Metallic MOS: Difference between revisions

In the first part of this discussion we’ll go over basic met-MOS behavior, deferring mathematical explanations until later.

A natural topic to begin with is generators, since we need a generator before we can generate scales.

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

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

Instead of every scale’s L:s equalling 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:  

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

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 L:s = δ_s and L:s = δ_s - 1, just like the silver generator’s, however — unlike the silver generator’s — it begins with L:s = δ_s - 1.

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.

As we’ve seen, the step sizes of metallic scales follow particular patterns. But not just the steps follow patterns — many other intervals of met-MOS scales do too.

We know that for golden scales:

Henceforth we’ll be referring to these types of recursive interval relationship patterns simply as ''interval patterns''.

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 L:s = φ).  

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

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,

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''.

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.  

Predictably, the aristocratic case combines the noble case and the beyond golden cases: at first, the ratios do not exhibit interval patterns, but eventually they do, and when they start to, they follow the interval pattern cycle for the appropriate metallic mean or isotope thereof.

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

Every metallic generator generates an infinitely long scale sequence.  

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.

Recall that in order to find a metallic generator, we find a value which splits an interval into two segments related by a metallic mean. When this technique was introduced earlier, we left the definition of “related by” vague. Well, now is the time to make it explicit.

[ math ] (num1 * metallic_mean + num2) / (den1 * metallic_mean + den2)

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, we approach num2/den2, and as it tends toward infinity, we approach num1/den1.  

[ math ]

To confirm that the weighted mediant formula gives the same result for the golden generator as we were using before, we can plug in 0/1 and 1/1 for our bounds:

[ math ] (1φ + 1) / (2φ + 3) ≈ 0.419821

In the generator diagrams earlier in this discussion — as we disclaimed at that time — the labels on the segments were not indicating length. We know now that what they were actually indicating was the mediant weight. For our purposes, mediant weight is more helpful information, so we’re going to continue labeling the segments this way.

== Stern-Brocot tree ==

We’ve left still another part about our instructions for finding metallic generators vague: the “an interval” part. It turns out that not just ''any'' interval can yield a metallic generator — only those whose bounding ratios are sequential convergents (or semiconvergents) of a continued fraction (<nowiki>https://en.wikipedia.org/wiki/Continued_fraction</nowiki>).

So to quickly find a unique metallic generator, we only have to choose a ratio from the tree, and then choose either one of its two parent ratios; the line connecting these two ratios will be our interval.

Finding by the child ratio is helpful when reasoning about the levels of the tree and its intervals. It’s just good recursive design.

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 0/1; if our level starts with 1/n, then the ratio is in the nth level.

Each interval in the Stern-Brocot tree offers two ''leans'': ''parentward'', and ''childward''.

[ math ] (0φ + 1) / (1φ + 1) ≈ 0.381966

The Stern-Brocot tree can be recursed indefinitely, so an infinite number of metallic generators exist. And since each parent ratio branches into two child ratios, each new recursive level of the Stern-Brocot tree offers the next power of 2 more intervals.  

Wilson documented (<nowiki>http://www.anaphoria.com/hrgm.PDF</nowiki>) noble scale sequences through the sixth level of the Stern-Brocot tree (or as he called it, the “scale tree” or “Peirce Series”), totalling 32 noble generators. He also recorded just the generator values down to the eleventh level for a total of 1024 generators (<nowiki>http://anaphoria.com/sctree.pdf?fbclid=IwAR2PahVuZJ18faXQA_IggdD52y9PWP4uyEeQALE8Q73MhIlploPYDinbAAk</nowiki>). Exploring generators beyond that was probably just not worth it, because their metallicity levels are too low. We will cut ourselves off at the seventh level in our scale trees, as we depict generators for the silver and bronze means, and their isotopes too.

While not every interval can make a noble number, every noble number can be found as the weighted mediant of many intervals. For example,

This solution will result in some generators being named for metallic means and others for isotopes. It will also result in some generators being named for parentward intervals and others for childward ones. We suggest this inconsistency is worth the otherwise simplicity of the solution.

Generators are equivalent to their ''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 > 0.5. Almost every interval we include does not even allow for that possibility, but one interval does threaten this: the interval 0/1 to 1/1. We include this interval because it occupies space between 0/1 and 1/2 — so has potential to find useful generators — but we have to be careful with it to avoid finding generators > 0.5. The method for this is simple. First, note that the unweighted mediant in the interval 0/1 to 1/1 is 1/2, or exactly 0.5. So if we want to avoid generators > 0.5, all we must do is make sure to weight more toward 0/1. Since of these two ratios 0/1 and 1/1, 0/1 is the parent ratio, weighting parentward is the solution.

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

First we need to review some MOS concepts. The mechanics of scale generation are such that — when iterating from one scale to the next densest one — all large steps in the preceding scale become one large step and one small step in the new scale.  

[ math ] L’:s’ = (L - s):s = (L - 1):1 = L - 1

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

Basically, we are subtracting 1 from the means until only their decimal part remains. One way to describe the metallic means, then, would be the set of values for which the reciprocal of their decimal part equals themselves.

And that is why the isotopes less than 1 do not work for finding generators: they are redundant with their respective metallic mean.

we’ve ended up splitting the period up into segments of the same length, just swapped which one is on which side.  

An understanding of continued fractions unlocks many insights about metallic means, and also many insights about MOS scales, so it should be no surprise that they are particularly rife with insights about met-MOS scales, the intersection of the two concepts.

First we’ll document some behavior of continued fractions. Then we’ll get into applications.

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 n if we base our noble on the n^th metallic mean. For example, our earlier example 0.226541 is [0; 4, <s>2</s>]  

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 g = [0; 2, 2, <s>1</s>] ≈ 0.419821.

[ math ]

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.  

If we look at the path that the generator ≈ 0.381966 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.  

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 1/5, 2/7, 3/8, and 3/7:

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Earlier we observed the interval patterns of met-MOS scales. Now we’ll explain why they are recursive in the way that they are.

We know that the golden generator’s L:s = φ, but we can also say this about them:

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 phi ratio to each other.

If the golden mean is the value for which a:b = (a+b):a, then the silver mean is the value for which  

As expected, 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.

For golden, silver, and bronze, we’ve prepared versions of the Stern-Brocot tree with generators depicted through the seventh level. The golden chart is nothing but an exact remake of the one Wilson created for his golden horograms; it was remade for stylistic consistency with the other two new ones here.

The bottom halves of each box are snippets of what such an equivalence looks like on the Scale Tree.

[ insert a table version of above chart here; too big to paste in Google docs ]

[ insert a table version of above chart here; too big to paste in Google docs ]

[ insert a table version of above chart here; too big to paste in Google docs ]

Including scale trees beyond bronze is outside the scope of this present work. However, an additional generator equivalence pattern diagram for the fourth metallic mean is illustrative of that meta-pattern as it continues to expand.

[ master table ]

== Golden Meantone ==

[ horogram for g = 0.292893, 7 iterations ]

Wilson gave names to a select few of the noble generators he described, and these have gotten some attention:

{| class="wikitable"

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In the section about weighted mediants we observed that, when we split an interval in two by its weighted mediant, the ratio between these two segments is not equal to the weight. We may wonder what the ratio is between the two segments, then.  

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 8/21 to 5/13, weighted by phi toward 5/13, the ratio between the two split segments would be 13φ/21 = 1.001640, making that split almost right down the middle; on the other hand, if we chose the interval 0/1 to 1/7, weighted by phi toward 1/7, the ratio between the two split segments would be 7φ/1 = 11.326238, extremely off.

Each new level of the Stern-Brocot tree introduces the next power of 2 more intervals, but this does not necessarily mean the next power of 2 more generators. Many potential new generators are equivalent to ones which have already been found in shallower levels. Some interesting patterns arise in the counts of truly new generators introduced per level. They are similar but different from one metallic mean to the next:

{| class="wikitable"

different generators.

'''[http://www.huygens-fokker.org/bpsite/833cent.html Bohlen's 833 cent scale]'''

'''[http://dkeenan.com/Music/NobleMediant.txt?fbclid=IwAR1kgaKREuE1eULDyAfWrVjntO1eGzmdYkIjGZvlycM5uLni_UETdF2wuX0 The Noble Mediant: Complex ratios and metastable musical intervals]'''

'''[[Golden Ratio|Golden Ratio on the Xen Wiki]]'''

Author’s site. Contains a rudimentary interactive MetMOS generator.

'''MOS generator (<nowiki>https://untwelve.org/static/javascript_demos/MOSring.html</nowiki>)'''

Helpful tool for generating horograms and quickly finding cardinality sequences.

'''Discussion about Continued fractions (<nowiki>http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html</nowiki>)'''

Another way of slicing and dicing the texture of rationals.

'''The Family of Metallic Means (<nowiki>http://www.mi.sanu.ac.rs/vismath/spinadel/</nowiki>)'''

Some fun and informative videos on metallic means.

'''cycle, interval pattern'''