I read Dave’s comment with great interest! It is fascinating to consider why we believe things to be the case as opposed to not being the case. At risk of being a bore as usual, I am fond of pointing out that the origin of the phrase “to prove” is in the sense of “to test” – as in “the proof of the pudding is in the eating”.

The potential asymmetries or differences in the proving analogy between music and mathematics do not trouble me much – mainly because I believe that in the ultimate explanation of the universe, time is an illusion. Consequently, the only thing that is “real” is that the composition, its derivation, and its multiple performances are related to one another by the laws of physics, and are consequently much more likely to be experienced. Cause and effect emerge only from the “arrow of time”, which arises (for some as yet unexplained reason) in our human brain.

I believe that the relationship between the encoding and the rendering, is the test, and the only reason we perhaps cannot easily perceive its mechanism is that our human memory of the “future” is so hazy and multi-faceted when compared to our memory of the “past” which is perceived as singular and sharp (even though it actually isn’t) by virtue of being constantly redefined and reinforced in the specious present.

Just as with traditional prescriptive music, all of the potential and actual renderings (performances) are in fact already encoded within the piece, and just as a gene is expressed (or not) in its ecological niche, then an n-p meme is expressed in its informational niche, giving rise to a rich variety of outcomes.

( A wonderful explanation of a timeless understanding of the universe is presented in physicist Dr. Julian Barbour’s book “The End Of Time”.

http://www.amazon.co.uk/dp/0753810204 )

I think your analogy to mathematical theory is quite apt in that there is so much synergy between maths and music on so many levels.

My background is in maths both at school and university leading on to a long career in computing and I still, to this day, marvel about some of the “proofs of theorem” that are so simple in execution yet so difficult to grasp in concept. As a simple example, take for instance, the proving of a factorial theorem where you generate a formula to add up the product of all the numbers between 1 and ‘n’. The solution of summing a divergent series was well know to the Pythagoreans in the 6th century BC and the numbers are known as triangular numbers:

See:

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

for a more visual representation.

The answer is (n*(n+1))/2 which almost any schoolboy will give you as an answer but how many know that there are many, many ways to come to the end result, just as there are many, many ways to come to the end performance in music.

How magical that the solution to the problem becomes instantly obvious once the numbers to be added together in the sequence are shown in a different form i.e instead of depicting 5! as 5*4*3*2*1 we realise that factorial (n+1) is simply factorial (n)* (n+1) where n is greater than 0 and factorial 1 ALWAYS equals 1.

See:

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

We can now redefine the problem as a triangle for each group of numbers (1+2+3+4) increasing in height, just like a pyramid in fact and similar in some ways to musical notation!! Music and maths definitely do have a special synergy in that the saying “simple is beautiful” was never so true. I leave it to the reader to work out the derivation of the end equation, just as the performer will do with a musical piece.

One other solution used to prove a theorem is using what is known as mathematical induction by which we:

1. Show something works the first time.

2. Assume it works for this time.

3. Show it will work for the next time.

4. Conclude that it works all the time (within certain pre-defined constraints).

Once the mathematical equations are established, there are a multitude of methods that can be used to confirm a formula by looking at the problem from different perspectives, similar to the musician interpreting a work. Many solutions, but one end result – once again as in musical composition.

Maybe you should incorporate these steps to your musical thinking/strategy a little more as I do believe that the concept fits in with your overall objectives, especially any future short pieces.

We always think of beauty in maths as being the simplicity and simplification of difficult concepts. The initial proof of the theorem (musical composition) is invariably difficult whereas the replication of further proofs (musical performance) once the goal has been well defined normally tends to be somewhat easier as well as providing a voyage of discovery to the listener. Composition or “Getting to the initial result” – is hard graft allied to a modicum of luck sometimes as you have no doubt discovered over the years.

There is however, in my humble opinion, one major difference between mathematics and music in that the mathematics always lead back to a common ground i.e the existence of the formula as opposed to the music which simply evolves as it is subject to outside forces (emotions, feelings, surroundings etc.). However, thinking in a different way about a problem often leads to miraculous discoveries – for the musical equivalent read “performances”. Maybe that is the dividing line between the elevation of music from a science to an art form.

Your own personal “base formula” (Eric’s formula) is undoubedly in your own head and consciousness, leaving the various “proofs” to both the player as well as the listener. The two are not so far adrift and appreciation of the end result can often not be quantified easily in words as I am finding out here, so less of my babble!

Meanwhile, I have now listened to the whole collection a number of times, sometimes enjoying certain compositions on differing levels depending on the listening environment and also my own level of concentration, just as i would do when analysing a good mathematical proof!

Keep up the great work and I look forward to more “proof by induction”.

ðŸ˜‰

Dave

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