ABRSM Grade 5 Theory Transposition by Numbers

One of the most difficult parts of the grade 5 theory exam is to get the intervals correct OK we can probably tell that it's a fifth but is it a perfect fifth or a diminished fifth or what?  Similarly with transposition again yes transposing a piece up or down by a minor 3rd say means that the notes are displaced up or down a line or space but what about the accidentals, even worse what if it asks for a key signature change. A while ago when I was first studying music with the OU I hit on the idea that a lot of the confusion could be clarified by using numbers instead of letters. This works really well for intervals and transposition

First number the notes of the Chromatic scale starting with middle C as zero

So we have the table

              Db     Eb        Gb     Ab     Bb
          C C# D D# E F F# G G# A A# B 
       
          0  1   2   3   4 5 6    7  8   9 10 11

Each number is the number of semitones above middle C that the corresponding note is.


Then for transposition for grade 5 the main ones are

Up or down a major 2nd (2 semitones) simply add or subract 2 modulo 12 to the numbers then use the first line to work out the notes


Up or down a minor 3rd (3 semitones) simply add or subtract 3 modulo 12 to the numbers then use the first line to work out the notes

Finally up or down a perfect 5th (7 semitones)

So for example suppose we want to transpose a given melody down a perfect 5th we subtract 7 then add
12 if the new number is less than zero or subtract 12 if the new number is greater than 12. 


             Db     Eb        Gb      Ab     Bb
          C C# D D# E F F# G G# A A# B 
       
          0  1    2  3   4 5 6    7   8  9 10  11

Performing the arithmetic gives for C 0 -> -7 then add 12 to get 5 (you only have to do this once)
then just start from 5 to give



              Db     Eb       Gb      Ab      Bb
          C C# D D# E F F# G G# A A# B 
       
          0 1  2 3  4  5  6   7   8  9 10 11
          5 6  7 8  9 10 11 0   1  2 3  4

So that we see eg that C goes down to F (5) or Ab goes to D (2) and so forth.

So writing this table out and doing the appropriate arithmetic will give you a fail safe method of
transposing accurately especially for those tricky accidentals.

Of course you have to remember if the original melody has a given key signature to take into account
the key notes. Thus for Bb major the accidentals are Bb and Eb so every time you see a B or an E remember
these are Bb and Eb

This also works for key signature changes so suppose I start in Eb and I want to go up a major second
Eb is 3 according to the table add 2 to give me 5 the key signature is now that for F major ie simply with Bb

In the next post I will show how a similar technique can be used for intervals    

On line piano tuition for Grade 1 and above

Well tomorrow I take my first official piano lesson with the aim
of doing grade 1 by June. I have been practicing on and off
for about 18 months now and I can't say it has been smooth sailing
to say the least. Inded I felt in a bit of a rut as I was playing
the Grade 1 pieces badly and never really completing them.

However over Christmas I found these amazing
on line videos by Alison Sparrow who has some really useful tips

http://www.youtube.com/user/theonlinepianotutor

Plus being very attractive to look at. Could be the Nigella
Lawson of piano and violin teaching.

Anyway her tip on practicing really helped me get to grips with
three of the Grade 1 pieces. When you try and set up a practice
session the temptation is to try to and play the piece as
a whole with the inevitable stumbling. Alison's method for which
I cannot thank her enough is to break it down.

At grade 1 the pieces are usually 16 bars what Alison suggests
is play the first 4 bar phrase 4 times. Then play the second
4 bar phrase 4 times then play the first and second phrases
together 4 times. Then move onto the third phrase play that
4 times, then the 4th phrase 4 times. Then put the 3rd and fourth
phrases together 4 times. Then finally play the piece 4 times
repeat this for about 2 weeks and you should have the piece
under your belt. Anyway it certainly seems to have helped me.

This is so obvious when pointed out to you but not at all obvious
when you are practicing on your own.

Another good website I have found is that of Shawn Cheeks
especially his 'boot camp' sight reading course. Shawn points
out that most musicians rely on their ear and memory. What
this misses is actually engaging with the music as it is written
as his career progressed he found it more and more difficult to
tackle the more difficult pieces. So he decided to go back to basics
and really learn the music. Have a look at his introduction
on You tube. Again I have been following his boot camp and
can see the improvement. Certainly spelling out the notes whilst
learning a new piece is really helpful

His philosophy is outlined here


http://www.youtube.com/watch?v=u3V-0iS8JMY

and you can follow the links to the other parts.



I've also put myself forward for grade 5 music theory this march

What i haven't done is any number theory or logic. I have
given myself 4 days  this weekend to complete the assignment
if i don't then I will probably quit I can't really say I'm
enjoying the course OK I haven't really put much effort into
learning it. Right now though my musical interests are
dominanting if I can achieve grade 5 and 6 theory and grade
1 and 2 piano by the end of the year and get back into general
relativity then I wont feel so bad about abandoning M381.

I guess the non stop deadline of assignments exams, followed
by going straight back to other assignments over the past two
yeara has taken it's toll there is a small chance I'll continue
but I can't see it.

Instrumentalism or Why we should Shut up and calculate.

Prior to the festive season, I got myself in one of my perenial debates on the physics forum about the meaning or not of quantum mechanics. My main protaganist will be well known to those who follow the debates. He claims to endorse the Copenhagen Interpretation but also denies that he is an instrumentalist. A somewhat inconsistent opinion in my view. Anyone it’s not my job to help him see the contradictions in his position.

I do want to make a defence of instrumentalism however, that is some what missed by self styled philosophers of physics, but is in fact the current practice of most physicists.

OK what is instrumentalism ? essentially it is the view that the main aim of science is to provide empirically adequate models of nature without bothering to much about how the underlying concepts used to make the predictions correspond to reality.

What do I mean by empirically adequate, it is essentiallly the condition that the predictions of the theory when instantiated in a concrete model give reasonable agreement with experiment. For the quantative sciences such as physics this makes the theory testable or at least a given model of a given phenomenon predicitable. If the model does not give reasonable agreement with experiment then one can try and make the model more accurate by including more terms in the model or trying another approach.

It is important to make a distinction between theories and models this distinction is often blurred. A theory is a set of general principles in physics, there are about 8 sets of general principles which have been discovered

Classical physics

Newton’s Laws of motion.
The macroscopic laws of thermodynamics.
Maxwell’s equations.

Modem Physics

Special relativity.
General relativiry.
Non relativistic quantum mechanics.
Statistical physics.
Quantum field theory.

Note I do not include speculative theories such as superstrings because so far there is not one concrete prediction that has come out of it. At this stage it is ‘Not even wrong’

In order to describe natural phenomenon, one takes one of the above set of principles appropriate to the phenomenon in question. Then with the aid of mathematics and empirical information, such as the masses of particles involved sets up the appropriate equations and solves them either analytically or for complicated problems one has to resort to computers.

So for example to model the properties of stars as they collapse, one needs a combination of Statistical physics and General relativity along with an appropriate equation of state. To model planetary motion one would use either Newton’s laws of motion or for say mercury one has to resort to General relativity. A simple model would neglect the interactions between the planets a more complicated model would include these. Deciding on what approximations are appropriate is a necessary skill of a good physicist.

As we apply the above general principles to more and more phenomenon we gradually begin to understand how nature works and whats more with the aid of mathematics can predict how systems will behave. We can predict the energy levels of a molecule or solid. We can predict the orbits of planets, we can predict the decay rates or scattering cross sections of particles. We can even predict the rate of expansion of the universe from it’s early stages by a combination of Einstein’s general theory of relativity. relativistic statistical physics and a knowledge of the basic particles involved.

All of this is so obvious to a practicing physicist, but so called self styled philosophers of science aren’t happy with this. For them the aim of science is not to make concrete predictions of phenomenon but to describe reality as it is initself. They want to concentrate on the meaning of the general principles but therein lies a problem because some of the concepts used can be quite obscure.

For example in classical physics the nature of gravitation remained obscure all one could say about it was that it obeyed an inverse square law. In thermodynamics the concept of entropy also remained obscure although it had a perfectly precise meaning in terms of a measure of heat transfer . Also whilst Maxwell’s equations involved electrical and magetic fields their real nature remained obscure and all sorts of weird and wonderful ideas about the ‘real nature’ of an electric field involving vortices and eddy currents in the Aether were prevalent at the time. However more concrete these models appeared rather than Maxwell’s equations they didn’t really add much to the understanding of electromagnetism and these were ultimately shown to be wrong as the Aether was proven not to exist. The situation in intepreting the nature of an electric field so exasperated Helmholtz that when asked what Maxwell’s theory was he replied that Maxwell’s theory was Maxwell’s equations. Quite similar to the attitude which I favour of ‘Shut up and calculate’ when it comes to interpreting quantum mechanics.

My protaganist in the OU debates on the fora doesn’t like the above view for him and many others like him such as Karl Popper this reduces physics to engineering (as if that were  a bad thing). Well I’ve got news for him and Karl Popper and other anti-instrumentalists most of what is published in physics journals today is an application of one of the 8 above sets of general principles to model a given phenomenon. Indeed it is only by continuing the process of detailed modelling that we understand how nature works. If that’s ‘just’ engineering so what ?

The rest arguing about the nature of gravitation, the wave function, whether or not particles are real or waves are, really doesn’t move us forward and to some extent it doesn’t matter, as I can still use the general principles coupled with empirical information to make concrete predictions about natural phenomenon. What other means of understanding nature do we have ? It is only by shutting up and calculating that we will get any where. The attitude prevalent amongst certain people that one should always be looking for more new general principles is misguided, the need for new general principles will be forced upon us when we are able to probe nature at more and more higher energies or shorter length scales.. Until then we should all shut up and calculate.