Metallic MOS: Difference between revisions

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, \overline{1}] ≈ 0.419821.

\begin{cases}

…

\end{cases}

</math>

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

…

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

…

=== Application: finding generator ===

We’ll use the example of the golden generator, with continued fraction [0; 2, 1] :

…

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.  

=== Application: finding tree level ===

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:

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 μ, or “mu” (μ because “m” figures so prominently in this domain: “m” for metallic, mean, or moment).  

This fact follows from one of the many ways of defining the golden mean: the value for which  

We can substitute into this equation our large and small scale step sizes in place of a and b, respectively, to see that

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

Continuing this recursive process, the case will be that  

We can visualize the interval pattern using what is called a horogram:

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 L:

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

The same will hold for the right side of the interval pattern, for s:

…

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

=== Beyond golden cases ===

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

Following the same logic as we followed for the golden case,

So, wherever we have a scale where L:s = δ_s, we’ll also see the interval pattern

Every other scale the silver generator generates has an L:s other than δ_s, namely, its isotope, δ_s - 1. These scales have a different pattern: L:s = (L+2s):(L+s) = δ_s - 1.

Due to this different pattern, we’ll see the different interval pattern

[[File:Silver horogram.png|alt=horogram for g = 0.292893, 7 iterations|right|396x396px|horogram for g = 0.292893, 7 iterations]]

We can use the horogram for the silver generator to see how its interval pattern cycle is length 2, i.e. that it alternates between two different interval patterns. If we want to understand the interval pattern for δ_s, 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 L in the horogram and observe how it gets split up as we iterate through the scale sequence. However, the complexity that silver introduces is that we don’t look to the next iteration to see the next entry in the interval pattern; we have to skip an iteration. So if we just look at all the odd rings, ring 1, 3, 5, 7, etc. then we’ll see the pattern. The same is true of s.

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

And now we’ll look not at the odd, but at the even iterations, rings 2, 4, 6, 8, etc. to see the pattern visualized.

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

here we do not see such overlapping; the pattern of intervals looks more 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 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 (L+s).

Finally, for the bronze ratio,  

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.

= Gallery of generators =

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.

== Golden ==

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

== Silver ==

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

== Bronze ==

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

== Beyond bronze ==

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 scale table ==

[ master table ]

== Golden Meantone ==

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

== Wilson/Pepper Fifth Tuning ==

The thinking behind this tuning is similar, except that the two steps in the ratio of phi with each other are the tone and the chromatic semitone.  

== Argent Temperament ==

[[File:Argent horogram.png|alt=horogram for g = 0.414214, 7 iterations|right|388x388px]]

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

|498.06

|}

To know how to define this scale in terms of the present discussion, we recognize:

# √2 is the first isotope of the silver mean, δs - 1.

== Imaginary ==

If Argent temperament splits the period into segments in the ratio of the silver ratio’s isotope, what if we split the period into segments in the ratio of the silver mean itself? That gives us a generator of ≈ 0.292893, which has been used by Billy Stiltner, who calls it Imaginary.

== Other Wilson scales ==

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

{| class="wikitable"

|name

== Split segment ratios ==

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.  

We’ll test this out on the example from before. We know that the weighted mediant formula with phi as weight, the interval between 1/3 and 1/2, and weight leaning toward the parent ratio (1/2) gives the value 0.419821. So our two segments are:

Their ratio is

Which we can see is the weight multiplied by the bounding ratios’ denominators:

In particular, we see that the denominator of the weighted ratio finds itself on the same side of the ratio as the weight.

The full derivation is here:

bounding ratio 1 = a₁/a₂

φa₂/b₂

The fact that the numerators do not figure into the result at all speaks to the impossibility of finding more than one interval on the tree with the same two bounding ratio denominators.

== Generator introduction counts ==

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"

|'''level'''

|60

|}

Each metal has one asterisked row. This row is the one where the pattern of increasing twofold gets temporarily impacted by a single subtraction of 1; thenceforth, twofold growth continues.  

But we don’t find generators for specific means or isotopes in isolation of their other isotopes; it is more helpful to consider the pattern of counts of new generator introductions for the entire family of values for each metal:

{| class="wikitable"

|'''level'''

|128

|}

And here we can see the pattern is even simpler. For gold it is the same, because it has no relevant isotopes. And the other metals simply map their metallic number over the values that gold finds.  

So by the 7th level, when looking at the bronze scale tree, we should see a total of  

different generators.

== Other metallic xenharmonic but not met-MOS ==

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

== Other met-MOS ==

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

== Other MOS ==

'''[https://untwelve.org/static/javascript_demos/MOSring.html MOS generator]'''

== Stern-Brocot tree ==

'''[[wikipedia:Farey_sequence|Farey Sequence]]'''

== Metallic means ==

'''[http://www.mi.sanu.ac.rs/vismath/spinadel/ The Family of Metallic Means]'''

= Glossary =

'''cycle, interval pattern'''