A Multidimensional Reality Made Of Music

The word 'dimension' is often used in the context of any of the three lines that are each mutualy perpendicular to each other. When we try to squez more than 3 lines to meet at a vertex, we find that we can't get to make the 4rth one perpendicular to all the 3 all at once. This has nothing to do with limited imagination. Rather the idea of 'space' we have in our consciousness is a 'space of geometry'However, we can always abstract necesary ideas from this 'space of geometry' and then form other 'spaces' such as space of colors, space of music etc. We abstract the concept 'orthonomal' from the concept 'perpendicular'. Then being perpendicular to another thing is just a special case of a more general idea of being 'orthonomal' to that thing. It is this latter general idea that is not limited to the space of geometry.We say, for instance, that a tone is orthonomal to the other tone that is one octave higher. In another language, we say that a sound that is one octave higher 'rans in another dimension' It is in this sence that music eventually forms a multidimensional reality of their own. Unlike the space of geometry though, space of music is not limited to only 3 dimensions. This is because there can be as many octaves as you can imagine. Each octave is orthonomal to all the other octaves.ORTHONOMALITYIf two quantities are orthonomal, then varying one does not result in varying of the other. The eastern direction is othornomal to the southern direction in that if you move strictly to the south, you don't move to the east. On the other hand, south-eastern direction is neither orthonormal to east nor to south in that moving in south-eastern direction results in moving both to east and to south.This orthonomality concept extends very well to vibrating things. You normally hear it said that two signals vibrating at different frequencies donnot interfere with each other in that they are off resonance. Compare this to two forces. If one force act to the south direction, then it does not affect the other force acting towards eastern direction. It is this idea that leads to the concept of dot product between two vectors, which I will latter elaborate on it.But does the vibration at some other frequency truely don't affect the other? The more accurate answer is that the effect a vibration has on another vibration diminishes gradualy as you increase the frequency then it the effect becomes completely zero at some frequency, then again the effect begines to gradually increas till it reaches maximum again at exactly twice the frequency. So the frequency which is one octave higher is once again at a resonant frequency to the other frequency. But another concept necesary in orthonomality 'coherence'. Coherent frequencies are also resonant. But resonant frequencies are not necesarily coherent.Coherence/decoherence is what gives rise to the notion of 'mutually perpendicular' is the space of vibrations. If we relied only on resonance, they yes, moving from one frequency all the way to twice that other frequency, hence once again to a resonant frequency, would indead bring the notion of 'turning by ninenty degrees' but would have landed us to the original axis in that the frequencies can now affect each other. It would be sort of like rotating by 45 degrees and then beginning a journey to return to where you started instead of proceeding to creat a 90 degrees angle. However, because frequencies that are an octave different are nolonger coherent, the various octaves are realy not on the same axis in the space of music.GEOMETRY FROM VIBRATIONSThe first thing we must begine investigating is the notion of 'angle' in the 'space of vibrations'. If a string vibrats at a certain amplitude, call it a, and then some other string vibrats at some other amplitude, call it b, then the two amplitudes can simply add up if bothe frequencies are the same and the vibration is coherent. However, if the frequencies are different, it is a bit trick. Much the same trick as adding to forces that are not both running along the same axis. We must talk of a component of frequency in vibration A that is the same frequency as vibration B.It is here where the concept of 'dot product' comes in. To understand it, consider a vibrating thing such as a string eventualy making the resonator to vibrate. Then it exerts a force. Since the string keeps changing direction rapidly, the force two must be vibrating. The vibration of the string is harmonic. That is to say that it is composed of frequincies that are interger multiples of a fundamental frequency. For simplicity, We are going to consider only two such frequencies. So if f is the fundamental frequency, the we will consider f and 2f. So our music space is a two dimensional one. If a string vibrates, its displacement, call it x, will be described by a sinosuidal wave. We say-x=-lcos(A)-kcos(2A) (I am using 'minus' because this browser does not type 'plus')l=amplitude of the wave vibrating at fk=amplitude of the wave vibrating at 2fA=2pfp=pieThe force F (not to be confused with f) too will vibrate at the same frequencies of f and 2f. So we have-F=-F1cos(A)-F2cos(2A)F1=maximum force for exerted by the string vibrating at frequency f, which is the same thing as saying the amplitude.F2=maximum force exerted by the string vibrating at frequency 2f.The energy is given by Fx (force times distance) so we have-Fx=-lF1cos(A)cosA-kF1cos(A)cos(2A)-lF2cos(2A)cos(A)-kF2cos(2A)cos(2A)Now recall that we said that waves of different frequencies are decoherent and waves of the same frequency are coherent. So when we want to do average energy, cos(A)cos(2A ) always equals to zero and average of cos(A)cos(A)=average of cos(2A)cos(2A)=1 so oure equation for average energy reduces to-Fx=-lF1-kF2Which is the farmiliar dot product of two vectors, F and x given by:-F=-iF1-jF2-x=-il-jkThis suggests that we think of two frequencies of a vibrating thing as two dimensions in a space of music.MUSIC AS A JOURNEYSo if I say doh, reh, mih, fah, soh, lah, tih, doh', I am always moving like a soldier in an 8 dimensional space considering only some 3 of them, doh, soh, lah, then doh coresponds to one step say eastwards, soh coresponds to a step say southwards and lah will corresponds to a step say upwards. Normally these 3 tones can create any other song. Compare this to the fact that you can always move to anywhere in a 3d space by using a combination of only 3 motions. So each and every song can be said to inhabit a unique place in a space of music. Infact it is possible that tunes are more of discoveries in this other space than invensions! Unwittingly, mucisians explores a space of music that pervades the whole space.IS OUR UNIVERSE INFACT A SPACE OF MUSIC?It is quite possible that the 'space of geometry' is infact a creation of mind by mapping what are infact frequencies as axi in a 3d space. Thinking this way makes the idea that our world is up to 26 dimensional makes perfect sense. If you doubt that this is possible, remember that 'creation of space' already happens when you dream. That the mind is capable of creating a vast space where infact there is non is not a far fetched idea at all.Quantum jumps backs up this idea. Why must motions etc happen in descrete jumps? If moving is understood as playing a guitar, then this makes perfect sense! To move 4 steps along the 'doh' direction, I must play doh doh doh doh. If each tone has amplitude l, then the total 'distance' would be 4l, i.e distance is automatically quantised. So if we simply understand the universe as the work of some realy good guitarist, then quantum mechanics becomes straight foward to understand!

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