Is there a direct mathematical equation?

[j_menz]

You might get a lot out of reading his "Godel, Esher, Bach" though; if you can be bothered wading through two inches of book, the core idea of which he could have written in about three pages.

Indeed. I got stronger arms, myself.

[ajspiano]

Here is a (crude) drawing that finally I did this morning:

[m1469]

The sort of nested, self-referencing hierarchy in your drawings is what Douglas Hofstadter thinks music, consciousness and almost anything is about, m1469. Might be, might not be, I wouldn't have a clue. You might get a lot out of reading his "Godel, Esher, Bach" though; if you can be bothered wading through two inches of book, the core idea of which he could have written in about three pages.

Do you mind explaining a bit more about what you mean by "self-referencing hierarchy"?  Does it basically mean that there is a hierarchy within a given model, and that this hierarchy is formed strictly within the model?  I am interested in more of that, but it seems I would want to read Hofstadter to get more of it.  Honestly, though I would be inclined to read a bit, the reality of me reading an entire book on it is very slim  :- ... at least at this point.  I remember the Esher drawing in reference to Bach being posted here ... I liked it, it sparked my imagination, and I could see a point, but it never seemed to fully do the job for me.  Perhaps I just couldn't comprehend it.

I am always fascinated when people exclaim something like in your last sentence, that the core idea of the book could have been written in three pages...

[m1469]

Is this yours?

[ajspiano]

Is this yours?

yes, as response to yours..

edit - and perhaps expansion on? i don't know. Perception of the idea is probably pretty personal.

[Bob]

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

[m1469]

yes, as response to yours..

edit - and perhaps expansion on? i don't know. Perception of the idea is probably pretty personal.

OK, thank you for both posting and for telling me about it.  It is an expansion on my drawing, yes, but what I've been thinking about is not fully represented by my little drawing (of course!) and your drawing is probably related to my bigger thoughts.  

Bob, you think I am talking about a Musical Fractal?  I do wonder how that could possibly be measured??

[ajspiano]

What about this one..



..where each of the 4 large notes represent a musical climax point, the development is ofcourse extremely complex and infinitely variable.. the fractal image is a cool representation of how things can develop from the original idea.. but in music there is the sense of "expression acheived", or an end point, a direction of motion.. rather than a never ending free expansion.

The fractal could represents the possibilities, but it doesn't represent the idea that we do actually choose a specific path in order to get from start to finish.

[m1469]

How do you suppose a fractal composition even begins?  I mean, what do you suppose is the starting point in conception?

[ajspiano]

How do you suppose a fractal composition even begins?  I mean, what do you suppose is the starting point in conception?

I doubt that one goes about composing a fractal composition..  rather that the possibilities of composition generally can perhaps be represented by a fractal algorithm.

I thought this was a telling paragraph.. 

Generation of fractal landscapes

A way to make such a landscape is to employ the random midpoint displacement algorithm, in which a square is subdivided into four smaller equal squares and the center point is vertically offset by some random amount. The process is repeated on the four new squares, and so on, until the desired level of detail is reached.

To me it seems related to the idea of motif variation..

Given a motif..
possible modal variations within a key - 7
inversion of all modal variations - 7
applied to all 24 keys (because we can modulate) - 322 variations
retrograde of all the above - 336 variations..

sequences of the above that are phrases constructed over 4 motif's - unfathomable.

So the motif is our "square" and its variations are all the smaller squares its subdivided into using different forms of displacement of the center..  or something..

I'm no maths wizz.

[Bob]

I think composers have been doing that for centuries, expanding and contracting things.  Some took melodies (I think), made that the bass line, and created new material over that.   Medieval music.

I don't think it can be measured with a ruler.  Schenkarian analysis is supposed to reduce things to their simplest form though, so you could compare Bach to Beethoven.  Style doesn't matter.  

And I remember a theory professor explaining, Schenkarian style, how a piece of music, Beethoven I think, had all these m2 intervals.  The whole piece could be reduced to m2 in terms of the melodies, the key centers, chord progressions, etc.... all boiled down to the m2 pattern.  

I haven't thought about that stuff in a while, but I think you can reduce pieces down to a a basic skeleton of the form and a basic skeleton of the melodies.  Something like that.  So they're on the same level as another piece.  But it can't really be measured.  It's a pattern.

And I remember a prof saying something about it's debatable whether composers were aware of those ur lines, if they started with something like that skeleton or interval and developed that or if they just wrote and that stuff happened to be there.

[ajspiano]

Personally I have an underlying structure that everything sits on top of..

for example, I hear/concieve the intended harmonic/melodic building block.. and I play all sorts of nonsense in, around and over it - but my conception or idea is always one specific thing and results in resolving to the basic harmony (or not resolving if that's the point I want to make), or getting to a point of melodic significance...

I've noticed how I understand this concept but others sometimes don't, because they hear what is absolute not what is possible. It happens if I play a pop song with my fiance.. she will say something like "that chord, its not right" (in reference to the official recording of it) and its because I'm playing a C6 instead of a C9 or something..  a chord derived from the underlying harmony which to me sounds right, because I hear possibility and choose one of the specific paths that fits..   

I think this can develop to really high levels too, and I notice it quite significantly with my recent explorations into baroque improv..  because I'm starting to be able to conceive variable harmonic/melodic paths for some of bach's inventions..  though it takes a lot of mental effort and I stall very quickly at this stage trying to keep playing and be aware of where I am and what can come next while maintaining proper use of his motif.

[ted]

Do you mind explaining a bit more about what you mean by "self-referencing hierarchy"?  

Bob's example of a fractal seems as good as any, m1469. It is a collection of abstract structures which contains structures similar to itself. Personally, while the Godel and Esher analogies are obvious, I have never thought the idea anywhere near as pertinent when it comes to music. Hofstadter is infatuated with the music of Chopin and Bach in particular, and seeks to explain its power (for him) through a lot of self-reference he perceives within it. I don't see it myself, but perhaps my fancy isn't strong enough.

[chopin2015]

One must build their own oscillator before one sees a mathematical equation to music. Crystal oscillators are in watches. Are they not? Anyways, why don't you take a vacation from classical music or maybe even go listen to some switched on Bach?

[Bob]

The homemade version I came up with was something like this...

For a melody, take out the repeating notes, take out the neighbor tones or passing tones, grace notes....   Rhythm doesn't matter for this. 

(Actually rhythm can do the same micro, 'regular,' macro pattern thing too like pitches.)

You end up with a simpler version of it.  Then you can take that and do the same thing -- remove repeated notes, neighbors, passing tones, etc.    Do that over and over until you get down to the most essential pitches.

Do that for phrases, for sections....

For chords... You can end up with the most essential chords of the piece.

Take the chords and the phrases and you'll see which notes are more important.  You can strip away the less important pitches from the pitch lines.

And then there's some idea Schenkar had (so said the prof) that... I think there would be certain "important" pitches during the piece that would lead the whole piece from Do up to V and then back to Do.  I think.  It was something like that.  Maybe that's the Ur line. Or if the composer was experimenting, maybe it doesn't go to V.  Maybe to IV or another pitch center.

I think that's what I did.  It was a long time ago.  It's not the official way.  I'm sure it's off and sloppy and incorrectly notated, but you can do that right now.  It does have that immediate payoff for the pieces you're working on -- And it really doesn't matter if you produce a perfectly notated analysis unless that's your goal, but it's probably not.

[chopin2015]

[m1469]

I will be interested to see what kind of bare bones structures could be found in pieces by Mozart and Bach and Beethoven.

[starstruck5]

I have been obsessed with this idea for a long time.  If music is the sun then its fuel is numbers.

I think you would be fascinated by the System Of Musical Composition by Joseph Schillinger. His Theory would definitely provide some answers to the questions you are asking.  I would say though that there is no direct Mathematical Equation which would define any complex piece of music -the equation is the music itself.

Check out though, the music of Daniel Leo Simpson -there is a video on You Tube which shows how he constructed a Piano Trio from some Schillinger principles -I am sure you will find it fascinating.  His Invention No6 in Bb for piano  is wonderful also -I am not sure if the System is still in print btw -it strectches to two volumes! However, you can usually find a good used copy somewhere -

[Bob]

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

https://josephschillinger.wordpress.com/introductio/


If you've got a few hundred bucks laying around...
https://www.amazon.com/Schillinger-System-Musical-Composition-Volumes/dp/B0006S7X8S/ref=lh_ni_t


https://forums.allaboutjazz.com/showthread.php?t=52537

https://www.schillingermusic.com/



https://www.amazon.com/exec/obidos/ISBN=0306707810/harrylydenA/
The Mathematical Basis of the Arts

Haha.  It might be this thread in book form.... sixty years ago.

[Bob]

What happened to this thread? 

I got the Schillinger books.

[the89thkey]

This thread had died---and you revived it. Not good to do.

[Bob]

*Bob pokes at the thread with a stick to see if it's still alive.*

[m1469]

Heyyyyyy!  YAY, this thread .  I was thinking about these drawings just the other day!  I am a little bit stuck in some strange vortex and slowly finding my way out  . 

[the89thkey]

This is not a real thread...it should probably be in the random discussions forum.

[Bob]

It appears to be some kind of solid.  *Bob continues poking thread with a stick.*

[m1469]

I'm in some really far away land, currently dominated by augmented triads, 7th chords, the realization that I think of chords way differently than I'm "supposed" to (and finally something makes sense for me regarding that), intervals of a third, and chord progressions.   I googled common chord progressions today and the first on the list was leading me to the forum, to your list about chord progressions, Bob  . 

[rachmaninoff_forever]

Has anybody mentioned Debussy in this thread?

[j_menz]

Has anybody mentioned Debussy in this thread?

They have now, but not hithertofore.

[look it up]

[zezhyrule]

Dat beautiful Golden Ratio.

[j_menz]

Dat beautiful Golden Ratio.

OK - I knew about that in art and architecture, but didn't know it had been applied to Debussy (and apparently Satie and Bartok). Hmm.... not sure I don't think it's not stretching matters.

(triple negative reflects indecision - changed mind 3 time while completing sentence; be warned, this is not a record)

[p2u_]

But, is it theoretically possible to actually directly translate a piece of music into one, giant(?) mathematical equation?

It's hidden territory because we would go mad if we did so, but there must be at least some truth in it, yes. If we attempt to find out for sure, then we should not only limit ourselves to the tone systems and chords, but also to the system of overtones, partials and harmonics. Certain music makes pain go away (depends on who is performing!), other music makes your heart go faster, punishment through sound makes you lose your mind, etc. This is not just pure emotions or temperament, not just that it distracts your attention, not just that endorphines are released, etc. It's the underlying mathematical and physical laws that make it all happen. But as I said: it's hidden territory and one should not attempt to go there.

Paul