Exploiting symmetry in reality


We explore the relationship of group generators where each element of the group is assigned a vector. Each successive element that is being generated by the generator is then set up to be orthogonal at step sizes of the generator. We then would like to find some structure in the space spanned by such vectors in a way that we can optimize the information content of moving in a disjoint manner between vectors in this space. Inspiration comes from the musical scale of 12 notes arranged by their frequencies, with a generator of 5. We hope such structures such as keys and scales can be imported from it into our system with the hope of increasing the information content of such progressions using algebraic information theory. It can be seen that permutations on an agents input and actions vector my be induced to have more information (be less random) this way.


The problem we hope to solve is one of , given a collection of objects where n can be selected at any one time. And we change the objects at each iteration or time step. How would we maximize the information content of such a progression?

The reason that this may be an interesting question is that we may optimize the input vector of an agent in reinforcement learning this way. Or, with little modification, generate new tactics for players in live games, or make for more efficient operations research at a mine or a shop or even a government. There has been such a problem that has been “solved” by artists all over the world. That is music theory. A scale of 12 notes is defined that form a group with a generator of 1 when describing the raw notes and 5 when describing the frequencies. This leads to a chromatic scale of 12 notes with each 7Th note being a 5Th apart in terms of frequencies. As you know this generator 5 induces a lot of structure in this system with the goal of maximizing the information content of the arrangements of chords and melodies that composers may come up with. We shall create a scale of 12 notes but rather than define a movement of a fifth in terms of frequencies we shall use vector notation and equate it to orthogonality. So every 5th vector is orthogonal in a cycle of vectors. That gives us a circle of vectors that correspond to the key signatures in a musical scale. Using the equivalence between this system and the circle of fifths we can use this to reorder the notes in ascending order according to their lettering. Automatically we can see that with this system certain masking out and masking in of notes as a preselection of permitted collections correlates with major and minor scales. In fact, it can be shown that adding notes from outside of-these scales in most cases correlates with selecting vectors that share a large component with vectors/notes already in the scale. And possesses less information content. In music, this shows up as disharmony. Of course, the structures we have built have a lot of nuance, it may be good to select a not from the minor of a major scale in a certain key in order to increase information content. In short with our system of vectors we can tackle the problem mentioned above using algorithmic information theory. Also note that when an agent seeks to optimize the information content of the changes in its input vector , it may choose to move in 5Th’s, where its input vector is one of 12. But occasionally more “surprise” can be generated by moving in 4Th’s for example. But the tension created by this must be resolved with a 5ths movement of some sort that is strong. This phenomena is the way we craft the exact nature of the information content found within the agent’s policy


A research objective would be to find out what sort of structures are defined or induced when there are more than one generator to the group.

Another research objective would be to find out is if we can apply a genetic algorithm to the actions vector of an agent in such a way that the structure in the state is maximized including the global reward in the fitness function as well

Another research objective is to find out if the information content found in the firing of neurons can be modeled with a fitness function and extend the input vector such that the extra dimensions represent nodes that can fire or not in harmony with reality.


Abstract—In music theory structure within a series of notes is induced by considering relationships between notes that have rational number with the lowest denominator. If we assign a vector to each note in a scale we see that if we arrange for this relationship to coincide with orthogonality that similar structures to the music system are induced in the span of these vectors.If we use the standard music system , every seventh note will be orthogonal to the first. Information content of progressions then depends on how wittingly you use its equivalent of the musical structure. We explore the uses of such a system when its generalised to more generators, graphs and networks. It can be seen that maximising the information content of the passage of input vectors in an agent makes it less random by definition though we will still need to guide this process with a reward function.

If we could assign in a hierarchical manner, orthogonalvectors to all the features in an image, we would have the mostfunctional type of input to feed to a neural network. That isbecause orthogonal features will have divergent effects on theoutput, while similar features would have similar effects onthe output. So functionality here is defined by the simplicityof the process of learning to classify features. But how wouldwe assign such vectors. We could learn them across the datasetby allowing gradient updates to effect the input. Anotherinteresting thing to consider where it would be easy to assignorthogonal vectors to a system is the musical system. Thisconsists of the chromatic scale (group) of 12 notes. With groupgenerators 1 and 7. We will not consider 1 here because itwill not work and is in some sense trivial. What we will dois arrange for each note to have a vector while in the cycleof notes, every 7th note is orthogonal to the rt. Why doesthis make sense? For one it is impossible to span the exactsame space by omitting any one of these vectors. So theyspan the space. And two, it leads to interesting coincidences.If we select a major scale from the twelve, we find that thenotes (chords) in the scale interact in an orthogonal way, bymaximizing the reach they have in the space with the fewestamount of linear combinations. That is, were we to selecta note from outside the major scale at random chances arewe could end up selecting a group of vectors that correlatetoo much. In music this shows up as dissonance. Thereare however rules of when such an act is permissible, andsurprisingly it leads to even more orthogonal constructs. Theserules are the laws of music theory. another coincidence withinmusic is that the information content of a progression isdirectly related to the amount of rules you consider whencoming up with a progression or melody. What that means forour system is that we can mask out a system of 12 vectors, ifwe divorce this from music, in such a way that we can generatemore informative progressions of selections of these vectors.This will work with systems with other generators. It is justthat we will form different scales and a different constructequivalent to the circle of 5ths in music theory that definespleasing (informative) progressions. In fact it is likely thatevery basis possesses an equivalent structure of keys, minors,majors etc. If it doesn’t have generators then the symmetry isnot highly functional and degenerate. How could we use this?For one we could generalize the musical scale from a series ofnotes, to a graph or network. Then factor in these principlesinto the learning structure of a deep neural network. Sinceweights are continuous we believe that We would representgroups of notes in a chord *(the equivalent of concepts) bydensities in a continuous field in the weight space, or inthe activation’s. This addition could come in the form of adifferent form to the loss function or a regularization to thelearning process.This might even lead to a new architecturethat modulates inputs before giving out outputs. One area thiscould be useful for is an RL agent. Given the elements in theinput vector, one would like to optimise the information foundwithin it as it changes. Were we to extend the input vector,the extension could be defined as an automaton. The firingsequences would be defined as chords along the extensionin much the way we have described. Given that the humanbrain acts as a n automaton , with neighbourhood defined byconnectivity. perhaps we could emulate the firing distributionof the structures found in the extension so that they areisomorphic with ones found in the brain. The problem withthis is we would need to discover such structures in the brainand model their firing distribution in order to approximateit. A better model would be just to learn firing distributionsthat maximise the information flow within the extension to theinput vector. as a side this theory could be useful for operationsresearch and even developing new strategies for live sports. Itmight also be useful to model other elements in mathematicssuch as numerical series and sequences, and might have solvedifficult problems in these fields


The western 12 note scale and intervals are not an absolute thing about music and sound. You could just as easily use the pentatonic scale. There are other systems and scales.

I would be cautious about using that as a starting point to find the truths of nature.

Indeed the pentatonic scale has significant history - 30,000 years

Ice-age musicians fashioned ivory flute


“They don’t seem to follow a diatonic scale, he notes, but rather the rules of the pentatonic scale that predominates in Asia.”

Something to be said for this. Why pentatonic? We are talking Paleolithic time (Neanderthals living in the cave next door). This is pre-language or very early language time. Then look at the pre-writing narratives (epic songs) all done in Dactylic Hexameter, which sets nicely to a pentatonic scale. Here is the ancient flute (see ‘Ice-age muscians’) being played along with a reading in the ancient greek of the beginning of the Illiad. Okaaay, maybe not–can’t upload a sound file.

Of course, correlation does not equal causation, but it is awfully coincidental to the overall topic here.

What i am saying is that the whole structure relevant to music theory, i.e. all the scales included. It can be generalised to uneven tempered scales or a basis with more than twelve notes. Youre not seeing the significance of what i am saying. The point of music theory is to maximise ratios of notes with the smallest rational integer denominator. If we replace this restriction by identifying those fundamental notes with the smallest rational ratio to be those that are most orthogonal we are free to use any musical structure, andy scale (which will essentially be a masking out of basis vectors) to maximise the in formation content of the functions expressed within this new basis. Note all scales are made with the circle of fifths in mind if you replace movements of fifths to be those that lead from one vector to an orthogonal one, all this structure emmerges.

Note that you can generalise this to graphs and networks, as well as cellular automaton. The point being we could learn the firing rules of the automaton so that they model the firing distributions of parts of the brain or you could simply learn an equivalent or better distribution of firing responses by maximising the information content of the firning of the automaton, then use this as an extension to the input vector.

If you are aware of music theory, the circle of fifths induces a phenomenal amount of structure in the chromatic scale with the purpose of increasing information content. If we identify the circle of fifths with orthogonal stepwise moves where fifth steps lead to the exact same notes as orthogonal ones this structure arises within the span of these basis vectors. In music theory you are not restricted to keeping in your scale, or even in your key. In fact complex information is generated by knowing the conditions when breaking these rules leads to a relationship between chords with the smallest rational integer denominator. i mentioned the circle of fifths. we could have other circles if we explore a different number of notes to say a circle of 9. then every 13th note will be orthogonal and we rewrite new laws and scales for this sysntem. Infact any basis has this structure simply because we can perform a change of basis into one that has.

The circle of fifths is responsible for ALL scales.

The pentatonich scale is a structure in music theory…i never said that we need to use a particular scale to maximise information content. We have to use ALL structures and devices, such as tritone substitution etc, in order to maximisie the passage of information in music. if an equivalent structure can be setup by a particular basis of vectors then all the structure present in the whole of music theory can be used in the space to maximise the functionality of functions. Or their information content. Note we can generalise a series of 12 notes to one of any number of notes, but we will always know how to decompose it into its structures given that. it may even have more generators the special things arise. It could also be generalised to graphs and cellular automata.

Please confirm if it is clear and if there are any questions

I can’t speak for anyone else but I have been practicing circle of fifths fretboard exercises in my practice sessions since the 1970’s. I am well aware that circling and resolving (or not to increase tension) to octaves and fifths is the essence of solo’ing.

This is the first time I have seen it explained in the way that you describe it. It would be interesting to take these concepts and apply them to a music writing app.

Is there some sort of resonance in play (:roll_eyes:) between the playing of an ancient 5-note flute and the cortex? We are talking (:roll_eyes:) pre-language or proto-language timeframes here. So one of the ancients notes (:roll_eyes:) that when they blow into a tube it makes a sound. Then they realize that if the tube has perforations, possibly cracks, when those are covered it changes the pitch of the sound. Voila, the flute is born. Did they create a five-note simply because a longer tube was unavailable or did they do it because the note sequences and their variants resonated with them?

(:roll_eyes:) unintentional puns

I am aware of musical theory, but does this actually mean anything? The natural relationships between musical notes are explained as simple frequency ratios. Octaves are simplest at 2:1, fifths are next at 3:2, then 4ths, 3rds and so on. The 12 tone chromatic scale and the well-tempered scale are recent constructs to fit the natural intervals into a keyboard-like (chromatic) layout. I doubt there are any other choices that do that but do feel free to go looking. Either way I doubt it has much to offer for AI.

For one you could learn the firing distribution of a cellular automaton so that it replicates HTM theory in fact if you are learning it something better could arise

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You could optimize the information in an input vector through actions cognizant of the structure in the input

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You could optimize the assignment of resources in a function this way too

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Over drinks with Roger Brockett at the WASHU Workshop on Brain Dynamics and Neurocontrol Engineering, he told me that it was all just ‘cycles in cycles’ and my point is that when cycles align there’s your resonance. I suppose what I am saying is that perhaps the beats and tempo as well as the pitch that humans either find pleasing or useful correspond to cortexual resonances.

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I am afraid this may have gone totally over your heads.lol Do you understand that if we set the circle of fifths to coincide with orthogonal vectors then linear combinations made with the laws of music will be ones that share the least components and thus have less interference? Do you see how using these laws creates functions that are more functional this way?

Yep. I was sort of hoping Grossberg (@Stephen_Grossberg) would jump in and help us out.