The 200 six component tuples we are playing with are distributions. Regardless of the Fortean, Hansonian, Consonancy ordering of their components, they are frequency distributions of interval classes found between all note pairings in a scale, or in a chord, or in any other bunch of (up to 12) distinct pitch classes you may care to think of.
We may have arrived at this point by counting things as devoid of musicality as convex polygons built from lines joining the cardinal points  adjacent or skipped  of a 12 hour clock face, but the musical interval classes being counted in these 'interval vectors' are, nonetheless, real musical objects. They are as apprehendable as any of the many other musical experiences (in melody, in timbre, in rhythm) can be. A major second and a tritone really do sound different to our ears and may conjure different feelings within us (depending upon how 'real' you regard biochemical reactions generated by tiny synaptic jolts). But does this mean that sets of 'densities' of these differing experiences have any musical meaning? It seems doubtful.
Interval Vectors aren't vectors
In order to make this case, we should point out that the term 'interval vector' is rather (read 'terribly') misleading, for in no sense are these objects vectorlike. A vector, after all, is an object which has something called 'direction' and 'magnitude' and is embedded (in one of two ways) in some kind of 'dimensioned metric space' where the components of the vector tell you how far we've gone in this direction of that space, how far we've gone in that direction, etc.
If interval vectors were truly vectors, we could say things like 'take six steps in the 'perfectfourthiness' direction, five along the majorsecondish, four along the minorthirdish, three along the majorthirdy, two along the minorsecondish and finally one step along the tritonic axis and you'll be at your destination. But what destination? And whence did we start this journey? And what was the utility of that particular route (which just happened to be the 'vector' describing the diatonic scale)? It's gibberish. Adding or subtracting one interval vector to or from another is meaningless.
One could go on. There's nothing contravariant or covariant about these objects. There's nothing looking like it might be some analogue of an oriented surface. The fact that in this purported sixdimensional space, particular 'magnitude and directions'  for example <676746> might be possible (but is not), and that others such as <281651> could never exist. Which is to say that this space (potentially containing 7×13^{6}, over two and a half million, points within it) is a tad sparse and disconnected.
But this may seem pointless  and cruel  dead horse flogging since we already know what these objects are. Despite the somewhat careless application of a term such as vector, which in the nonmathematical world might be more loosely applied to anything at all with numbers attached to components, with 'measurable bits', those who termed it thus did know exactly what it was they were dealing with because they had calculated it as what it is, as a distribution.
So, having got that out of the way, we don't need to waste our time looking for vectorlike analogs for things like dot or cross products within music. We can instead concentrate any possible timewasting on the distributional natures of these objects. Indeed the very first thing we should notice is that  since they are frequency distributions, which is to say counts of occurrences of things (specifically interval classes) in a population of those things  an interval 'vector' (we shall soon dispense with the scare quotes as a lost cause) is a statistical object associated with a particular pitch class set (or family thereof, of which we know that there are 200).
Probable Consonance
We could visit our usual 'go to guy'  the diatonic scale  but for a change we'll have a look at the Hungarian Minor scale. We'll graph out everything we know about the scale:
2131131 

(Fortean) Interval Vector: <424542> (Our) Consonance: 424542  
1131213 
The first row shows the polygonal representation of the Hungarian Minor scale and a musical presentation based on a tonic A. The scale's interval sequence (semitone step count from note to successive note) is wholetone, semitone, sesquitone, semitone, semitone, sesquitone and final semitone returning us to the (octave above) tonic and is presented under that symmetric (and therefore properly invertible) scale as the label 2131131. The next row shows the interval vector for the pitch class set represented by this scale, and also our nonvectorylabel (which we're calling consonance) which just happens to look identical. Finally the third row shows the same pitch class set  represented both polygonally and musically  in its prime normal form where the interval transitions (1131213) are at their most leftwise compact.
So what does that middle row tell us, musically, about this scale, or indeed about any heptachord comprising all seven notes spread (however narrowly or widely) across a musical rendering thereof?
Suppose you filled a bucket with hundreds of examples of these pitch classes, perhaps instantiated as small lengths of magnetic tape upon which are recordings of clarinets, flutes, trombones, ocarinas, cellos and kazoos playing only the notes shown in the class (but from any octave). The only requirement is that the bucket contains the same quantity of each pitch class so that if you dip into the bucket and pick out a piece of tape, it's equally likely that you'll find yourself holding (in this case) an A, B, C, D#, E, F# or a G. You'll certainly never pick an F or a C# and the chances of picking any one of those seven is exactly one in seven. Build melodic sequences, or chords (as you wish), out of these pickings and we'll end up in the 1950s, where the aleatory fauna of musique concrète roam free.
If, on the other hand, you were to pick out two pieces of tape then you're going to be selecting something a bit more musically useful, viz an interval. You may find something 'nice' like a perfect fourth or a perfect fifth (in the guise of an example of interval class 5) or you may be lumbered with a less agreeable minorsecond or (god forbid) the devil's interval of a tritone. The difference now is that  rather than writing down the notes as you pull them out of the bucket  you instead apply the interval that you've just picked. You can elect to do this either melodically, by applying the interval class to transition to your next note. Or you may choose to do it harmonically by building up a chord using the interval class as a guide to the next note in the stack. Being interval classes you can allow dice to decide the actual instantiation of the interval class (maybe a major seventh instead of a minor second, perhaps a thirteenth instead of a minor third drop) or use your musical skill and judgement to perform the application.
But we're not concerned here with the particulars of any application of these randomly choosable intervals  we leave that up to the individual composer. We concern ourselves only with how often the interval classes turn up. In the Hungarian Minor case, you're going to get a class 5 19% of the time and a class 4 a bit more often at about 24% of the time. This contrasts with the diatonic scale where you'd expect to get a class 5 29% of the time and a class 4 rather less often at about 14% of the time. The exact proportions are of course given by the interval vector. For every 21 draws you make of 2 pieces of tape on average (like cointossing it's not guaranteed) you'll draw 4 of IV (class 5), 2 of II (class 2), 4 of iii (class 3), 5 of III (class 4), 4 of ii (class 1) and 2 tritones (class 6).
(As is often the case, the devil's in the detail. The above exercises assume that you put the pieces of tape back into the bucket, and reshuffle, after every draw. Otherwise the proportions will gradually get more distorted as fewer pieces of tape remain. And in the case of picking out pairs of tape lengths, inevitably you'll find yourself with pieces of tape with the same pitch class on each, representing interval class 0 or unison. These are regarded as 'uninteresting' and are disregarded but still replaced. So if you're goint to actually do this random interval selection from a bucket, it would be much more efficient to use hundreds of plastic counters with numbers 1 to 6 written on them, in the proportions dictated by the interval vector.)
Probability
From the preceding discussion, it's possible to see some real applicable musical aspects to what is  after all  the decidedly abstract and unmusical activity of the counting of coincident vertices of rotated versions of triangles, pentagons, nonagons, etc lined up against themselves. I.e. from something purely geometrical we may generate aleatoric music of different  and, what is more, predictable and under our control  flavours. Whether or not you regard aleatoric music (harmonically or melodically) as music is, naturally, up to you as an individual (be you listener or composer). But no matter how you feel about it, it remains the case that the aleatory natures of pieces of randomly generated music will be distinguishably (but not necessarily recognisably, for other reasons) diatonic, or Hungarian Minorish, or pentatonic, or bebop dominant flatniney accordingly as the their respective interval class vectors are employed in their generation. It is also the case that  say for example in the diatonic case  you will be unable in general to discern its mode. Aleatory Aeolian is going to sound the same as aleatory Dorian or aleatory Phrygian  precisely because they all share the same interval class vector and because picking out intervals at random is not, in general, going to present you with a predictable tonic pitch class, even if you decide, say, that the first of the first two notes you pick out is the tonic.
It is but a small step from a frequency distribution to a probability density function. We've just seen this  for example, the probability of picking out a pair of tape fragments separated by an interval class 5 from a bucketful of uniformly distributed tape fragments from a particular key in a diatonic scale is 6/21. If the tape fragments are instead uniformly distributed from a particular key in a Hungarian Minor scale, the probability drops to 4/21. If instead we consider major thirds (interval class 4) then the situation is reversed as you're more likely to draw a major third from a Hungarian Minor bucket than a diatonic one (the diatonic probability being 3/21 and the Hungarian Minor 5/21).
It's perhaps of some small interest here that wherever you have reliable probability distributions of things from different populations of those things then it's possible  via Bayesian Modelling  to swap the prediction around. Which is to say that rather than predict what you're likely to get when you select from a known bucket, you can instead select from an unknown bucket and deduce the provenance of the bucket from the samples you take. For example, suppose you pick out a class 2 interval from an unknown bucket where you know there's a 50/50 chance it's one of our two buckets. Then the probability that the bucket is full of diatonicity is (1/2)×(4/21)/(10/21) = 0.2 and the probability that you have a bucketful of Hungarian Minorness is (1/2)×(6/21)/(10/21) = 0.3. Of course with one draw you cannot be certain, but with many draws you can accumulate lots of evidence to support one conclusion or the other (with decreasing probabilities that you're still wrong, regardless of that evidence!).
Information and Entropy
The interval vector (derived, if you need reminding, from autocorrelation) can be said to inform any music which conforms to its pitch class set (or, more accurately, to any of its sibling pitch class sets). The sequence of intervals presented in a piece of music thus conditioned, whether that sequence be presented vertically with chords or horizontally with melodies, or polymelodically with concomitantly implied harmonies as in counterpoint, or via any other construction, can be said to constitute a particular message. We don't pretend (program music notwithstanding) that the message has any linguistic equivalent, but only that it's a particular assemblage of symbols  representing intervals  which might also be used to describe something presented in a spoken or written language. We dare only to suggest that a musical message constructed from one pitch class set will be at least measurably, if not always discernibly, different to one constructed by a different pitch class set with a different autocorrelation. We won't, at this stage, even suggest a motive for why we may wish to do this, other than hint at the possibility of distinguishing between pieces of music that many people are more likely to prefer.
A single number would be useful. At the moment we are faced with 200 distinguishable and uniquely identifiable (by Forte's interval vector or our consonance label) groups of pitch class sets. Even taking into account that many of these are musically 'uninteresting' (e.g. when used for describing scales, perhaps we won't be interested in scales with fewer than five pitch classes), that's still an awful lot of distinguishable objects (e.g. 153 scales of degree 5 and up). Far too many for a human being to apprehend. Claiming that, for example, the human brain is capable of distinguishing millions of colours doesn't alter the fact that we have names for very few of them  which must indicate something about the relatively low importance we attach to such impressive powers of distinction. On the other hand  if namability is to be the deciding factor, perhaps most of us could name 150 animals? How about half of that again  say 60  animals we might like to eat (vegetarians need not stop reading; that was only an example). Despite the impressive discriminatory powers of the brain, its capacity for distinguishing value is probably not quite so remarkable. The attention you will pay to bothering about naming something is likely to be proportional to the value you assign to it, and the constant of proportionality is probably closer to 1 than it is to 10, unless it's your living.
Shannon Entropy
In communications theory there is the idea of an alphabet of symbols from which you may compose messages in a given language which employs those symbols (e.g. the dotdashes of Morse Code, the onoff voltage levels in bit transmissions or of course an actual alphabet of letters we're using here). In a given language, those symbols will have a certain frequency of appearance in typical messages (e.g. in English, the letters E, T, A, O ... etc tend to turn up a lot more frequently than do J or Q).
If you have a set of probabilities { p_{i} } associated with each of those distinct (i, of N) symbols (with the usual stipulation for discrete probabilities:
 0 ≤ p_{i} ≤ 1
 Σp_{i} = 1
for each (i) of the N symbols, then the Shannon Entropy is defined as:
 H = Σp_{i}log_{2}(1/p_{i})
which is basically a weighted average of the logarithm of the frequency of each symbol (in thermodynamics, logarithms of numbers of possibilities or frequencies are entropic quantities). Note that this (since the logarithms are base 2) tells us the average number of bits  as an absolute minimum  needed to represent each distinct symbol of the alphabet. Naturally we may elect to use more bits per symbol than this minimum if we wish, but we'll never be able to get by with fewer bits (for arbitrary messages).
For pitch set classes in general we are required to support an alphabet of 6 symbols, one to represent each interval class. Without regard to any particular scale then the probability of each interval turning up is 1/6. We have what must be an extremal entropy H = 6×(1/6)log_{2}(6), approximately 2.585. As it happens, there are two scales (or chords) which (along with their inversions and four 'modes' each, thus 16) have the required consonance (i.e. 111111) and these  perforce order 4  pitch class sets are fairly well known for what they are, the 'all interval tetrachord':
111111 <111111> 
1245 

1326 
In a particular pitch class set such as that which may be used to cast the aforementioned Hungarian Minor scale, its consonance (424542) allows us to calculate its Shannon Entropy as
 (4log_{2}(21/4) + 2log_{2}(21/2) + 4log_{2}(21/4) + 5log_{2}(21/5) + 4log_{2}(21/4) + 2log_{2}(21/2))/21
 = (12 log_{2}(21/4) + 4 log_{2}(21/2) + 5 log_{2}(21/5))/21
 = (21 log_{2}(21)  12 log_{2}(4)  4 log_{2}(2)  5 log_{2}(5))/21
 = (21 log_{2}(21) 24  4  5 log_{2}(5))/21
 = log_{2}(7) + (21 log_{2}(3)  5 log_{2}(5)  28)/21
 ≈ 2.506
Rather than simply presenting the final answer (of about 2.5 bits per symbol) you will have noticed that we've gone through a fair amount of what might appear to be unnecessary detail. The reason for this exposition is to draw your attention to the fact that we will need to know only three (base 2) logarithms for any heptatonic scale  notably log_{2}(3), log_{2}(5), and log_{2}(7). This is because there are invariably 21 (= _{7}C_{2} = 7×6/2) distinct note pairs in any heptatonic scale and because the frequency of any interval class therein may never exceed 6 (if your scale has 7 notes in it, you'll never find a pair of notes more than 6 semitones apart  there's simply no room!). This means that any probabilities turning up will always be factorisable in terms of these three integers (we hardly need add that factors of 1, 2, 4, or 8 are minimally problematic with respect to base 2 logs).
In fact for any heptatonic scale, its interval class entropy will always be of the form
 H(S_{7}) = Lg(7)  (2n_{2} + 3n_{3}Lg(3) + 5n_{5}Lg(5))/21
where we have introduced the notation S_{7} to represent any pitch class set of order 7; where n_{5}, n_{3}, and n_{2} are integers, and where we have replaced the slightly cumbersome log_{2}(x) with Lg(x) (a reasonably common notation within communications theory).
For comparison, we can calculate the entropy of the diatonic scale (consonance 654321, remember) as:
 (6Lg(21/6)+5Lg(21/5)+4Lg(21/4)+3Lg(21/3)+2Lg(21/2)+1Lg(21/1))/21
 = Lg(7)  (5Lg(5)  12Lg(3) + 16)/21
 ≈ 2.398
 (note: n_{2} = 8, n_{3} = 4, n_{5} = 1)
It will probably, by now, be no surprise to learn that of all the heptatonic scales this diatonic interval entropy (of just under 2.4 bits per interval class) is the minimum value possible. But what may  slightly  surprise is that this entropy turns up for another scale, the one with consonance 254361 (aka Forte <654321>). This should not actually be so surprising as we can instantly see that this other scale's Fortean interval vector (which we already know is just a permutation of our consonance nonvector) matches our diatonic consonance (654321). We show both scales below (in their prime forms).
654321 <254361> 
1221222 

254361 <654321> 
1111116 
As their interval class distributions are essentially identical  being, as they are, simple reorderings of each other  their entropies must match. Of the 66 heptatonic pitch class sets (or scales, or heptachords, as you will) which are known to settle into 35 distinct autocorrelative groups (each with their characteristic consonance or interval vector), those 35 distributions turn out to produce only 12 distinct entropies. The following table shows those 12 values in descending order, along with the triplet of n_{k} values discussed above.
entropy  n_{2}  n_{3}  n_{5}  consonances covered 

2.5702  12  4  0  434343, 344433 
2.5468  17  6  0  434442, 444342, 442443 
2.5295  9  5  1  443532, 543342, 533442, 453432, 435432, 433452, 353442, 343542, 335442, 544332, 445332, 443352, 344532, 344352, 345342 
2.5216  3  0  0  336333 
2.5121  1  4  2  532353 
2.5061  14  7  1  454422, 424542, 254442 
2.4876  20  7  0  444441 
2.4702  12  6  1  434541, 544431, 344451 
2.4528  4  5  2  554331, 354351 
2.4157  16  5  0  424641 
2.4031  9  2  0  262623 
2.3983  8  4  1  654321, 254361 
The above image shows the entropies (in minimal bits required per interval class symbol per consonance) for all 66 heptatonic pitch class sets, ordered by descending consonance. Because all 66 are shown, some of the labels along the xaxis are duplicated  thus demonstrating how frequently each entropy arises. The twelve horizontal coloured lines mark the twelve entropy values, as noted in the preceding table.
The following table shows the entropies for all pitch class sets. Extending the purely heptatonic case, it’s easily shown that a generalised interval class distribution entropy will always look like
H(S_{n}) = 2(2n_{2} + 3n_{3}Lg(3) + 5n_{5}Lg(5) + 7n_{7}Lg(7)+ 11n_{11}Lg(11))/n(n1)
S  H  n_{2}  n_{3}  n_{5}  n_{7}  n_{11}  dist  consonances covered 

12  2.5503  30  0  0  0  6  6CCCCC  CCCCC6 
11  2.5503  25  0  0  0  5  5AAAAA  AAAAA5 
10  2.5672  60  30  8  0  0  588888  888885 
10  2.5468  52  24  9  0  0  488889  888894, 888984, 889884, 898884, 988884 
9  2.5635  23  18  0  2  0  466677  676764, 766674 
9  2.5577  8  16  0  0  0  466668  668664 
9  2.5386  30  19  0  3  0  366777  667773, 677673, 766773, 767763, 777663 
9  2.5328  15  17  0  1  0  366678  676683, 686763, 876663 
9  2.5221  24  9  0  0  0  366669  666963 
8  2.5644  28  1  5  4  0  355555  555553 
8  2.5567  6  4  0  4  0  444466  464644, 644464 
8  2.554  21  3  3  4  0  345556  456553, 546553, 554563, 556453, 556543, 654553 
8  2.5436  14  5  1  4  0  344566  644563, 654463 
8  2.5246  13  3  0  3  0  344467  464743, 474643 
8  2.5235  24  2  4  4  0  255556  555562, 565552, 655552 
8  2.5216  4  0  0  4  0  444448  448444 
8  2.5131  17  4  2  4  0  245566  456562, 465562, 545662, 546652, 566452, 645652, 656542, 665542 
8  2.5045  23  0  3  3  0  245557  545752 
8  2.4941  16  2  1  3  0  244567  465472, 765442 
7  2.5702  12  4  0  3  0  333444  344433, 434343 
7  2.5468  17  6  0  3  0  234444  434442, 442443, 444342 
7  2.5295  9  5  1  3  0  233445  335442, 343542, 344352, 344532, 345342, 353442, 433452, 435432, 443352, 443532, 445332, 453432, 533442, 543342, 544332 
7  2.5216  3  0  0  3  0  333336  336333 
7  2.5121  1  4  2  3  0  233355  532353 
7  2.5061  14  7  1  3  0  224445  254442, 424542, 454422 
7  2.4876  20  7  0  3  0  144444  444441 
7  2.4702  12  6  1  3  0  134445  344451, 434541, 544431 
7  2.4528  4  5  2  3  0  133455  354351, 554331 
7  2.4157  16  5  0  3  0  124446  424641 
7  2.4031  9  2  0  3  0  222366  262623 
7  2.3983  8  4  1  3  0  123456  254361, 654321 
6  2.5559  3  2  3  0  0  222333  322332, 332232 
6  2.5232  8  4  3  0  0  222234  224223, 224232, 224322, 234222, 322242, 324222, 422232 
6  2.5056  1  1  3  0  0  123333  233331, 333231, 333321 
6  2.4729  6  3  3  0  0  122334  223431, 232341, 233241, 322431, 323421, 342231, 432321, 433221 
6  2.4662  5  5  2  0  0  222225  225222 
6  2.4402  11  5  3  0  0  122244  142422, 241422, 242412, 421242 
6  2.4226  4  2  3  0  0  113334  313431 
6  2.3899  9  4  3  0  0  112344  143241, 443211 
6  2.2892  5  2  3  0  0  23334  323430, 343230 
6  2.2566  10  4  3  0  0  22344  420243 
6  2.1493  5  4  2  0  0  12345  143250, 543210 
6  1.9219  3  0  3  0  0  3336  303630 
6  1.5219  6  0  3  0  0  366  60603 
5  2.5219  1  0  2  0  0  112222  122212, 212122, 212221, 222121 
5  2.4464  3  1  2  0  0  111223  113221, 121321, 122131, 122311, 123121, 131221, 211231, 213211, 221131, 221311, 223111, 231211, 311221, 321121, 322111 
5  2.3219  0  0  2  0  0  111124  114112 
5  2.3219  0  0  2  0  0  22222  220222, 222220 
5  2.2464  2  1  2  0  0  12223  032221, 122230, 202321, 212320, 232201, 322210 
5  2.171  4  2  2  0  0  11233  132130, 310132, 332110 
5  1.9219  2  0  2  0  0  2224  202420 
5  1.8464  0  1  2  0  0  1234  032140, 432100 
5  1.5219  4  0  2  0  0  244  40402 
4  2.585  3  2  0  0  0  111111  111111 
4  2.2516  2  2  0  0  0  11112  012111, 102111, 110121, 111120, 112011, 112101, 121110, 210111, 211110 
4  1.9183  1  2  0  0  0  1122  012120, 021120, 101220, 102210, 122010, 200121, 201210, 210021, 212100, 221100 
4  1.7925  3  1  0  0  0  1113  101310 
4  1.585  0  2  0  0  0  222  020202, 200022 
4  1.4591  2  1  0  0  0  123  020301, 021030, 030201, 321000, 
4  0.9183  2  2  0  0  0  24  4002 
3  1.585  0  1  0  0  0  111  001110, 010101, 011010, 100011, 100110, 101100, 111000 
3  0.9183  1  1  0  0  0  12  002001, 010020, 020100, 210000 
3  0  0  0  0  0  0  3  300 
2  0  0  0  0  0  0  1  000001, 000010, 000100, 001000, 010000, 100000 
The table contains 55 distinct entropic values amongst 64 rows distributed between the scale orders as follows
S  #H  note 

12  1  Same entropy as for S_{11} 
11  1  Same entropy as for S_{12} 
10  2  
9  5  
8  10  
7  12  
6  13  
5  8,9  2 identical entropies for 2 distribution patterns 
4  7  
3  3  with zero entropy case for augmented tritonics (3 identical intervals) 
2  1  Ditonic scales have no entropy as there’s only 1 interval, no choices! 
Apart from the peculiar case amongst the pentatonics where the same entropy value turns up twice (since a probability distribution with 5×(2/10) has the same entropy as one with 4×(1/10), 1×(2/10), 1×(4/10)) the entropies in the different orders are distinct except for the values in bold red which occur – in pairs – in different orders. These duplicates arise because the probabilities (though differing in detail because they’re in different orders) are, ratiowise, identical (e.g. pentatonic { 4/10, 4/10, 2/10 } = hexatonic { 6/15, 6/15, 3/15 } ).
Here's a graphical representation of the interval class entropies, shown in descending order for each scale order.
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