Chord complexity: Difference between revisions

Benedetti height:

Weil height:

The use of either the geometric mean<ref>We also note, strictly speaking, that the original formulation of Benedetti height for dyads is equal to the product rather than the geometric mean, although the geometric mean ranks chords of the same size identically to the product. However, the geometric mean version, with the additional step of taking the <math>N</math>th root of the product, has shown up in many "natural" settings such as Harmonic Entropy, and can be thought of as generalizing the expression to measure any number of tones in a similar way, and the additional adjustment of dividing by <math>N^{1/s}</math> calibrates chords of different sizes relative to one another.</ref> or maximum has a pretty long "folklore" history of being used to evaluate the complexity of a chord; such expressions routinely show up in the computation of [[Harmonic Entropy]], for instance. These expressions are the same, but simply multiply the result by an extra normalizing term of <math>1/N^{1/s}</math>. This normalizing term doesn't affect the rankings for chords of the same size, but does affect how chords of different sizes scale in complexity with regard to one another. There is one free parameter <math>s</math> which can be used to adjust this scaling between chords of different sizes; we suggest setting <math>s=1</math> as a good default value. We also note that we get the usual raw geometric mean and maximum as <math>s \to \infty</math>.

One simple function, which meets both of our simple criteria, is simply to assign the <math>n</math>th harmonic a weighting which is some power <math>s</math> of <math>1/n</math>, called the '''rolloff''', and then sum together the weights to get a strength for the overall chord. Thus, we have

We also note that this function is even defined for infinite chords as long as <math>s > 1</math>:

This type of infinite series can be thought of as a type of [[Wikipedia:General_Dirichlet_series|general Dirichlet series]], with the caveat that the numerators are all equal to 1 and we only care about real values of <math>s</math>.

Now, we note that function is inverted, so that it is a simplicity rather than a complexity. To correct this, we simply take the reciprocal:

We call <math>D_s(\mathbf{C})</math> the '''Dirichlet complexity''' of our chord <math>\mathbf{C}</math>, with the free parameter <math>s</math> choosing the rolloff. In general, we will view <math>s = 1</math> as a decent choice, given that we typically only care about finite chords.

To see this, the first thing we should note is that the Dirichlet complexity can be viewed in terms of the [[Wikipedia:Generalized_mean|power mean]] of the coefficients of the chord. The power mean is defined

Thus, we can define the Dirichlet complexity in terms of the power mean:

We also note that, as long as we only care about chords of some particular size <math>N</math>, it makes no difference if we multiply by <math>N</math> and raise the entire thing to the power of <math>1/s</math>, as neither affects the chord rankings within each chord size. So we get

And now we need only use the proof that it is well known that the power mean tends to the geometric mean as <math>p \to 0</math>, the minimum as <math>p \to -\infty</math> and the maximum as <math>p \to \infty</math>. Thus, since we have flipped the sign so that <math>D_s = M_{-s}</math>, we have the aforementioned result, but with <math>s \to \infty</math> being the minimum and <math>s \to -\infty</math> being the maximum.

Note that some version of this also holds when extending to multiple chords of varying size, at least for <math>s \to \pm \infty</math>. To see this, we note that we can still raise things to the power of <math>1/s</math> without affecting the result, but we can no longer multiply by <math>N</math> as that now affects the rankings. So we still have the identity

As <math>s \to \pm \infty</math>, that <math>\frac{1}{N^{1/s}}</math> term tends to 1, so that it cancels out and we are simply left with the minimum and maximum of the chords.

In general, we will derive the following expressions for the '''Generalized Benedetti Height''' or '''Generalized Tenney Height''' of any chord, in such a way that chords of different sizes can be reasonably compared:

is the "Benedetti" version. The numerator is the geometric mean, and the denominator normalizes by the size of the chord. The logarithmic "Tenney" version is as follows:

In both cases, the free parameter <math>s</math>, which is derived from the original expression, now only determines the way that differently-sized chords scale relative to one another. The results for some value of <math>s</math>, when comparing chords of different sizes, will closely resemble the relative scaling of chord sizes in the Dirichlet complexity of equal value <math>s</math>, but without the caveats regarding span.

We will likewise derive the same expressions for the '''Generalized Weil Height''' of any chord:

or the logarithmic version

where the <math>s</math> parameter has the same interpretation as the above.

Interestingly, the Generalized Weil Height for all harmonics from <math>1</math> to <math>N</math> has the following form

so that in fact, we note that if we set <math>s=1</math>, the Generalized Weil Height of the first N harmonics are all equal to <math>1</math>! This is a very unique and interesting property and we will talk about it in the section called [[Chord_complexity#The_Bar|The Bar]] below.

For dyads, note that we have the following property for Weil height:

The first term on the right hand side is the Tenney height, and the second term is the span. As a result, we can see that the Weil height is equal to the Tenney height plus the span, so that it can already be viewed as an alteration of the Tenney height with even greater emphasis placed on small intervals.

We have the following generalization for larger chords, where we assume without loss of generality that we have <math>x_1 \leq x_2 \leq \ldots \leq x_N</math>:

Note that the latter terms are the spans of the upper subdyads of the chord. Thus, we have the same basic principle.

We can use this to define the [[Tenney-Weil Height]] (or Benedetti-Weil height, if you prefer) for the chord, with a free parameter <math>k</math> interpolating between the two (and thus determining just how much we care about the span). We will define this as follows:

so that for <math>k=0</math> we get the Tenney height, for <math>k=1</math> we get the Weil height (with even greater emphasis on smaller intervals), and for other values of <math>k</math> we can get values in between or beyond.

To start, let's look at some dyad <math>a:b</math>, where we can assume without loss of generality that <math>a \leq b</math>. The the Dirichlet height of the dyad, then, is

We can multiply both numerator and denominator by <math>(ab)^{-s/2}</math> to get

By using the hyperbolic trigonometric identity <math>\cosh x = (e^x + e^{-x})/2</math>, that denominator can be rewritten in terms of the <math>\cosh</math> function as follows:

Now, we note that <math>\log(b/a)</math> can basically be thought of as a function of the span of the dyad. The span in cents would be <math>\text{cents}(b/a) = 1200\log_2(b/a) = 1200\log(b/a)/\log 2</math>, so we have <math>\log(b/a) = \text{cents}(b/a) \log(2)/1200</math>.<ref>In fact, this can also be thought of as a representation 'of' the span in terms of a different unit: rather than cents, we are using "nepers", where one "neper" is equal to <math>1200\log_2(e) = 1731.234</math> cents, rather than the typical units of cents or octaves - perfectly legitimate, if not a bit strange, and used rather frequently in the writings of the late [[Martin Gough]].</ref> Thus, the above expression is a monotonic function purely in terms of the span. Putting it all together, we have

This shows us how this metric relates to the Benedetti height. The numerator is the geometric mean raised to the power of s, but the denominator is an exponentially increasing monotonic function of the span! This is the basic issue: if we view Benedetti height as a decent barometer for how things should scale, then relative to that, intervals are being rewarded for having larger spans.

So what we will do is simply modify our formula so that the behavior for relatively small intervals is preserved across the entire interval spectrum, thus "span-correcting" our original formula. Doing so, we simply keep the numerator (the "complexity" part) the same, while pretending that we have always plugged 1/1 into the denominator. Thus, we simply get

We are mostly done here, although the result is a bit nicer if we raise the whole thing to the power of <math>1/s</math>, which doesn't affect the results in any way. If we do, we get our original result for dyads:

Now, as we will see, this basic principle also works very well for general N-ads, giving us a formula with the same basic properties as the original when comparing between chords of different sizes, while also span-correcting for very large intervals in the same way.

However, there is a very simple and elegant proof - one so simple that it seems almost tautological - which can prove our statement for arbitrary N-ads, both for Weil and Tenney height. To see this, we will look at our original definition of Dirichlet Complexity:

Since this is basically just a harmonic mean divided by <math>N</math>, we can rewrite as

where <math>\text{harmean}</math> refers to the harmonic mean. Then, although it seems somewhat spurious to do so, we can rewrite this as

where <math>\text{geomean}</math> likewise refers to the geometric mean. This, of course, trivially cancels out to give us our original expression, although we may wonder why we have introduced the geometric mean at all.

We can then do the same span-correction procedure as before, where we want the behavior for "small" intervals to be exhibited across the entire spectrum, but with the same properties in the way we compare chords of different sizes. So if we simply just "pretend" 1:1:1:...:1 is being plugged into the denominator no matter what, we get

and once again, raising to the power of <math>(1/s)</math> to get a slightly nicer looking result, we get our original expression for the generalized Benedetti Height:

Now, as a last note, we can easily see that our choice of the geometric mean above was somewhat arbitrary. The main important point is that this quotient of two power means gives us something which already is, perhaps non-obviously, already a non-trivial function of the spans of the subdyads of the chord. But, we could have chosen any mean which has the property that it is always greater than or equal to the harmonic mean. For instance, the maximum function, which can be viewed as the power mean as <math>p \to \infty</math>, also has the same property. If we did the above with the max function instead, we'd have instead gotten our expression for the generalized Weil height:

These are both useful as the dyadic versions are norms on monzo space, which makes them easy to prove theorems about. And in general, since the Weil height can itself be looked at as a version of the Benedetti height with even greater span-correction, we can take some average of both to determine how much we care about the span. The easiest way is to take a weighted average of the logarithmic versions of these two height functions, which corresponds to a geometric mean of the non-logarthmic versions, giving us our original expression for the Tenney-Weil height.