T’s siblings were of various ages and sizes, and as such, pretty much all kept to their own business. Like his siblings, though, he went through many phases of development. Some were awkward, some were unpleasant, and others were rather fine experiences.

Like any being, T also supported a slew of organisms and bacteria upon himself. To some extent, he must have known this was the case, but I suppose it’s not something you pay much attention to most of the time. During puberty and all the hormonal changes that go with it, T seemed to have some problems with temperature fluctuation. Alternating between extreme hot and cold, T was somewhat sickly in his earlier years, especially compared with his oldest brother, who had always been a mighty beast. There always seems to be one – the guy who has it all: looks, charisma, strength and a nonchalance that somehow made him irresistible. His eldest sister was similarly fortunate. She was the prettiest girl that T had ever laid eyes upon, and apparently this went for most other people too. She had a flurry of followers jumping for her attention at all times, like moths to a flame. Her shiny rings and bangles caught the eye of every observer.

But T was special in his own way. He had a burning hot core, but kept a cool head. He was the most accommodating of them all, although he didn’t much realise this himself. Because of his relative quietness, he spent most of his time with Looner, his pet.

When you do stop to think about the organisms that cover your body (you are forgiven if you don’t do this often), you might consider the relationship to be somewhat symbiotic. Like the cleaner wrasse munches all the nasty stuff on larger fish, some of your bacteria seem to live fairly happily in your presence. You give them a nice home and they do their best to keep it in good shape. Like yin and yang, a balance is created whereby excess resource is a nuisance for one party, but a gift for the other. For years, this was the situation for T and his inhabitants. Although they were not consciously aware that this was the case, something of a ‘circle of life’ was in operation. Mother would provide nourishment to T, which would allow T to grow and flourish. A healthy T meant a healthy biome for his parasites. Over time, he grew older and stronger, overcoming his early illnesses.

—–

T’s health had been magnificent and stable for some time now. Because of this, one particular strain of his resident organisms had also been able to flourish and reproduce fruitfully. The organisms were not just maintaining the status quo any more. These guys had become T’s pit crew. As long as T was kept happy, the organisms seemed to enhance him. They had connected up networks and created new nerve centres. They were able to use hairs as building blocks and devised methods for growing new ones. They’d found ways of covering vast distances to move to bits of T that not even T could reach (one of the serendipitous consequences of this was that T never suffered from an itch ever again because it was scratched before the itch even registered).

T and Looner both had great imaginations and a playful innocence about them. In one of their favourite games, T would look up and face his mother whilst Looner hid behind him, completely out of view. He would then pop out of nowhere and give mother a short fright. They also tried to do this the other way around on occasion too, but as you would expect, being about 50 times the size of his pet, T was a little more difficult to hide entirely. Still, his juices would often slosh around at the excitement from all the fun and games. Even the organisms seemed to get a kick out of it themselves, wondering whether some strange mystical force was at work in the yonder, causing these bizarre sudden movements and occurrences.

Time cantered along.

The dominant organisms continued to grow wiser to their environment. They survived on simpler organisms with whom they shared T’s person. An intricate web, a fusion of science and nature they had created. As they evolved, their curiosity as to who and where they were grew, burning fiercely at their core like T. Unlike T though, they were not satisfied with stasis. Like all organisms, they had a very simple yet strong biological need to dominate and excel in their surroundings.

With any quest or motive, there will always be unintended consequences. T was a finite being. He couldn’t support an infinite variety or number of organisms. As the dominants grew exponentially more powerful and intelligent in relation to the other organisms, they ended up eliminating many other strains entirely. This is the natural order of life, you understand. The fittest and most adaptable organisms survive; inferior ones cannot cope and become extinct. However, on occasion, some of the organisms that were wiped out, although not strong, were quite helpful to T’s well-being. Some, for example, lived off particles and smaller bacteria that emitted rather foul smells, ensuring T maintained his pine-fresh scent.

T, though, didn’t complain much when he developed issues like a chronic unpleasant odour and what seemed like some form of halitosis. The benefits he had been feeling from all the new life and optimisation of resources the dominant strain had brought about quite comfortably outweighed those little negatives. In any case, his siblings were too caught up in their own lives to notice, and Looner’s adoration for T was unconditional.

Unfortunately, the magnifying glass often showed less harmony. The dominants, having asserted themselves upon T, had split into sub-strains. Some came from lower down T’s body, and others were from near the top. Because of the abundant spread of life, a lot of these sub-strains had never encountered one another. As a consequence, many saw each other as threats and decided that they should conquer the other strain to claim T’s body for themselves. Although some of the wiser old cells realised this, the lack of any inter-generational history meant that most of the organisms naively saw versions of themselves with even slight variations as threats to their survival. There was little appreciation or thought given to the fact that they may once have originated from the same small cluster of primordial life, and that the future was there for all to share.

Time galloped on.

The infighting of the dominants had made T worse for wear. In destroying each other, the organisms had inadvertently damaged much of their environment. T would be sluggish and bedridden for extended periods of time, and this inevitably took its toll on the organisms too. More and more weaker strains were rendered extinct, and the dominants were deprived of adequate food and a hospitable environment. Something, they realised, had to change.

Although the occasional power struggle between essentially selfish organisms was inevitable, the dominants soon sussed that they needed to play nice with one another in order to survive. They began to organise themselves much more logically. More senior and experienced beings took the lead in directing and managing the activities of their locality. On the surface, this manifested itself in the occasional wolf-whistle directed T’s way and hushed comments about his buff appearance.

This era of relatively peaceful progress was fruitful for the dominants. It seemed that they had learnt to concentrate on the more important issues at hand: how to allocate patches of T and the food he provided to them via his excrement to the increasingly growing population. You see, when organisms aren’t killing each other, they tend to divide and reproduce most rapidly. They also managed to develop various new ways of utilising the resources they had to squeeze out every last drop. Some of the more adventurous ones even jumped from T to Looner and back again during one of their play sessions. Others had tried before, but failed to get close enough to Looner. Many were lost in the blackness of the abyss. The successful ones found that Looner wasn’t particularly pleasant to spend any extended period of time on, so no immediate efforts were made to repeat the feat.

Time raced into darkness.

T was seriously ill. The dominants had mutated and reproduced, unrecognisable from what they once were. They had lost sight of all other objectives. Overwhelming dominance is what they desired. It wasn’t a sudden change in perspective, of course. The underlying motivation had always been there. What they had created was a form of distraction, to channel themselves into something other than uncontrollable growth. But now, T was barren.

In their vast numbers, the dominants had no time for contemplation. It was kill or be killed – the primitive existence from which they once originated. They were a cancer. And thus, we come to understand that power is absolute, but its orientation is relative and expressly changeable. The same civilisation that had contributed so much to T’s health and well-being was now tearing him apart at the seams, piece-by-piece.

Looner noticed oddities during this time, though clearly didn’t realise that anything was going on as such. He often found he had to claw at himself. It felt like ticks. This irritation was, in fact, being caused by a few runaway organisms who realised they would not survive for long on T. But it was a fruitless endeavour, and all who tried eventually perished; the barren and murky wastelands failed to facilitate for strangers.

The tumours destroyed everything in their path. Different groups of dominants metastasised in different directions, and in obliterating one another, they were slowly grinding away at the very essence of T. Ironically, by trying to live forever, they were killing their home – the place that was keeping them alive. There was not even but a hair left upon T’s being. All moisture had dried up from within him. His skin oozed and blistered, hot red liquid spewing from the wounds of pillaging and plunder. There were still individuals on T that seemed to have avoided the mindless corruption and single focus. But to expect them to reverse the devastation of an uncontrollable army was asking too much. The wise reflections of a few were no match for the sheer force of many.

These dominant organisms, once so advanced in intellect and gifted in survival had gotten the better of themselves. Power begat power; greed begat greed. The once efficient systems they had created made successive generations so confident in their abilities that they kept consuming, kept growing without pause for thought. The majority had not faced any challenge for survival in an age, so indeed, what reason did they have to research new realms of optimisation and progression?

—–

Inevitably, the day came when there was nothing left. The dominants had outsmarted themselves. Their seemingly perpetual growth machine had run out of fuel. The tumour had been cut off at the source. By this time, all inferior organisms had long perished. But now, the dominants too began to falter. The cancer regressed and receded. The organisms, once fruitful, successful and content had suffered from their own excesses.

They perished.

T’s long howls of pain had become shorter, until they finally stopped. Though alive and burning at the core, he had transformed into a zombie; a shadow of his former self. His visage was unrecognisable. All essence and spirit had seemingly left, along with the microscopic life that had once accompanied him. Of all the history and progress the dominants had created for themselves, all the improvements they had made to T’s health and function – no trace or memory remained.

Was there a purpose to their presence? Would T have been better off if they had never been? Was that short, intermediate period of contentment worth the initial struggle for survival and the eventual capitulation? Did it really change anything in the grand scheme of things for T or his siblings?

T’s mother beamed on. It was the only thing left to do.

But ask any world-class or well respected guitar player about playing, and I’m sure they will agree that technique comes second to musicality, and developing an instinct for creating music, not just sound. And if you start to focus on technique for a while, it can be very easy to get trapped in a cycle of trying to develop flawless physical technique whilst ignoring the creative mental processes that you need to make something sound pleasant.

I’ve always had an instinct for musicality, but I too became somewhat trapped in scales and speed. I decided to think about how to start expanding my mental capacity for music creation. No doubt, there are many psychological methods that will help with this. However, I’ve decided upon a list of 3 that allows enough room for growth, but without overwhelming myself (and you) with too much information. The first tip is somewhat specific to guitar, but the other two can be applied to most any instrument you play.

1. Use the Whole Neck!

This first tip is probably the most practical one. I found when analysing my own playing that I’d focused a lot on the standard scale shapes that are 2 or 3 notes per string and progress in a ‘vertical’ fashion from lower to higher strings. Your fretting hand adheres to a kind of ‘invisible box’, from which your fingers don’t deviate much. This makes sense, of course, because it is efficient in that your hand and fingers don’t have to move very far.

But I noticed two consequences that seem to emerge from this approach. Firstly, you quite often end up playing through these shapes in a somewhat linear fashion, because that’s what is easiest for your brain and fingers to process. Secondly, because of this, a lot of the notes you play may be a bit one-dimensional – the tonality and dynamics of each note are often sacrificed because of the way some of the notes fall on the neck. For example, you might be unlikely to add vibrato to notes that always land on your little finger if you adhere to strict scale fingering guidelines, because that finger is usually weak and therefore a bit lazy.

There are a couple of solutions to this. The first is to learn how scale patterns join up with one another along the neck and to try and play scales on just one string horizontally along the neck, rather than down the strings. To start with, you might want to try to play a pentatonic lick over say 2 strings that you would normally play across 4. This way, you will be incorporating a lot of sliding along the neck and this will in turn give you the opportunity to add further dynamic flourishes to the notes you play e.g. by adding in large bends.

One important skill that this forces you to learn, among others, is eye training. I find that this is somewhat akin to driving. When you drive, you look at where in the road you want to go and your hands will naturally steer you in that direction by adjusting the wheel until your body direction lines itself up with your eyes.

The guitar is no different. If you’re not used to making 6 or 7 fret jumps along the neck in either direction, what you will likely find is that you will miss the fret you want to land on. The key is to train your eyes so that you’re always looking one note ahead of where you are playing. Once you master this, you will find that your hand will always follow and land where you’re looking.

Another useful thing to practice, and one which has quite a lot of coverage on the internet already (though it tends to be focused around technique) is string skipping. There is no reason why you need to play notes up and down a scale. You can jump from very low notes to ones which are next in the sequence but a couple of octaves higher, for example. This allows you to add a bit of spice to a seemingly standard scale fragment or lick.

So the main message for this point is: don’t confine yourself to standard hand positions. Experiment and use the entire length of the neck you have been blessed with!

2. Play Outside the Scale

Here’s something that may be a little surprising (maybe less so if you listen to a lot of improvisational jazz). Sometimes you can play notes that are totally ‘wrong’ in theory, in the sense that they are not ‘supposed’ to be played over the chord or key that you are currently playing in because they will lead to a dissonant sound. This is certainly true most of the time.

However, time is the most important factor here. If you play a dissonant note on a downbeat when a chord is being resolved to its root, for example, then it will sound horrible. Birds will screech, glass will break etc etc. On the other hand, sneaking a couple of chromatic notes in between 2 ‘strong’ notes (ones that are in the key you are playing in) might sound pretty cool if you make sure that you land on a ‘good’ note on the downbeat, or other strong point in the rhythm you’re playing.

It’s important not to go overboard with this idea. There’s a fine line between sounding cool and sounding like you can’t play. But if you can execute it correctly and at the right times, then sticking in unconventional notes will help make your music stand out, if you’re going for that sort of thing of course. Steve Morse, Greg Howe and Guthrie Govan are all great examples of players that seem to do this extremely well, if you need a little inspiration. (I’ve actually used this kind of idea in the music I’ve created to date. Go to the Auditorium to listen.)

3. Listen to Other Instruments

Here’s a suggestion for you. If you predominantly like playing rock guitar, try listening to some jazz saxophone (John Coltrane, maybe)! And then try to play it! This is definitely the most abstract of my 3 tips, but as such, I think it may well be the most useful.

The problem with learning any instrument is that, more often than not, you are merely letting the instrument learn you! What I mean by that is there is a danger of looking at how to play the guitar and doing things that are specific to the guitar, learning lessons that show you how to get sounds that other guitar players are making and so on. This is fine up to a point. Everyone needs to be able to handle the instrument so that it feels natural to them.

But, it is important not to get in the mindset of ‘I’m playing a guitar solo’. You need to think ‘I’m playing a musical solo’. Think about the sounds you want to make rather than the sounds you can make just because they are standard ones for your instrument. For the guitar, I just need to point towards the number of solos or riffs that go over the same old pentatonic patterns time and again. There are some great examples of doing this well and effectively of course, but I think that these are the minority rather than the majority of cases.

What listening to different instruments will do is expand your auditory horizons. For example, piano players find it easier to jump across octaves and play the same patterns, so you’ll notice this a lot in piano-centric music. To adapt techniques from other instruments to guitar is often quite difficult from a technical standpoint, but even attempting to do so will stretch you in ways that you aren’t used to. And this is always a good thing when it comes to learning and development.

A similar thing goes for listening to new genres. They might use scales that are unfamiliar to you. They might play in strange time signatures, or have exotic rhythms. All of these things will help you see music in a different light than you are used to, and by pushing you outside your comfort zone a little, you can start to broaden your vocabulary. Over time, you’ll start to notice that you incorporate some of these new ideas into your natural style, and that’s when the creative juices will hopefully start to flow.

One final thing related to this is something I remember from a Steve Vai interview. He often sings a melody (in his head or out loud) to a chord progression or idea he has in mind, or has recorded. By singing (or if you have a terrible voice like me, by thinking about what you want to sing) you aren’t bounded by the limitations of your instrument. That way, you can come up with things that you wouldn’t otherwise do if your technical knowledge and note vocabulary wasn’t necessarily of the highest standard.

Once you have the idea, then you can start to try to play it on your instrument. Listening to a voice is the same as listening to a different instrument. When trying to play a sung piece on the guitar, you will probably run into new patterns and finger positions that you haven’t encountered before, or are not very comfortable with.

———————

By using these 3 ideas over time, you will make the unfamiliar familiar. Eventually, the ideal outcome is for you to be at the point where you can think something or hear something and play it on the guitar (or your chosen instrument) as if it was an extension of your own body, in the same way that you can repeat words that someone else has said to you or that you have just read. I hope this helps to give you a fresher perspective on playing guitar.

Check out my video below if you want to see the kinds of things I’ve been talking about in action.

Image from Eurogamer

I wanted to make a quick point about the current trend in console gaming. This generation of consoles (i.e. the PS3 and Xbox 360) have all touted ‘High-Definition’ as their key selling point and progression point over the previous generation. Both of these consoles are capable of outputting at 1080p resolutions.

As this generation seems to be nearing the end of its lifespan, there are a couple of things that I’ve noticed.

Resolution

It seems that the vast majority of games are rendered on consoles in either 720p or ‘sub-HD’ resolutions (which are lower than 720p). I suspect this is due to a combination of at least two factors. Firstly, despite the strong hardware specifications of the consoles at release, games technology has progressed to the point where console hardware simply does not have the raw horsepower required to render graphics at high resolutions without a serious performance hit. This is a problem that cannot really be solved because of the fact that consoles have to stick with the same innards for their entire lives.

The other reason may simply be laziness on the part of developers. We have seen games like Gran Turismo 5 achieving incredible amounts of performance on the PS3. However, this game took a huge amount of time and a lot of money to develop – something which a lot of developers do not have. The latter is especially telling when companies are churning out incremental updates to franchises every year (think FIFA and Call of Duty). Unfortunately, because this is where the big money is, developers are much more willing to maintain the status quo and not really expend much effort to push any boundaries, simply because there is no financial incentive to do so.

The thing that I’ve realised the most whilst playing many of these games is that graphical sophistication isn’t much of an issue to me. Judging by the success of the Wii, it probably doesn’t concern more casual gamers either. I would rather take a game with lower graphical standards but with a higher resolution. In my opinion, acuity is more important from a functional perspective than perceived realism. Great games are great games at the core. This is why a lot of PS2 games that have seen HD face-lifts have been popular. The graphical complexity may be lower than today’s standards, but make a hit PS2 or PSX game sharp enough to run in 1080p and I would be sure that the game would still be a hit today.

Frame Rate

Coming to the main point then, although I think resolution is more important than graphical quality, I think frame rate is more important than both.

Have a look at the first video at about 1:11, and the second at 4:41.

Ignoring the fact that the resolution in the second is clearer because the game has been upscaled to 720p, notice the added smoothness when the camera pans in first-person view (admittedly it’s a bit tricky to see in a YouTube video that is probably rendered in 30fps anyway!). This is because the original MGS3 ran at 30fps on the PS2, whereas the HD version runs at 60fps on the PS3. Having played the original to death, playing the HD version proved to be a great experiment. The added acuity was nice, but by far the biggest improvement in the game was down to the improved (and consistent) frame rate. There’s almost no visual ‘tearing’ when the camera is moved quickly and almost no stutter in busy sequences. The controls feel much more responsive as a result – you don’t get that feeling of ‘lag’ as the screen catches up to your movements.

And this made the game infinitely more ‘playable’. I didn’t care that the grass doesn’t look as realistic as in the latest Crytek games, or that some of the textures look blocky. I just liked the game responding accurately to my commands, and not having the distinctive blur and tearing that you get with low frame rates.

Yet, the vast majority of games on the PS3 and 360 run at a maximum of 30fps (and if you’ve played Skyrim on the PS3, that ‘maximum’ is almost never attained). When there are large action sequences, the rate often slows and games end up stuttering. It’s quite frustrating, and I think that this should absolutely not be allowed to happen. Because developers are making games for multiple platforms, they tend to aim high and then ‘cut down’ the performance to suit inferior hardware. But the experience on inferior hardware ends up becoming inconsistent.

It would be far better if developers made sure that games have a minimum rather than a maximum frame rate. Also, I’d like to see far more games running at a constant 60fps. I think this is vastly more important to the overall experience than maintaining flashy effects on hardware that is inevitably going to struggle with it if it is not coded efficiently.

PC users have had the luxury of being able to tweak their games to achieve a balance between performance and visual quality. Unfortunately, console gamers are stuck with the lowest common denominator in many cases. My key message here is that smoothness and responsiveness trump ‘being pretty’. Nice graphics look great in still images, but games are dynamic. The philosophy around game design maybe needs to take this into account a little more in future than they seem to do currently.

It’s quite possible that in your early school days, you had to draw a line of best fit through some data by eye, with a pencil and ruler. Not necessarily a bad way to do it, but it’s not going to be the optimal line every time (unless you’re extremely lucky). Of course, these days, you can just ask a computer program to do it for you – no doubt many of you have used the ‘add trendline’ feature in MS Excel.

Depending on how advanced your mathematics/statistics education is, you may have drawn a ‘line of best fit’ (or regression line) using a formula. This formula gives you an easy way of working out the exact slope and intercept of your regression line, given information about the data you’ve collected. But have you ever wondered why this works? Where does this magical formula come from?

Some Terminology

I cooked up a graph to help you understand some of the things that I’ll be talking about. Firstly, we call y the regressand (or dependent variable) and x the regressor (or independent variable). This means that given some value of x, we want to see how the value of y depends on it.

If it helps, you can give x and y names. For example, x could represent years of experience at knitting socks, and y could represent the overall quality of sock produced by a particular granny. Each point on the graph represents the experience and output quality information of a specific granny.

Here’s where it gets interesting. There must be some underlying true relationship between x and y (it could be that there is no relationship, but that still counts). We don’t know what that true relationship is, because we only have a sample of n data points. Or, we don’t know the real relationship between years of granny’s knitting and quality of sock, because we can’t track down every single granny in the world that knits. That true relationship could be linear, or it could be something far more complicated. However, suppose I have magical powers and actually know the true relationship in this instance. I’ve represented that with a blue line.

Of course, not every point lies on that line. This is because there are probably other factors which we can’t observe that cause a ‘disturbance’ to the data. Your granny’s knitting ability may not only depend on years spent knitting, she may have had joint replacements that affect her motor skills or she may have inherited an instinctive genetic ability to perform a cross-stitch. Either way, the difference between the actual data observations and the true relationship is known as the disturbance (or error) term.

Let’s denote the disturbances of the true line as ε, the intercept (where it crosses the vertical axis) α and the slope of the line β. Then, for each observation i=1,2,…,n (or each granny), the true relationship between x and y is given by:

The equation of the true line is given by the first two terms on the right hand side. The disturbances are what takes us off the line and to a particular data point.

Now, as I said before, in reality, we wouldn’t know what the true relationship was. We wouldn’t know the ‘true line’, so we have to estimate it. This is what we call the ‘line of best fit’ or regression line.

In the graph, the fitted line is in red. As you can see, the line is a bit different to the true line, but this is because we are working from a sample and don’t have all the data. The fitted line allows us to predict, on average, what we can expect y to be given an x value. In terms of grannies, our line tells us that for a granny with p years of knitting experience, we can predict that she will produce socks of quality q. The equation for our fitted line is:

The ‘hat’ symbol on the letters means that they are estimates of the true values. Of course, if we wanted to get the actual data points, we’d have to move off the line. The distance that we move off the line is known as a residual. The residual is basically an estimate of the disturbance term, and so it’s not quite the same as the disturbance itself. So, to get the original value of y, we must add on our residuals to the fitted line for each observation i:

How can we draw a line that fits the data best? There are many ways of doing it, but the general idea is always the same. We want to try to draw the line so that we minimise the distance between our line and all the data points that we have. This, by definition, will be the line that best fits the data. The method we will use is the most common one, the method of least squares, invented by Gauss in the 18th century.

The idea is this: to get the line as close to all the data points as possible, we want to minimise the residuals. However, if we just add up the residuals, they’ll cancel each other out, because some will be positive (when the data point is above the fitted line) and others will be negative (when the point is below the fitted line). So we instead take the squares of the residuals and minimise the sum of them all. Nice neat idea.

In summary then, what we need are estimates for α and β such that the residual sum of squares (RSS) is minimised. All of a sudden, our problem has become a 2-variable optimisation problem, and so we can use calculus to solve it.

Deriving the Ordinary Least Squares (OLS) Estimators

The thing we want to minimise is the RSS, which is:

Here, I’ve just substituted for ε by rearranging the previous fitted equation. We can continue simplifying by explicitly squaring what’s inside the brackets:

Note that to simplify, I’ve used some shorthand to represent the sample means of x and y:

To minimise the expression for RSS that we’ve obtained, we partially differentiate with respect to the variables we want to get optimal values for:

The first simplifies nicely to give us the expression for the OLS estimator of α:

The second requires a little more fiddling:

Substituting for α:

Rearranging:

Or equivalently (and a bit more neatly):

And so, we’re done! We now have the OLS estimators for the case where there is one regressor. This analysis can be extended to allow for any number of regressors with the use of vectors and matrices, but I won’t go into this here since I think we’ve done enough for one day. Happy line drawing!

The Greeks were obsessed with the beauty of numbers. They liked everything to work out nice and neat, and this meant that their proofs and theorems were very elegant on the outside (though they incorporated a lot of imaginative and abstract thinking behind the scenes).

Unfortunately, this meant that they got a little irritated when things didn’t work out so nicely. One such instance came down to Pythagoras when, after championing his theorem about the sides of a right-angled triangle, he came across a problem in trying to work out the length of the hypotenuse when the other two sides were of length 1. That problem was √2. ‘Haha’, I thought, when I first heard that story many years ago. ‘Stupid Pythagoras’. Yet, the puzzle was not an easy one to solve.

Of course, you probably know that √2 is an irrational number, because this would have been something that you were told. This means that you cannot express it as a nice fraction, and, as a result, writing it out as a decimal number is also futile because it will go on forever.

However, to actually show that it is impossible to write out √2 explicitly is easier said than done. When I thought of it as being a nice proof to look at, it turned out to be a lot more difficult to crack than I expected! The proof emphasises the beauty of number theory, and will hopefully also make the point that there’s a lot more going on behind the things we take for granted than it may seem.

The Hard Bit

There is a preliminary proof that I need to do in order to prove that √2 is irrational. I’ll try to take this slow, because conceptually, it is not so easy.

I need to show that:

For any positive integers p and q, we have that p² ≠ 2q²

Eh? ‘Why should this be the case?’ you say. Good question. It’s not obvious, which is why we want to prove it. Try putting in some numbers for p. 3² = 9 but  9/2 = 4.5 which we cannot obtain by squaring a number. 4² = 16, 16/2 = 8 and 8 is not a square number either. Try this with more numbers if you want. Eventually, you might resign yourself to the fact that it probably cannot be done.

To prove this, we are going to assume that it is true first of all. In other words:

Suppose p² = 2q² for some positive integers p and q

If this is true, then it means that p² is even (because it is a multiple of 2). But if p² is even, then it means that p must also be even. It is quite easy to prove that squaring an odd number will always give you an odd number, so I won’t do this here (although if anyone is having trouble with this then leave a comment).

Let’s call the smallest value of p such that the above relationship is satisfied P. So, for this P, we have that:

P² = 2Q²

for some positive integer Q.

P is even, as we know that all values of p are even. Therefore, since even numbers are just multiples of 2, we can write:

P = 2M

for some positive integer M.

Now we can substitute in for P in the original expression, so that:

(2M)² = 2Q²

4M² = 2Q²

2M² = Q²

Can you see what this tells us about Q? Well, Q² is a multiple of 2, so Q² must be even. And from what I said before, if Q² is even, then Q must also be even. So, just like we did with P, we can write Q in terms of some positive integer N:

Q = 2N

Now, we can substitute that into the previous expression involving M and Q:

2M² = (2N)²

2M² = 4N²

M² = 2N²

Want to guess what’s coming next? M² is a multiple of 2, so M² is even, and therefore M is also even! Does this expression look familiar? The relationship between M and N is exactly the same as the relationship between P and Q.

This means that M is another possible value for p. But what is M? Let’s look back up our equations. We defined M such that P = 2M. Hence:

P > M

Uh oh! We have a contradiction! We defined P as the smallest possible value of p. But M is also a possible value of p, and M is smaller than P. So P cannot be the smallest possible value of p!

If we use this argument recursively, it will go on forever. This means that there is no smallest value of p because we can always find one smaller. But if you have a set of numbers, there has to be a smallest one… unless the set is empty! Therefore, we have just shown that the set of numbers that satisfies p² = 2q² is empty. In other words, this relationship is impossible to satisfy!

The Easy Bit

That was probably a bit difficult to get your head around (if not, then you should take up mathematics at a high level if you aren’t doing so already!). Now, we can prove that √2 is irrational, and this is actually quite easy in comparison.

A rational number is one we can write as a fraction p/q for some integers p and q. Of course, since √2 is positive, p and q must also be positive. So what we want to prove is that:

There exist no positive integers p and q such that √2 = p/q

As we did with the previous proof, the way to approach this is by contradiction.

Suppose there exist some positive integers p and q such that √2 = p/q

Squaring both sides gives us:

2 = (p/q)²

2 = p²/q²

2q² = p²

Aha! We have proved earlier that there cannot exist positive integers p and q such that this equality is satisfied. Therefore we have a contradiction. Hence, our assertion that √2 = p/q must be false. Of course, this means that √2 is not a rational number. Thus, √2 is an irrational number!

And with that, we’ve opened up a whole new realm of numbers – ones that we need to use but that we can’t write out explicitly. Isn’t maths exciting?

The Problem with iWeb

iWeb was actually one of the reasons that I was able to start this website with very limited prior knowledge of web design. I liked it. It’s very unintimidating for the newcomer, and that’s clearly what Apple were going for. I’m much happier and more comfortable when I can see how everything is going to look and how everything connects visually. I also like making little tweaks on the fly. For this iWeb was much more accessible to me than, say, something like Dreamweaver. It’s effectively a website-oriented version of PowerPoint, or a desktop publishing program.

Pretty on the outside. Horrible under the hood.

However, quite early on, I spotted a few very strange behaviours when I tried to create content. iWeb likes to give you the freedom to do anything. “How wonderful! Why can’t I do that with practically any other website design application?” I thought to myself. Turns out there’s a reason why you can’t.

Firstly, I was using some very cool fonts for parts of my site. Indeed, the in-built theme I started out with used Futura for pretty much everything. Unfortunately, this is not a web-safe font (and most non-Mac users wouldn’t have the font on their system anyway). Luckily, I changed this quickly, but I was rather confused as to why there was not a restriction put in place regarding font selection. What was even worse is that I started putting drop shadows on titles, and that screws things up to the point where text is saved as an image file! My first site creation attempt was a slow and buggy mess – and that was before I tried viewing it on Internet Explorer 6!

Secondly, the impression I got that I could drag and drop anything anywhere was misleading. Once the site was published, I ended up with shapes being split up and scattered across the page, and images overlapping text instead of the text wrapping around it.

Both these cases are examples of the wool iWeb puts over your eyes with regard to how web pages actually function. Because web pages are designed in terms of style sheets and frames, manual placement of individual elements wherever you want for each individual page should not be possible without some horrible and inefficient coding. And horrible and inefficient it all was.

iWeb treats each page as an individual – not as part of a template that runs throughout your site. This means that to create new pages in a unified style, you had to keep duplicating pages. This wasn’t much effort in itself, but because iWeb treats each page as unique, even if it is following exactly the same style as all your other pages, it means that common elements are stored and loaded separately for each page.

As an example of how silly this is, consider my signature image in the header of this page. It appears in every page (and in the same place) so the smart thing would be to remember this positioning and have one copy of the image on the server. Then, that image could be referenced for each page load. Because it is the same image, your browser wouldn’t have to keep reloading it, as it would be stored in your local cache. Instead of this process (which is what happens now), iWeb would save a new copy of that header image for each and every page I had on my website! This means that every time you navigated to a new page in my site, it would have to pull out the same image but from a different location. As far as your browser knows, this is a completely separate image, so it has to fully load it each time! This happened for every single image element in my pages, meaning that my site was horribly slow to load and used up far more storage space than it should have.

Finally, iWeb was designed to tie in closely with MobileMe. It meant that Apple wanted you to pay an extortionate yearly subscription fee for hosting your site. In return for this, your site would function with all the bells and whistles that they intended. I had no intention of doing this, of course, so I had to sacrifice automatic FTP updating and things like an inbuilt commenting system – something that is practically essential in a blog-based site.

Alternatives to iWeb

I stuck with iWeb for a long time for a few reasons. Firstly, I’d reached a steady state where I’d learnt how to get what I wanted out of the program by invading it a little. I discovered the deeply buried theme files and edited them to my liking. I’d learnt what fonts to use and how to avoid putting things in my pages that would create major slow-downs. The limitations and inefficiencies were frustrating, but I somehow hoped that some of these issues would be addressed even slightly in future program updates. Once I heard that iWeb was pretty much dead, and that there was therefore no potential for improvement, I thought it would be a wise time to ditch it altogether. I knew the process would be painful initially, but I was looking forward to having a much cleaner and more functional website.

Hence, I started looking at some alternatives. I wanted something that would allow me to create and maintain a site with dynamic content easily, which meant something with a large visual editing component. However, with the knowledge I’d amassed over time regarding web design, I was less afraid of dabbling with a little code if necessary.

Desktop Based Applications

Naturally, the first applications I tried to look for were direct iWeb equivalents. I tried a few of them out to see how easy it would be to customise my site to my liking, and what sorts of new things I’d be able to do (as well as the obvious issue of how easy it would be to transfer over a whole other website).

RapidWeaver

This was the first application I found when searching for an ‘iWeb killer’ and it’s one that I’ve come across in the past. It is less of a WYSIWYG experience compared to iWeb, but one that allows your end product to be more standards compliant. However, I found it to be way too fiddly to get things looking right. Whilst there are a lot of customisation options, you are still bound to built-in themes. These can be edited in a similar way to iWeb’s themes – via the back end. However, this usually involves editing chunks of code, which can become tricky and unintuitive.

Failing this, there are a lot of ‘plugins’ that can be downloaded for RapidWeaver that allow you to add tables and frames to your page. However, I found it strange and silly that many of these things aren’t built into the application itself. As an example, without the ‘Stacks’ plugin, there is no easy way of arranging your page into columns or framed segments. This wouldn’t necessarily be a problem on its own, but everything that even marginally improves RapidWeaver’s functionality must be paid for. This includes themes. Just to recreate my iWeb site on RapidWeaver, I would probably have had to spend in excess of $100. iWeb, whilst unwieldy, at least did it all for free.

Yup, time to empty your wallet.

Sandvox

Put simply, Sandvox is ‘iWeb Plus’. It’s pretty much exactly like iWeb, but with a few more features. It also generates cleaner code. For someone familiar with iWeb, it was instantly usable. However, I came across niggles in designing pages that were somewhat similar to RapidWeaver. Still, I liked the application, but I never really felt like it was much of a step up from iWeb, and there wouldn’t have been much point in expending the effort of migrating my site to practically the same program. It would have been like moving from solitary confinement to solitary confinement in a room that was a couple of square metres larger. Sandvox doesn’t really give you very many possibilities of expansion beyond the iWeb experience, and is certainly less advanced than RapidWeaver.

Goldfish and Freeway Pro

These applications gave me a much more desktop publisher-like experience than the former two. This was nice, as it appeared to give me more flexibility. However, I found that the freedom was actually a bit too great. I felt like I was designing isolated sheets of A4 rather than a contiguous, flowing web page. These tools were probably not designed for dynamic content such as blogs, and so it seemed like I could only really create a blog by creating individual static pages and manually linking pages together. This was actually a step down from iWeb in terms of functionality (though the page generation may well have been cleaner)!

Remote Solutions

RapidWeaver seemed to be the best option as far as the desktop clients were concerned, but I still wasn’t prepared to put my faith in it entirely. As I had pretty much exhausted the list of programs that I was after, my next step was to look at content management systems that were remotely stored and edited, rather than locally on my computer. I never really liked the idea of online editing, but there’s no harm in giving these solutions a try.

Blogger, WordPress.com, Tumblr etc.

I’ll give a brief mention to these ‘ready-made’ blogging infrastructures. I considered them when I first created this website, but discounted them almost immediately because I didn’t like the lack of control I was afforded. I would be lumped in with millions of other users with no real means of differentiation or sense of identity. A blog, which to me is a form of expression, becomes redundant when you surrender the extent to which you can express yourself. This is the reason why I still look down upon these platforms, and why I wasn’t going to consider using them this time either.

WordPress

I’d mucked about with WordPress in the past, and although I liked it, this was at a time when I was less comfortable at fixing and playing around with code directly to get what I wanted. At the time, WordPress was on version 2, and quite a bit less intuitive than it is now. (Note that WordPress is a content management system that is found on WordPress.org, but WordPress.com is a closed version of WordPress that is designed to function as a fixed, ‘ready-made’ blogging tool like Blogger)

Once I tried it again, I knew it was the right direction for my site. It is completely flexible, as you are directly in touch with the source files right from the get go. At the very least, you need to understand how everything pieces together so that you know where to look if something goes wrong, or if you want to change anything. The great thing about this is that it forced me to get my hands dirty and play around with code. By doing this, you get a real sense of the things you should and shouldn’t do when designing your site.

Like RapidWeaver, WordPress relies on themes and plugins. However, because WordPress is open-source, there is a huge support base for all related things on the internet, and almost every add-on or bit of code is freely available. Once you get more proficient at PHP and CSS editing, the only real limit to the things you can do is your knowledge of the code and your imagination. This is the kind of open-ended potential I was after. If you visited the old site, hopefully you’ll appreciate the difference!

Drupal and Joomla seem to be the two main contenders to WordPress’ online content management system throne. I haven’t really looked into these two enough to be able to comment on them, but I do feel that these types of open-source, remote packages represent the best option for those looking to make an individual and powerful website that is very easy to maintain and update with rich content.

What next for Apple and website creation?

One word – iCloud. Apple announced this as the replacement to MobileMe a few months ago, and it was this that effectively signalled the end of the road for iWeb. As I’ve discussed earlier, iWeb really creates horrible websites from a usability standpoint, and this goes quite against Apple’s own ‘clean yet functional’ design philosophy. Although you’ll still be able to use iWeb to create your own sites as long as you have the application on your Mac, the fact that Apple have stopped including it in new Macs with the release of OS X Lion suggests that they really want you to stop using it, and won’t really offer any support to people who run into problems with it.

One option is for Apple to call it a day when it comes to web/blog designing tools. They may just accept that there are enough good alternatives out there for them not to get involved. But does that really sound like Apple to you? After all, if anybody is capable of consolidating available technologies into a neat and elegant package, it’s Apple.

With iCloud, I think Apple may well go the way of self-contained blogging hosts like Tumblr and Blogger. By ditching the application-based method of content creation, they can make sure that everything runs optimally and smoothly within their own environment. This includes things like proper indexing, search engine optimisation and so on. With no software on the user end, it makes it easier for Apple to keep themselves up-to-date, and also gives them the degree of control that they desire over how people should and shouldn’t go about doing things.

Apart from the fact that they are easily capable of creating something that looks and feels a lot more polished than something like Blogger, the other big draw to a platform created by Apple is down to the fact that so many people are already part of the iTunes infrastructure as a result of the iPod, iPhone and iPad. Because of this, and because of the syncing features that Apple are planning to offer with iCloud, they can allow you to access this repository of data to add dynamic content to your website. For example, people could have their top 10 most played songs on iTunes listed on their front page, and this list would be dynamic as it would pull data from your desktop and mobile versions of iTunes. It would allow you to put a map with a live geolocation flag indicating your position via your iPhone. It would allow you to post new content easily from any iDevice. It could pull up your current availability from iCal and the calendar on your iDevice.

Yes, all of this technology is currently available, but it will be much easier to implement successfully if everyone is already part of the shared information infrastructure – which is certainly the case with iTunes.

All I can say is – watch this space. If this is the direction that Apple choose to go in, then it could even give social networking sites like Twitter a run for their money. Whilst I’m happy with my decision to switch to WordPress, I’m also interested (as always) to see how the future of content creation will unfold.

Not only do they paint a very interesting picture, I think they do a good job of summarising the headaches (and also some excitement through curiosity) that economists have been experiencing in the last couple of years.

There were two graphs that caught my eye in particular. The first one, looks at debt in the UK from private institutions and households, and compares this to public debt.

There are 3 obvious facts we can get from this graph at first glance. Since the global financial crisis:

1. Companies and households in the UK have massively increased savings (and most of this seems to have come from households).
2. The government has massively increased debt, almost to an equal magnitude as the private savings increase.
3. The underlying trend of a small financial surplus seems to have remained stationary for the rest of the world.

The second graph is a little simpler. It shows that real income in the UK has dipped a little since the crisis, but the overall trend, whilst flattening slightly, doesn’t appear to have been affected hugely. The big message here though, is that consumption has plunged since 2008, and has remained low since 2009. It is now well below real income, where it was previously roughly in line with it.

Why is this significant?

Well, the other bit of information missing here is the Bank of England’s interest rate. Interest rates have been at an historic low of 0.5% since March 2009 (see graph below). According to standard consumer theory, a lower interest rate reduces the yield we get on our savings, whilst simultaneously decreasing the cost of borrowing. So, one would expect that following such a dramatic reduction in interest rates, consumption would have risen (or at least not fallen as much). Indeed, there was a small recovery during 2009, but that seems to have worn off.

Mike vs Mike

It’s pretty difficult to think about the album without taking a close look at Mike Mangini’s new drumming, simply because Mike Portnoy had shaped Dream Theater’s rhythms for decades. Mangini gained a bit of fame in the 90s and early 2000s with Steve Vai. Whilst he is the better technician of the two by miles, I’m not a big fan of his tone.

Tone is something you hear guitar players talking about all the time. Every player wants to find exactly the right tone for his guitar, and this can involve years of finding the right combination of equipment. In addition to this, a lot of tone comes from the individual. How you move your fingers has a dramatic impact on how notes come out of the fretboard. Although it is not talked about as commonly, drummers have exactly the same individual influence on their tone too.

Portnoy’s fat, chunky style combines speed and decent technique with very solid, groove-oriented playing. I take particular enjoyment from listening to Dream Theater’s drum parts. They stick in the mind as much as a guitar solo would.

However, I cannot say the same of Mangini’s parts in A Dramatic Turn of Events. Firstly, his sound is very ‘weak’. This could be the result of a poor recording arrangement (the production values on A Dramatic Turn of Events are pretty bad, similar to those found on Octavarium). But if you listen to Steve Vai’s Alive in an Ultra World album, the drum tone is very similar. This leads me to believe that it is just a consequence of his playing style.

Superior technique does not a better drummer make. There’s something instinctive about the notes you choose to play or the drums you choose to hit when you hear some music, regardless of what you are able to play. I felt that Mangini’s note choice just didn’t seem right to my ears when I’d heard him before. In the opening seconds of the album, whilst I waited for the drums to kick in, I was left feeling quite disappointed when they actually did. More evidence that Mangini’s sound is not to my taste came my way many years ago when I saw Virgil Donati playing with Steve Vai on Live at the Astoria and really stamping his authority by improving on Mangini’s old parts in every way.

Throughout the album, the feeling I got from the percussion was very robotic. Almost like the drummer was a ‘hired gun’, given a wage for his labour and then sent home. The drums just do their job in keeping the songs going, and although they are flawless in doing so, they don’t add any new dimension to the music. This can be qualified by word that a lot of the drum parts were set before Mangini came into the band and so he may not have had much room for experimentation. If this is true, then many of my comments on the drumming may not hold. But if it is, then the question would be: why wasn’t Mike allowed to experiment a little more?

On a slight tangent, I was quite disappointed whilst watching the documentary that followed the band on their search for a new drummer. Virgil Donati was trying to innovate and improvise a lot of part to traditional songs, and it seems like this approach was shunned in favour of staying in a zone that is more familiar to the rest of the guys.

My view on the drummer issue is that Dream Theater are lacking without Mike Portnoy, and Mike Portnoy needs Dream Theater to showcase his talents to the fullest.

The Songs

On The Backs Of Angels kicks the album off with a trademark heavy and technical instrumental section after a bit of a build up. There are a few things that strike me about the sound of the album already. Firstly, as mentioned in the previous section, the production values are awful. I’m not sure what’s going on here, but in comparison with the last couple of albums, it’s a huge step down. Octavarium had a similar muddiness going on with it too. Secondly, the choice of drum grooves doesn’t please my ears too much. Finally, the lyrics and vocals seem a little ‘forced’.

However, this song is one of the better ones on the album. The guitar pre-chorus at 3:40 is beautifully executed, and Jordan Rudess’ synth and piano wizardry is breathtaking. It’s the closest thing to a modernised version of Pull Me Under that I can think of.

In general, this song is a good representation of the direction that DT has gone towards in the past few years. Good technical sections let down by a very stereotypical and cliched metal chorus, as well as John Petrucci trying a little too hard with his solos, to the point where some musicality is sacrificed in favour of the ‘look how fast I am!’ effect.

Build Me Up, Break Me Down starts off with a riff that sounds a lot like something out of an Alter Bridge song. This song is another good example of the run-of-the-mill metal that seems to have seeped into Dream Theater. Listening to the verse with its processed vocals and then listening to any song from Images and Words makes me want to cry a little. Nothing in particular to like about this song, apart from possibly the anthem-like chorus which, although a little cheesy, is somewhat infectious after a few listens.

Ahh, a piano opening that is reminiscent of Muse’s Butterflies and Hurricanes kicks off Lost Not Forgotten. The initial drum groove reminds me a lot of Virgil Donati’s work. It’s well executed, but the kick drums sound a little too thin and poppy, lacking any real body. The early harmonies and unisons between keyboard and drums really showcase what I mean by losing some of the musicality over just showcasing technique. The sweep picking sections remind me of DragonForce – or in other words, a pinball machine crossed with a Super Nintendo. This is not a good thing in my book (although the Super Nintendo had its fair share of brilliantly arranged music). The later guitar and synth solos are also quite forgettable, although technically brilliant as usual.

Halfway through the song, there is a really nice bass break and an uplifting staccato guitar and vocal melody. Speaking of which, John Myung really doesn’t seem to get to flex his bass muscles as much as he should these days.

This Is The Life is a powerful rock ballad, in the same vein as something like The Answer Lies Within from Octavarium. An emotive song, it sets a great mood. No objections here – this is one of the things Dream Theater continues to do very well.

You don’t particularly want to start a song with something that sounds rather like a frog vomiting quite violently. Bridges In The Sky, however, manages to do just that. I’m not sure what they were thinking here. The choir section following it does do a good job of setting a dark mood for the song overall. The riff is quite reminiscent of the 7 string chunkathon that is Dark Eternal Night. The vocal lines in this song are particularly strained, though this is likely the fault of the low key rather than James LaBrie’s voice.

Other than this, it is really a case (once again) of rock chorus and pretentious instrumental sections. The continuum solos are getting a little repetitive, Jordan.

Another Alter Bridge riff? Nope, this time it’s Outcry. It gets better (read: worse) when the drum machine kicks in, and Alter Bridge turns into Linkin Park. Another cliched chorus. Sigh. But then we have the extended instrumental section. Whilst another frenzied encounter, there’s something about this one that actually sounds musical. It is abstract, yet well orchestrated. Systematic Chaos – to use a cheap pun. Aside from the opening, Outcry actually develops into a solid effort, the kind that people have come to expect from Dream Theater. Maybe we’ve been spoilt by the abundance of them in previous albums.

Far From Heaven is a piano and vocal ballad, following the style of Wait For Sleep and Vacant. However, the lyrics aren’t as chilling and the delivery not as sweet. It’s pleasant, but very forgettable.

There seems to be pattern emerging in the intros to the songs on this album. They all seem to remind me of other bands! Breaking All Illusions starts with a string chord progression that sounds a lot like Coldplay’s Viva La Vida (Somewhat ironic, since this song also attracted a lawsuit from Joe Satriani for copying the lead from If I Could Fly). The chorus uses this chord progression too, and actually, hardly any of the choruses on the album have lifted the song in the way they’re supposed to.

However, as with Outcry, this song actually develops into something brilliant. In fact, it’s the stand-out masterpiece of the album by a long, long way. This is cemented by some of the best lyrics on the album and very addictive riffs. Truly progressive, the song shifts gears multiple times. As an example around the 5:00 mark, you hear a delightful Middle-Ages style keyboard interlude, followed shortly by piano and organ breaks, as well as a disco-style guitar groove.

But by far the best thing in this song, and on the entire album, is John Petrucci’s guitar solo. He always seems to pull one out of the bag when you start losing faith. This is the Dream Theater of old. This is what we all like to hear.

In recent years, JP’s tone has become cleaner to the point of becoming slightly sterile. I can understand why to some extent: because he’s utilising more speed and technique these days, ‘clean distortion’ is what’s required to stop the notes bleeding into each other. However, I really miss the character of the ‘dirtier’ crunchy distorted tone that he used on older albums.

We end the album with Beneath The Surface, another melodic ballad. This one incorporates some very nice guitar work, and the vocals are up to par. The synth solo sounds a little out of place on this track, and the strings are a little over the top later on in the song. It is a beautiful way to end the album though.

Overall Thoughts

For me, Dream Theater peaked around the Six Degrees of Inner Turbulence album 10 years ago. Octavarium was their last true masterpiece in my opinion. Since then, a lot of the traditional progressive elements that defined Dream Theater have been washed away.

A Dramatic Turn of Events is a mixed bag. At times, it brings back some old-school progressive moments, but these moments are marred by completely forgettable sections that get lost in the sea of modern metal material. Dream Theater has certainly lost some of the production and ‘big picture’ orchestration that I think Mike Portnoy brought to the table. Without him, the technique remains, but it seems to be lacking direction.

But things were sounding a bit fatigued to me even in previous efforts when Mike was still around. Maybe, as Mike felt prior to leaving the band, the machine is a little tired of churning out albums at regular intervals. His suggestion of a 2 or 3 year break doesn’t seem like a bad idea. Even if the talent is still there, sometimes you just need time for reflection before you get hit by inspiration once again.

Physically, you are training your muscles to develop ‘muscle memory’. In other words, by repeating the action over and over again, you are essentially ‘programming’ it into your muscles so that it feels natural. If you have gotten this far with any bit of physical skill, you might realise that you don’t actually need to think as much about what you’re doing once you get quick enough.

This is what I’ve been thinking about, and I want to offer a few of my thoughts on the matter. Apologies in advance for the lack of coherence to follow…

What goes through our mind?

The point that interests me the most is the one I just mentioned. I find it a little strange that we actually think less about something the better we get at it. Surely, in the process of acquiring greater skill, wouldn’t you think that the brain is doing more work because it has to handle more information in a given space of time?

What seems to be going on here is a form of subliminal processing. It reminds me a lot of reading. Remember when you were a kid and learning to read? You might have followed each word with your finger and pronounced each word carefully by combining the individual letters in it. How about now? I’m sure you can just look at the word and subconsciously know what it is without actively having to look at each letter. In fact, you may have come across the famous Cambridge study which found that people only need the first and the last letters of a word to be in place to recognise the rest of it. For example:

The fcat taht you can udnretnasd tihs whiutot mcuh dfiflcituy is qtiue azmiang, dno’t you tnihk?

The same sort of thing seems to be going on when you’re internalising some sort of muscular motion by way of playing the guitar.

Fast Hearing

Returning back to music (and I’ll use the guitar as my example here), suppose you’ve just learnt a 4 bar riff inside out. You can play it at practically any tempo without any errors. If I asked you to play it now, would you sit there and think ‘hmm, I need to play an A and then move my finger 2 frets to play a B…’? You might have done that when you were first learning it, but you wouldn’t do that any more. Instead, your fingers would automatically get into position and your brain would do the processing in the background. You would have become so efficient at the process that the amount of conscious thinking that is required by you is minimal.
And in fact, I think that this is a very important step in being able to play something quickly and accurately. The brain just isn’t fast enough to actively think about each note separately. You will only become fast when you can relegate those thought processes until they become as natural as breathing. Think of The Matrix if you want an exaggerated version of the kind of thing I’m thinking of! I would conjecture that the faster the player, the lower the level of conscious activity in the brain will be. Efficiency of the mind, if you will.

The same goes for hearing music. When you play the notes, you will likely only ‘zone in’ at key points in the riff. You might consciously pay attention once each bar at places where your brain deems it necessary to focus in order to get your bearings right. It’s somewhat of an ongoing calibration process that only needs to come to the forefront when something significant (physiologically or musically) is about to happen in the piece. As a result, you won’t actually ‘hear’ each note, but you’ll intrinsically know whether it’s right or not.

Ever notice how the listeners of more complex genres of music are largely comprised of musicians in their own right? I think this is far from coincidence and links into this subconscious ‘fast hearing’ phenomenon. Where the untrained ear just hears a blur or flurry of random, seemingly unrelated notes, those who have this ability to process quickly in the back of their minds seem to be able to pick out the relevant parts and make sense of the aural information coming their way. I know that in my case, I’ve gone from thinking that a Red Hot Chili Peppers solo was unfathomably fast and incomprehensible to being able to mentally pick out runs that Dream Theater use in their music, just by internalising a lot of the extraneous information that comes with playing certain things on the guitar.

OK, so is there a point here?

Well, yes and no. I’ve pointed out something that I find quite fascinating: the way our mind becomes proficient at something by effectively minimising the level of conscious effort required. To me, it seems like a very fundamental optimisation exercise, and is a great example of how nature seeks to simplify and streamline everything as much as possible (which, as a side note, I think is why mathematics captures natural processes so effortlessly).

Although this information is not going to make you the next Michael Angelo Batio, I think that understanding the principles by which our minds work is important if we want to achieve a higher level of proficiency in anything we choose to do. At the very least, I hope this brain dump has provoked a bit of curiosity!

Once upon a time, the great country of Englonia was ruled by the benevolent leader Neelios. He was famed for his mathematical methods of problem solving, and peasants and landowners alike would come to Neelios for justice and fairness to be calculated for them. Of course, his success was, in part, due to the strong support he received from his five noble and wise aides.

Drove ‘the Car’ was in charge of healthcare and had almost supernatural healing powers for those especially in need. Nijller was a master craftsman. There wasn’t an arm, a leg or an antique table that he couldn’t slice up and restore to its original glory. Natalya was a master of communication and instilled a great love for language throughout the nation. Legend has it that she was once a cat. However, to this day, nobody knows the truth behind the mystery or where she actually came from. V-Ray was a cyborg. A keeper of the peace and master of justice, he enforced and optimised the law. Finally, Aware was a master of aesthetics and design. He also moonlighted as the court jester.

After many years of harmonious leadership, Neelios unfortunately became very ill. According to Drove’s diagnosis, there wasn’t much time left for him. Not wanting to leave things unsettled, Neelios had to decide how the 48 counties of Englonia would be allocated before he died. With his last gasp, he announced how the country would be split:

“I have decided what will become of Englonia after I depart. I give V-Ray control of Greater Londonium, our nation’s capital.

Of the remaining 47 counties: Drove shall receive half, Nijller will receive a third, Natalya shall get a twelfth and Aware will get a sixteenth.”

And with that, Neelios passed on. His corpse was traded overseas for chocolate.

—————

Everyone was very excited by the prospect of ruling their own lands. Whilst V-Ray toddled off to Londonium, the remaining four were left to think about the delegation process. They sat around a round table and contemplated over cappuccinos and chocolate croissants. After a short while, Nijller scrunched up his face. The others asked him what was wrong.

“Well” said Nijller, “I’ve just calculated our shares of the counties. It doesn’t look good.”

Drove agreed, but Natalya and Aware weren’t too sure why. Maths wasn’t their strongest suit. Drove explained the issue to them.

“Our shares of the 47 counties were: 1/2, 1/3, 1/12 and 1/16. However, when we work out how many counties that means for us each: I would end up with 23.5, Nijller would get 15.66666…, Natalya would get 3.91666666… and Aware would get 2.9375!”

Natalya sniggered because apparently Neelios wasn’t as great at calculations as he thought. But amusement quickly turned to bickering and frustration as the quartet argued about how this would work.

—————

Whilst Nijller was trying to cut Cornwallinium into thirds, Aware (who was quite unaware of the sheer magnitude of the task surrounding the others, and who had said nothing thus far) exclaimed,

“What if we ask V-Ray if we can borrow Londonium for a bit?”

“You’ve gotta be kidding me!” cried Drove. “You think we’re going to kick him out of his only county? The one Neelios specifically held for him? I don’t think that will go down too well…”

But Aware was on to something.

“We’d give it back to him tomorrow!” he replied.

Whilst Drove and Nijller proceeded to give Aware wedgies for his absurdness, Natalya began to think about his crazy scheme.

“Hang on!” said Natalya, with some authority. “If we borrow Londonium, then we’d have 48 counties to divide right? I think that could make things easier.”

She called V-Ray to explain the situation.

“Okay, I’ll let you borrow Londonium. But if I don’t get it back tomorrow and find out you’ve divided it among you, I’ll kill the lot of you.” V-Ray said, worryingly calmly.

He was a cyborg after all. Nobody wanted to mess with those bionic biceps. Natalya agreed.

After calming down Nijller and Drove (and sorting out Aware’s underwear), Natalya proceeded to explain why she had done what she did (it apparently wasn’t because she used to be a cat).

“We have 48 counties now. Drove, you get half so that’s 24 for you.”

“Great!” said Drove.

“Nijller, you get a third, so that’s 48/3 = 16.” Nijller could dig that.

“I get a twelfth so that’s 48/12 = 4 counties, and Aware gets 48/16 = 3 counties!”

Natalya was delighted that her plan was so effective, and blew raspberries at Drove and Nijller. The pleasure was short-lived, however, as the fear of death by V-Ray’s v-ray set in.

Nijller started doing some sums.

“24+16+4+3. That’s exactly 47 counties! Perfect! There’s exactly one left over!”

Natalya quickly called V-Ray to return Londonium to him. All that was left was for the four to decide who got what. At least they had a lot more chocolate to help them through that decision…

—————

And so, the problem had been solved. The famous five lived happily ever after and made their counties even better than they were before. Englonia had turned into a haven, and it was all thanks to some clever mathematics, a little bit of inspiration and a lot of chocolate.

The End

I’d noticed recently that a few people had mentioned the expensive cost of housing in relation to the average salary in their area. So I was interested to find out which places were particularly bad in the grand scheme of things.

To do this, I picked some major cities from around the (reasonably developed) world. I used payscale.com to get data for the median salaries for these cities, and predominantly numbeo.com for data on housing.

Methodology

Initially, I thought I’d find out the average house price for each city by looking at country-specific statistics websites. Unfortunately, what seems to be a fairly easy figure to work out is very difficult to come by, especially in the less developed countries within my sample! Many countries don’t seem to have great statistical reporting, and many others had independent firms with research on house prices that you had to pay vast amounts of money to see. For the data I did find on average house prices, it was often unclear how it was calculated, so it was difficult to compare figures credibly between countries.

Luckily, I managed to come across a source (Numbeo) which gave average house prices per square metre for individual cities. This way, I could compare equally sized properties between them. I’m still not too sure about the accuracy of the data sets, but this was the fairest comparison I could manage with the data I had.

Numbeo gives two prices per square metre: one for properties inside the city centre and one for properties on the outskirts. Not to choose one over the other, I averaged the two to come up with an overall average per square metre price for each city. A little research showed that a good ball-park figure for the average size of a property would be 100m². Of course, the average property size will vary between cities. For example, congested Tokyo has very small apartments compared to the much larger places that you’d find on average in many US cities. Still, using a fixed area makes it easier to compare costs of living more accurately.

Once I had the average price of 100m² worth of housing in each city, I simply divided it by the median annual salary. This gives us a nice index which tells us roughly how many years you’d have to work in a particular city to be able to afford to buy 100m² worth of property.

Results

Salary

The top 3 cities in my sample by median salary:

1. Geneva, Switzerland – £77,313
2. Zurich, Switzerland – £76,588
3. Tokyo, Japan – £48,728

I was shocked by the gulf between the Swiss salaries and everyone else. However, everything is famously expensive in Switzerland, so taking into account food, utilities, transport and housing, they may be justified such a high salary.

The bottom 3 cities in my sample by median salary:

102. Karachi, Pakistan – £8,579
103. Dhaka, Bangladesh – £7,883
104. Colombo, Sri Lanka – £6,847

I didn’t include many African countries in my sample. Clearly, many of their salaries are likely to be lower.

Housing

The 3 most expensive cities in my sample by average cost of a 100m² property:

1. Hong Kong – £1,302,967.50
2. Tokyo, Japan – £909,432.50
3. Rome, Italy – £768,687.50

I was quite surprised at the fact that Rome beat out London, Paris and Singapore.

The 3 cheapest cities in my sample by average cost of a 100m² property:

102. Karachi, Pakistan – £35,460
103. Lahore, Pakistan – £28,694.50
104. Dhaka, Bangladesh – £18,464.50

Ratio of salary to housing

The 5 cities with the greatest proportional disparity between salary and housing:

1. Hong Kong – 43.98
2. Singapore – 25.55
3. Rome, Italy – 25.45
4. Taipei, Taiwan – 25.13
5. Moscow, Russia – 20.76

This means that, theoretically speaking, it would take you 44 years of working in Hong Kong to buy 100m² worth of property! Of course, this ridiculous figure is the reason why apartments in Hong Kong are so tiny. This is the reason why I would have preferred to get actual median house price data. 100m² is simply not a realistic property area for Hong Kong. Clearly, actual average home prices would be lower to reflect the smaller average size. Nonetheless, it is a rather frightening statistic (if not of living costs, then certainly of the extreme density and lack of space).

The 5 cities with the smallest proportional disparity between salary and housing:

100. Montreal, Canada – 3.55
101. Istanbul, Turkey – 3.25
102. Atlanta, USA – 3.21
103. Ankara, Turkey – 2.83
104. Chicago, USA – 2.69

So it seems North America and Turkey are the way to go if you want to live comfortably!

A final interesting statistic to come out of this was the difference between inner city and outer city housing prices. In almost all cities, the city centre was more expensive to buy property in than the outskirts. However, in 3 cases (Johannesburg, Durban and Santiago), property costs more outside the city centre than inside. One explanation may be that the inner city areas in these places have higher levels of deprivation than the outer city areas, when the opposite may hold in most other places. However, I do not know enough about these places to make an accurate judgement. If you have any thoughts or experience in this area, feel free to leave a comment.

The full dataset can be downloaded here: salary vs house price

Here’s a video of me playing Joe Satriani’s classic – Always With Me, Always With You. Enjoy.

A bit of history

Marcus was a prized disciple of the famously foul-mouthed Gordon Ramsay. Under Ramsay, Marcus opened up Petrus at The Berkeley. It was a very successful restaurant, gaining 2 Michelin stars. However, Marcus and Gordon had a rather high-profile falling out. Because the restaurant was owned by Ramsay’s company, he still owned the rights to the name. Marcus managed to buy out the restaurant in a legal battle, but lost the Petrus name to Ramsay. As a result, he renamed the restaurant after himself and Ramsay opened a new Petrus down the road.

The Environment

As you’d expect from a restaurant situated within a hotel that charges around £250 a night, ‘posh’ and ‘pretentious’ was the atmosphere that you got. It was a great experience, although I did feel a little out of place there, especially when your average hotel resident was walking around casually in a 3 piece suit.

The restaurant itself is actually a modestly sized room. It was nicely decorated, but didn’t feel over the top. The overall atmosphere was cozy and understated. A pleasant place to enjoy your meal; some might expect a little more for the kind of status it holds.

The Service

If the decor was subtle, the service was definitely not. I was escorted to the toilets because they were not so obvious to find. When I came back, my napkin had been replaced and neatly folded. If my glass of water dropped to beneath about 25% full, someone would magically appear from nowhere to top it up. I’m not used to this. At first, I felt compelled to thank them each time, but eventually forced myself not to to save energy and awkwardness. I guess some of the more affluent diners were used to this and so had higher expectations, therefore saying less.

Everyone was delightfully friendly, even when we started asking seemingly unorthodox questions about what some of the terms in the menu meant (speck is ham, apparently), and why they did some of the things they did. I suppose they are paid to behave in this way, but my expectations are usually not that high so I was impressed nonetheless.

The only criticism of this type of attention is that if you’re not used to it, it can seem overbearing, therefore making it difficult to relax. To start off with, there was someone at our table every time I tried to start a conversation. I don’t see why all the top restaurants can’t tone it down at least a little.

Starter

With each of our 3 courses, we were first given a mini ‘experimental’ dish to try out, which was a nice little bonus. First up for us were some sesame ‘crisps’ with an aubergine dip. Nothing special, but they would have been fine if they didn’t contain the salt content of the Dead Sea. After most of the water had left my body via osmosis, we were brought a selection of interestingly flavoured breads along with butter. I love bread, and so I lapped this up. They didn’t quite have that warm and ‘freshly baked’ taste that I’ve experienced before, but it was exceedingly good bread.

For our actual starter, we both ordered the pea ravioli with bits of apricot and ham/bacon. Oh, and pea flowers with a pea sauce. Garnished with peas. Obviously, they had a posher name for it, but I don’t have the foggiest idea what it was. Note to posh restaurants: name your dishes things so that people don’t have to ask what’s in them. Or put in a layman’s translation.

The presentation was nice, but I’m not that fussed about what it looks like. Taste is paramount. I tucked in. Oh look, more Dead Sea salt. I’m sure the ravioli would have been nice but all I could taste was salt, and I believe it came mostly from the sauce. As you can see, the sauce was difficult to avoid. It also had a strange aftertaste to it that I wasn’t too keen on. This was definitely the worst dish of the 3, and I did mention the excessive saltiness to the waiter.

Main Course

The experimental dish before the main came in the form of a cappuccino-like object made from sour cream and other foamed delights. I’d always imagined that I’d only be drinking food when I end up feeding through a straw in hospital, but once again, the world of fine dining breaks down all preconceptions. The texture was nice, being that I enjoy frothed coffee-like beverages, but I’d rather have it have tasted of coffee too, rather than some sour (and honestly, slightly salty!) stuff.

For the main, I plumped for pollock and Mr Fonz chose the chicken. My favourite course. The fish fillet was in a nice neat little cube surrounded with greenish beans and other garnishes. The pollock was cooked perfectly – the best cooked fish I think I’ve eaten. It had a perfectly crispy skin on top. No harsh flavours, no excessive saltiness. A very well prepared dish. Apparently, the chicken was also very well cooked. Thumbs up here then.

Dessert

The experimental dessert dish looked like a strip of toothpaste. Except it was pineapple flavoured ice-cream, and was extremely refreshing. Not much to say here. A nice, but small, treat.

For dessert, both of us chose the custard tart. I can’t remember what the other option was, but I remember it sounding pretty boring, so this one was a no-brainer. As you can see, I’d say this was the nicest looking dish we were faced with:

The custard tart was served with some strawberry (or raspberry) sauce, dollops of cream and strawberry ice-cream. Finally, the green blobs you see were tarragon flavoured ice-cream. Yes, you heard me correctly. The custard tart was very nicely done – it had a gentle flavour and lovely, smooth consistency. For me, the best part of any tart is the pastry, and again, this was done to perfection. Solid enough to hold together and add crispiness, but brittle enough to break and crumble. However, it did seem a bit ‘boring’ for a dessert in this sort of establishment, and considering that this was the better of the choices, it was a little disappointing. I guess that’s what we got for taking the so-called ‘cheap’ lunch option.

The one thing neither of us could stomach was the tarragon ice-cream. Up until this, although I’d had issues with saltiness and some of the flavours, there wasn’t anything that I’d left. I really tried to like it, but in the end, my tongue could take no more. As it melted, I frantically tried to stop it from running into my precious tart with mixed success. Mr Fonz also left his share, and we both shared our puzzlement at this choice of flavour with the waiter. Politely, I exclaimed “I think it’s a bit of an acquired taste”. That it was, but I’d like to see some statistics on what percentage of diners had genuinely ‘acquired’ a taste for it.

Finally, we were brought a selection of freshly-made chocolates. We tried each of the 7 or 8 flavours in a mini tasting competition and concluded that the banana caramel truffle was the unanimous winner. Some of the flavours were quite ridiculously overpowering (fennel?!), and I think this sums up the whole dining experience. Simple and effective ideas were somewhat tarnished by needless pretence (by way of adding strong flavours) where it wasn’t needed.

Oh, and finally finally, we were given the painful bill. I must say though, that by London prices, it wasn’t bad. Also, I appreciate the practice that some of the more exclusive restaurants are employing – to add a 12.5% service charge automatically to the bill. I don’t see why waiters aren’t paid a fixed fee like any other profession. The process of tipping seems haphazard and unfair.

In summary…

Marcus Wareing at The Berkeley turned out to be a mixed bag of delights and, well, salt. It was certainly a unique culinary experience. But I have been to places that are cheaper and have left me more satisfied in the stomach (notable example: Pearl by Jun Tanaka, which I think dominated Marcus Wareing in every way). Sure, a lot of what we paid for were the frills that came with the dining, and they were entertaining. I’m also sure that the reputation and Michelin rating added a premium to the price. For me though, none of the frills are worth it if the core – the food – is not up to par. After all, the food is why I go out to eat in the first place.

To try to balance the argument, I’d considered that maybe the full dinner and a la carte menus had better dishes. From what I read, the dishes did indeed sound more interesting. But, as Mr Fonz was quick to point out, a highly acclaimed restaurant should have the same excellence in taste throughout its inventory.

Marcus Wareing at The Berkeley. A restaurant whose food was preceded by its reputation.

The type of shuffle in the photo above is called a ‘riffle shuffle’. I’m sure you’ve all seen it before. You split the deck into two parts and interlock the parts together by using a fanning motion. The way most people do it in general is quite randomly and haphazardly, which is fair enough because the idea is to mix the cards up as much as possible.

However, you can be a bit more precise. You can perform a ‘perfect’ riffle shuffle by splitting the deck into 2 exact halves (26 cards each) and interlocking the cards so that the cards from the 2 halves alternate into the final pile. To be a little more precise, if we call the two halves A and B, then we want to combine the cards so that every other card in the deck after the shuffle is from pile A.

Now, you may already have realised that there are exactly 2 ways of doing this. Suppose half A is the top half of the unshuffled deck and B is the bottom half. Then you can perform the shuffle with either:

1. The top card from B ending up on the top in the shuffled deck
2. The top card from A ending up on the top in the shuffled deck

Situation 1 is called ‘middle card top’, and 2 is called ‘top card top’, because these are the cards that will end up on the top of the shuffled deck.

It may seem like a trivial point, but magicians will know that this is not so! If we keep repeating the ‘middle card top’ shuffle, then we’d have to do it 52 times to return the deck to its original order. However, if we repeated the ‘top card top’ shuffle, we’d only have to repeat it 8 times to get back to the starting deck! This seems a little surprising!

Can we prove formally that this will happen every time without physically shuffling the cards and making a mess? Of course we can!

Middle card top

Let’s give each card in our unshuffled deck a number from 1-52 i.e. the top card is numbered 1, the second 2 and so on.

After a middle card top shuffle, the 27th card will be the top card in the shuffled deck. The 28th card will be in position 3, the 29th in position 5 etc. Therefore, the ordering of the shuffled deck after one iteration is:

27, 1, 28, 2, 29, 3, 30, 4, 31, 5, 32, 6, 33, 7, 34, 8, 35, 9, 36, 10, 37, 11, 38, 12, 39, 13, 40, 14, 41, 15, 42, 16, 43, 17, 44, 18, 45, 19, 46, 20, 47, 21, 48, 22, 49, 23, 50, 24, 51, 25, 52, 26.

The clever bit is that we don’t have to repeat this 52 times! We can see how the cards will cycle after shuffles by noticing that the 1st card becomes the 27th. The 27th card then becomes the 40th according to the ordering above. If we follow the above chain, we can write down the cycle in the following form:

(1, 27, 40, 20, 10, 5, 29, 41, 47, 50, 25, 39, 46, 23, 38, 19, 36, 18, 9, 31, 42, 21, 37, 45, 49, 51, 52, 26, 13, 33, 43, 48, 24, 12, 6, 3, 28, 14, 7, 30, 15, 34, 17, 35, 44, 22, 11, 32, 16, 8, 4, 2)

What does this tell us?

The first thing you should notice is that we have one big chain of numbers that loops around. The length of this loop is 52. This means that to get card 1 back in the top position, it has to travel through each of the other positions in the cycle. In other words, we have to repeat the shuffle exactly 52 times to get each of the cards back to their starting positions.

Now that we know what to do, it should be easier to look at the second shuffle.

Top card top

After one iteration of this shuffle, we get the ordering:

1, 27, 2, 28, 3, 29, 4, 30, 5, 31, 6, 32, 7, 33, 8, 34, 9, 35, 10, 36, 11, 37, 12, 38, 13, 39, 14, 40, 15, 41, 16, 42, 17, 43, 18, 44, 19, 45, 20, 46, 21, 47, 22, 48, 23, 49, 24, 50, 25, 51, 26, 52.

Writing this in cycle form, we can immediately see that there will be multiple cycles, because the first card always stays on top. Similarly, card 52 is still card 52 after an iteration of the shuffle. Writing out all of the cycles gives:

(1) (2,27,14,33,17,9,5,3) (4,28,40,46,49,25,13,7) (6,29,15,8,30,41,21,11) (10,31,16,34,43,22,37,19) (12,32,42,47,24,38,45,23) (18,35) (20,36,44,48,50,51,26,39) (52)

We can see that there are two cycles of length 1, one cycle of length 2 and six cycles of length 8. The lowest common multiple of 1, 2 and 8 is 8. Therefore, to reach the starting position, we only need to perform 8 shuffles of the top card top kind.

Remember folks, dexterity and maths are the key to becoming a successful close-up card trick expert!

First things first – let me say that LA Noire is an enjoyable experience with a gripping plot. I had a lot of fun playing through it. That said, as a piece of interactive gaming, it falls short in a couple of key areas.

Open world?

LA Noire is a Rockstar game at heart. It uses the same engine and open-world mechanic that made the Grand Theft Auto games so rich and lasting. But you have to understand why GTA games ‘work’. Sure, they have a vast map area to explore, but the key to longevity and fun comes from the missions and objectives that it has you do. During the course of a GTA game, you work for different people located around different areas of the map. Each phase of the game helps you get familiar with a part of the map. Other ‘side-missions’ are designed well too – again, because they take advantage of your environment.

Like GTA, LA Noire has a huge map to explore. But the point is, none of the cases you have to solve involve exploring any of it. You just go from crime scene to various locations until you close up the case. Yes, there are things like hidden cars to find if you so wish, thereby allowing you to explore the map further, but why should you bother? Unless you’re someone who likes to tick everything off a list, there is absolutely no value in the game to finding anything ‘hidden’, in that you gain no advantage by doing so.

Driving from location to location is fun at first. You soak up all that 1940s LA atmosphere and enjoy the erratic yet sluggish handling of the classic vehicles. But very, very quickly, it gets boring and I found myself skipping all the journeys after not very long (thankfully it gives you the option to do so).

All of this leads me back to the title of this article. Heavy Rain – the PS3 exclusive released last year – was a revolutionary mystery solving interactive movie-like experience. It based its gameplay on a variety of context-sensitive QTEs (quick-time events). You know: press the correct button when it flashes on screen and sometimes choose from different actions/responses to situations.

Heavy Rain’s gameplay had its criticisms too, but it knew what it was. The game was totally linear in that it popped you in different scenes according to the story. The problem with LA Noire is that, at its core, it is a mystery solving, linear interactive story just as Heavy Rain was. But rather than accept this fact and optimise the gameplay experience to focus just on the things you can do in each scene, they had to give you the illusion of an open world sandbox type game, which it is most certainly not. As a result of this, the crime solving, clue gathering scenes which make up most of the game are so one-dimensional in gameplay terms that it is unbelievable. Walk around a room or area, wait for the controller to vibrate and press X. Rinse and repeat until the music stops so you know you’ve found all the clues. Then move on to the next area.

The compelling plot is what holds the thing together, but without that, the game would be a joke. Also, most of the locales you visit (especially in the homicide portion of the game) are so similar that it’s unbelievable. If Team Bondi had canned the whole open map and spent that development time on making each scene unique and fun to play, the game might just have been a classic.

Interrogation

Other than clue finding, the other main game mechanic in LA Noire is performing interrogations on suspects and witnesses. The big draw to this feature, and indeed, the thing that makes it all possible is the amazing facial motion capture system. For the first time in a game, faces look believable and lifelike. The acting is also excellent, and this helps to immerse you in the plot and characters further. LA Noire has definitely raised the bar on facial animation, and I hope that more story-driven games will use the technology in future.

With that said, it is unfortunate that the interrogations are repetitive and give you the same three options each time: Truth, Doubt and Lie. The idea is that the subject can either tell the truth or lie/withhold information. If you have proof they are lying, you select ‘Lie’ and choose the piece of evidence that is relevant. If you have no proof but don’t trust them, you’d select ‘Doubt’.

Unfortunately, the 3 options are limiting. Shifty eyes mean that the subject is probably lying, and apart from the odd few difficult suspects, it’s usually fairly easy to tell. However, this doesn’t mean you’ll get it right! As Yahtzee explains in his excellent Zero Punctuation review of the game, the problem is that the options are completely ambiguous. Sometimes you know that the guy is lying and have loads of evidence against him, but selecting the wrong one will have the game shoot you down like the loser it thinks you are. Other times, you might feel that part of their statement is inaccurate so you choose Doubt, only to have your character go completely schizophrenic with accusations. In reality, there could be more than one right answer, but the game is quick to make it clear that there is only one correct path. Not that it makes a whole lot of difference if you get things wrong anyway – the game still progresses quite normally.

This is another area where LA Noire could have learnt from Heavy Rain. I appreciate that it may be more difficult from a development point of view, but having situation specific responses a la those found in RPGs would really have made interrogation more interesting and given the acting a real chance to shine through. If they’d removed all the peripheral rubbish and stuck to making the core gameplay a bit better, it may not have been much of an extra development burden.

They might also have been able to make the game longer. At about 12 or 13 hours, it really doesn’t compare to other Rockstar games that use the same open world mechanic. But it is at heart a different game to GTA IV or Red Dead Redemption, and so they shouldn’t really be compared. That’s why I think that they should have approached the game more like Heavy Rain, and less like a cookie-cutter Rockstar title (hell, they already stole a trademark plot device from it). Maybe it would have elevated LA Noire from a decent game to an all-time great.