Category Archives: Mathematics

England’s World Cup Failure: A nerd explains why all the pundits are wrong

worldcupfootballEvery time I hear another pundit explaining their theory behind England’s failure at the world cup I get all hot and bothered. My wife could literally not care less about football or my feelings on the subject, so I thought I would share my them with you ūüėČ

You see, it seems that football does not benefit, as cricket and baseball do, from that important type of pundit, the statistician.

For if they did, they would realise that failing to make it into the last 16 this year is not a failure at all.

Why? Because we have to remember that there are, at time of writing, 209 national men’s teams registered with FIFA – and that FIFA estimates that 250 million people play the beautiful game. So just getting to the finals is a real achievement.

On the other hand you can argue that England should be in the top flight – it has a decent population, it has money to spend and many aspirational heroes.

Well.. it does do well, currently, England is ranked 10th in the world, and has often been higher. But does that mean it should always reach the last 16 of the world cup? No.

Take a step back. Even with so many teams, that world ranking should mean a team like England should make it ‘usually’, but certainly not always. Indeed, it has made the last 16 every time since 1958, indeed it’s a surprise to me that they have run so long without missing out.

It has always been known that football has a ‘luck factor’ indeed this is one of its best features – upsets happen – and that is why the league has a round-robin design, and also why some tournaments are done by knockout – the league aims to find the best teams, tournaments aim to find the best moments.

England did not play badly this world cup. The goals conceded were really pretty darn good, and England had more shots on goal than their opponents. But for the rub of the green, they could have been through and lauded by all. So how can the pundits have such strong opinions?

Easy. Because it’s their job to sound like they know.

WorldCupRankingsPlot

Addendum
So how have teams like Brazil, Spain, Germany, Italy, France and the Netherlands managed to spend so much time in the top 3? If it was all luck, they would not. Well, this means that there are actually recipes for better performance, elusive but real…

However, just as the government would hate for you realise they do not control the economy, football pundits and administrators alike would not like you to know (or indeed to know themselves) that this recipe is largely outside of their control – the biggest factors being: population size, other games to play, weather, virtuous circles (inspiration, promise of fame, etc) and last but certainly not least, luck.

Requirements for Promoting a New Scientific Theory

I have been reading some pretty strange stuff about Gravity recently. It seems there are some pretty odd folk out there who have taken thinking about physics to a new, possibly unhealthy, level.

Gravity: It's the Law

Basically, they are crackpots. Well I guess it’s a slippery slope – one day you wonder why the¬†earth¬†is sucking down on you, the next you decide to spend some time on the knotty question. Soon¬†enough¬†you think you’ve got it, it is clearly that the earth is absorbing space which is constantly rushing down around us dragging us with it. Or similar.

So yes, its true, Einstein did not ‘solve’ Gravity, and there is still fame and fortune to be had in thinking about gravity, so this is the example I shall use today.

The trouble with Gravity is that Einstein’s explanation is just so cool! He explained that mass warps space and that the feeling of being pulled is simply a side effect of being in warped space. It sounds so outlandish, but also so simple, that it has clearly encouraged many ‘interesting’ people to have a crack at doing a better job themselves.

So, as a service to all those wannabe physics icons, I offer today a service, in the form of a checklist Рwhat hoops will your new scientific theory have to jump through to get my attention, and that of the so-called ivory tower elite in the scientific community?

Requirement 1: Your theory needs to be well presented

presentation counts!Yes, it may sound elitist to say, but your documentation/website/paper/video should have good grammar. Yes, yes, one should not use the quality of one’s english to judge the quality of one’s theory, and I know prejudice is hard to overcome, but this is not my point. My point is that in order to understand a complicated thing like a physics theory it needs to be unambiguous. It needs to be clear. It needs to use the same jargon the so called ‘elite’ community uses. Invented acronyms, especially¬†those with your own initials in them,¬†are out.

Requirement 2: Your proposal needs to be respectful

Image courtesy of Wikimedia Commons

Image courtesy of Wikimedia Commons

Again, this is not about making you bow to your superiors in the academic world. Indeed in the case of Gravity, the physics community is one of the most humble out there. While I agree academia is up it’s arse most of the time, this is about convincing the reader that you know your stuff. In order to do that, you need to show that you know ‘their stuff’ too. If you have headings like “Einstein’s Big Mistake” it is a bit like saying to the reader ‘you are all FOOLS!’ and cackling madly. Don’t do it!

Respect also means you need to answer questions ‘properly’. That means clearly, fully, and in the common language of the community. You cannot say “its the responsibility of the community to test your theory”. This is a great way to piss people right off. It is your responsibility to make them want to. This usually means dealing with their doubts head-on, and if you can do that, I promise you they will then want to know more.

Requirement 3: You need to develop credibility

Sorry, as you can see we have yet to consider the actual merit of the theory itself. I wish it were not so, but we are humans first and scientists second. We cannot focus our thoughts on a theory if we doubt the payback. And if you say¬†that aliens came and told you the scientific theory, then people are unlikely to keep listening, unless, perhaps they’re from Hollywood.

But seriously, credibility is the hidden currency of the world, it opens doors, bends ears and gets funds. It takes professionals decades to build and it is really rather naive to waltz into a specialism and expect everyone to drop their tools and listen to you.

That said, the science world is full of incomers, it is not a closed shop as some would you believe. If you follow requirements 1 and 2, and are persistent (and your theory actually holds water) then you are very likely to succeed.

Penrose_triangleRequirement 4: Your theory needs to be consistent

I have seen some pretty strange stuff proposed. Gravity is a¬†manifestation¬†of the flow of information, or the speed of light is determined by a planet’s¬†density. Find your own at crank.net. Let’s look at this peach as an example:¬†http://www.einsteingravity.com/.

This exhibit is great example of how not to go about promoting your theory. “Monumental ¬† Scientific ¬† Discovery ¬†!” it screams across the top, then the first claim is this:

1) The Acceleration of earth’s Gravity x earth orbit Time (exact lunar year) = the Velocity of Light.
(9.80175174 m/s2 x 30,585,600 s = 299,792,458 m/s)

Now this is rather remarkable. Can it really be that you can calculate the speed of light to 9 significant figures from just the earth’s gravitational¬†acceleration¬†and the length of a year? Intuitively I suspect you could (eventually), but then I started to think, well, what if the earth was irregularly shaped? The gravitational constant is actually not all that consistent depending on where you are either. So I checked, then I noticed he said ‘lunar year’. What? Why? What is a lunar year? Then I calculated that the time he used was 354 days, which isn’t even a lunar year. Add to that that he gives the¬†acceleration¬†of gravity on earth to 9-figures despite the fact that nobody knows it that well (like I said it is location dependent). Does he do the same test for other planets? No. Well what if they have no moon!

So, 0/4 for on our checklist for einsteingravity.com!

Requirement 5: The theory needs to be be consistent with well-known observationsevidence

Now if your theory has got past requirements 1-4 , well done to you, you will be welcome to join my table any time. Now is when you may need some more help.

Once a theory is consistent with itself, it now needs to agree with what we see around us. It needs to explain apples falling, moons orbiting, light bending and time dilating. This is the hardest part.

It cannot leave any out, or predict something contrary to the facts. It cannot be vague or wishy-washy. It needs the type of certainty we only get from the application of formal logic – and that old chestnut – mathematics.

No you cannot get away without it, there is no substitute for an equation. Equations derived using logic take all the emotion out of a debate. And they set you up perfectly for requirement #5.

crystal-ballRequirement 6: The theory needs to make testable predictions

If your theory has got past the 5 above, very nice job, I hope to meet you one day.

We are all set, we have a hypothesis and we can’t break it. It has been passed to others, some dismiss it, others are not so sure. How do you create consensus?

Simple, make an impressive prediction, and then test that.

Einsteins field equations for example, boldly provide a ‘shape’ of space (spacetime actually) for any given distribution of mass. With that shape in hand you should then be able to predict the path of light beams past stars or galaxies. These equation claimed to replace¬†Newton’s simple inverse square law, but include the effects of time which creates¬†strange¬†effects (like frame dragging), which, importantly could be, and were,¬†tested.

The beauty of these equations, derived via logical inference from how the speed of light seems invariate, and now proven many times, is that they moved physics forward. Rather than asking, ‘what is gravity’, the question is now ‘why does mass warp space’. It’s a better question because answering it will probably have implications far beyond gravity – it will inform cosmology and quantum theory too.

Conclusion

So if you are thinking of sharing with the world at last your immensely important insights, and want to be listened to, please remember my advice when you are famous and put in a good word for me in Stockholm. But please, if, when trying to explain yourself, and are finding it tough, please please consider the possibility that you are just plain wrong…

————-

Jarrod Hart is a practicing scientist, and wrote this to shamelessly enhance his  reputation in case he ever needs to peddle you a strange theory.

Further reading:

Elegant Maths! If you can follow this it might blow your mind…

To most people, maths is just something we learned in order to avoid being ripped off. To some, maths is an essential tool, helpful in modelling plague outbreaks or cracking encryption ciphers.

However, for an elite few, maths is simply a parallel universe and they are its explorers.

Today let us discuss what I consider perhaps the most beautiful discovery to date. But first, some introductions…

Part 1: Consider, to start, the circle

If you have a wheel a metre across, it will roll out about¬†3.14159…metres each revolution. This number, which we call ŌĬ†turns out to be some sort of fundamental property of ‘space’.

The¬†Greeks¬†were not very happy about the ‘messiness’ of this number. They preferred numbers that could be expressed as fractions – while 22/7 was close to¬†ŌÄ, it was not exact and they lost a lot of sleep trying to find a neat way to write¬†ŌÄ.

Mathematicians have since grudgingly accepted that it cannot be written as a fraction, and indeed it cannot be written down at all because it has no ‘pattern’ and never ends, those digits just keep coming at (almost) random! Here are the first 100…

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 ….

wife of pi

The wife of pi…

Understandably, they decided to call this sort of number¬†‘irrational‘.

Part 2: Consider now, ‘powers’

Mathematicians may work tirelessly on some very pointless looking things, however, they are still fairly lazy when it comes to writing stuff down. They like shorthand. So rather than writing 3+3+3+3+3 they invented ‘multiplication’, giving them 5×3.

Likewise, rather than writing 3x3x3x3x3 they invented ‘powers’, so they could write 35.

Of course they then¬†realized¬†these tricks could be extended past ‘whole’ numbers. 2.5×3 is 7.5. But what about 32.5?

It works of course, the answer turns out to be about 15.59 plus change.

But what does it mean? 32.5 is three, times by itself, 2.5 times! or 3x3x30.5. What on earth is that?

Well it turns out, when you ponder this (maybe I should say, if you ponder this), that¬†30.5¬†is the same as¬†‚ąö3. So ‘root three’ is three times itself half a time…

[pregnant pause]

Ok, let’s look at it another way

Consider, for example 32x33, which is the same as (3×3)x(3x3x3) which is the same as 3x3x3x3x3 which is the same as¬†¬†35¬†, so¬†3(2+3).

So using that logic…

3 = 31 = 3(0.5+0.5) = 30.5 x 30.5

And what times¬†itself¬†is equal to 3? Well¬†‚ąö3! So¬†30.5¬†is¬†‚ąö3…

It makes sense now, and we can even get used to saying things like 31.9 x 30.1 = 9.

Of course, these fractional powers also commonly yield those ‘messy numbers’, so abhorred by the Greeks. ‚ąö3¬†is, roughly:

1.73205080756887729352744634150587236694280525381038062805580…

The logic follows through for negative numbers. 3-2  is just 1/32 which is  1/9.

Part 3. Now consider ‘e’

y=e^x. The slope is always the same as the value! This has the interesting effect that the tangent to the line always intercepts the y axis precisely 1 unit back…

y=e^x

Here is a third sort of messy number, one which the Greeks are probably glad they missed. We have Leonhard Euler to thank for discovering this one.

He noted there was a number ‘e‘ giving an equation of the form y=ex¬†(see the graphs pictured), where the slope of the curve is the same as the height of the curve at each point.

Strange and pointless sounding perhaps but pretty simple. So¬†y=2x¬†doesn’t work,¬†y=3x¬†doesn’t work, but by trial and error you can find¬†a value for a that works, which is, roughly:

2.71828182845904523536028747135266249775724709369995…

It too has no pattern and no repeats so is also¬†‘irrational‘. This number has a whole book written about it, for those who are keen.

Part 4. Now consider ‘i’

The last piece of the puzzle now.

Consider the equation 3 + x = 0

Now solve for x. Seems pretty easy, but really you are cheating. There is no number that solves that equation. Really, to solve it you had to ‘invent’ the concept of a negative number.

Ok. Now consider the equation x2 + 1 = 0

Ah. Trickier! However it turns out that we can do the same trick; this time we simply invent another sort of number – the ‘imaginary’ number. Now if you’ve never heard about these numbers before, you may think I’m joking. Alas I am not. This is what mathematicians have been up to for the last few hundred years, just making stuff up as they go along.

So we define i as¬†‚ąö-1, or a number, that when multiplied by itself, yields the more respectable -1.

Aside: Just as -1 is a number which, when multiplied by any negative number renders it decent (i.e. positive) once more.

So i2 + 1 = 0 and the equation is solved. It turns out mathematicians were suddenly able to solve loads of really tedious equations using this trick, which made their entire week.

So we have now got i! At last we are ready to put the puzzle together.

Simplicity emerges from the complex…

So, now I ask, what happens if you raise e to the power of i?¬†What does it even mean? It means, e times itself ‚ąö-1 times. Ouch. Nonsense surely?

Well it works out at roughly 0.540302306 + 0.841470985 i, which is a right mess, something they call, for fairly self-explanatory reasons, a complex number.

So now lets stick our old friend from the circle,¬†ŌÄ, in there¬†and see what happens:

What is¬†eiŌÄ?

Surely an even bigger mess? I mean its all these messy irrational¬†numbers combined with this home-made¬†imaginary¬†number…

Google can do it for you… and the answer is…..

-1.

So there you have it, all these messy numbers –¬†ŌÄ, e and non-integer powers¬†combine with i and the answer pops out as -1.

Blows my mind.

—————–

For more info on whay the heck this is, look up¬†Euler’s Identity!

Musical Notes Explained Simply

Have you ever wondered how the musical notes we use were chosen?

I mean when I was growing up I was learning one thing in music class ¬†(do-re-me-fa-so-la-ti-do!) and another in science class (440Hz) and never the twain did meet…

So what gives? I always suspected the musical community were being scientific, but their language was all Greek to me.

Years passed and only rarely did I get the chance to wonder at this question – and meantime¬†my science education was getting the upper hand – I learned how sounds travel through the air and how the ear works –¬†how deep, low notes are the result of compression waves in the air, perhaps a few meters apart, while higher pitched sounds where compression waves much more tightly packed, perhaps millimeters apart. I also learned a note could have any¬†frequency, and so no reason to pick out any ‘special’ frequencies.

However,  just recently I realized, in a flash of light, that with an infinite number of notes to choose from, musicians had very deliberately selected only a few to make music with, and I suddenly wanted to know why. Was it arbitrary? Was it the same in different cultures? Why did some notes seem to go together and others seem to clash? And of course, as The Provincial Scientist, I wanted to know if our early musicians had done well in their choices.

As it is now the era of the internet I set about to find out more and thought it was so interesting, it would be a crime not to report what I learned on my blog. So here is what I learned…

In Search of Middle C

The best place to start is probably a vibrating string. The vibrating string is clearly key to pianos, harps, guitars and, of course, the entire ‘string’ section of an orchestra. If you stretch a string and pluck it, you are starting an amazing process – as you pull on the string, you create tension, you literally stretch the string and store energy in the fabric of the string. When you let go, the string shrinks under that tension, which pulls it straight. Alas, when its straight it has picked up some speed and the momentum keeps it going until the string is stretched again – thus the string swings back and forth – and it would continue forever were it not for frictional losses – energy is lost in heating the string, but some is also lost in buffeting the air around the string. The air is pushed then pushed again with each cycle creating compression waves that ripple out into the room – and into our ears.¬†Thus we hear the string.

You can see the vibrating string doing it’s magic here:

You can see in the video that the string swinging back and forth is an awful lot like a wave moving up and down the string! Indeed it is!

The speed at which the wave moves (or string vibrates back and forth) – and thus the note we hear – is determined by a few simple factors – the tension in the string, and the weight of the string and the length of the string. The greater the tension, the greater the force trying to straighten the string, but the greater the weight, the more momentum there is to make it stretch out again.

It is therefore easy to get a wide range of notes from a string, start with a long, heavy wire and only tension it enough to remove all the slack. The note can then be gradually increased by decreased the length or the weight of the wire, or by increasing the tension. These are the tricks used in pianos, guitars and so on.

So far so good. But if you have several strings to tune up, what notes should you pick – from infinitely many – to make music with?

The human ear is an amazing device and can hear notes ranging anywhere from 20 to 20,000 compressions per second (the unit for per second is called Hertz or Hz for short). That is a lot of choice!

As I am sure you guessed, the key is to understand why some notes seem to ‘go together’, and the answer lies back in the vibrating string.

Overtones of Overtones

Firstly, it turns out that when you pluck a string, you actually get more than one note. While the string may swing back and forth in one elegant sweep, it may also get shorter waves, with half or a third or quarter the wavelength hiding in there too. This video shows how one spring can vibrate at several speeds:

Although the video shows the string vibrating at one speed each time, it is actually possible for a string to carry more than one wave at a time (this amazing fact deserves its own blog posting, but we will just accept it for now).

So when a string is plucked, the string ‘finds’ ways to store the energy with vibrations – it finds a few frequencies that carry the energy well, these are called ‘resonant frequencies’, there will be several, but they will all be multiples of one low note. As these higher notes are all multiples of a single low ‘parent’ note, they also have consistent frequency relationships between one another, 3/2, 4/3, 5/4 and many many others.

String Harmonics

So clearly, once you have one string, and you want to add a second, you could tune the second string to try to match some of the harmonics of the first string. The best match is to pick a string whose fundamental note is at 2x the frequency of the first string. This string’s fundamental note will match the first string’s 2nd harmonic (also called its first overtone). The second string’s harmonics will also perfectly match up with pre-existing harmonics from the first string. The strings are what is called consonant, they ‘go together’.

Now although the second string will have some frequencies in common with the first string, it turns out that there is an even stronger reason why these notes will go together – it is because when you play several strings at once, you are no longer just playing the strings, the instrument you are playing is the listener’s eardrum. The eardrum will vibrate with a pattern that is some complex combination of the wave-forms coming from the two (or more) strings.¬†When you add two notes together, it is like adding two waves¬†together¬†and you get an¬†interference¬†pattern¬†– the interference may create a nice new sound:

If we add a low note (G1) to a note one octave higher (G2) we get a totally new sound wave.

If, as in this example, one string vibrates at exactly twice the frequency of the other, the two notes will combine to make a handsome looking new waveform, with ‘characteristics’ from both the original waves – but if the frequencies are not a neat ratio, you will get something a bit messy:

This waveform may not repeat, and is unlikely to be consonant with any other notes you may care to add.

Sometimes, when your second string is fairly close in frequency to the first (say 1.1 x the first string’s frequency) then a second¬†phenomenon rears its head, beating. This leads to the creation of entirely new (lower) frequencies that the ear can hear [click here to listen to a sample]. The sum now looks like this:

Beating can sound awful, though of course, the skilled musician can actually use it to create useful effects.

Beautiful Ratios

We have seen that once you have selected one note, you have already greatly reduced the ‘infinite’ choice of other notes to use with it – because only some will be consonant. Although the best consonances are at exactly 2x the first frequency, we see that once you have picked two strings, the choice for the third string is more limited. Should you be consonant first the first string or the second? Can you be consonant with both? You can be fairly consonant with both, but only by being 2x and 4x their respective frequencies. If you picked all your strings as multiples of the first string, the ‘gaps’ between the notes would be very big, akin to playing a tune with only every 12th key on a piano. So how can we fill in the gaps?

Well, early thinkers quickly¬†realized¬†that you can’t actually select a perfect set of notes – some combinations will mesh well, others will be just a little bit odd. This¬†realization¬†was probably a bitter pill for early musician-scientists to swallow.

In the end, they came up with many competing options, each designed  to maximise the occurrence of good ratios  Рa good example is the just intonation scale:

Note: C D E F G A B C
Frequency ratio to the first note: 1 9/8 5/4 4/3 3/2 5/3 15/8 2

Here, the musician picks two notes that are consonant (C and the next C one octave higher) and then divides the gap into seven steps. Each note is a special ratio of the lower note Рwe get neat ratios of 5/4, 4/3 and 3/2 showing up which is good, however the ratios between adjacent notes are much less pleasing!

Aside: You will also see that the steps from¬†B to C¬†and E to F¬†are rather small! Now take a look at your piano and note these notes correspond to the white keys on the keyboard that have no black keys between them! This is no coincidence…

Is the ‘just intonation’ division perfect? No, the notes are not all consonant! Remember than with 8 notes in this group, there are 7+6+5+4+3+2+1=28 ratios (or note pairs), and there is no known way to choose them to all be consonant. That is why, although most musical¬†cultures divide their music notes into ‘octaves’ (nicely consonant frequency doublings), there have evolved many different ways to make the smaller divisions.

Western music has tended to divide the octave into 7 notes (the heptatonic scale) , you could really use any number.¬†Let’s stick with 7 for now.

Another popular way to divide the octave is the Pythagorean tuning:

Note: C D E F G A B C
Frequency ratio to the first note: 1 9/8 81/64 4/3 3/2 27/16 243/128 2

This scale is based on prioritizing the 3/2 overlap of harmonics and moves three notes very slightly.

It is key to remember there are dozens of ways to do this, depending on what you are trying to optimise – do you want to match the greatest number of harmonics, or some smaller number of stronger harmonics? It may even be that personal taste could come into play.

The Wonderful Piano

Have you ever wondered why you hear someone is playing something in C-minor or F-major? What is the deal there? Well, these are also ‘scales’ – alternative ways to cut up the octave, but from a specific family that lives on the piano.

You see, the piano could also divide the octave into 7 notes, and indeed it was once so divided, but with time musicians realised they could open up more subtlety in their music by adding in more notes. So they decided to add the ‘black notes’, the extra black keys on the keyboard!

So in addition to the 7 notes A,B,C,D,E,F & G, they added C#, D#, F#, G# and A# – they called them ‘half tones’ or accidentals. Of course, there are already two half steps (B-C and E-F) which is why there is no B# or E#. These extra notes gave us 12 smaller steps, and of course choosing 12 consonant notes was even harder than choosing 7!

So, after some hard thinking by scholars including ¬†J.S. Bach, a very sensible decision was made – to divide the octave into 12 ‘equal’ steps, which gives us the so-called ‘equal temperament‘, the most popular way to tune a piano. To do this, each note is 21/12¬†or 1.05946… times higher in frequency than the last one, such that twelve steps will eventually give you a doubling.

However, our musical notation is older than the piano and generally only allows for 7 notes per octave, so how do you write music for 12?

Despite that there are 12 notes, composers have tended to still feel some combinations of 7 notes ‘go together’ better than others and so have persisted to write music using only 7 notes, though of the many hundred’s of ways you could choose the 7 notes, they have selected 12 combinations, the 12 “Major scales“:

The Major Scales (down the left). Each uses only 7 of the 12 notes on the piano keyboard. The shaded vertical lines correspond to the black keys on the piano.

Personally, realising what these scales were was a breakthrough for me. Looking the above map helped me to realize several things:

  1. Many long pieces of music will completely ignore nearly half (5/12ths) of the keys on the piano! To play a tune based on a certain ‘scale’ is sometimes said to be played in that ‘key‘.
  2. The scale of C-Major ignores all the black keys, and is probably the oldest/original scale.
  3. Each scale is displaced 5 ‘steps’ from the previous scale (there is a #1 beneath each #5)
  4. The empty squares occur in vertical groups of 5, and move up 5 spaces each time you move a column to the right.

Aside: Note that there are also the 12 “minor scales“. These scales actually use the same 12 subsets of keys as the major scales, but are ‘shifted’ ¬†– they have a different starting point (base note, or ‘tonic‘). ¬†This may seem a trivial change, but because the gaps (steps in frequency) are not all evenly sized in these scales, the major and minor scales have their two ‘small’ steps in different places, which is supposed to change the feel or mood of the music (or even the gender!)

The Number 5

The number ‘5’ in the pattern we saw above was noticed by musicians long before me, and it shows up in other places too.

For example, we saw in the ‘just intonation’ scale above, that the note G had a frequency ratio of exactly 3/2 with the note C. This means that when you hear both together, every third vibration of the higher note will coincide with every second vibration of the lower note. They are thus highly consonant – and they are 5 steps apart on the stave. ¬†This¬†relationship¬†is called the ‘perfect 5th‘. It is again no coincidence that the 5th note of each scale is the base note (tonic) of the next scale. Stepping in 5’s (ratios of 3/2 in frequency) 12 times takes you through exactly 5 octaves and eventually back to the first scale.

Of course, the scales repeat for every octave, so you don’t really need top go up 5 octaves! This cycling¬†behavior¬†allowed the¬†invention of a learning tool called the¬†‘circle of fifths‘, which helps us to understand ¬†the relationships between the scales.

Yet another aside: The ‘perfect fifth’ is called perfect if it is truly a ratio of 3/2 – but recall that pianos have their 12 notes ‘evenly spaced’ (a geometric progression) so the ratio of G to C on the C-Major scale will not be exactly 3/2 – it is actually 0.113% off!

But What About Middle-C?

Ok, so we have seen how some notes ‘go together’, and how once you have one note, you have clever ways to find families of notes that compliment that note – but that leaves just one question – how do we pick that first note?

The leading modern convention is use the note A that comes after (above) middle-C, and to set it at 440Hz exactly.

The question is, why?

Well firstly, I shall point out that the 440Hz convention is not fully accepted. For example, anyone who wants to hear, for example, the Gregorian chants the way they originally sounded, would need to use the conventions of the time. Thus there are pockets of musical tradition that do not want to change how their music has always sounded.

However, when it comes to performing a concert with many instruments, it is useful if they all adopt the same standard. The standard is thus sometimes called the concert pitch, and though 440Hz for A is common, this number has been seen to vary from 423Hz to as high as 451Hz.

So the short answer is, there is no really good reason; the choice of 440Hz really just ’emerged’ as a more common option, and when they¬†standardized¬†they rounded it off. While this answer is ultimately trivial, I find a little amusement in the fact that all the music we hear sounds the way it does for no particular reason!

Conclusion

Before I go, there is a video I want you to look at. I think it shows beautifully how 12 different frequency oscillations can exhibit some beautiful harmony (or harmonics!)

All Done! Ready to Read Some Music?

The next step is to learn to read musical notation – luckily someone has already written an excellent tutorial with pretty pictures.

All I can hope is that the weird things they teach you in this tutorial will be a little less weird now we have covered the baffling origins of the notes!

Jarrod Hart (Los Olivos, CA, October 2011)

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A couple more useful references:

http://www.mediacollege.com/audio/01/sound-waves.html

http://www.get-piano-lessons.com/piano-note-chart.html

http://www.thedawstudio.com/Tips/Soundwaves.html

Exceeding the Speed-Of-Light Explained Simply (and the Quantum riddle solved at no extra cost)

It has recently been in the news that some particle may have exceeded the legal speed limit for all things : 299,792,458 metres per second.

Of course, this will probably turn out to be a bad sum somewhere or perhaps waves ganging up, but the whole hubbub has raised my hackles, and here’s why.

Because Albert Einstein at no time said what they say he said (see here for example). They misunderstand relativity! Things can move at any speed we want, and I will try to explain the fuss now.

So let’s get to it!

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First, we have to consider the way space warps when we move.

The problems started when people realised that light always seems to have the same speed, regardless of the speed you were moving when you saw it. This seems to be a contradiction, because surely if you fly into the light ever faster, it will pass you ever faster?

Well the tests were pretty clear, this does not happen. The speed is always c.

For several years, people were unsure why – until they were told by Einstein in 1905. In the meantime, another ponderer of the problem (Lorentz) decided to write down the maths that are required to square the circle.

The so-called Lorentz equations show, unequivocally, that space and/or time need to warp in order for relative speeds of c not to be exceeded, even when two items are going very close to c in opposite directions to one another.

So something needed to give, and it was space and time!

So, newsflash! it was not Einstein that first published on space and time warping. His contribution (along with Henri Poincar√©¬†and a few others)¬†was to explain how and why. His special theory¬†showed that because there is no ‘preferred’ frame of reference, a speed limit on light was inevitable. The term ‘relativity’ come from this – basically he said, if everything is relative, nothing can be fixed.

==============

Ok, so we have some nice observations that nothing seems to go faster than the speed of light  Рand we have a nice maths model that allows it. So why do I persist in saying things can go faster than the speed of light?

Let me show you…

There is a critical difference between ‘going’ faster than light and being ‘seen to be going’ faster than the speed of light, and that is where I am going with this.

So lets take this apart by asking how we actually define speed.

If a particle leaves point a and then gets to point b, we can divide the distance by the time taken and get the mean speed (or velocity to be pedantic).

The issue with relativistic speeds are that the clock cannot be in both point a and point b. So we need to do some fancy footwork with the maths to use one or other of the clocks. So far so good. This method will indeed never get a result > c.

The nature of space forbids it – if the Lorentz transformations that work so well are to be taken at face value, then for something to exceed c by this method of measurement, is much the same as a number exceeding infinity.

So all is still well. Until you ask, what about if the clock is the thing that travelled from a to b?

In this case, the transformations cancel! The faster the movement, the slower time goes for the clock, and you will see its ticks slow down, thus allowing its speed to exceed c.

The clock will cover the distance and appear to have tavelled at c on your own (stationary) clock, but the travelling clock will have ticked fewer times!

If you divide the distance by the time on the travelling clock, you see a speed that perfectly matches what you would expect should no limit apply. Indeed, the energy required to create the movement matches that expected from simple Newtonian mechanics.

The key point here is that while the clock travelled, the reader of the clock did not. If you do choose to travel with the clock, you will see it tick at normal speed, and see the limit apply – but see the rest of the universe magically shrink to make it so.

Some have argued that I am not comparing apples with apples, and that by using an observer in a different frame to the clock I am invalidating the logic.

To those who say that, I have to admit this is not done lightly. I have grown more confident that this inference is valid by considering questions such as the twin paradox over and over.

The twin paradox describes how one twin who travels somewhere at high speed and then returns will age less than his (or her) stationary twin.

Now if we¬†consider a¬† trip to Proxima Centauri (our nearest¬†neighbour) the transformations clearly show that if humans could bear the acceleration required (we can’t) and if we had the means to get to, say, 0.99c for most of the trip, that yes, the round-trip would¬†take over 8 years and no laws would be broken. However the travellers themselves will experience time 7 times slower (7.089 to be precise). Thus they will have aged less than 8 years. So, once they get home and back-calculate their actual personal speed, it will exceed all the live measurements.

This has bothered me endlessly. Although taken for granted in some sci-fi books (the Enders Game saga for example) this clear ‘breakage of the c-limit’ is not discussed openly anywhere.

Still uncertain why people were ignoring this, I read a lot (fun tomes like this one) learned more maths (Riemann rules!) and also started to look at the wider implications of the assertion.

On the one hand, the implications are not dramatic, because instant interstellar communication is still clearly excluded, but that whole issue of needing a 4 years flight to get to Proxima Centauri is just wrong. If we can get closer to c we can indeed go very far into the universe, although our life stories will be strangely punctuated, just as in the Ender books.

But what about the implications for the other big festering boil on the body of theories that is physics today – quantum theory?

Well, if one is bold enough to assert that it is only measurement that is kept below¬†c¬†and not ‘local reality’, then one can allow for infinite speed.

In this scenario, we are saying measurement is simply mapping reality through a sort of hyperbolic lense¬†such that infinity resembles a limit. Modelling space with hyperbolic geometry is really not as unreasonable as all that, I don’t know why we are so hung up on Euclid.

With infinite speed at our disposal, things get really interesting.

We get things like photons arriving at their¬†destination¬†the same tme they leave their source. Crazy of course… but is it?

Have we not heard physicists ask – how is it the photon ‘knows’ which slit is blocked in the famous double slit experiment? It knows because it was ¬†spread out in space all the way from it’s source to it’s final point of absorption.

If you hate infinities and want to stick with Lorentz, you can equally argue that, for the photon, going exactly at c, time would stand still. Either way, the photon feels like it is everywhere en route at once.

If the photon is indeed smeared out, it probably can interfere with itself. Furthermore, it is fitting that what we see is a ‘wave’ when we try to ‘measure’ this thing.

A wave pattern is the sort of thing I would expect to see when cross sectioning something spread in time and space.

Please tell me I’m wrong so I can get back to worrying about something useful. No, don’t tell me – show¬†me – please! ūüėČ

Pet Peeve of the Week: Starfield simulations are always wrong, and here’s why…

Ok, if you don’t know what a starfield simulation is, lets sort you now – look at the video below first.

Ok, for those of you without youtube, think then of the screen savers on early windows PC’s – you may recall the screensaver that makes it look like you are flying through space – this “stars flying by” visual is the thing I am talking about. If you are interested, you can presently download this screensaver here.

Now when this screen saver came out, I’ll admit I was still a bit of a nerd – with a thing for both astronomy and for computers, so I set out to make my own. What I learned along the way initially puzzled me then annoyed me and then made me give up in disgust.

Ok, so before I tell you the ‘big secret’ of what annoyed me so, take a look at this animation:

I think you’ll agree it’s quite good – yes the stars are not perhaps as pretty in their distribution as some of the pictures from the Hubble¬†(see below) but that is quite¬†forgiveable.

Despite the boring uniformity of the stars, I want to draw your attention to the complexity involved in creating this animation. Just ‘guessing’ the paths of the stars by having them start small, somewhere near the middle, and then gradually grow and swing to one of the edges will not do. I tried this, trust me, it looked crap.

No, it turns out the only way to make this look decent is to do the honest thing and create a virtual 3-d world and then place the stars in it, then fly the camera through the space and have the computer figure out the paths for all the stars. Sound tricky? Well it bloody well was in 1995 when I tried it, though I reckon it’s easier now. I used POV-Ray¬†to render hundreds of stills and then tried to create a loop to make an animated gif. It was only like 200-200 pixels and it took days to render but it eventually finished and looked – absolutely nothing like the windows screen saver.

You see, I made the school-boy error of distributing stars ‘realistically’ in my 3-D space – I put them proper distances apart, randomly, and I gave them¬†realistic ‘sizes’ (relative to the inter-star distance). Instantly I had my first problem. The stars were all too small to even be detected by the renderer. Ok, so it turns out stars don’t work like normal things, their apparent size is not due to their actual size but a combination of their brightness and their distance. Fine. So I had to make them far bigger so they could be seen (which is utterly wrong wrong wrong to my purist heart).

Ok, so now I had spots. Did we get that sense of flight? No.

The next issue was that you needed only a few stars to create a ‘busy frame’ (say 20 stars), but most of them were stupendously far away and would stubbornly refuse to budge. The only option was the put absolutely bazillions of stars in the field so that at least a few were nearby enough for you to ‘swoosh’ lavishly past. Of course, to get that many stars, the whole view has to be completely plastered with stars – to the point of being a plain white screen. So I had to do another fudge – I had to create a sort of ‘fog’ that filtered distant light. This meant the viewer would only see nearby stars. Wrong wrong wrong again!!!! I happen to know from my own space travels (on spaceship earth) that we can see rather far without trouble, and thus this fog effect is a terrible hash.

However, I was getting somewhere with the sim. It looked like dots moving now. They did not get any bigger as they got closer, but they did move faster and get brighter, due to the fog. But damn, all the ‘nice’ starfield sims did¬†have the stars actually getting bigger, so now I increased the size of the stars again – so big that the stars were¬†literally¬†only a few dozen¬†diameters¬†apart and hey presto, it now looked good.

binary stars

Stars are not happy bedfellows!

Now think about that Рthe stars were only a few dozen diameters apart. The earth is actually about one-hundred sun-diameters from the sun; so what we are talking about it a super dense space, rammed with stars. Wrong wrong wrong. Stars that close tend to get involved in all-out gravity war (see the picture!)

So it occurs to me that the nerdy folks who have a hand in creating those ‘nice looking’ simulations are probably aware of their dirty little crimes. These simulations are not simulations at all, they are but an ‘artist’s renditions‘. Now that is an insult of great proportion to any red-blooded computer programmer. All I can say is, you should have formatted that floppy when you realised what you were doing and moved on with your life. It’s too late now – I know your crimes and will not let you sleep easy tonight.

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Update, 2011…

Ok, I have that out of my system. The question is (it should be burning your lips): what does superfast space flight look like then?

To answer this well, you simply need to put more effort into the simulation – you need to consider the great¬†asymmetries¬†in the star distributions – think how small they really are, then think about their clusters, then spiral arms, then galaxies, then clusters of galaxies, then…

I have referenced this video before and I do it again unashamedly – take a look, because they have already done what I suggest…

I think the makers should get a Nobel prize.

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Update 2012…

Ah, I am not alone in my nerd-dom. Now you can fly around in a pretty darn impressive virtual universe and see for¬†yourself¬†how the stars really actaully fly past. Happily, the results are not at all like most starfield simulators. You can fly vast distances with the sky literally ‘unmoved’. It is only once you come near a star or star cluster that those few will move, and only when you are moving stupidly fast yourself (like 2 parsecs per second) in a dense part of a galaxy, will you get anything like the old Windows starfield effect. My inner nerd feels justified. You can run the simulator on your own PC, get it at:

http://en.spaceengine.org/

Or read about it at io9:

http://io9.com/5924776/new-simulation-is-as-close-to-traveling-through-space-as-it-gets

Amazingly, this has been in the works for some time – this video from the sim was uploaded in 2009 already, it gets to the starry stuff in the second half:

Fun Physics Questions: Does time flow in baby steps?

Question: is it possible time flows in little steps?

At some small scale, could it be, that time is simply a ‘symptom’ of a sequence of events, or states, that there is no actual time passage ‘between’ those states?

This scenario has interesting implications – it suggests life is a bit like a movie – a series of pictures on a strip of celluloid, or pages in a book, and like a book, while the story may unfold to you at whatever speed you read it, it does not matter how fast you read the story itself still has its own pace.

This doesn’t mean the book has to be pre-written, it can still unfold with utter unpredictability, the book is unfinished if you like – the¬†important¬†point is that we are stuck experiencing the passage of time at a rate determined internally – by the rate of chemical reactions in our brains. The drum beat of those reactions would feel the same no matter how fast or slow they seems to an outside observer. They could even be paused for a few minutes – we could not tell!

Now physicists studying energy balances of sub-atomic particles have seen that energy often seems to come in little chunks (the ‘quanta in’ quantum), and that can imply that time may also be chunky (maybe¬†Planck time?); alas, time chunking has contradictory implications –¬†contradictory¬†to common sense anyway- like infinite energy flux, not to mention infinite speeds, but hey if you can just get your head around some of the workarounds physicists have dreamed up (quantum tunnelling for example) everything’s all right again. I am personally highly suspicious of workarounds, and that is what I think they are!

Anyway, even if you try to get away from quantum weirdness, you get sucked back in – take for example this geometrical example. Consider the relative positions of three point objects (small particles?) moving freely in space: they could, for an instant, line up perfectly, but if your measurement were infinitely accurate, this could only occur for an infinitely small duration so long as the particles are moving. If you try to explain this by saying¬†space¬†is divided up into chunks (like ‘snap to grid’ in MS Powerpoint) you get into geometrical issues that three points cannot always be¬†integer¬†increments apart ¬†(nor even rational increments apart) without breaking the most basic number axioms.

So even if space isn’t chunked, it turns out you can appeal to¬†the uncertainty principle, which handily says you can only measure the position of anything infinitely accurately if you allow its momentum to be anything at all, including infinite – and infinite momentum is exactly what you (temporarily) need if you are bold enough to let time ‘leap’.

So none of these issues with time chunking turn out as solid proofs against the possibility, they just make things more slippery!

Aside: rather than a book, I like to think of our universe as being a bit like a computer program ¬†– I like to think about Pac-man when it plays itself in ‘demo mode’ – in demo mode, used to allure people at the arcade, the computer controls both the ghosts and pac-man. In the computer, a sequence of commands is run in the CPU and the speed of the computer¬†(like the reader of the book)¬†controls the rate at which we ‘see’ the ghost-chase on the screen, but this speed is invisible to pac-man himself – yes the ghosts chase faster across the screen, but he can run faster too.

Question: Does a time-increment universe allow time travel?

Well I don’t think we can ‘skip’ events out (we have to experience them all), but if we can go somewhere where events are more or less ‘dense’, maybe we can. We will not feel the difference, we will not get any extra life-span, our cells will age just the same – but if a friend had gone to another place is space-time, where events have bigger gaps, he may have aged at a different rate, and when you meet your friend again one of you will have time travelled forward and the other backward relative to one another.

Is this really possible? Well, yes, I think so – this model ties in very well with relativistic time travel: if you assume events are more spaced out (less dense, with bigger ‘leaps’ between them) in areas with more mass nearby. or when moving vary fast, it maps perfectly.

Conclusion

That’s it for now! Of course, maybe time does not leap, I don’t know, but its something I love to think about! Please let me know your thoughts…

Death by KPI. The unintelligent design of the modern company… and what to do about it.

KPI!

A beginner’s guide to KPIs…

Background, Context and all that…

Some of you luckier readers will be wondering what on earth a KPI is. Alas, many of my readers will know, and rarely will they have a happy tale to tell about them. Let me tell you mine.

You see, to me the story of the KPI is none other than the story of the modern trend to remove the human element (that most fallible of elements) from big business. I propose that there has crept up upon us, starting when we came down from the trees and now coming to its final fruition with the industrial revolution, a situation in which the workings of our society, our organisations, governments, armies or companies, are simply too complicated to be designed or managed by any one person, whatever force his or her personality might possess.

No, the time of the one big man, the head honcho, the brain, scheming away in his tower, is over – and we enter the era, nay the epoch, of the human hive.

For now we see that in order to achieve great things, it is the ability to sort and organise mankind, rather than the ability of each man on his own, that matters most.

I could wander from my thesis awhile to describe a few side effects, for example, to point out the age of ‘middle management’ is here to stay, or to¬†philosophize¬†about the world where no one is actually ‘driving’ and the species is wandering like a planchette on a Ouija board and how this explains why no one seems to be able to steer us away from the global warming cliff just ahead…

But no, I will not be pulled off course, I will return to our good friend, the KPI.

So it seems we have these complex organisms, such as the venerable institution that is the company, that have evolved to survive in the ecosystem that is our global economy. Money is the blood, and people, I fancy, are the cells. And just as no particular brain cell commands us, no particular person commands a company. Just as the body divides labour among cells, so does the company among its staff. We train young stem cells into muscle, tendon and nerve. We set great troops of workers to construct fabulous machines to carry our loads just as our own cells crystallize calcium to make our bones.

Having set the scene thus I must move on, for where are the KPIs in all this?

As clever as the cells of our bodies may be, to choose depending on the whims of circumstance to turn to skin or liver or fat, that is but nothing compared with the cleverness with which the cells are orchestrated to make a cohesive body, with purpose and aim,  with hopes and dreams, and of course sometimes even the means to achieve them. And the question is: how is that orchestration achieved?

Being a devout evolutionist, its is clear to me that it was by no design, but rather by the constant failure of all other permutations that led to the fabulously clever arrangement, and so it is with the organism that is the company.

It is this conviction that leads me to claim, contrary to the preaching of many business schools, that good companies are not designed, but evolve, and by a process of largely unconscious selection. Like bacteria on a petri dish, companies live or die on their choices, and by every succeeding generation, the intelligence of these choices is embroidered into the DNA of the company.

Yes, I am saying that the CEO of Microsoft, or Rio Tinto of Pfizer, is no more sure of his company’s recipe for success than any one cell in your brain is at understanding how it came to be that you can read these words. Ok, maybe that is unfair. They probably have a good shot at reproducing success in other companies, but they only understand how the¬†company¬†works, not why.

It is well to remember that very few of the innovations present in a modern successful company were developed within that company – the system of raising money from banks or through a system of stocks and shares, the idea of limiting liability to make these investments more palatable,¬†of development the of modern contract of employment to furnish staff; nay the very idea that a group of people can actually get together and create the legal entity that is the ‘company’ has been developed over centuries.

Even within the typical office we see many innovations essential to run a business that could never have been ‘designed’ better by single mind – the systems to divide labour into departments – finance, marketing, sales, R&D, logistics, customer service; the reporting hierarchies and methods for making decisions; the new employee checklists, the succession plans, the new product stage-gate system, the call-report database, the annual budget, the balance sheet, the P&L – these are all evolved and refined tools that incorporate generations of brain power.

Whether it is the idea of share options or the idea of carbon copies, the list of machinery is endless and forms the unwritten DNA of the modern company. The company of today has little in common with the farmstead of 300 years ago, and indeed, just like the bullet train, it would not work too well if taken back in time 300 years. It only works in its present setting. It is part of a system – an ecosystem.

Now a business tool that has been evolving for some time, slowly morphing to its full and terrible perfection is the Key Performance Indicator or KPI.

The Key Performance Indicator

There is a saying in engineering circles: you cannot improve what you do not measure.

This philosophy accidentally leaked to the business community, probably at Harvard, which seems a great place to monetize wisdom, and so there is presently a fever of ‘measurement’ keeping middle managers in their jobs, and consultants on their yachts.

This sounds reasonable – but let’s pick at it a little.

When the engineer installs a sensor in a reactor to measure its pressure, it is usually just one element in a holistic system of feedback loops that use the measurement in real-time to control inputs to that reactor. Thus we can see that measurements in themselves are not enough, there needs to be an action that is taken that affects the measured property – a feedback loop.

Likewise any action taken in a complex system will tend to have multiple effects and while you may lower the pressure in a reactor, you may raise it somewhere else, thus the consequences of the action need to be understood.

And lastly, if the pressure in the reactor is now right, the value of the knowledge has diminished, and further action will be of no further benefit.

So just like a body, or a machine, a company needs to be measured in order to be controlled and improved, and it occurs to me that these key measurements, the KPIs, need to fulfil a similar set of requirements if they are to be of value.

A few years ago I decided to write a list of KPI must-haves, which I present here:

  1. the property measured must be (or correlate with) a company aim (eg profitability)
  2. measurements need to taken to where they can be interpreted and acted upon, using actions with predictable effects, creating a closed loop
  3. the secondary effects of that action need to be considered
  4. repeat only if the benefits repeat too

Now let’s take a look at some popular KPIs and see if they conform to these requirements.

Production KPIs

There are a multitude of KPIs used the ‘shop floor’ of any¬†enterprise, be it a toothpick factory, a bus company, a florist or a newspaper press. Some will relate to machines – up-time metrics, % on time, energy usage, product yield, shelf life, stock turns – indeed far too many to cover here, so I will pick one of my favourites.

Availability”¬†is a percentage measure of the fraction of time your machine (or factory or employee) is able to ‘produce’ or function. If you have a goose that lays golden eggs, then it is clear that ‘egg laying¬†activity’ will correlate with profits, so point #1 above is satisfied, and measuring the egg laying frequency is potentially worth doing. If you then discover that your goose does not lay eggs during football matches you may consider methods to treat this, such as cancelling the cable subscription. A few trials can be performed to determine if the interference is effective, proving #2. Now all you need to do is worry about point #3; will the goose fly away in disgust at the new policy?

What about #4? Indeed you may get no further benefit if you continue to painstakingly record every laying event ad infinitum. KPIs cost time and money Рthey need to keep paying their way and may often be best as one-off measures; however, what if your goose starts to use the x-box?

So that is an example of when the KPI ‘availability’ may be worth monitoring, as observing trends may highlight causes for problems and allow future intervention. So surely availability of equipment is a must-have for every business!?

No. There are many times when plant availability is a pointless measure. Consider for example an oversized machine that can produce a year’s supply in 8 minutes flat. So long as it is available for 8 minutes each year, it does not matter much if it is available 360 days or 365 days. Likewise, if you cannot supply the machine with raw materials, or cannot sell all the product it makes, it will have forced idle time and availability is suddenly unimportant.

The simplest way to narrow down which machines will benefit from an availability or capacity KPI is to ask: is the machine is a bottleneck?.

For other types of issues, let’s look at some other KPIs.

Quality KPIs

Quality KPIs are interesting. Clearly, it is preferable to ship good products and have happy customers. Or is it?

In any production process, or indeed in any service industry, mistakes will be made. Food will spoil, packaging will tear, bits will be left out. It is now fashionable to practice a slew of systems designed to minimise these effects: to detect errors when they are made, to re-check products before they are shipped, to collect and collate customer complaints and to feed all this info back into an ever tightening feedback loop called ‘continuous improvement‘.

KPIs are core to this process and indeed KPIs were being used in these systems long before the acronym KPI became¬†de rigueur. To the quality community, a KPI is simply a statistic which requires¬†optimization. The word ‘Key’ in KPI not only suggests it represents a ‘distillation’ of other numerous and complex statistics, but implies that the optimising of this particular number would ‘unlock’ the door to a complex improvement.

Thus, a complex system is reduced to a few numbers, and if we can improve those numbers, then all will be well. This allows one to sleep at night without suffering ¬†nightmares inspired by the complexity of one’s job.

The reject rate is a common quality KPI Рit may encompass many reasons for rejection, but is a simple number or percentage. It is clearly good to minimize rejects (requirement #1) and observing when reject rates rise may help direct investigations into the cause thereof satisfying requirement #2.

However, from rule #2 we see that this KPI is only worth measuring if there will be follow-up: analysis and corrective action. This must not be taken for granted. I have visited many plants that monitor reject and when asked why, they report that head office wants to know. What a shame. Perhaps head office will react by closing that factory some time soon.

The failure to use KPIs for what they are intended is perhaps their most common failure.

Another quality KPI is the complaint rate. Again, we make the assumption that complaints are bad, and so if we wish to reduce these we should monitor them.

Hold the boat. How does the complaint rate fit in with company aims? We already know that mistakes happen, but eliminating quality issues is a game of diminishing returns, so rather than doggedly aiming for ‘zero defect’ we need to determine what complaint rate really is acceptable.

So here is another common KPI trap. Some KPIs are impossible to perfect, and it is a mistake to set the target at perfection. Think of your local train service. Is it really possible for a train system to run on time, all the time? The answer is an emphatic no!

The number of uncontrolled inputs into a public transit system – the weather, the passengers, strike actions, power outages and the like will all cause delays, and while train systems can allow more buffer time between scheduled stops to cater to such issues, this type of action actually dilutes other aspects of service quality (journey frequency and duration). Add to that finally the fact that a train cannot run early so losses cannot be recovered.

The transport company will of course work to prevent delays before they occur, and lay on contingency plans (spare trains) to reduce impacts, but the costs and practicality mean that any real and meaningful approach needs to accept a certain amount of delays will be inevitable. A train company could spend their entire annual profit into punctuality and they would still fall short of perfection.

So it is with most quality issues,¬†the law of diminishing returns¬†is the law of the land. Thus the real challenge is to determine at what point quality and service issues actually start to have an impact on sales and cashflow. This is another common pitfall of the KPI…

The correlation between the KPI and profitability is rarely a simple positive one, especially at the limits.

Some companies get no complaints. Is this good? No, often it is not! This company may be spending too much money on QC. The solution here is to work with and understand the customer Рwhat issues would they tolerate and how often? If you did lose some customers by cutting quality, would the financial impact be greater than the savings?

However, and on the other hand, don’t make the opposite mistake: once a reputation for poor quality is earned, it is nearly impossible to shake.

Financial KPIs

As companies become bigger, there is a tendency to divide tasks according to specialized skill and training. Thus it can happen that the management of a big mining company may never set foot in a mine, may not know what their minerals look like, nor may they know how to actually dig them up or how to make them into anything useful. In other words, they would be useless team members after a nuclear apocalypse.

However, this is no different from our brain which is little use at growing hair, digesting fat or kicking footballs.

It is thus necessary that the organism (the company) develops a system to map in a flow of information from its body to its mind and then another mapping to take the decisions made in that mind and distribute them to the required points of action.

Indeed to blast in a quarry, or to kick the football are indeed best done by organs trained and capable and no less important than the remote commanders in the sequence of events.

And just as our bodies have nerves to transmit information about the position of our lips and the temperature of the tea to our mind, so the company has memos, telephones and meetings. And KPIs are the nouns and subjects in the language used.

Furthermore, some KPIs need to be further distilled and translated from the language of the engineer (Cpk), the quality manager (reject rate) or the plant manager (units shipped) to that of the accountant (revenue) , the controller (gross margin) and eventually that of the general manager and the shareholder (ROI). This is perhaps the main duty of middle management, bless their cotton socks.

Unfortunately, the mapping of everyday activities to financial KPIs is fraught with danger. The biggest concern comes from the multiple translation issue. That is to say, KPIs can suffer from a case of Chinese whispers, losing their true meaning along the way, resulting in the worse result: a perverse incentive.

Yes, ladies and gentlemen, this does happen.

Let’s say you want to improve your cash situation. You may choose to change the terms in your sales contracts for faster payment, in essence reducing the credit you allow your customers. This may have the desired effect, lowering the KPI ¬†called “receivables” and this looks good on the balance sheet – but let’s look at requirement #3 in the KPI “must have” list. What are the ripple effects of this move? It is¬†clear¬†this will not suit some of your customers, who, considering recent economic trends, probably also want to improve their cash situation; thus you may lose customers to a competitor willing to offer better terms.

And so we see the clear reason for perverse incentives is the consideration of KPIs individually instead of collectively. There has to be a hierarchy upon which to play KPIs against one another. Is revenue more important than margin? Is on-time shipping a part-load preferable to shipping “in full” a little late?

So we see again that the systems used to distill company indicators the choice of which decisions are centralized and which are localized need to be developed and constantly refined using an iterative process. The art of translating the will of shareholders into a charter or mission statement and then translating that into targets for sales, service and sustainability is a task far too complex to perfect at first attempt.

KPI Epic Fails

While on the subject of KPIs I cannot resist the opportunity to bring to mind a few fun examples of KPIs gone badly wrong.

The Great Hanoi Rat Massacre

The French¬†administration¬†in Hanoi (Vietnam) were very troubled by the rat population in Hanoi around the start of the last century, and knowing as they did about rat’s implication in the transmission of the plague, set about to control the population. A simple KPI was set – “number killed” and payments were made to the killers on this basis. There was immediate success with rats being brought in by the thousand and then the tens of thousand per day. The administration was pleased though somewhat surprised by the sheer number. There surprise gradually¬†transformed¬†into¬†disbelief as time wore on and the numbers failed to recede.

You guessed it. The innovative residents of Hanoi had started to breed rats.

The Magic Disappearing Waiting List

Here’s a more recent example from the National Health Service in the UK (the NHS).

A health service is not there to make a profit, it is there to help the population, to repair limbs, to ease suffering, to improve the length and quality of life Рand to do this as best it can on a finite budget. So the decisions on where to invest are made with painstaking care Рand needless to say, KPIs are involved. Not only big picture KPIs like life-expectancy, or cancer 5-year survival rates, but also on service aspects, such as operation or consultation waiting times.

It will therefore not be surprising to you to learn that the NHS middle management started to measure waiting times and develop incentives to bring these down, or even eliminate them. This sounds very reasonable, does it not?

Now ask yourself, how do you measure a waiting time? Say a surgery offers 30 minute slots – you may drop in and wait for a vacant slot, but as the wait may exceed a few hours it is just as well to book a slot some time in the future and come back then. So one way to measure waiting times is to measure the mean time between the call and the appointment. This of course neglects to capture the fact that some patients do no actually mind the wait and indeed may choose an appointment in two weeks time for their own convenience rather than due to a lack of available slots. Lets put that fatal weakness aside for the minute as I have not yet got to the amusing part.

After measurements had been made for some time, and much media attention had been paid to waiting times, the thumb-screws were turned and surgeries were being incentivized to cut down the times, with the assumption they would work longer hours, or perhaps create clever ‘drop-in’ hours each morning or similar.

Pretty soon however, the results started coming in, the waiting times as some surgeries were plummeting! Terrific news! How were they doing it?

Simple: they simply refused to take future appointments. They had told their patients: call each morning, and the first callers will get the slots for that day. This new system meant nobody officially waited more than a day. Brilliant! Of course it is doubtful the patients all felt that way.

How The Crime Went Up When It Went Down

If you work for the police, you will be painfully aware that measuring crime is difficult. And so it is with the measurement of many ‘bad’ things – for example medical misdiagnoses or industrial safety incidents.

Let’s look at workplace safety; while it may be fairly easy to count how many of your staff have been seriously injured at work, it is much harder to record faithfully the less serious safety incidents – or more¬†specifically, the ones that might have been serious, but for reasons of sheer luck, were not. The so-called ‘near-misses’.

Now to the problem. Let us say you are a fork-lift driver in a warehouse and one day, it a moment of inattention, you knock over a tower of heavy crates. Luckily, no-one was around and more luckily, no damage was done. So what do you do? Do you immediately go to the bad-tempered foreman with whom you do not get along and tell him you nearly killed someone and worse, nearly caused him a lot of extra work? Or do you carefully stack the crates again and go home for dinner?

The police suffer a similar predicament. The reporting of a crime is often the last thing someone wants to do, especially if they are the criminal. Now let’s say you are an enterprising young administrator just starting out in the honourable role as a crime analyst at the Met. You want to tackle crime statistics in order to ensure the most efficient allocation of funds to the challenges most deserving thereof. Do bobbies on the beat pay for themselves with proportionately reduced crime? Does the ‘no broken windows‘ policy really work? Does the fear of capital punishment really burn hot in the mind of someone bent on murderous revenge? Such are the important questions you would wish to answer and you have a budget to tackle it. What do you do?

You set out to gather statistics of course, and then to develop those tricky little things, the KPIs.

Now let’s say a few years pass, and after some success, you are promoted a few times and your budget is increased. Yay! You have always wished for more money to get more accurate data! Another few months later and the news editors are aghast with the force. Crime is up! Blame the police – no, blame to left – no blame the right! Blame the media! It’s video games – no, it’s the school system!

No, actually it’s a change in the baseline. ¬†The number of crimes recorded most likely went up because the effort in recording them went up. The crime rate itself may well have gone down.

The opposite can also happen. Say you run a coal mine and you will be given a huge bonus if you can get through the year with a certain level of near misses. Will you really pressure your team to report every little thing? I think not.

So the lesson here is: watch out for a KPI where you want the number to go one way to achieve your longer term aims, but where the number will also depend on measurement effort.

The Profit Myth

Most KPIs are dead dull. The very mention of KPIs will elicit groans and be followed swiftly by a short nap. The ‘volume of sales’ KPI is no different. The issue with the volume KPI is probably made worse by the clear fact that actually thinking about KPIs is a strong sedative. Surely selling more is good? Well, if you can fight through the fog of apathy, and actually think about this for a second, it is easy to see this is often not so.

To see why, it is important to understand¬†price elasticity. It if often true that lowering price will increase sales, so an easy way to achieve a target ‘volume’ (number) of sales is to drop price. That way to can make your volume KPI look good, as well as your revenue KPI, and so long as there is still a positive margin on all the units sold your earnings KPI (aka profit!) will see upside. There can’t possibly be any downside, can there?

Of course, astute financial types can find this fault easily, and perhaps the question of ‘how’ would make a good one for a job interview. It turns out more profit is not always good (seriously!). Whether more profit is really worth taking depends on the ratio of the increased earnings to the increased investment that is needed to make them.

There is another KPI I mentioned earlier the “ROI” or¬†return on investment. When I discussed it I implied that it was only of interest at the GM level, but really the reason only the GM tends to see this KPI is because it is difficult to calculate, and often only the GM has the clout to get it –¬†but it should be considered by all. To me, it is the king of KPIs for a publicly listed company. And it turns out the ROI¬†may actually go down with increasing profit.

If making more widgets requires no further investment, then the maths is easy, but that is rarely true.

The question is this: is it better to have a large ‘average’ business or a smaller one with higher profit margins? It turns out, from an investor’s perspective, that the latter is fundamentally preferable.

The ROI treats a business a bit like a bank account: asking what interest rate does it offer? The business should be run to give the highest interest rate‘(%), not the highest interest ($). It is always possible to get more interest from a bank account – just put more money in the bank.

Translating a Mission Statement into company KPIs

I mentioned above that the ROI is a pretty darn good KPI – so can we use it alone? Of course not. Recall that the KPIs are mere numbers we measure that try to tell us how we are doing against the company mission statement, and while the company mission statement may unashamedly describe vast profits as a goal, this is almost universally not the whole story.

The organism that is the modern company has one particular need besides profit today, and that is profit tomorrow. The ROI does not capture this need, so more KPIs are required, and trickier ones Рones that capture sustainability, morale, innovation and reputation (brand value). This is the turf of the mission statement.

Even though the more cynical of my readers will know the mission is usually ‘make lots of dosh’, and anything beyond that is window dressing, I would venture that the mission statement is the first step to figuring out which KPIs to put first.

Let’s dissect a few examples.¬†In my own 2-minute analysis I decided they fall into four types:

  1. To appeal to employees:
    McGraw Hill:
    We are dedicated to creating a workplace that respects and values people from diverse backgrounds and enables all employees to do their best work. It is an inclusive environment where the unique combination of talents, experiences, and perspectives of each employee makes our business success possible. Respecting the individual means ensuring that the workplace is free of discrimination and harassment. Our commitment to equal employment and diversity is a global one as we serve customers and employ people around the world. We see it as a business imperative that is essential to thriving in a competitive global marketplace.
  2. To appeal to customers (aka the unabashed PR stunt)
    A recent one from BP:
    In all our activities we seek to display some unchanging, fundamental qualities – integrity, honest dealing, treating everyone with respect and dignity, striving for mutual advantage and contributing to human progress.
    I couldn’t leave this one out from Mattel:
    Mattel makes a difference in the global community by effectively serving children in need . Partnering with charitable organizations dedicated to directly serving children, Mattel creates joy through the Mattel Children’s Foundation, product donations, grant making and the work of employee volunteers. We also enrich the lives of Mattel employees by identifying diverse volunteer opportunities and supporting their personal contributions through the matching gifts program.
  3. To appeal to investors. This is usually a description of how they are different or what they will do differently in order to achieve big dosh.
    CVS:
    We will be the easiest pharmacy retailer for customers to use.
    Walt Disney:
    The mission of The Walt Disney Company is to be one of the world’s leading producers and providers of entertainment and information. Using our portfolio of brands to differentiate our content, services and consumer products, we seek to develop the most creative, innovative and profitable entertainment experiences and related products in the world.
  4. For the sake of it – some companies clearly just made one up because they thought they had to, and obviously bought a book on writing mission statements:
    American Standard’s mission is to “Be the best in the eyes of our customers, employees and shareholders.”

Now a great trick when analysing the statement of any politician, and thus any mission statement, is to see if a statement of the opposite is absurd. In other words, if a politician says “I want better schools”, the opposite would be that he or she wants worse schools, which is clearly absurd. Thus the original statement has no real content, it is merely a statement of what everyone would want, including the politician’s competitors. Thus to judge a politician, or a mission statement, it is important to look not at what they say, but at what they say differently¬†from the rest.

Mission statements seem rather prone to falling into the trap of stating the blindingly obvious, and as a result become trivial, defeating the point. Such is the case with American Standard. Of course you want to be the best. And of course it is your customers, employees and shareholders who you want to convince. Well no kidding!

So discounting those, we can see that a good mission statement will focus on difference.¬†If we look at CVS, their mission is to be easy to use. This may seem like a statement of the obvious, but I don’t think it is – because they have identified a strategy they think will get them market share. Now they can design KPIs to measure ease of use. This is the sort of thinking that led to innovations like the ‘drive-thru’ pharmacy.

If we look at Disney, you can go further. “…[To] be one of the world’s leading producers and providers of entertainment…”¬†OK, so they admit being #1 is unrealistic, and if you want to be taken seriously, you need to be realistic. But if you are one of many, how do you shine? “Using our portfolio of brands to differentiate”¬†They realise they can sell a bit of plastic shaped like a mouse for a lot more than anyone else can. There is a hidden nod to the importance of brand protection. So KPIs for market share and brand awareness fall right out. They finish off with “the most creative, innovative and profitable entertainment” Well you can’t blame them for that.

Very rarely, you see a mission statement that not only shows how the company intends to make money, but may also inspire and make pretty decent PR. I like this one from ADM:

To unlock the potential of nature to improve the quality of life.

I have no idea how to get a KPI from that though!

Summary

In this article I have tried to illustrate how measuring a KPI is much like taking the pulse of a body – it’s a one-off health check, yes, but more importantly it can be a longer term measurement of how your interventions are affecting company fitness in the longer term.

I also try to describe some common pitfalls in the use of KPIs and presented four simple tests of their value:

  1. the KPI must correlate to a company’s mission
  2. the KPI must form part of a corrective feedback loop
  3. perverse incentives can be avoided by never considering any single KPI in isolation
  4. repeat the treatment only if the benefits repeat too

I have personally used this checklist (with a few refinements) over the years to some good effect in my own industry (minerals & materials) and though I am confident many of my readers will have more refined methods, I live in hope that at least one idea here will of benefit to you.

Stuff I Wish I Had Read When I Was Younger

Over the years I have supervised and mentored several PhD students, and recently our firm started to award scholarships to undergrads, and I was asked to support one such scholar. These scholars are from the best and brightest and so I got to thinking…

Graduates today have it tough, competition is tough, people work longer and harder than ever and stress is hitting us earlier and earlier in life – or so it seems. I would argue that, to some real extent, things have always been getting worse, and therefore by induction, we can prove that they have haven’t really changed at all.

No, the graduates of today have unparalleled opportunity to learn, to travel and to experience. The brightest graduates have the world at their feet and will be its commanders when we are are all retired and done for.

So what could I do to support this scholar? In the end it was easy – I asked myself – what do I know now that I wish I had known sooner? Most of this is in attitudes and is deep in my psychology, and is the result of direct experience – but it turns out that a healthy chunk of my scientific learning experience can be re-lived – by reading some of the books I think steered my course.

So I made a point to summarize some of the best science related books I have read (and some of the most useful internet resources I have found), and dumped the list complete with hyper-links in an email to the scholar. I hope she goes on to be president!

Now having gone to the effort, it would be a crime to keep this email secret, so here it is, (almost) verbatim!

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As promised, here is a list of useful resources I wish I had known about when I was an undergrad. I am glad I got round to this, it should be useful for several other students I work with, and has also led to me revisiting a few things! I think I may brush it up and pop in on my blog if you don’t mind…obviously I won’t mention you!
Anyway, back to the business. To me, science is not all about chemistry, molecules, atoms, valence electrons and so on. To me, is is the process of trying to understand the world, and this set of materials I have hand picked, should you get through even a part of it, will not only educate but inspire.

This may not be the very best list, and I am sure there are many great books I have not read, but I have stuck with ones that I have, so you will have to rely on other people for further recommendations.

Jarrod’s reading list: science/psychology/economics & so on

  • I’ll start with something really easy, relevant and engaging – an excellent (if quirky) summary of material science:¬†The New Science of Strong Materials – Prof Gordon ¬†has written another on Structures that is also worth reading.
  • Ok, this next one is not a book, but a paper; I like it because it shows that many stuffy professors are wrong when they prescribe boring scientific prose for papers. This paper uses the criminal “us” and “we” and discusses subjects as if with a friend. Shocking form, especially for a junior scientist. This paper by an unknown, changed the world.
  • Guns, Germs and Steel” – this is large-scale scientific thinking at its best- the book looks at how we can explain why the world is the way it is (especially the inequality) by looking at how technology spreads through societies.
  • Mistakes were made…but not by me” – this is required reading if you want to work with other people, so its basically for everyone then…
  • Then to take it to the next level – “How the mind works…” – Stephen Pinker‘s other books are also good if you like this one.
  • “Flatland”, (full text here) was written in 1884, and is essential reading because it defines the cliche “thinking outside of the box”.
  • To make your upcoming economics courses more interesting, first read this easy-to-read popular book: “The Undercover Economist“.
  • Also, Freakonomics– it’s shameless self promotion by egotistical authors, but hell they are smart, so put up with it.
  • The Tipping Point – ¬†Malcolm Gladwell is a current thinker I really like; he’s not satisfied to focus on one thing for very long – his other books are on totally different stuff, but are equally thought provoking.
  • The selfish gene” – Obviously I would firstly recommend “On the Origin of Species”, (full text here) but if you are short of time (which you should be as an undergrad), you can learn most of the basics, and also get updated (well up to the 1970’s at any rate) by reading Dawkins’ classic.
  • I couldn’t ignore statistics, so I will include two – one classic, “How to Lie with Statistics”¬† and a more modern one “Reckoning with Risk“, they are quite different, but either will get the important points across.

Alas, books are perhaps becoming obsolete, so I better include some other media:-

  • The first one is so good I can’t believe its free – try watch at least one a week, but the odd binge is essential too. http://www.ted.com/
  • Next, an excellent physics recap (or primer) – but ¬†you need lots of time (or a long commute!) to get through this lot – look on the left menu for Podacts/Webcasts on this webpage: http://muller.lbl.gov/teaching/physics10/pffp.html – I cannot begin to praise the worthwhileness of this enough. It used to be called “Physics for future presidents” because it teaches you enough to understand the risks of nuclear energy, and the likelihood that we will all run our cars on water – and let you know when you are being duped or dazzled by big words.
  • When I was somewhat younger there was a TV show called Cosmos, hosted by Carl Sagan, you may know of it. You could watch in now here, though obviously it is dated, so perhaps you shouldn’t; the reason I mention it, is because it was key in creating a generation of scientists, people who were inspired by Carl to be inspired by the universe. The previous generation had the space race and the moon landings to inspire them, but since then science has been on a downhill, with 3-mile island, global warming, etc, etc, and we have had no more Carl Sagans to cheer for us; Cosmos was a rare bit of resistance in the decline of the importance of science in society. You may also know that there have been battles in society (well in the circles on intelligentsia at any rate) about science – on the one had the ‘two cultures debate‘¬†and more recently, the ‘anti-science’ movement (suggested in books like “The Republican War on Science“. I do not wish to indoctrinate you, but rather make you aware that being a scientist used to be cooler and used to be more respected and something is indeed rotten in the state of Denmark.
  • Getting back on track, here is an excellent guide to critical thinking (something else sadly lacking in the world) – don’t read it, listen to the podcast versions (also on itunes):
    “A Magical Journey through the Land of Logical Fallacies” – Part 1 and Part 2
    I think this should be taught in school. Brian Dunning’s¬†other Skeptoid podcasts put these lessons into practice showing how a scientific approach can debunk an awful lot of the nonsense that is out there (alternative medicine, water dowsers, fortune tellers, ghost hunters, etc etc).
  • If you do happen to have any time left, which I doubt, there are several other podcasts on critical thinking – that use a scientific approach to look at the world and current affairs: –

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Postscipt – Dear readers, please feel free to append your own recommendations to my letter in the comments section below. If there is one thing I know well, and that’s how little I know. I feel I only started to read ‘the good stuff’ far too late in life, and so those with more years than me (or better mentors), please do share. But bear in mind, this is principally a science oriented list, and is meant to be accessible to undergraduates – I left out books like¬†Principia Mathematica (Newton) because it is really rather unreadable – and the Princeton Science Library (though awesome) is probably a bit too intense. Also,¬†in the 30 minutes since I sent the email, I have already thought of several others¬†I sort of, well, forgot:

That’s it for now…