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it is said that music and mathematics go together, and it so happens that the best mathematician and the best musician I have ever met a happen to be the same person. But they go together as opposites; no musician or mathematician has ever been able to explain the beauty in music in terms of mathematics. it is as if mathematics uses exclusively one hemisphere, and music uses exclusively the other, and this is the connection: a connection of no connection.

play the note middle C on a piano.

now turn on your recording device and sing some melody that revolves around this note, finishing on this note. Something like Frere Jacques, maybe.

now play it back one note at a time and try to match the notes against your piano notes.

what do you notice?

You will notice that the octaves match up, but nothing else does. everything else is a little bit off.

the vast majority of piano players are not even aware of this discrepancy; very few people will be able to tell you exactly why it exists. but behind this barely perceptible phenomenon is something of fundamental musical significance.

you could get out a spanner and adjust all of the notes to fit around C ie to fit the notes you are singing. but then that say your song becomes more adventurous, it revolves for a few moments around G before dropping back into C... (ie it modulates) you would have to get out your spanner in the middle of the song and instantaneously retune around G now.

if you are playing on a violin on the other hand, you don't have this problem.

  • free pitch instruments: voice, horn, trombone, violin, ...
  • fixed pitch instruments: bells, flute, guitar, harp, piano, ... these all have to be tuned

and we can split the second category into:
  • instruments that only play in one key ( church bells, wooden flutes, panpipes, ... ), which can be tuned in Just Intonation,
  • fully fledged instruments ( guitar, pop, metal flute, piano, ... ) that can play in any key.

So, the problem for the second category is: how to choose a set of pitches which lets you play whatever you want? and the answer is, there isn't. because you could keep on and on modulating, bringing in new pitches. so it becomes a question of compromise. a huge number of systems came out -- with all of them you could only modulate so far in one direction, then you had to retrace your steps. people started bringing out piano keyboards with extra rows of keys just to give a bigger range of modulation. all sorts of nonsense...

12-TET

until 12-TET. 12 tone equal temperament. ( 'equal' as in ' equal ratio ' ie the ratio C:C# equals the ratio D:D# etc, and once you have done this 12 times you are back to C, just an octave higher ).

in this tuning system you modulate enough times and you come back to where you started. It is like an ant who walks to the edge of his piece of paper before falling off. by wrapping the piece of paper into a band, he will never fall off.Composers such as Chopin and Rachmaninov go crazy with this, and the result is utterly fantastic.

so that is what everyone is using today. 12-TET. Everything is a bit off, but it is tolerable. And you can play anything. It is suitable for imperfectly expressing any kind of music, though it will not handle anything perfectly.

you may remember the sharp ugliness of having to play 'Three Blind Mice' or 'Mary had a Little Lamb' on a piano or recorder during primary school. it actually sounds all wrong. this happened to me, aged 5. I instantly took an aversion to both recorder and piano, and was declared unmusical. But maybe I was simply reviled by the off-pitch 12-TET sounds.

( Technical note: this is a tricky one. very simple diatonic music sounds bad on any 12-TET tuned instrument for this reason. an electric guitar may be an exception: just the sound from that instrument is inherently so sweet, you could play just one note all day and be happy. Many people do. note that a simple wooden flute, though fixed pitch, will be tuned to just intonation. a kindergarten plastic recorder should be also, but 99% of the time I bet won't be -- that is a guess, I would like to know the truth on this )

if any of my family or teachers had been musically aware, they would have the sense to encourage me to sing, and only pick up an instrument once singing was established. this is the only sane way to aspire to musicality.

' if you wish to play, you must first learn to sing ', Chopin

now, the great thing is you don't need to understand any of this 12-TET theory to be musically fluent.





this blind girl has taught herself. her mother is unmusical but bought a keyboard for her daughter -- a good gift for a blind girl.

this is how I would like to learn.

it is so simple. Her inner ear hears the musical note she wants, and she can connect this with a note on the piano that is close enough.

this is what piano playing should be about; but nobody has the patience to train the ear. The voice and the ear are completely neglected. Instead the hand, the eye and the logical mind are employed to teach the child to memorise a sequence of finger work or read from a written score. this will achieve results far faster. after a few months you can be fooled into thinking that your child is some sort of prodigy, whereas in fact nothing musical is transpiring.


the brain chooses the path of least resistance to satisfy its reward system. sure, this gives a quick fix hit. but it doesn't go anywhere, it creates an illusion, it creates a charlatan. it is a very psychologically damaging path to put a child along. It is plain wrong. I can guarantee that the child in this video will be completely incapable of expressing his own musical thoughts, he has simply been trained to mechanically transfer the musical thoughts of others, like a windup musical box, or a player piano.