is it the answer to the riddle of the universe ? or just a key to human understanding ?

This article would discuss how in music we encounter the Carthasian mind–body dualism as a disharmony in the form of ⁵³¹⁴⁴¹⁄₅₂₄₂₈₈

our mind would forever try in endless circles to impose order on chaos, but the mind always translate to body as similarity and not identity, approximation rather than unification

Let talk about music theory ! did you know that most (western) musical interments are not really in perfectly in tune ? there are small deviations from perfect ratios. Similar to cinema, even though the cinema is screening at 24 frames per second we conceive it as legitimate movement, or a led light blinking at about 100 times per second seems solid to us, the gap is just to small for us to recognize and therefore we ignore it. The reason its quite an important subject to discuss because it reflects perfectly the scope of human understanding and perception. Most musicians are not aware of the fact that they are never in perfect tune.

perfect ?

The approach to perfect tuning is supported by cosmology, music tuning seems to be created under the influence of the cosmic order, reflecting a system of perfect order, together with rational math and numbers. Time was relating to a flat linear movement of the stars and not for example to the general theory of relativity. (planets>time>measurement>frequencies)
Today with modern understanding of physics one knows that a musical instrument (with more than an octave) would never produce what we call 1:1 perfect tuning. One would not be able to perceive perfect tuning because it will always go through a medium – an instrument, air, the vibration of the ear – these are factors which would impair what we call a perfect tuning. Think about measuring the frequency of tones within a time framework, general relativity suggests that a time framework is relative therefore we are lacking the validation of absolute measurement. (think about the Doppler effect).

Psychologically we are believing in the integrity of perfection, a film represents perfect motion, the light is always on, the tune is harmonious and perfectly overlaps, but the only fact is that it isn’t. We can call it Logos, rationality, linearity, stability, solid, harmony, negentropy, objectivity, mind on the other side there are irrationality, exponentiation, fluid, disharmony, entropy, subjectivity, relativity. The model of Darwinism shows us a way to look at patterns and order as means of survival and sustainability, as the only thing that enables long-term-existence. It is true to identify order with the most superior element of existence, the ability to sustain life itself. It is important to understand this statement in the correct way, one can see that the element which creates order inside our minds from the meaningless nothing outside our mind is just an illusion provided by existence, which is only promoting its sustainability by offering incentives. Like someone once said “Life without music would be a mistake”

Ohne Musik wäre das Leben ein Irrtum

entropy/meaningless > existence > survival/sustainability > mind/tool > incentive> order. Psychologically and biologically it is suggested that having capability of perceiving complex patterns offers more incentives that ranges from the biological simple survival to the psychological feeling of order and stability and to the transcendent existential function that goes beyond human and beyond rational understanding, through life affirming instincts, feelings and emotions.

imperfect

Today in western music, for the standard tuning we use something different than a Pythagorean tuning (Pythagoras c. 570-495 BC). While the Pythagorean tuning was thought about in the constraints of a purely rational world of whole numbers, what we use today is based on irrational numbers and advanced math, it is no longer a naive pattern of thought which sought to achieve wholeness. Tiny corrections compensate for rational-number ratios.

Note : We actually don’t know what the ancient Greeks had in mind. There is an options that they were not naive and quite aware that their Logos is not meant to be as a 1:1 representation of reality but just the model of the human rational. The ancient Greeks discussed the lack of possibility to touch reality in philosophy while modernity does it with advanced science and art. For sure we modern people are a lot more materialistic, which suggests the quite valid option that we were and still are naive or blind to the nature of reality especially since the time of Socrates until the time of Nietzsche, which reached its peak with the ideas of the enlightenment period.

The ancient Greek mode of thinking was trying to find perfect relation and ratios, though existing within ourselves as representation of reality, it cannot be achieved in external reality. Another way to say it is to say we discovered that everything has more layers of complexity on top of it. Pythagoras achieved it within his Pythagorean theorem which can only offer a down-scaled or projected view of reality, it is only accurate and perfect when it is tested by its own axioms : like the sum of the angles in a triangle will always be 180° which today we know works in a very limited spectrum of action, since reality has more than two dimensions. How can we look at it from a progressive point of view ?

a new definition of perfection and wholeness

Parallel to the progression of philosophy, psychology, science, math and our cultural description of the world, we are in the period of shifting between two modes of perceiving the world around us:

The Greek Logos – suggested only one possible description of reality, based on rationality, whole numbers and cosmic order, all scales from planets to atoms and via sound waves work in an identical way. The basic element is reachable and possible to predict. This paradigm reached its peak in science with Newton’s laws and Naturalism in art representing a deterministic course in which everything can be known and predicted by using the right formula. All the relations between objects around us are linked through sets of rules.

The modern period – a more complicated description of reality was possible that was still in a way coherent with its previous paradigm. After Darwinism, Nietzsche, Einstein’s theory of relativity, the aim is not to capture reality as a wholeness, because its proved impossible (so far). The aim became then to build various models which represents various scopes ranges and scales rather than one coherent theory, not perfection, but rather a high degree of accuracy in relation with reality that allows us to work and to progress. As an example Karl Popper’s Falsifiability offered a methodology which aims for progression and development rather than a megalomaniac target oriented methodology, that something new which is constantly dynamic replacing old models with new ones. The Nativity of the Greek Logos is gone, the death of the subject, but existing side by side with Entropy (a term borrowed from late 19th century theories) proves to provides endless possibilities as an anti-naturalistic approach.

The Pythagorean comma as a test case, the details and how do we go on from there ?

In its basis this tuning takes two ratios into account, one is 2:1 which is an octave and the second is 3:2 which is a perfect fifth, for example: 2:1=C2 and C1 and 3:2 are G2:C2. They sound most harmonic to us, because we recognize the relation between them in the most easiest way, so further 4:3 (perfect forth) 5:4 (maj 3rds) are harmonic and aurally pleasant.
If we instinctively manage to find an aural pattern we can feel it, harmony expresses itself as a bodily feeling which indicates the recognition of a pattern, the simpler the relation the stronger the feeling.

The technical part of the theory

What Pythagoras tried to do is to link the two 2:1 3:2 ratios into a spectrum of notes which can then be used to create variation which is sufficient to build upon to create music. The three main condition were these :

1. Subdivision between an octave. meaning a step range between C1 and C2.
2. The perfect fifth 3:2 as one of the step subdivisions within an octave.
3. Continuity and overlap between octaves meaning C1:C2:C3:C4:C5:C6:C7 would be the ratios of 1:2:4:8:16:32:64:128 if you go up an octave it is double the frequency of the original one.

Then Pythagoras decided to create 7 octaves with 12 equal subdivisions, through the use of perfect fifths 3:2 to find all the notes between octaves.
There are two ways to view this, linear and exponential way.

Linear thinking :

The reason why I use the term linear because of the method it was used to self approve itself. In this way playing the perfect fifth (C1 to G1, 3:2) 12 times going up should give us the 7th octave above, and going through the 12 tones. This is how we get the circle of fifths.

This would work perfectly, not only that 7 octaves are more or less the tonal hearing spectrum range of a person (not tonal being 10), the division of 12 main notes in an octave in which one is the perfect fifth would have to close perfectly with 7 octaves, meaning the playing 12 times the perfect fifth above each other should match 7 octaves, but it doesn’t, because :

(³/₂)¹² ≠ (²/₁)⁷

playing the perfect fifth 3/2 ^ 12 times is not equal to playing an octave 2/1 ^ 7 times. wait…what ???

The scheme above is misleading, it represents relations which we did not agree upon, so it is a matter of confusion. The relation from C1 to C8 is not linear but exponential. Therefore, it cannot be drawn as a liner scheme. The scheme then only relates to how to mistakenly imagine these relations. The distances between every two notes are then not linear.

Every Pythagorean ratio, 1/2, 2/3, 3/4, 4/5 defines a step from our base note C1. Although we recognize the numbers they do not overlap. We have to choose or a system of 1/2, or a system of 2/3, but they do not overlap.

This means that we cannot draw a Pythagorean scheme for all notes, it does not exist. We have two contradictory definition of a step size, two systems without an overlap (that our common sense might not understand) that are supposed to make one system.

What we call harmony can be misleading : it’s not cosmological, but only referring to one center, ratio is harmonious while only one proportion exists, we recognize that repetition, and apparently it doesn’t have to be accurate because what we call accurate is not something of the real world, it only our bad habits.

Exponential thinking :

The way how note subdivision are created is a part of a much bigger topic : how we conceive wholeness and how we see ratios.
The systems of ratios of octaves and perfect fifth’s are not linear, meaning they have different rules of construction of a system, different definition of how to construct an element compared to the linear one. something in the core of understanding is non-intuitive and complex.

Linear	Exponential	Logarithmic
Rational Thinking with numbers	Octaves	Perfect fifths
n, n+1, n+1+1	n, n*2, n*2*2	n, n*3/2, n*3/2*3/2
	n, n+n, n+n+n+n+n	n, n+1/2n, n+n+n+1/4n
2,3,4,5,6	2,4,8,16,32	2,3,4.5,6.75

Exponential is a specific type of logarithmic, what we can see that with Logarithmic thinking every increment or reduction is dependent not on a fixed value but a changing one, there is fluidity. It would be easier to look at this graph and realizing that a division of 3 equal parts (of the perfect fifth) are easy and straight forward on a linear line but on a logarithmic one it is more complicated. It would be easier to project our problems into the linear rational realm and then switch back to logarithmic. its what we do in math.

Octaves are divided in such a way :
2:2²:2³:2⁴:2⁵:2⁶:2⁷
Perfect fifths are divided in such a way :
3/2:3/2²:3/2³:3/2⁴:3/2⁵:3/2⁶:3/2⁷:3/2⁸:3/2⁹:3/2¹⁰:3/2¹¹:3/2¹²

Though looking at these ratios 2:1 and 3:2, gives us the feeling of a 1:1 rational system, simple numbers and relations, when translated to real-world tonality they are becoming two separate logarithmic systems therefore when they came across each other its not necessarily a meeting point on rational description like 12 octaves which are 84 notes (but almost !) nevertheless, it does not prevent us at all from using it like it is perfect, with tiny adjustments.

what can we learn from this ?

First time when I read about the Pythagorean comma, I was trying to look for an explanation for the mysterious Pythagorean comma. How does something so theoretically clear does not settle with reality (linear vs. exponential thinking) ? its not for nothing that only around 2400 years after it was thought about that it was explained through advanced math, before that musicians didn’t really know what was wrong with their musical interment (twelve-tone equal temperament is only being used since the 18th century) it is still a subject well discussed in theses and in articles, check out the Wikipedia of https://en.wikipedia.org/wiki/Just_intonation or https://en.wikipedia.org/wiki/Equal_temperament.

We found tracks on a lonely island, we searched for its maker only to find out it was our own.

Irrationality = lacking a ratio

The Pythagorean triangle is the cross road of rationality with irrationality, a triangle with two legs of 1 gives us a hypotenuse of √2. The triangle points to the two systems that don’t mix (the Cartesian duality).
The root of two is the exponential equivalent of one. Exponentiallity is a system of relations, it contains objects the relate to one another with rules. Linearity is a rational system, the basic object is 1, the relation between objects is 1. Exponentiallity changes the relation between objects based to their own identity. It is where 1+1≠2. Linearity reflects the subject sanctifying the number one. Systems are linear and deterministic.

Pythagoras found a relation between the numbers 3,4,5. They are related to each other in special ways as a triangle can show. As we already must know, the emergence of patterns doesn’t necessary have reasons (in causation sense) but for us they are useful.

3 :	4	: 5
20 :	21	: 29
119 :	120	: 169
696 :	697	: 985
4,059 :	4,060	: 5,741
23,660 :	23,661	: 33,461
137,903 :	137,904	: 195,025
803,760 :	803,761	: 1,136,689
4,684,659 :	4,684,660	: 6,625,109

The Pythagorean theorems show us the power in simple basic thinking but also its flaws, the triangle and its Euclidean geometry exist primarily in the human mind, same goes for the cosmological view of music, and the same thing applies for modern physics (in my opinion), the only way we can validate reality is with the same systems we made up to define it. All we see and experience including music is bound to the framework of time, time we cannot define accurately, it is external to us. Music is the perfect example, outside the human mind it is just a vibration. Inside our mind it is pure beauty only because it is a play of proportions and relations, which is the root of how we give value and definition.

When modern math tries to find the beauty and logic of music with irrational numbers it fails (unless you are a real geek), it fails because it defines music as frequencies, it tries to describe meaningless, irrational number, irrational is the lack of ratio and proportion, lack of value and definition. Its not to condemn the use of irrational numbers in the help of which we build advanced tools, but to use it in the right way which is not through a preference of rational or irrational thinking but the acceptance of the true nature of both.