Glenn Gould and Andras Schiff are good of course (they are probably more popular too), but my personal taste probably lie toward that of Igor Levit.

Another pianist I admire is David Fray, who did wonderful interpretations on the keyboard concertos.

]]>Let’s start with Bach’s 2-part inventions. And the easier ones first.

The ones that I can play HT (hands together) slightly fluently together are no.1 in C, no. 8 in F, and no. 13 in a.

Difficulties that I have with these pieces (which probably will be true for all the Bach pieces):

- counterpoint (to play different voices convincingly and beautifully on the same instrument, e.g. left hand for one voice, right hand for another, unlike many other pieces which only require the left hand to be an accompani·ment to the right hand)
- the many modulations which mean the main key of the piece changes a lot, so a lot of accidentals come and go, which may be frustrating at times
- awkward fingering: unlike other composers who are more pianist composers like Chopin, a lot of Bach’s keyboard music has very awkward moments for fingers
- ideas are dense and there is not much room for error
- there is also not much room to rest too, e.g. one of the Bach’s pieces has triplets on the RH throughout almost the whole piece (WTC I’s d minor prelude)

I need to master and memorise these first three pieces before I feel comfortable to move on from them. This will take effort and time.

]]>When I was teenager, I was obsessed with classical music, and would spend all my allowance money on boxes of classical music. But somehow things got in the way and throughout the transition to adulthood (I can’t say I am a full adult yet, I am always childish) I forgot about classical music. Pop and country music somehow dominated my mind whenever music was ever involved with my life.

Anyways, I have decided that maybe my next blog posts will be about my journey back to classical music, especially Bach’s music. I play the piano, so I will focus on his keyboard works first. Like the Well-tempered Clavier, Inventions & Sinfonias, Partitas, French Suites and so on.

Few things that help me with this journey:

- Yamaha electronic keyboard (unfortunately I do not have the resources for a Steinway, although I have the fortune to practice on one for a few years, years ago, but this electronic keyboard will be sufficient especially the touch is in fact quite similar to a real one – the keys are fully weighted)
- Spotify (subscribing to Spotify means that I have the whole library of classical music in my hands; how times have changed)

https://gowers.wordpress.com/2011/11/06/group-actions-i/

It is quite abstract and group actions will lead to the orbit-stabiliser theorem, which cause be some dizziness.

I need to understand group actions so I can understand Sylow’s theorems.

]]>Algebra and analysis.

Algebra and geometry.

Algebra and number theory.

What is algebra? It comes from arabic, meaning the **reunion of broken parts**. “In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.” Wikipedia.

So far in reading algebra, I have been reading about groups, different types of groups, subgroups. Basically a lot of definitions and theorems and proofs regarding groups which follow from the group axioms.

Fairly abstract.

Groups, along with rings and fields (which I have not read yet) are descriptions of the simplest algebraic structures which allow equations like polynomial equations to be solved.

My life is broken, can algebra help to reunion the broken parts of my life?

]]>And then it gets tough, and some things like abstract algebra gets very abstract quickly, the learning curve is fairly steep for someone with no guidance.

I then try to skim through it and see if I can find any beacon of light where I can latch onto or motivate me to persevere through the text.

For modern/abstract algebra, this will have to be **Lagrange’s theorem** which states that for any finite group G, the order of every subgroup H of G should divide the order of G.

I understand what it means, but I don’t think I truly appreciate it yet. Which means I will have to backtrack and see if I can get a better intuition through means of maybe drawing some pictures or construct a more complete picture (story) of group theory.

]]>A lot of algebraic structures, like the addition and multiplication of real numbers or the rational numbers have the **associative property**, which is often taken for granted.

It basically says that given a binary operation ∗ , is associative if for all three elements x, y, and z in a given set S, (x ∗ y) ∗ z = x ∗ (y ∗ z). Then the products can be unambigiously written as x ∗ y ∗ z.

From this, the general associative law follows in that for any number of elements, their products can be unambiguiously written without any parentheses.

This can be proven by mathematical induction, but it is not the easiest one to formulate.

As a non-math major, it took me some time to absorb this fact.

Without the associative property, as the number of elements increase, the number of possible products increase quickly (look at the Catalan numbers for a formula).

Operations that don’t have associativity include subtraction, division and exponention.

For example, we can see that (2/3)/5 does not equal 2/(3/5). The first equates to 2/15 and the second equates to 10/3.

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