56 versus 200

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 vector-like. 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 six-dimensional 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 7×136, 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 non-mathematical 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 vector-like 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:

scale #2873
2131131
scale #2873
(Fortean) Interval Vector: <424542>
(Our) Consonance: 424542
scale #3692
1131213
scale #3692

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 non-vectory-label (which we're calling consonance) which just happens to look identical. Finally the third row shows the same pitch class set - re-presented 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 minor-second 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 coin-tossing 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 poly-melodically 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 dot-dashes of Morse Code, the on-off 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 { pi } associated with each of those distinct (i, of N) symbols (with the usual stipulation for discrete probabilities:

  • 0 ≤ pi ≤ 1
  • Σpi = 1

for each (i) of the N symbols, then the Shannon Entropy is defined as:

  • H = Σpilog2(1/pi)

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)log2(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>
scale #3344
1245
scale #3344
scale #3232
1326
scale #3232

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

  • (4log2(21/4) + 2log2(21/2) + 4log2(21/4) + 5log2(21/5) + 4log2(21/4) + 2log2(21/2))/21
  • = (12 log2(21/4) + 4 log2(21/2) + 5 log2(21/5))/21
  • = (21 log2(21) - 12 log2(4) - 4 log2(2) - 5 log2(5))/21
  • = (21 log2(21)- 24 - 4 - 5 log2(5))/21
  • = log2(7) + (21 log2(3) - 5 log2(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 log2(3), log2(5), and log2(7). This is because there are invariably 21 (= 7C2 = 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(S7) = Lg(7) - (2n2 + 3n3Lg(3) + 5n5Lg(5))/21

where we have introduced the notation S7 to represent any pitch class set of order 7; where n5, n3, and n2 are integers, and where we have replaced the slightly cumbersome log2(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: n2 = 8, n3 = -4, n5 = 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 non-vector) matches our diatonic consonance (654321). We show both scales below (in their prime forms).


654321
<254361>
scale #3434
1221222
scale #3434

254361
<654321>
scale #4064
1111116
scale #4064

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 nk values discussed above.

entropyn2n3n5consonances covered
2.570212-40434343, 344433
2.546817-60434442, 444342, 442443
2.52959-51443532, 543342, 533442, 453432, 435432, 433452, 353442, 343542, 335442, 544332, 445332, 443352, 344532, 344352, 345342
2.5216300336333
2.51211-42532353
2.506114-71454422, 424542, 254442
2.487620-70444441
2.470212-61434541, 544431, 344451
2.45284-52554331, 354351
2.415716-50424641
2.40319-20262623
2.39838-41654321, 254361

Heptatonic Pitch Class Set Entropies by Consonance

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 x-axis 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(Sn) = 2(2n2 + 3n3Lg(3) + 5n5Lg(5) + 7n7Lg(7)+ 11n11Lg(11))/n(n-1)

SHn2n3n5n7n11distconsonances covered
122.5503-3000066CCCCCCCCCC6
112.5503-2500055AAAAAAAAAA5
102.5672-6030800588888888885
102.5468-5224900488889888894, 888984, 889884, 898884, 988884
92.563523180-20466677676764, 766674
92.5577816000466668668664
92.538630190-30366777667773, 677673, 766773, 767763, 777663
92.532815170-10366678676683, 686763, 876663
92.5221249000366669666963
82.564428-1-540355555555553
82.55676-4040444466464644, 644464
82.55421-3-340345556456553, 546553, 554563, 556453, 556543, 654553
82.543614-5-140344566644563, 654463
82.524613-3030344467464743, 474643
82.523524-2-440255556555562, 565552, 655552
82.5216-40040444448448444
82.513117-4-240245566456562, 465562, 545662, 546652, 566452, 645652, 656542, 665542
82.5045230-330245557545752
82.494116-2-130244567465472, 765442
72.5702-124030333444344433, 434343
72.5468-176030234444434442, 442443, 444342
72.5295-95-130233445335442, 343542, 344352, 344532, 345342, 353442, 433452, 435432, 443352, 443532, 445332, 453432, 533442, 543342, 544332
72.5216-30030333336336333
72.5121-14-230233355532353
72.5061-147-130224445254442, 424542, 454422
72.4876-207030144444444441
72.4702-126-130134445344451, 434541, 544431
72.4528-45-230133455354351, 554331
72.4157-165030124446424641
72.4031-92030222366262623
72.3983-84-130123456254361, 654321
62.5559-32300222333322332, 332232
62.5232-84300222234224223, 224232, 224322, 234222, 322242, 324222, 422232
62.5056-11300123333233331, 333231, 333321
62.4729-63300122334223431, 232341, 233241, 322431, 323421, 342231, 432321, 433221
62.4662-55200222225225222
62.4402-115300122244142422, 241422, 242412, 421242
62.4226-42300113334313431
62.3899-94300112344143241, 443211
62.2892-5230023334323430, 343230
62.2566-10430022344420243
62.1493-5420012345143250, 543210
61.9219-303003336303630
61.5219-6030036660603
52.521910200112222122212, 212122, 212221, 222121
52.44643-1200111223113221, 121321, 122131, 122311, 123121, 131221, 211231, 213211, 221131, 221311, 223111, 231211, 311221, 321121, 322111
52.321900200111124114112
52.32190020022222220222, 222220
52.24642-120012223032221, 122230, 202321, 212320, 232201, 322210
52.1714-220011233132130, 310132, 332110
51.9219-202002224202420
51.84640-12001234032140, 432100
51.5219-4020024440402
42.58532000111111111111
42.25162200011112012111, 102111, 110121, 111120, 112011, 112101, 121110, 210111, 211110
41.9183120001122012120, 021120, 101220, 102210, 122010, 200121, 201210, 210021, 212100, 221100
41.7925310001113101310
41.58502000222020202, 200022
41.459121000123020301, 021030, 030201, 321000,
40.9183-22000244002
31.58501000111001110, 010101, 011010, 100011, 100110, 101100, 111000
30.9183-1100012002001, 010020, 020100, 210000
30000003300
20000001000001, 000010, 000100, 001000, 010000, 100000

The table contains 55 distinct entropic values amongst 64 rows distributed between the scale orders as follows

S#Hnote
121Same entropy as for S11
111Same entropy as for S12
102 
95 
810 
712 
613 
58,92 identical entropies for 2 distribution patterns
47 
33with zero entropy case for augmented tritonics (3 identical intervals)
21Ditonic 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, ratio-wise, 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.

Interval Class Entropies

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