Metallic MOS: Difference between revisions

But phi is only the first of an infinite continuum of such metallic means which can be used to generate scales offering interesting musical possibilities. And while some attention has been given to silver scales, what we seek to do here is centralize all met-MOS knowledge and generalize principles across all of the metallic means.

The golden generator’s L:s sequence is simple. Every L:s ratio is phi:

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.  

But that’s not all. Due to the mathemagic of phi, we also get a recursive interval relationship pattern:

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.

And as we can see, the ratio between those two lengths is phi:

The golden generator is essentially the noble generator for the interval 0/1 to 1/1, which — being the root of the Stern-Brocot tree — is as golden as we can get: we see L:s = φ and phi’s distinctive interval pattern from the very start.

is very close to the root of the tree; it has initial L:s ratio of φ + 2, then attains 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 L:s = φ. This corresponds to it being the noble generator between 5/18 and 3/11, an interval which lies five levels deeper in the Stern-Brocot tree than the interval from 0/1 to 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.

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  

So, in the case of phi:

For example, the golden mean does have an isotope, ≈ 0.618034, however, because phi minus one is the inverse of phi,

We’ve stated that 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 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 [0; 1] we get the L:s sequence  

In other words, any interval in the scale which spans exactly one large and one small step is phi times the size of one large step.

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.

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.

Abstractly speaking, Golden Meantone’s generator is a noble generator weighted by phi from 1/3 toward 1/2, ≈ 0.419821, 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 phi 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 phi from 3/7 - 2/5, ≈ 0.413254.  

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 √2 with each other. The justly tuned versions of these intervals, 3/2 and 4/3, respectively, are remarkably close to this already, only off by a fiftieth of a cent:

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 phi and having the generator to period ratio be phi 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.

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:

We can also infer why the ratio worked out to exactly phi in the case of the entire period: both of the denominators of 0/1 and 1/1 are 1, so the scalar on phi 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 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.