than <span><math>δ_s</math></span>, namely, its isotope, <span><math>δ_s - 1</math></span>. These scales have a different pattern:

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 <span><math>δ_s</math></span>, we’ll look at the right and left sides separately, as we did with the golden:

We’ll repeat the technique we used for the golden case: find any <span><math>L</math></span> 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 <span><math>s</math></span>.

And if we want to understand the interval pattern for <span><math>δ_s - 1</math></span>, we’ll look at the right and left sides separately:

There’s something a bit different about the interval pattern for <span><math>δ_s - 1</math></span> from the other two we’ve looked at so far. The interval patterns for <span><math>δ_s</math></span> and <span><math>φ</math></span> exhibited overlap, i.e. we saw something like

The reason the other cases exhibited such overlapping is that the small step size of the next ratio in the equivalence pattern became an <span><math>L</math></span>, which is the same as the <span><math>L</math></span> size of the preceding ratio. However, for the silver mean’s first isotope here, no such link exists, since s is substituted not for <span><math>L</math></span>, but <span><math>(L+s)</math></span>.

Another way of looking at this is: for <span><math>δ_s</math></span> and <span><math>φ</math></span>, it was the case that both <span><math>s</math></span> and <span><math>L</math></span>’s interval sequences were the same, just offset from each other by a step. Whereas for <span><math>δ_s - 1</math></span>, <span><math>s</math></span> and <span><math>L</math></span>’s interval sequences are completely different.

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

The thinking behind [[Golden Meantone]] is to put the whole step and half step into the ratio of <span><math>φ</math></span> with each other. Most discussion of Golden Meantone assumes a twelve-note scale that spans an octave.

Abstractly speaking, Golden Meantone’s generator is a noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 13</math></span> toward <span><math>\frac 12</math></span>, <span><math>≈ 0.419821</math></span>, which by design was the one chosen for all noble generator examples in this discussion.

The thinking behind this tuning is similar, except that the two steps in the ratio of <span><math>φ</math></span> with each other are the tone and the chromatic semitone.

Abstractly speaking, Wilson/Pepper Fifth Tuning’s generator would be the noble generator weighted by <span><math>φ</math></span> from <span><math>\frac 25</math></span> to <span><math>\frac 37</math></span>, <span><math> ≈ 0.413254</math></span>.

Most discussion of Argent Temperament — like Golden Meantone — assumes an octave period; the thinking behind it is to put the fifth and the fourth into the ratio of <span><math>\sqrt{2}</math></span> with each other. The justly tuned versions of these intervals, <span><math>\frac 32</math></span> and <span><math>\frac 43</math></span>, respectively, are remarkably close to this already, only off by a fiftieth of a cent:

# <span><math>\sqrt{2}</math></span> is the first isotope of the silver mean, <span><math>δ_s - 1</math></span>.

So in our terms, this would be an isotopic generator, and again, by design, this was the isotopic generator chosen for examples in this discussion, <span><math>≈ 0.414214</math></span>.

It may seem odd that the most popular use of the silver mean uses its isotope rather than the mean directly. However, if we consider the ratio of the generator to the period here, that ratio is the silver mean. In the golden case, there was no difference between these two conceptions; both splitting the period into two segments in the ratio of <span><math>φ</math></span> and having the generator to period ratio be <span><math>φ</math></span> produce the same result. So while in this discussion from the beginning we put things in terms of splitting intervals (in order to smoothly transition from the golden generator into noble generators), it is probably the case that those who first brought us the Fibonacci generator and Argent Temperament were thinking in terms of the ratio of the generator to the period.

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 <span><math>≈ 0.292893</math></span>, which has been used by Billy Stiltner, who calls it Imaginary.

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

We’ll test this out on the example from before. We know that the weighted mediant formula with <span><math>φ</math></span> as weight, the interval between <span><math>\frac 13</math></span> and <span><math>\frac 12</math></span>, and weight leaning toward the parent ratio <span><math>\frac 12</math></span> gives the value <span><math>≈ 0.419821</math></span>. So our two segments are:

The full derivation follows. With lower bounding ratio <span><math>\frac{a_1}{a_2}</math></span> and upper bounding ratio <span><math>\frac{b_1}{b_2}</math></span>, we have a mediant of <span><math>\frac{φa_1 + b_1}{φa_2 + b_2}</math></span>. So then the segment from the lower bounding ratio to the mediant has length

We can also infer why the ratio worked out to exactly <span><math>φ</math></span> in the case of the entire period: both of the denominators of <span><math>\frac 01</math></span> and <span><math>\frac 11</math></span> are 1, so the scalar on <span><math>φ</math></span> was 1.

And from this we also ascertain that weighted mediants sometimes fall toward the edges of the interval and sometimes toward the middle. I.e. if we choose the interval <span><math>\frac{8}{21}</math></span> to <span><math>\frac{5}{13}</math></span>, weighted by <span><math>φ</math></span> toward <span><math>\frac{5}{13}</math></span>, the ratio between the two split segments would be <span><math>\frac{13φ}{21} ≈ 1.001640</math></span>, making that split almost right down the middle; on the other hand, if we chose the interval <span><math>\frac 01</math></span> to <span><math>\frac 17</math></span>, weighted by <span><math>φ</math></span> toward <span><math>\frac 17</math></span>, the ratio between the two split segments would be <span><math>\frac{7φ}{1} ≈ 11.326238</math></span>, extremely off.

In fact, every odd power of <span><math>φ</math></span> will be equivalent to a higher metallic mean. Whichever metallic mean that is will be the sum of two Fibonacci numbers <span><math>F_n</math></span> and <span><math>F_{n-2}</math></span> (e.g. 4 = 3 + 1, and 11 = 8 + 3). For the alternative expression of the higher metallic mean as a constant plus some coefficient on the golden mean, the values of the constant and the coefficient will also be values from the Fibonacci series (e.g. 1 & 2, 3 & 5).

For the golden mean, <span><math>m = 1</math></span> and <span><math>n = 1</math></span>; for the fourth metallic mean, <span><math>m = 4</math></span> and <span><math>n = 2</math></span>.

If we use <span><math>φ^2</math></span> as our generator (note: not as a logarithmic generator, but as an acoustic one, i.e. we repeat the generator by multiplying it, not adding it), then all of the pitches in our scale will be goldenish means (relatively speaking; if we multiplied every one by <span><math>φ</math></span>, preserving their ratios, they would be). Equivalently, we could include goldenish means until we found scales with exactly two step sizes, then divide every pitch by their shared factor of <span><math>φ</math></span>.

The silverish means can also all be expressed in a ratio form, but where the golden mean uses <span><math>\sqrt5</math></span>, they use <span><math>\sqrt2</math></span>.

So, if we use <span><math>δ_s^2</math></span> as our generator, then we get silverish scales, whose pitches are all silverish means.

And if we use <span><math>δ_b^2</math></span> as our generator, we can generate bronzish scales.

* '''cycle, <span><math>L{:}s</math></span>:''' The periodic part at the end of an  sequence, cycling through the isotopes of the given metallic mean.

* '''generator, bronze:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to <span><math>[0; 4, \overline{3}] ≈ 0.232408</math></span>.

* '''generator, complement:''' A generator <span><math>g</math></span>’s complement generator is equal to <span><math>1 - g</math></span>.

* '''generator, golden:''' The generator found using the weighted mediant formula on the period interval with the golden mean, equal to <span><math>[0; 2, \overline{1}] ≈ 0.381966</math></span>.

* '''generator, silver:''' The generator found using the weighted mediant formula on the period interval with the silver mean, equal to <span><math>[0; 3, \overline{2}] ≈ 0.292893</math></span>.

* '''sequence, <span><math>L{:}s</math></span>:''' For a given scale sequence, the corresponding sequence of their  ratios.