Of the 555 sonatas he wrote for keyboard (typically harpsichord or fortepiano), I have (unsurprisingly) listened to but a small fraction of them – a few of which I can happily call true masterpieces. Quite possibly my favourite is his Sonata in D major, K. 491, which I have chosen to share today.

Listen or Download Here

This wonderful recording by Raffaele Terlizzi, is again of very high quality and free to download. Although I’m hardly informed to judge, it would seem that this performance was made on a fortepiano (an instrument of Scarlatti’s day, as opposed to the more modern pianoforte) – either way, it gives a feeling of contemporary authenticity. On the subject of online recordings, you can find a immense number of great recordings of piano/keyboard music available for download from the Piano Society website (which I somehow managed to miss out from my previous list of resources for free clasical music).

Let me conclude with a interesting anecdote of Scarlatti’s meeting with Handel. Most unfortunately to the world of music, Handel never did manage to meet the other giant born in 1685, despite several attempts.

It was in 1708 or 1709 that Handel and Scarlatti, then in their early twenties, were involved in a celebrated keyboard contest in Rome. According to the surviving (and not always reliable) account, Handel was declared to be superior on the organ, while Scarlatti matched or perhaps even surpassed Handel’s skill at the harpsichord. Scarlatti is said to have realized the musical possibilities of the organ only after hearing Handel play; regardless of the reliability of such a statement, it certainly confirms Scarlatti’s love for the instrument. In any case, the two became good friends and Scarlatti followed Handel throughout Italy in order to listen to him and learn from his mastery of the organ.

(Extracted from the text of The Great Contest: Bach, Scarlatti, Handel.)

Before I continue too far on such a tangent myself, let me present the focus of this post, that is one of the paradoxes with which I became acquainted during one of those many unusual English classes. I have forgetten the precise details, but the following I think is a half-way accurate rendition of what was then told (though the many embellishments may differ to those of my former teacher). As you may guess from the title, I term this little problem the “Traveller’s Paradox”, though I don’t think it has any conventional name, and has undoubtedly been repeated in many varying forms.

Warning to unsuspecting readers: the following situation is presented in the fairy-tale style. I was told it this way, so that’s how it’s going to get repeated – can’t please everyone!

A wearied prince has travelled many leagues on his quest to reach the all but forgotten castle that is the target of his quest; the location of the the legendary gem that is the final chance to save his kingdom from ruin. The pale sun is gradually vanishing behind the horizon as the young man approaches a great fork in the road ahead, far beyond which he can glimpse the tall spires of the castle that is his destination. Having sought his goal by his wits and instincts alone thus far, he is now unsure of which path to take from this point. Alas, time permits him to delay no longer if he is to succeed in his quest, and he must make the choice of road before darkness falls.

Approaching the split in the wide path, he scarcely notices two giants standing motionlessly at the side of the path. In this moment the prince recalls the words of the sage whom he had oft consulted; these twin giants, identical in appearance, were the only living creatures who know the safe pafe to the fabled castle. Yet he has a dilemma, for one of them always lies while the other always tells the truth, and what is worse, no-one may ask either of them but a single question.

Making the wrong choice of path will lead inevitably to peril and the ultimate failure of his quest. Only one of the roads provides a safe route directly to his goal. Not willing to let the fate of his realms rest in the hands chance, the prince knows he must ask one of the giants the question that will tell him the safe path – but which giant, and what question?

Now, before continuing, do ponder for a while the nature of this paradox, if you are not already. I do promise  that this dilemma does in fact have a sane solution, and unlike other paradoxes, is not a fundamental contradiction of logic and reason. Read on only when you wish to see the answer, and more relevantly, the formal (mathematical) method we can use in treating such a paradox.

Mathematical (or formal) logic is arguably at the heart of mathematics itself, and is to many the foundation of all science, philosophy, and general reason. Logic is unfortunately not such an intuitive thing to we as humans (without exception) – at least, beyond the very superficial level. Certainly, what has developed into the field of formal logic (in particular modern research into higher-order logics) would make little to zero sense to anyone discovering it upon the first time. It does not take much consideration to realise that the human mind, not unalike to those of other creatures, is one tailored by the eons of evolution to the tasks of survival and continuation of the species. Indeed, we were never remotely designed to unravel the mysteries of the universe, and it is only through our higher level capacities developed through other means that we may begin to do so. (That is at least the brutal atheists approach, and I am among those who would argue the point at a higher level.)

Without assuming too much prior knowledge of fundamental mathematics, specifically formal logic, I will now introduce an approach to resolving in a (simple?) mathematical manner what initially appears to be a very counter-intuitive situation. Do not fret though, for I myself have barely touched the surface of these areas. Still, for the sake of conciseness, I am going to assume you either have an elementary knowledge of formal logic, or can look up the ideas involved, where necessary. If you haven’t yet seen it, I mentioned a great tutorial for starting out in a previous post. (The Wikipedia pages may be enough if you just want an overview though.)

Firstly, let us formulate the given problem in the language of propositional logic, which involves nothing more than the manipulation of true and false values, in essence. There is no fool-proof way of doing this translation, of course, but read on and I think you will see it all works out pretty well.

Note that I use for the truth value and for the false value here.

To begin let us define the propositional variables and functions we will be using:

By arbitrarily labelling the pair of giants, let represent the truth-telling one and the lie-telling one.

Note that $\equiv$ is not actually a symbol in propositional logic, but just syntax for defining a variable.

Now, using this notation, and carefully analysing the situation (problem) presented in the above text, we can see that the problem is to find a formula for that satisfies a proposition that represents the problem. First, observe that whichever giant you ask, the reply you get will be either or in response.

The proposition that defines the situation is then:

In words, that is: asking the particular question to either giant, you will get the same response (yes/no answer) every time, and this answer will also indicate that you should take path (if true, the “first” path; if false, the “second” path).

Unsurprisingly perhaps, there is no well-defined procedure for finding the correct formula for . (There are in fact a number of perfectly valid/equivalent solutions that are relatively simple.) Now, since is simply a function of the propositional variable , we can quite easily factor out the dependence on and still write without loss of generality as:

If one wanted to be utterly rigorous, one could then take this formula for , substitute it into the problem definition, and use the rules of inference (for whatever formal system) to manipulate it into a form that explicitly gives and . On the other hand, I’d rather not lose any readers I still have at this point, so let’s do things the slightly more intuitive way by using some simple human analysis!

The propositional operator , while defined as the bidirectional application of the logical implication operator (i.e. ), can also be seen as representing equivalence of two formulas. (This can indeed by proven formally, though again it’s not really worth the space here I feel.)

Taking the original proposition, we can see (again, with a bit of higher-level analysis) that it implies both:

and

By straightforward substitution for , and a bit of simplification, we can then deduce that:

With the understanding of as representing identity, as previously stated, we can substitute the values for and back into the previous formula to get the following.

Problem solved… right? Well no, not quite actually. If we had mechanically applied the rules of inference from the axioms and the formulaic representation of the problem, we would undoubtedly be able to stop at this point (requiring a good few more pages of derivation however). Since we have not been completely rigorous, we must now prove that this definition of is indeed a correct solution to the problem. Let us substitute the formula back into the problem definition and see.

Hence, we now know that the above solution is correct. What remains is only to translate this formula into plain English. Not so trivial, I think we would agree. The first thing to note is that the parameter is unknown to the prince who asks the question. Well, enough hints…  let’s see if anyone can figure it out first (no cheating), and I’ll update the post with an answer in a few days.

Listen or Download Here

(Played in full Burghard Fischer.)

Side note: Although I suspect no-one has realised yet, I’ve evidently adopted more of a bi-weekly schedule for these posts. No excuses here, though given my track record in this respect I’m liable to continue as long as no-one comments! If any of you readers are lurking out there, do please let me know; saying that, my obsessional need to record my thoughts and opinions probably will likely keep me active enough.

Now to the point of my post:  I have recently found a marvellous resource on the subject of formal logic; this website provides a gentle yet thorough introduction to both propositional and predicate logic (among other things such as set theory and recursion), along with interactive exercises. Between spending a few hours over a couple of days reading through it and a bit of discussion with someone who knows the subject has given me a pretty firm grasp of the conepts and maths behind it – at least I’d like to think. I know (perhaps foolishly) plan to go on to learn about such crazy topics as higher-order logic, categorical logic, type theory, model theory, and so on. Finding such an elucidating resource for those might be rather more challenging, though I have a few books in hand that I may review if all turns out.

Here is the relevant C# 3.0 source code from my sample project that contains all the functionality relating to Markov chains and Markov generation. A rather nieve implementation, I would freely admit, but a simple and effective one, I’d like to think.

MarkovChain class

using System;
using System.Collections.Generic;
using System.Collections.ObjectModel;
using System.Diagnostics;
using System.Linq;
using System.Text;

namespace MarkovChainTextBox
{
    // Represents a Markov chain of arbitrary length.
    [DebuggerDisplay("{this.nodes.Count} nodes")]
    public class MarkovChain<T>
    {
        private static readonly IEqualityComparer<T> comparer = EqualityComparer<T>.Default;

        private readonly Random random = new Random();

        private List<MarkovChainNode<T>> nodes;
        private ReadOnlyCollection<MarkovChainNode<T>> nodesReadOnly;

        public MarkovChain()
        {
            this.nodes = new List<MarkovChainNode<T>>();
            this.nodesReadOnly = new ReadOnlyCollection<MarkovChainNode<T>>(this.nodes);
        }

        public ReadOnlyCollection<MarkovChainNode<T>> Nodes
        {
            get { return nodesReadOnly; }
        }

        public IEnumerable<T> GenerateSequence()
        {
            var curNode = GetNode(default(T));
            while (true)
            {
                if (curNode.Links.Count == 0)
                    break;
                curNode = curNode.Links[random.Next(curNode.Links.Count)];
                if (curNode.Value == null)
                    break;
                yield return curNode.Value;
            }
        }

        public void Train(T fromValue, T toValue)
        {
            var fromNode = GetNode(fromValue);
            var toNode = GetNode(toValue);
            fromNode.AddLink(toNode);
        }

        private MarkovChainNode<T> GetNode(T value)
        {
            var node = this.nodes.SingleOrDefault(n => comparer.Equals(n.Value, value));
            if (node == null)
            {
                node = new MarkovChainNode<T>(value);
                this.nodes.Add(node);
            }
            return node;
        }
    }
}

MarkovChainNode class

using System;
using System.Collections.Generic;
using System.Collections.ObjectModel;
using System.Diagnostics;
using System.Linq;
using System.Text;

namespace MarkovChainTextBox
{
    // Represents a node within a Markov chain.
    [DebuggerDisplay("Value: {this.value == null ? \"(null)\" : this.value.ToString()}, {this.links.Count} links")]
    public class MarkovChainNode<T>
    {
        private T value;
        private List<MarkovChainNode<T>> links;
        private ReadOnlyCollection<MarkovChainNode<T>> linksReadOnly;

        public MarkovChainNode(T value)
            : this()
        {
            this.value = value;
        }

        public MarkovChainNode()
        {
            this.links = new List<MarkovChainNode<T>>();
            this.linksReadOnly = new ReadOnlyCollection<MarkovChainNode<T>>(this.links);
        }

        public T Value
        {
            get { return this.value; }
            set { this.value = value; }
        }

        public ReadOnlyCollection<MarkovChainNode<T>> Links
        {
            get { return linksReadOnly; }
        }

        public void AddLink(MarkovChainNode<T> toNode)
        {
            this.links.Add(toNode);
        }
    }
}

As usual, I am keen to hear any sort of feedback about what is a useful little piece of code. Undoubtedly, markov chains have a number of pretty interesting applications in science and the computer world, so it would be cool to hear if other people are using this for different purposes…

The quote I wish to share here is undoubtedly one of those that should be meaningful to any man or woman alive today. The so-called “Litany against Fear” is a passage of text that first appeared in Herbert’s original Dune novel near the start of the book (when Paul undergoes his test of “humanity” under the threat of a Gom Jabbar). Call this passage a litany, mantra, or what you will – to me it is something that has significance well beyond it words.

I must not fear.
Fear is the mind-killer.
Fear is the little-death that brings total obliteration.
I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past I will turn the inner eye to see its path.
Where the fear has gone there will be nothing.
Only I will remain.

While I would not be inclined to say I use it in the same way the fictional Bene Gesserit do, it is nonetheless a surprisingly effective passage to recite in ones mind at various times. What is more, the Litany against Fear is not simply confined to combat feat; to me, it is equally effective to resist any other “overwhelming” emotion, be it anger, hatred, regret, or obsession. Perhaps “resist” is the wrong word here even; the way the Litany helps to overcome fear is somewhat less obvious and forceful, not to be easily expressed in words, I think.

Any regular readers, please excuse this brief diversion into emotional and philosophical. Juxtaposed with all my technical and scientific posts, it may not fit in very well, but alas, this blog has never had much cohesion and is rather a brain dump for me. And while I may be unique in this opinion, I actually find that discovering the ongoing variety of thoughts in someone’s mind is far more interesting than consuming formalised articles.

Again, not to give too much of a preamble, but this composition does merit a quick introduction. Concerto No. 8 from Antonio Vivaldi’s Opus 3 (known as L’Estro Armonico) is a piece somewhat overshadowed in the common repertoire by the concertos from The Four Seasons. In fact, in my opinion it is in several of Vivaldi’s other works that more rightly deserve their places at the pinnacle of Baroque music (most notably several of the concerti from L’Estro Armonico and his Gloria in D major). As it was, the great Bach himself took great inspiration from this set of concerti, transcribing several of them for other instruments – some accalade. (Funnily enough, “l’estro” means “the inspiration” in Italian.)

Apart from this particular concerto being a wonderful example of musical creativity and artistry, I think it is perhaps its enormous vivacity that sets it apart from the others. Full of energy, yet equally sophisticated – a defining mark of Italian Baroque music, in particular Vivaldi’s, in my view.

Listen or Download Here

1st Movement, Allegro

2nd Movement, Larghetto e spiritoso

3rd Movement, Allegro

This recording, by the Advent Chamber Orchestra, is as far as I know public domain, and pretty darn good given that. Anyway, enjoy.

As it is, I am fortunate enough that all my lecturers have decided to hand out printed (and generally pretty complete) notes for the subjects this term. Me, being the lazy student I am, am finding it a good excuse to not bother with any notes for the current term. Saying that, at least it’s letting me focus more on actually understanding the (painfully opaque) material of the current courses, which is surely a good thing!

Well, at the least, I hope I have give some thoughts to anyone else considering a similar plan for lecture notes. It would be actually be quite satisfying to hear if there are any humanities students out there that have adopted the reST/PDF approach. Equally so, I would  be rather surprised if any science guys have managed to make the thing work with an Eee PC.

Philistines, proceed with caution… Experts equally so, perhaps, since I am but a dilettante here!

Unsurprisingly, my first “gem” of the series will be a Baroque piece. I have chosen the little known piece, a Viola Concerto in G major, written by a one Georg Philipp Telemann. Telemann, a contemporary of Johann Sebastian Bach and resident of Germany (the Holy Roman Empire at that time), was in fact much more highly regarded than Bach (whose music was considered to be turgid and old-fashioned) in his day, and yet far less since – a fact that rather surprised me upon reading it.  He was, however, a most prolific composer, rivalling his other contemporary in Italy, Antonio Vivaldi, in this respect. Despite his fame being somewhat diminished by time, he was without doubt a musician of great talent, the Viola Concerto among his finest works.

Since I am neither inclined nor qualified to launch into a theoretical discussion of this piece, I hope you will simply hearing its beauty. Hearing and admiring such a work is, in my opinion, a very personal thing that should only be done through oneself, gradually, and in a holistic way.

Listen or Download Here

1st Movement, Largo

2nd Movement, Allegro

3rd Movement, Andante

4rd Movement, Presto

Note: This recording, while quite decent, is not my preferred one, and I have included it mainly because is is Open Audio. My favourite recording of this concerto is actually available free from the Lancashire Sinfonietta website – you just need a quick registration before you can download the complete work.

I still remember the sensation of hearing for the first time a particular phrase near the end of the Allegro (you’ll know which I mean one when you listen). A feeling of recognition that is, but alas, I cannot remember the source. If anyone could suggest where I might have heard this before, that would much alleviate this ongoing irritance!

Expect (nay, rely upon) my second post in the series next week. Until then, I hope I have given some of you enjoyment in sharing this first piece. Comments and suggestions welcome, as always.

The channel is currently the most popular WPF-related channel on any IRC network (of which I know), and sees regular activity, though not quite as much as we hope, hence this post! While the channel averages around 20-30 users at any time, it has not quite seen the growth it deserves as WPF gains more and more popularity. Hence, I urge anyone interested in the subject to at least pop in and check out the channel, perhaps idle around for a while. We’re a friendly lot, I promise, and will be glad to help out any newcomers!