Chord complexity: Difference between revisions

The use of either the geometric mean<ref group="note">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 ''N''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 {{nowrap|''N''^1/s}} 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 {{nowrap|1/''N''^1/s}}. 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, ''s'', which can be used to adjust this scaling between chords of different sizes; we suggest setting {{nowrap|''s'' {{=}} 1}} as a good default value. We also note that we get the usual raw geometric mean and maximum as {{nowrap|''s'' &rarr; &infin;}}.

If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for ''n''/''d'', any function of ''n'' and ''d'' that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of ''n''*''d'' and {{nowrap|log(''n'' {{asterisk}} ''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|log(max(''n'', ''d''))}}, which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

If we do care about modeling the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.

The former is basically a stack of three 4:5:6 chords on top of one another, and thus has lots of simple subdyads, subtriads, subtetrads, etc., whereas the latter has been formed by simply subtracting 1 from each note in the former chord. Thus, the latter is "simpler" from the standpoint of the heptadic complexity, but doesn't have many simple subchords at all. And at least to the ears of this author, the former seems to clearly sound much "justier" than than the latter&mdash;and in a very immediate way&mdash;even though the latter is less complex from a "heptadic" standpoint.

Now, of course, we have cheated somewhat&mdash;note that we have extended the chord in such a way that the differences between each frequency ratio are 2, making this an isoharmonic chord, which are known to strongly exhibit "periodicity buzz." Still, though, this general principle seems to hold to some degree, even if some of the notes are moved around by 1 here and there to form a non-isoharmonic chord, and it works well enough as a basic guiding principle to be viewed as significant, at least in the view of this author.

So we would at least like some kind of reasonable starting point in modeling this phenomenon, so that we can compare chords of different sizes.

One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency&mdash;the GCD of the notes of the chord&mdash;and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:

1. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.

2. Given some fundamental frequency ''f'' and a chord built from ''f''{{-'}}s harmonics, adding ''another'' note from ''f''{{-'}}s harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

Strictly speaking, this phenomenon&mdash;the reinforcement of virtual pitch&mdash;is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good-enough" rule of thumb which is simple enough to be worth modeling. (As we will see, we depart from strict adherence to this criterion anyway.)

And now we need only use the proof that it is well known that the power mean tends to the geometric mean as {{nowrap|''p'' &rarr; 0}}, the minimum as {{nowrap|''p'' &rarr; &minus;&infin;}} and the maximum as {{nowrap|''p'' &rarr; &infin;}}. Thus, since we have flipped the sign so that {{nowrap|''D_s'' {{=}} ''M_&minus;s''}}, we have the aforementioned result, but with {{nowrap|''s'' &rarr; &infin;}} being the minimum and {{nowrap|''s'' &rarr; &minus;&infin;}} being the maximum.

Note that some version of this also holds when extending to multiple chords of varying size, at least for {{nowrap|''s'' &rarr; &#177;&infin;}}. 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 {{nowrap|''s'' &rarr; &#177;&infin;}}, that {{nowrap|1/''N''^1/''s''}} term tends to 1, so that it cancels out and we are simply left with the minimum and maximum of the chords.

For {{nowrap|''s'' &rarr; 0}}, on the other hand, the {{nowrap|1/''N''^1/''s''}} term tends toward infinity, and what we are left with is a ranking which is basically equivalent to the geometric mean for each chord type, but where all triads are ranked better than dyads, all tetrads better than triads, etc. It turns out, however, that we have another useful relationship to the Tenney height, which we will look at next.

We have the following generalization for larger chords, where we assume without loss of generality that we have {{nowrap|''x''₁ &le; ''x''₂ &le; ... &le; ''x''_''N'':}}

To start, let's look at some dyad ''a'':''b'', where we can assume without loss of generality that {{nowrap|''a'' &le; ''b''}}. The the Dirichlet height of the dyad, then, is

Now, we note that {{nowrap|log(''b''/''a'')}} 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 group="note">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 {{nowrap|1200 log₂''e'' {{=}} 1731.234{{cent}}}}, rather than the typical units of cents or octaves&mdash;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

Now, it is interesting to look at how the above expression scales for intervals that are fairly small. It is relatively easy to see that this denominator will be maximized when {{nowrap|''b''/''a'' {{=}} 1}}, meaning the span is zero, so that the cosh term equals 1 and thus the denominator is 2, leaving only the numerator of {{nowrap|(''ab'')^''s''/2}}. For relatively small intervals, we'll get something close to this, meaning span is irrelevant&mdash;perhaps what we want.

However, there is a very simple and elegant proof&mdash;one so simple that it seems almost tautological&mdash;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:

To see this, we can easily note that if we multiply all of the coefficients in our original chord by two, for instance, going from 4:5:6 to 8:10:12, the geomean/harmean quotient remains unaltered, as the multiplier cancels out&mdash;so the only thing we care about is the general shape of the chord, not the absolute values of the coefficients (unlike with Tenney height, for instance). And we also note that this quotient is maximized, again, if all of the numbers are equal&mdash;and if they are, that means that the chord we are evaluating is 1:1:1:...:1, which is of minimum span. Once any of the numbers increase from this, the geometric mean becomes larger than the harmonic mean, so this quotient will become larger than 1. As a result, it is easy to see that this quotient is a monotonically increasing function of the spans of the subdyads of the chord. And since we are ''dividing'' by this result, this means we are, once again, dividing by some monotonically increasing function of the span.

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 {{nowrap|''p'' &rarr; &infin;}}, 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:

This looks slightly neater at first because many similar chords of similar size are ranked together&mdash;but in this situation, the problem is that it is ranking many of these chords the same. So we have 1:2:5 and 3:4:5 ranked the same&mdash;either a blessing or a curse, depending on how you look at it (the latter is smaller in span, but more complex, and it balances out).

The results seem reasonably sensible to me, although with a few little caveats here and there&mdash;we have 1:7 ranked above 4:5:6:7, partly because there is no notion of octave-equivalence involved, and partly because Tenney height may not be prioritizing small-span intervals quite enough. But this is at least ballpark-sensible. We can tweak it slightly by looking at the Tenney-Weil norm with {{nowrap|''k'' {{=}} 0.5}} and {{nowrap|''s'' {{=}} 1}}:

Thus, for {{nowrap|''s'' {{=}} 1}}, we have an increasingly simple set of chords from {{nowrap|1 &rarr; 1:2 &rarr; 1:2:3 &rarr; 1:2:3:4 &rarr; ...}}, which we view as an interesting feature of this system, and a relatively simple "bar" which scales chords in a reasonably sensible way.

Note that for the Weil height, on the other hand, we get some additional ways to express this bar, because we have that the Weil height of 6:6:6:6:6:6, 1:2:3:4:5:6, and 1:1:1:1:1:6 are all the same thing. This may seem strange, but it's simply what results from taking the max of the elements in the ratio. One way to look at it is that the spans of the subdyads of 1:1:1:1:1:6 are much larger than those of 6:6:6:6:6:6&mdash;you have five 6/1 dyads in the first chord, for instance, whereas the second chord is all unisons&mdash;and with the max function, these things simply balance out with the decreased complexity and they are ranked the same.

This would appear to be raising the bar&mdash;when written this way, now a large chord has to be much simpler in order to have the same complexity 1&mdash;but because the other bar is also equally valid, it doesn't really make much difference either way. The Weil height simply ranks lots of things as equal in complexity, so we're really talking about a difference within the rankings of chords that are the same size, without really any significant change in large-scale behavior between chords (if the first effect is accounted for).

These metrics are fairly useful as a starting point&mdash;from some basic first principles we have derived a fairly neat way to normalize the Tenney or Weil heights to compare chords of different sizes in a reasonably sensible way. We would like to build on this to derive a better metric for large chords.

There are two basic ways to proceed with this: The first, simpler method is to simply look at all of the subchords of our chord and evaluate the simple complexities of each. The result will be a vector of '''subcomplexities''' of our chord. We can then integrate these together into a notion of the scalar complexity of the chord by taking some monotonic function of the resulting vector of subcomplexities&mdash;such as a p-norm or power mean&mdash;to get a measure of the recursive complexity of some chord of size N. Then we can scale the result so that it is meaningful to compare chords of different sizes, as we did before.