Saturday, 10 September 2016

And we're off S383 First impressions

So the website for S383 opened this week and I got to look at the TMA's and the pdf of the course units.

S383 Unit 1 is a pretty good summary of General Relativity and the Schwarzschild metric and the Friedmann equations. It skimps on some details for example there is not a full blown derivation of the Friedmann equations from General relativity but enough is given so that you could in principle derive the results  for yourself (Good luck with that one I would estimate a fairly intensive month to get it exactly right if you don't die of boredom :) )

S383 Unit 2 is a summary of observational cosmology starting with the big bang and then an overview of modern ideas in cosmology centering around inflation. From what I can tell there is not a great deal of detail here.  However the author seems to think inflation is still quite speculative and wants to concentrate on more tangible things. The course then goes on to look at Galaxies, the evidence for black holes gravitational lensing and the Lyman alpha forest. All interesting and active areas of research and good to get an overview

S383 Unit 3 is called extreme environmental astrophysics and is concerned amongst other things with the formation of acretion discs near a black hole and many other things which I will cover in more detail as time goes on.

Looking at the TMAs that are available it has to be said that they don't really do justice to the depth of material covered in the course I dare say the electronic tmas will cover more topics. This is based only on 3 of the TMAs the TMA for the second block being an extended one and so not issued yet.

Ok so initial impressions are that this course covers a lot of interesting material but not necessarily at a mathematical depth that would satisfy more mathematically minded people. I guess one would have to look elsewhere for that.

One place to look is the Cambridge Part III courses examples sheets

http://www.damtp.cam.ac.uk/user/examples/indexP3.html

Scroll down to the bottom and you will find example sheets for three of the core courses in relativity that are offered. Also most generously Dr Baumann has published his lecture notes on Cosmology

http://www.damtp.cam.ac.uk/user/db275/Cosmology.pdf


More than enough to satisfy those wanting a more mathematical approach I hope to do at least some the examples associated with Cosmology and maybe some of the General Relativity example sheets alongside my work on S383.


As an aside I decided to leave MST326 for this year, I may as I have 2 years left do it in conjuction with the new MST327 course

http://www.open.ac.uk/courses/modules/ms327

next year and round off my second open degree with M303 the year after or vice versa Who knows







Monday, 29 August 2016

Decisions again MST326 0r Not MST326

As well as doing S383 I have a week to decide whether or not to do MST326 again
I hadn't realised that if you fail as I did you can retake the whole module again at some point and not get your marks downgraded

Pros

I have revised my interest in partial differential equations

If I get say grade 2 along with a grade 2 in M303 I could start the MSc in 2 years time

If I decide not to do it then I will have to wait 2 years before taking it again I can't see me doing this alongside M303 next year so its now or never

Cons

Money and time as always :)

It is a tough one

Speaking of Partial Differential equations I am currently trying to understand the solution of Schrodinger's equation in parabolic coordinates. I have seen it mentioned in a few books but the solutions are only sketched out. It turns out that it is possible to solve Schrodinger.s equation in 11 of the orthogonal coordinates but most courses just concentrate on the well known spherical polar coordinate system. It would be an interesting exercise over the next few years to see how feasible it is to solve Schrodinger's equation in all eleven coordinate systems.

https://en.wikipedia.org/wiki/Orthogonal_coordinates

and with the power of computer graphics plot out the solutions Watch this space (but don't hold your breath)

Sunday, 3 July 2016

Update S383 and M303

Hi sorry for not blogging for a while other priorities and so forth
Anyway I got bored with linear statistical modelling so couldn't motivate myself to complete it My track record for dropping out of OU courses isn't looking so good.
However I have decided to register for S383 Relativistic Universe

http://www.open.ac.uk/courses/modules/s383

which starts in September

I then intend to do M303 which I have been lucky enough to get the materials for

http://www.open.ac.uk/courses/modules/m303


I will then have enough credits to get my second Open university Open degree
After that who knows I believe it is possible to study modules separately rather than as part of a degree as I have no inclination to do a whole load of level 1 courses for the sake of a degree
I am tempted to essentially build up a portfolio of second level courses followed by a third level course

Topics I would like to study include Chemistry, Economics and Biology and I dare say completing the third level philosophy course would be useful as well. I think it would be quite satisfying to do a second level course followed by a third level course I'll see how things pan out after M303

In the mean time almost by coincidence there is a coursera course on Galois theory starting up looks quite intimidating but I have the OU M303 books to give me background in ring and field theory

https://www.coursera.org/learn/galois#syllabus

At any rate I shall get some idea of what the topic is all about.




Thursday, 3 December 2015

M346 Linear Statistical Modelling TMA01

So finished the first TMA01 for the linear statistical modelling course M346
This course is dull as dishwater but a necessary evil for me to complete my maths and statistics degree

The first question covered the usual stuff t tests, various distributions and chi squared and F tests and a bit of hypothesis testing.

All quite mind numbingly tedious you look up the formulae in Unit 1 and apply them

Question 2 covered how to use Gen Stat to test for normality and how to use the tools to calculate various probabilities

Question 3 covered linear regression the first a straightforward application of linear regression to a data set and the other requiring a transformation of the variables. You are taught how to interpret the linear regression output from GenStat

Really am going to die of boredom from this course but wont give up as I need the qualification

Not even sure what this is doing in a maths course as there are no derivations or proofs in the course

Addendum 16th DEC 

Got the tma back on Monday 4 marks short of distinction
lost marks on the transformation question as I didn't think of transforming the x variable and made a silly mistake in the linear regression question as I used the prediction of mean instead of future prediction. Still my tutor has provided a really useful summary of the correct answers

Probably heading for grade 2 if I don't die of boredom before I aim to do 1 questions from the TMA's per week, Might even finish the 2nd TMA over the Xmas holidays





Monday, 30 November 2015

Grade 6 result

So got the percentage mark for my Grade 6 theory result it was a respectable pass 72% but way short of a merit so glad to have passed but slightly disappointed the mark wasn't higher. I will know in 2 weeks just what the breakdown in marks was and round about the end of January ABRSM  will publish the 2014 papers along with the model answers. So I will be able to learn from the experience

Grade 7 next depending on how it goes I will try and do it in June but at least by November of Next year

Wednesday, 4 November 2015

Grade 6 Music Theory Debrief

Well yesterday I braved the gloomy Fog of Edinburgh and went to the darkest frontier of Sighthill to sit my grade 6 Music theory exam. I think it went reasonably well although there were one or two tricky bits. The exam consists of 5 sections and I had prepared myself by working through quite a few past papers upto 2014 over the past year or so. Also with the help of Victoria Williams my  music theory website and Udemy videos

http://www.mymusictheory.com/



https://www.udemy.com/u/vickywilliams/

A hint for those doing Udemy courses if you invest in one  then practically every week they will have course offers between £8 and £15 pounds. For grade 6 I did the Grade 6 melody course and the Figured bass course. Definitely worth investing in if you are contemplating doing the Grade 6 exam

So to the exam itself there are 5 questions

1 Given a melody can you work out an appropriate chord sequence or alternatively given the start of a melody with figured bass can you complete it.

I had got into the habit of doing the first question. To start with it is usual to start a melody with either an anacrusis leading onto the tonic chord or just usually the tonic chord so the first chord is either the tonic (Ia or ia) or if an anacrusis is used (Va-1a)

Usually the piece  ends with a perfect cadence (Va-Ia) if the melody ends with a descending scale figure ending on the tonic of the piece then it is possible to use the cadential 6-4 formula (Ic-Va-Ia)
So that is usually 3 or 4 chords that you can write down almost automatically.

The rest of the chords are worked out by first writing each possible chord that would fit (at this level there are only 3 that would work) and seeing if a logical sequence can be fitted in
 For example if in the middle of the piece 3 notes form a descending scale passage leading to the tonic then a good fit is a sequence in the opposite direction ascending upwards

Suppose we have (in the key of C major) the notes E D C then in the bass we would have C D E and the chords would be Ia -viiob - Ib or vice versa if the melody was ascending

Then it is usually possible to find a sequence which follows part of the circle of 5th's progression namely

I-IV-vii-iii-vi-ii-V-I

So to some extent this question is a bit like a Sudoku puzzle you have certain clues and you have to fill in the rest of the grid. There are certain rules which one must follow, Never have the bass line or any of the other lines move in parallel fifths or parallel octaves with each other, So if the soprano line is G and you have chosen to use the tonic chord C if the next note in the soprano line is A do not make the note in the bass line D (a ii chord) One way to avoid this as far as possible is to ensure for the most part that the bass line moves in contrary motion to the soprano line.

Also never double the third of a major triad and a few other rules all explained brilliantly by Victoria Williams in her notes, sticking to these and learning a few basic chord progressions, usually gets you through. For this question you don't have to fill in the actual chords but just give the letters. One thing I find helpful in these exercises is to fill in the bass line making sure it doesn't give rise to parallel fifths or octaves with the Soprano line. If I were doing a full harmonisation I would always write the bass line first. Then fill in the other notes. One thing that is slightly odd about the ABRSM exam is that you can only label the chords by their letters or fill in the notes, So if you do leave the bass line in you will get marked down, So you end up rubbing out the bass line. A bit pointless in my opinion

The actual question worked out OK I think but I was a bit repetitive using far more Tonic and Dominant chords than I would normally expect to and also repeating the Ia-viib-Ib chord progression twice, I'll wait and see what the examiner thinks,

The second question was figured bass a topic which can be quite confusing but again explained brilliantly by Victoria in her notes and video. This is even more like a Sudoku puzzle in that the figures on the bass line tell the arranger or accompanist the type of chord to use, So example if there are no figures underneath the chord then the chord is in root position, If the note has a 6 below it is in first inversion and finally if it has 6-4 then it is in second inversion

Thus if the bass note were C with no figures under it would mean a chord of C major in root position C - E -G. If a figure of 6 were below it would be an A minor chord (A-C-E) in first inversion i.e with the note C as the bass note. Finally if 6-4 were the figures under the note then it would be the F major chord (FAC) in second inversion. Also there may be an accidental included. So the first thing to do is to work out which notes the figures are telling you to use. After this is done write out a soprano line which is interesting again trying for contrary motion with the bass line as much as possible and again making sure that the third of a major triad is not used.

Then comes a slightly trickier part filling in the Alto and tenor parts making sure that there are no parallel fifths or octaves between all 4 of the other parts. The only way to fully check this is to systematically work out the intervals between all the other parts. In the context of exam pressure this is really quite a tedious exercise to say the least. I generally trust to judgement and intuition hoping that provided I have no parallel fifths  or octaves between the Bass and Soprano lines no others will creep in. Generally speaking provided the alto and tenor parts just move to the nearest available note in the next chord then things should work out, But I have in my practice been known to let the odd parallel fifth creep in. I suspect the same would happen here. In order to make the Soprano part a bit more interesting then it is possible to fill in gaps with quavers so that suppose in the soprano part you had a descent from D to B then make the D a quaver and put in another quaver before the B.

The question was slightly odd in that there were quite a few quavers in the bass line needing a different chord. However provided I got the correct notes and didn't have to many parallel fifths or octaves then I should be OK (fingers crossed anyway)

The third question is to complete a melody of which the first few bars are given again Victoria explains how to go about doing this very well in her Grade 6 video. This often involves a modulation in the last few bars. If you plan out in advance the key points then it helps give a structure. Generally speaking a melody will have 8 - 10 bars and will move from the tonic chord to an imperfect cadence V on the 4th bar and then back to the tonic of either the same key if there is no modulation or to the new key. So again the first note of the 4th bar will be a note which fits the Vth of the given key and the last note will be the tonic of whatever key you have ended up with.

The examiner is looking for you to take what ever is given and use it again with some possible variations in rhythm. I generally try and sequence the second bar in the third bar by raising it up a tone and ensuring a movement by step to the dominant note in the first half of the 4th bar. Then using a logical sequence such as the circle of fifths to get to the modulation section usually with an inversion of the movement in the first phrase so if the first phrase has an ascending passage I will have a descending passage in the second phrase, If there is a large upwards movement then will have downwards leap and so forth finally if there is a modulation ending with a pivot chord from the old key to the new key and then V7-I in the new key. Where in the penultimate bar or half bar the notes of the new key are emphasised.

The question was a bit odd in that it involved a rapid scale like figure occupying 4 hemi-semi quavers and also a modulation from a minor key to the dominant of that minor key. Also despite being in 6/8 time the harmony of the second bar was static just oscillating about 1 note. The modulation caused me a bit of trouble but I managed to scrape something together hopefully it will be enough to get me through,

The last two questions are usually a bit more straightforward than the first 3 and I usually do them first to get them out of the way. They involve asking questions about 2 extracts of music the first one being a piece for piano or with piano and another instrument or voice. and the second a short orchestral extract.

In these questions you are generally asked to identify various chords and intervals the meaning  of certain terms such as Etwas Geschwind (Somewhat agitated, which I was before the exam), Also can you spot certain things such as a triad in first inversion and so forth.

The orchestral score question usually involves asking you transpose the music written out for horns in F or clarinets in Bb to how they would actually sound. If a horn in F plays what is notated as C it actually sounds like F so  you would transpose down a 5th,(7 semitones), Similarly for clarinets in Bb a notated C sounds like Bb  you transpose down a 2nd (2 semitones). So the easiest way to do this is to write out the notes of the chromatic scale of C major and underneath write out the notes of F major underneath it for horns in F or Bb for clarinets. Then write out explicitly the notes for each part as given and then under them the corresponding transposed notes. This can be a bit time consuming, also and I never got the hang of  this, if the original note has a flat attached do you transpose with the original note or with a sharp instead. Technically the notes would sound the same so Db is equivalent to C# but an interval with either C# or Db would have a different name. Thus Db - A is a 5th but C# - A is a 6th. As the interval questions on the orchestral score often involves one of the transposing instruments. It's not clear whether or not after you have made the correct transposition that you will get the correct interval


Anyway that was that I think I got most of the last two questions correct and hopefully will have done OK on the first 3. We'll just have to see. I await the examiners report with interest.

Next year provided everything went reasonably well I hope to do grade 7. For now upto Xmas I need to focus on my statistics course.

If you took the grade 6 exam or anyother grade theory exam I hope it went well for you as well


As a final comment it may seem to the average person that this approach takes the creativity out of music. However just as physicists or mathematicians can be creative within the context of a given framework (i,e the laws of physics or axioms of mathematics) so in music the laws of harmony provide a framework whereby some one can be creative. Of course if you are a Mozart or Beethoven then you can use the framework to construct quite different types of  pieces but the idea that is often floated that composers don't use the laws of harmony but just write whatever comes into their head is quite a misconception. They will usually have an idea of a melody either by picking notes out on a piano or something that they hear in their head, But those fragments will be developed into complete pieces only by following a given structure. I find it quite remarkable that provided you follow the basic laws of harmony and form you can write down a competent piece of music and that this craft can be taught.

Wednesday, 7 October 2015

A bit more on Bell

Some readers of the last post on Bell  may claim that I have missed the point. What my last account fails to do they would say is take into account the effect of the polarisation settings and the fact that they can be changed during the time of flight. So that as the usual story goes if the polarisation of the left detector is changed then the polarisation of the photon (or electron) will change in accordance with that polarisation but because the other photon must have opposite spin then it will change its polarisation accordingly.

I think from a statistical point of view this is misguided. A polariser acts as a filter and for a given polarisation angle only a fraction of the particles in the beam will be able to pass the polariser in a accordance with Malus's Law the probability of passing through the filter being proportional to the square of the cosine of the angle between the polariser setting and the photons polarisation angle. On emergence the photons that pass through the filter will have the polarisation of the polariser.

However some photons will not pass through. The polariser has destroyed any correlation there might have been with an EPR pair. Similarly at the other end a different number  of  photons will also pass through. The number of photons that pass through both ends being dependent on the polariser settings All this can be calculated from the basic laws of quantum physics. But given that polarisers will destroy any prior correlation. I can't see why it is inconsistent with the idea that before 1 or both of the photons hit the polarisers they were emitted from the source with perfect anti-correlation.

Clearly if one thinks the Bell State applies to single pairs alone then one would have to invoke some spooky action at a distance, But from a statistical point of view all we have is the probability of the photons electrons or whatever emerging through the polarisers.  One has to imagine the pairs being randomly emitted from the source with all sorts of polarisation angles, each pair being anti-correlated especially as the Bell state is rotationally invariant. However it is not clear, if one is just concerned with the overall transmission probability of photons through the polarisers which is all one can measure, that the photon that passes through the left polariser is one that is paired with the photon that passes through the right hand polariser. Because the polariser obviously destroys any correlation. Imagine if for example there was only 1 polariser say the left side then clearly the polarisation of the other photon would not change.

Given that in order to verify the statistics many events have to occur then there is no way of ever detecting a single pair of photons.

On that basis then I still feel justified in denying the need for any form of super-luminal signalling between a single pair of photons.