Saturday, 27 October 2012

A224 Intervals What do you call the musical Interval Db-D# ?

Well I have started in earnest on A224 music. One of the early Chapters is on the nomenclature used to describe muscial intervals. Eg given two notes how are we to describe them. The first starting point is to take the lowest note say C and then if the next note is G just count the letters CDEFG there are 5 so this is a fifth in fact a perfect fifth .
This is relatively straightforward. The problem comes when one of the letters is sharpened. In the case of C - G if we sharpen the G then we G# and because this is a semitone higher the fifth is said to be Augmented if the note was Gb a semitone lower then this would be a diminished 5th. However on a keyboard Gb is the same as F# so the same sounding interval has a different name C-F has four letters and so is a fourth (again a perfect fourth) so C-F# is now called an augmented fourth. The perfect intervals relate to only fourths and fiftths. Other intervals C-D a second or C-E a third and are both called major C-Db or C-Eb would be called minor intervals and C-D# is an augmented interval.

There are two main ways of looking at intervals the one in the text bases the intervals on which position they are in the scale. The lower note of an interval being treated as the tonic (First note of the scale) However I find this confusing as one has to remember all the sharps and flats in a scale. D-F# for example is a major third as the scale of D major has F# included in it. But D-F is a minor third. One can see how this could add to confusion.

A much simpler way is to just count the number of semitones between each interval and use the following table

  • P1, d2 = 0 semitones
  • m2, A1 = 1 semitones
  • M2, d3 = 2 semitones
  • m3, A2 = 3 semitones
  • M3, d4 = 4 semitones
  • P4, A3 = 5 semitones
  • A4, d5 = 6 semitones
  • P5, d6 = 7 semitones
  • m6, A5 = 8 semitones
  • M6, d7 = 9 semitones
  • m7, A6 = 10 semitones
  • M7, d8 = 11 semitones
  • P8, A7 = 12 semitones
The abbreviations are P perfect, M major m minor d diminished A Augmented.
The number corresponds to the number of letters thus C-A is a 6th the notes between the letters are
C# D D# E F F# G  G# A which is 9 so this is a major 6th. Once one has got the hang of this it is quite straightforward.

Putting my (failed) mathematicians hat on the obvious thing to do would be to draw a sort of Cayley table listing the lower note in the columns and the upper note in the intervals count the number of semitones between each note and then use the table above to label the interval. Starting from C there will be 21 entries as each note will have three forms eg Cb-C-C# however once one does that it soon becomes apparent that there are gaps. Db -D # is a good example Db-Db is a perfect unison, Db-D has one semitone and so is an Augmented unision but what is Db-D# ? it must be a unision as it has the same letters but it can't be a perfect or augmented unison. So I propose to call this a doubly augmented unison.

A similar problem occurs for say Db-B# this must be a form of 6th. there are 11 semitones between Db and B# but the only sixth's that have a name are the Major 6th 9 semitones and the Augmented 6th which has 10 semitones. Thus again I propose to call this a doubly augmented 6th. This is not a term I've seen used in any music textbook that I have, so I might have invented a new musical term.

When I've completed the laborious task of labelling all the intervals in this manner I'll publish the results.

Thursday, 18 October 2012

MS324 Exam Debrief

Well I feel a lot happier about this one than last weeks topology. However I still think I'll be borderline grade 2/grade 3. OK so the questions

1) 3 first order differential equations I think I got most of this correct

2) A tricky one on Fourier transforms the first part was OK but the second and third parts were tricky I couldn't see how to do the second part although I'm sure it will seem obvious on reflection. Left this question and then went onto the second part

7) A question on Lagrangian mechanics a pendulum attached to a massive spring. This question had appeared before but was a bit fiddly still got most of the marks for this one

5) The inevitable question on waves on a rectangular boundary again this question appeared before. Straightforward but tedious and a long time to write out. Got say 3/4 of the marks for this one

6) One dimensional diffusion equation for a bar. Again this question has appeared before but only once (2006) so I guess some people might have been taken by surprise by this one. The past two years have been a heat conduction type equation for a cylinder or sphere and generally a lot easier to get full marks for. Time was running out and as the question involved a Fourier series I got the method but my constants came out all wrong. So reckon about 1/2 marks for this one

So back to part 1 question 3 A probability question on recurrence relations which I left and question 4 a tricky one on variational principles. The first part involving a tricky taylor expansion upto 4th order however the second part involved solving a fairly simple differential equation and in the last 10 minutes I think I got this out so say 5 marks

So overall estimate say
13 for question 1, 4 for question 2, 18 for question 7, 14 for question 5 10 for  question 6 and 4 for question 4 to give

13+4+18+14+10+4 = 63 so if the examiners are feeling generous they might bump up the marks a bit and I might just scrape grade 2.

Anyway it's over I'll post full reviews of both M338 and MS324 later on this month.

Managed to have a few pints in Diggers with another guy. One of the problems with this course is the lack of face -face tutorials so it is difficult to get to know people. He's doing fluids next year along with me so hopefully we can meet up again.

Sunday, 14 October 2012

MS324 Past Exams and Revision Strategy

Well there aren't that many past papers here only 4 But even within those papers there appears to be quite a strong variation.

Question 1 in earlier years seemed to be 3 part questions on First order differential equations.
This seems to have crystallised to a longish question on 1 differential equation. I was caught out by
a part question in 2005 covering a technique which I had forgotten

Question 2 is usually on Fourier Transforms and the translation theorem the later part of this question tends to be a bit tricky so probably best avoided

but again 2005 differed as it had a question on D'Alembert's solution to the Wave equation

Queston 3 is on probability alternating between a continuous probability distribution or a recurrence relation type question for random variables (One of my weak points)

Question 4 is on Euler Lagrange equations for a given Functional

Then Question 5 is on the Wave equation for a rectangular membrane, This type of question has occurred in all 4 papers and should be a banker question but it is actually quite time consuming

Question 6 is on the heat equation usually this is in cylindrical coordinates and reduces to 1 dimensional form
However in 2005 there was an exception as it asked for a Fourier Series type solution

Question 7 is on Lagrangian mechanics involving usually a pendulum with a variable support. (I think the past papers have exhausted all the possibilities ) so they might decide to give a completely different type of problem here.

We can get our marks from any part of the paper and those people who are quick could probably get over 100% However my practice seems to be showing that I'm quite slow. Some of the latter parts are really ludicrous for just a few marks.

So my strengths are
a) Differential Equations, Lagrangian Mechanics and the Euler Lagrange equation, The wave equation and the Heat Equation

So I'll do the part 1  questions on differential equations and the Euler Lagrange equation first. If I work quickly I should be able to do most of these questions  in the first 3/4 hour

Then the Lagrangian Mechanics Question, the 2 dimensional Wave equation and the Heat equation again 3 questions in hopefully no more than half hour per question, leaving 3/4 hour to tackle the other two questions. As it is easier to pick up marks in the earlier part of the questions If I find I'm getting stuck I'll move on. The Fourier Transform question is usually quite straightforward in the first part but gets trickier afterwards again similarly with the probability question. If it's a recurrence relation I'll probably leave it.

My first attempt this morning got me borderline grade 3 grade 2 as I mucked up the Equations of motion and made the mistake of getting bogged down with the Fourier transform question and made some stupid mistakes. I must admit this harum scarum test of ones ability is a bit unfair. It all hangs on whether you can do enough in the exam. I appreciate that one must have some guarantee that one has worked independently but the move away from taking the average of one's TMA score and the Exam when I first started my OU life 12 years ago to basing it on your worst score of the two is quite unfair.

I predict borderline grade 2 grade 3 for this one simply because I just cannot work quickly enough to answer the questions accurately enough.

Why do we put ourselves through this torment  ?

Friday, 12 October 2012

What Next after Topology

The question was asked in the Topology forums what happens next. What is the point of say Cauchy series and completeness and connectedness. Here is my personal take on where I want to go next.

The next step is to finally tackle functional analysis this is a combination of metric space theory and linear algebra. The concept of a vector space of linear algebra is extended to spaces of functions, orthogonality being defined by an integral over the functions. The simplest example being the trig functions which satisfy

$$\int^{\pi}_{-\pi}sin(nx)sin(mx) dx = 0$$ .

if m is not equal to  n or $$\pi$$ if m = n

The Integral can be seen as a type of scalar product or Norm for the trig functions.
Thus after suitable normalisation the set of functions sin(mx) m = 0,1.... can be said to form a set of orthonormal functions. Also sin(mx) is an eigenfunction of the differential operator

$$\frac{d^2}{dx^2}$$ with eigenvalue $$-m^2$$ so that the solution space mapped out by the differential equation

$$\frac{d^2y}{dx^2}+ m^{2}y = 0$$ is precisely the set of orthonormal trig functions.

The generalisation of this to all sorts of differential operators and the abstract study of such relations forms a large part of functional analysis.

The OU MSc course does a course on this and the recommended text book is by Maddox which is out of print. I managed to get a copy from Amazon but it is really dry, more importantly from my own perspective, namely trying to understand the applications of Pure Maths to physics there is nothing on which most physicists would call the essential point namely the spectral theorem for both finite and unbounded operators. This is important for understanding quantum mechanics.

A more accessible book on Functional Analysis is Kreyszig

This has an introductory chapter on metric spaces, open sets compactness and convergence but also(eventually !)  proves the Spectral theorem for unbounded operators which is important in understanding quantum mechanics.

The definitive book on the applications to quantum mechanics being Von Neumann's book

But some understanding of Lebesgue Integration may be necessary the Old OU course used to be based on Weir

Unform convergence and Cauchy sequences as I discussed with Neil and the other Edinburgh M338 er's in the Diggers after the exam  play an important part of showing rigourously why Fourier Series converge. The MSc course in Approximation theory covers some of this. It's covered in Kreyszig as well.

Topology is of course used in General relativity especially the proof of the singularity theorems the definitive book on the subject being Hawking and Ellis.

However the book is extremely concise some familiarity with differential geometry in it's modern form is needed and an ideal book would seem to be the second edition of O'Neill's book on differential geometry

If that is considered to simple then either read chapter 2 of Hawking and Ellis or

A slightly more accessible account of the singularity theorems seems to be given by Wald

There appears to be an extension of the notion of compactness to differerntiable manifolds called para compactness
And this would appear to be an essential condition for defining a Riemannian Manifold.

Another area of study is the relationship between Lie Groups and Topology a book on which is given by Gilmore

And there are books on the application of Topology to physics the most accessible of which is probably by Nash and Sen

So plenty to build on from the topology course  is a stepping stone to understanding the pre-requisites to understand the pre-requisites for books like Hawking and Ellis or Von Neumann

I intend to tackle Kreyszig and O'Neill and possibly Weir next

In the mean time I shall have my pudding and get stuck into to some past papers for MST324 a weekend solving differential equations both partial and ordinary. Doing Fourier transforms and solving problems in Lagrangian Mechanics absolute Bliss. I hope I can get up to speed.

Wednesday, 10 October 2012

M338 Exam debrief

Well this will be quite short as predicted in the previous post I really did not do well on the exam time just run out and I got bogged down on basics. I managed 4 out of 8 questions in part 1 and 1 more or less full question in part 2 and bits of a question that should have been a doddle

OK question 1 was as expected a relatively straightforward question on the interior and exterior of two sets.
Think I got most of this but due to nerves etc I took 25 minutes on this

Question 2 was on whetther or not a set defined a topology on a finite sets again I think I got most of this out another 20 minutes on this

Question 3 was showing that certain edge equations could be reduced to canonical form. It looked like this would take ages to work out so I moved on, with the hope that I could return to it later

Question 4 was a question on a shape with a twist in it and you had to divide it by drawing veritices etc a question I never got the hang of so moved on

Question 5 was one which should have been straightforward involving the subdivision of a sphere in terms of octagons and triangles. However normally this type of question just involves one type of polygon. I Didn't have time to think through the implications of this so moved on

Question 6 was about compact sets again I left this as I was never that clear about this type of thing

Question 7 was about sequences in a topology think I got most of this correct

Question 8 was about similarity relations for a fractal again I think I got most of this correct

So maximum possible marks is 32 say 24 after dropping two points per question

Then part 2

Question  9 was on edge equations and reducing them to canonical form something that should have been a doddle. But the twist here was that the edge equations were separated out. So before one could even start one had to combine the equations to get a single equation. I did this then got the Euler characteristic but completely forgot how to obtain the boundary nuimber for this type of question. And I could not reduce to the equation to canonical form. Part 2 of this question was similar involving 4 equations. Again I could get the Euler characteristic but not much more. So a maximum of 4 marks here.

Question 11 was on metric spaces and sequences defined on the metric space. The metric was quite straightforward and I got most of the first two parts correct. Namely calculating the metric for two values for 2 marks and then showing that the metric was indeed a metric space. 8 marks. One then had to calculate the metric for various open balls defined on the metric a maximum of 5 marks so say 10 marks for this question overall.

Time ran out I couldn't be bothered to chase after the remaining marks
so a reasonable estimate is 24 + 4 + 10 = 38 marks. You need 40 marks to pass so I have definitely failed this one.

Hopefully I can move on and do better with my waves course. If I get offered a resit I will probably take it. In the long term scheme of things Topology doesn't matter. I'm a bit more familiar with the language of topology and that should give me the confidence to look at say the applications of topology to physics. But this has been a nightmare best forgotten.


Tuesday, 9 October 2012

Going to Fail M338

Well tomorrow is M338 exam day. I feel that I deserve to fail and I probably will. Still I'll turn up and see what happens. OK what went wrong. I basically switched off. I thought I might be able to salvage something by concentrating on Block B, But the exam questions seem to bear little or no resemblance to what is in the course material or if it does then there are very little guidelines as to what one is supposed to do, OK so a typical question will ask us to subdivide a solid with twists and deduce it's characteristic equations and say its boundary number. In order to calculate the Euler characteristic one must work out the number of vertices the number of edges and the number of face. The first two are relatively straightforward but the last one seems to defy all sorts of logic. Again there is normally a question on embeding a network within a rectangle representing say a torus or a projective plane but again  the course material gives no general guidance as to how this can be done apart from here is the diagram and here is the embedding. Fine I'll leave it to those who can intuitively understand what is going on for me it just leaves me baffled. Again in the second part there is usually a question relating the forms of surface described in connected sum form  to another descriprtion in connected sum form . Again little or no guidance has been given as to how to tackle this type of question.

My banker questions from part 2 will be deducing the form of a solid from vertex insertions and possibly one on Metric spaces and whether or not sequences defined on the metric space are Cauchy. But the examples apart from the one in 2010 are such that one has to really think on ones feet as to what the metric really means.

So I should get more or less full marks for one long question in part 2 that is say 14 marks. There are two possibly 3 part 1 questions I can get more or less full marks for thats 24 marks then I'll be really stretched to get more. I could aim to delibrately fail in the hope of getting a resit next year but there seems to be some ambiguity as to if I did that I would only qualify for a straight pass. So I might just scrape a pass but anything more I wont deserve. Fortunately I don't have to count this course for anything for my second open degree so it's best quietly dropped whatever the grade whilst I include other stuff.

Good luck to those who have got more out of this course than I have. But I never really got in to it and so deserve to fail.