Musical Notes Explained Simply

25 10 2011

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.

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  – a 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 – we 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





Bacteria don’t think.

8 06 2013

It seems a statement of the obvious, but bacteria don’t think.

Yet bacteria get around, and indeed are remarkably successful. Same with viruses. So thriving as a species does not require planning, studying, concentration and imagination, all the things we humans are so arrogant about.

And I’m not just talking about surviving, I’m talking about achieving the incredible. Think of the fungi that take control of ants, get them to climb up to a good spot and hunker down so that when the fungus bursts out from the corpse like a slow motion firework (see the picture) it’s got a fair chance of spreading its spores.

20130608-110232.jpg
Did the fungus plan it? Of course not, the trick ‘evolved’ as the most successful of many different permutations, via, of course the process of natural selection.

So what?

Well, what about humans? We like to think we are the pinnacle of evolution with our big brains and our consciousness and our self-awareness. Our abilities to plan, co-operate and imagine have led us to dominate the planet. Or have they?

Could it be, that just as no ant envisions the design of the anthill, none of us can claim to have masterminded very much? Yes perhaps a building, a harbour or a town’s zoning, but who can claim to have masterminded New York or world trade or democracy?

Surely these ‘real’ achievements are not ours to claim, but should also be laid at the door of the power of evolution, of the unstoppable force of trial and error, of the natural emergence of order from the chaos?

See more about the Cordyceps fungus:
http://www.bbc.co.uk/nature/life/Cordyceps





Requirements for Promoting a New Scientific Theory

25 04 2013

I have been reading some pretty strange stuff this week 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 – what 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 does 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…

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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:





Leveraging the Inventiveness in your Mind

15 02 2013

There are some tasks our brains find hard. We cannot remember long numbers or calculate square roots and we learn information at such a low rate, it takes a lifetime to fill up our hard drive/brain.

illusion

The impressive visual tools in our brains are fun to trip up.

We are fooled by simple magic tricks, our memories can change and we constantly lie to ourselves in order to avoid cognitive dissonance.

Yes, we are pretty awful, and it’s pretty amazing we manage to get through the day. The reason we do is that our brains were not designed to remember long numbers or to calculate square roots, we were designed to …get through the day.

Thus it’s no surprise that we can spot tigers hiding in the shrubbery, and judge someone’s intent from the curl in the corner of their mouth – things computers can’t even dream of!

evolution

Amazing Things the Brain Can Do

There are some really remarkable abilities the evolutionary arms race has given us. Consider for a moment how hard it is to teach these skills to a computer:

  • Facial recognition (from any angle!) – and similar advanced pattern recognition
  • Theory of mind – our ability to realize that others have motives and intentions and the ability to guess them reasonably well
  • Inventiveness – our ability to make connections from disparate fields

Much has been said about these skills, and in particular, much value has been placed on theories about our inventiveness – if only we can understand how we invent, we can unleash a torrent of innovation!

torrent_of_ideas

The ideas usually run something like this: the human mind is so highly integrated that many concepts are forced to overlay one another so connections are inevitable – while others suggest the mind reviews new learning each night during sleep and tries to spot patterns, suggesting our innovative spark is really just our pattern recognition skill in disguise [1].

While I suspect there is truth to both theories, there is probably more to it than that…

Another Amazing Skill Often Overlooked

Now – if you have ever caught a child being naughty, you may have been lucky enough to see another remarkable human talent…

Lying.

naughty-baby

Lying is tricky. Lying requires amazing computation – it needs theory of mind, it requires creativity, and does its invention under pressure.

Lying requires creating an entire alternate reality that fits the evidence but makes you look innocent of all crimes! It’s so hard that young kids don’t always get it quite right, but at some point most of us master the art. Our brains can also be switched to this mode of inventive overdrive in another way: when we attempt to explain incomplete data.

The most common opportunity to fit a narrative to incomplete data is when we recall faded memories – it turns out many of  us can bring out our internal Dr. Seuss when recounting our roles in past events.Dr-seuss-oh-the-thinks-you-can-think1

And because we all like to think of ourselves as pretty darn awesome, our memories cannot contain any information that could contradict this most evident truth. Thus when we recall situations when we did something downright shameful, our brains become positively electrified and we will magic up perfectly good reasons for what we did out of thin air.

Almost everyone can do it. However, if you ask us to write a short bit of utter fiction, our ability instantly vanishes.

writers_block

Leveraging Brain Power

So the question is this… how can we tap into these remarkable abilities? Do creative people already do it?

I, for one am going to try!

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[1] How the mind works – Steven Pinker





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

11 02 2013

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 π 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 π, it was not exact and they lost a lot of sleep trying to find a neat way to write π.

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 √3. 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 √3! So 30.5 is √3…

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. √3 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 √-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 √-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, π, in there and see what happens:

What is e?

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 – π, e and non-integer powers combine with i and the answer pops out as -1.

Blows my mind.

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





L’Aquila Earthquake: Science collides with the law…

22 10 2012

Today we heard that a group of seismologists have been jailed over the advice they gave prior to the 2009 quake in L’Aquila in Italy which killed over 300 people.

Is this good and right?

Well let’s start by imagining you lived in l’Aquila and a series of tremors had the town worried. A group of experts comes to town, and after a long meeting, come out and declare the risk is miminal. So rather than evacuating, you go home, rest assured, only to lose your family two days later.

I would want blood. I would want to lash out. My family could have been saved!

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So at first glance this judgement may seem like a last sad ripple from a tragic event, however, it isn’t. Upon further reflection, this judgement has serious implications for the relationship between science and the law.

I have always been troubled by how a guilty verdict means ‘you did it’ even if you didn’t, turning ‘beyond a reasonable doubt’ into no doubt at all. I have previously blogged about pragmatism in law, and how you cannot be 80% guilty and thus serve 80% of a sentence.

With such pressures on the law to come to a clear conclusion, it was only a matter of time before a court was asked to decide on whether scientific advice was correct…

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So today, when a court decided that scientists were ‘too reassuring’ to the public, my alarm bells were set ringing and still haven’t stopped.

Armed with the hindsight that an earthquake in fact did occur, it is natural that the families of the victims are appalled by the advice they got. However, in this case it is critical to also put yourself into the shoes of the scientist – in the days before the quake.

They must have honestly doubted a quake would happen. If you had expected a quake you would surely have said so!

In their statements they confirmed a quake was possible but unlikely. So what we have to ask is this: based on the data they had, was that conclusion faulty?

To answer that you only need to ask – do all tremors lead to quakes? Well no, most don’t. So the quake was by no means likely, let alone inevitable. They were not covering up. None in the group was suppressed or censored. I can only conclude that they, after years of chasing  tremors, had come to discount the value of tremors as indicators.

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And today, a judge decided that their advice was not only wrong, but criminally wrong.

The problem here is that science is not a PR exercise. If scientists put spin on data, they lose credibility. If they cry wolf, they lose credibility. The only safe way to do science is to stick to the facts. The facts did not indicate an imminent earthquake. The judge does not seem to realise: scientists cannot, and do not claim to be able to, predict earthquakes.

Perhaps the judge should read Taleb’s treatise on rare events, and he’d seen that just because something is unlikely (as the scientists said) does not protect us from the possibility it may still happen.

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So what we have seen today is this: scientists, giving their edified analysis, have been thrown in jail. The mob are satisfied, but you should be distinctly worried.

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See also: Letter from the AAAS to the Italian president





Global Warming Canaries, Anyone? A short intro to why most scientists are worried about climate change…

18 08 2012

The current state of arctic sea ice (see graph below) sends a chill down my spine.

Image

So what it says is that the ice is melting furiously, and looks like it’s not yet slowing down even though the days have started to draw in.

However, any scientist will tell you that no single data point can be used as evidence of global warming, there are simply too many fluctuations for anything to be concluded over anything but the longest timescales. We cannot simply look at the mean temperature for a hot year and say, there you go, global warming!

Now, the issue is, there are well-known cycles over pretty much all timescales – this pretty much undermines all serious attempts at prediction.

So, what to do? Well all is not lost; there are still some clever little leading indicators we can look at to give us that sobering wake up call.

#1:  CO2

Firstly, we know CO2 concentration is up, no doubt or argument, this can be seen in the famous Hawaii data above, complete with the seasonal ‘breathing’ by global plant-life. The argument is about whether the greenhouse models that say this will result in warming will turn out right. I honestly don’t know, but I wouldn’t even have to wonder if the CO2 levels weren’t going up, would I?

#2: A Record Breaking Rate of Record Breaking

Secondly, rather looking at averages or ‘new records’, we can look at the frequency of records. So rather than saying, “we just had the hottest summer ever in some parts of the US, there’s the proof” we can look at how often records are set all over the world – hottest, coldest, wettest, dryest and so on. This approach creates a filter; if it shows there are more records being broken on the hot side than the cold side, could this be an indicator? I hope not, because there are.

Again, it could be part of a long-term cycle that could bottom out any time now. But on the other hand, if it was going the other way, I wouldn’t have to hope, would I?

#3: Sea Ice

Now the sea ice. The sea ice is another proxy for temperature. The reason it’s interesting to climatologists is because it is a natural way to ‘sum-up’ the total warmth for the year and longer; if ice is reducing over several years, it means that there has been a net surplus of warmth.

CryoSat – The European Space Agency’s Sea Ice Monitoring satellite launched in 2010 – (Image credit ESA)

Today we are seeing a new record set for minimal northern sea ice. And not only is there less area of ice, but it is thinner than previously realized and some models now suggest we could be ice-free in late summer in my lifetime.

Now if that does not strike you cold, then I didn’t make myself clear. This is not some political posturing, not some ‘big-business’ spin, nor greeny fear mongering. It’s a cold clean fact you can interpret for yourself, and it could not be clearer.

So is it time to panic?

Well it can still be argued the melting is part of a cycle, it could of course reverse and hey, no biggy. After all, what does it matter how much ice there is?

Well, yet again, I hate to rely on the ‘hope’ that it’s a cycle. Because if it continues, the next effect will be felt much closer to home…

Sea Level

Sea level is the ultimate proxy for warming. Indeed, sea level change can be so serious, maybe it is the problem rather than the symptom. If the ice on Greenland and Antarctica melt, the rise in sea level would displace hundreds of millions of people and change the landscape so dramatically it’s a fair bet wars and famine will follow. Now that is serious.

So have we seen sea level rise? Well, yes. Here’s the plot:

Now, it looks pretty conclusive but hold the boat. Some say’s it’s proof of warming but not everyone agrees. It’s true it could again be a cycle. Also, the sea level rise is fairly gradual; what people are really arguing about is whether we should expect it to speed up. If temperature goes up a few degrees it could go up 5 or 10 times faster. The speed is the issue. Humanity can cope if the level goes up slowly enough, sure, countries like Tuvalu will be in big trouble either way, but countries like Bangladesh and cities like New York and London will only be in real trouble if the rate increases.

Actual Canaries

Canaries taken into mines in order to detect poisonous gases; the idea being they would suffer the gas faster than the people and if the canary dropped, it was time to vacate. Do we have systems that are hypersensitive to climate change?

Yes! There are many delicately balanced ecosystems that can can pushed over a tipping point with the lightest of touch. Is there an increase in the rate of species loss, or an increase in desertification? Yes!

We can also look at how far north certain plants can survive, how high up mountains trees can live or how early the first buds of spring arrive.

Again, these indicators fail to give solace. Everywhere we look we see changes, bleached coral, absent butterflies, retreating glaciers.

The conservative approach is to ascribe these changes to the usual cut and thrust of life on earth; some take solace from the fact that humankind has survived because we are the supreme adapters and that the loss of species is exactly how the stronger ones are selected.

Yes, we are great at adapting, however, to kill any complacency that may create, consider the following: for humans just ‘surviving’ is not the goal, that’s easy, we also need to minimize suffering and death, a much tougher aim. We’ve also just recently reduced our adaptability significantly by creating ‘countries’. Countries may seem innocuous, but they come with borders – and mean we can no longer migrate with the climate. Trade across border also needs to be of roughly the same value in both directions.  While some countries will actually see productivity benefits from global warming, most will not, and without the freedom to move, famine will result. Trade imbalances mean inequality will become extreme. The poorest will suffer the most.

So for now changes are happening, and advances in agricultural technology are easily coping; however, because ecosystems are often a fine balance between strong opposing forces, changes may be fast should one of the ropes snap.

Conclusion

Looking at the long history of the earth we have seen much hotter and much colder scenes. We have seen much higher and much lower sea levels. We are being wishful to assume we will stay as we have for the last 10,000 years. It may last, or it may change. Natural cycles could ruin us. And mankind is probably fraying the ropes by messing with CO2 levels.

Can we predict if we are about to fall off of our stable plateau? No, probably not. But is it possible? Heck yeah.

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If you liked this, you may like these earlier posts on the subject of global warming:

  1. What does the earth’s history tell us about climate? And how can we find out if our house will be one that sinks should sea levels rise? Find out here!
  2. Can we  change the planet’s dangerous behavior? Read my call for a study in mass behavior.




Clever Invention!

13 07 2012

Ever had a tedious job where you just know there’s gotta be an easier way? Well farmers who spend all day driving around switching on and off sections of their irrigation systems have long been frustrated by the shortcomings of remote control valves. Can you imagine little air lines or wires running back and forth all across the fields? Wouldn’t last a week!

Anyway, clearly the guys at ColtValve have obsessed about this and come up with an ingenious solution – check out the video – I pretty sure its not magic, but it might as well be! “Look mum no wires”.

There’s something so reaffirming about great ideas, something that reminds us of the power of human ingenuity, and the reminder that some idea of pure cunning may be coming to you at any moment…