Maths in Music

[russ]

Theres probably a really good topic on this already, but im sorry, i couldnt find it. I needed some help, because im doing this project at skool and i need to identify the mathematical aspects in music. I was thinking of researching about how Bach was supposedly a mathematician. Help in any way would be great - its also kind of urgent.

Thanks for any help, Russ.

[Tash]

the ratio stuff between notes (ie. tones, semitones, intervals etc), with all the logs and stuff agh i'm so vague we learnt about it last year and i can't find the notes on it. if someone can elaborate that'd be good! otherwise i will attempt to find my notes.
also the possibility that bartok made use of the golden fibonacci series thing in some of his works he has key parts in the music set in this same proportion as the golden section
sorry i'm really vague my notes are even vaguer i have no idea how i managed to pass musicology last year! but if you're interested in the interval stuff look for books on the science of music and that has all the detail. hope that helps in some really vague way!

[Dazzer]

mozart and the golden ratio.

[claudio]

that's a huge subject and i am sure there is lots of information on the
internet. if you can read german try: https://www.kzu.ch/fach/mathe/Unterricht/mathemusik/MatheundMusik.htm#Down1

for an english speaking page try: https://www.math.niu.edu/~rusin/papers/uses-math/music/

seriously: speak to your teacher or try to narrow down your topic yourself; e.g. pytagorean theory of music, euler math, theory of cords, differences in well temperament and natural temperament tuning, etc. otherwise you get crazy because there are so many aspects.

[xvimbi]

I agree with claudio; you will have to narrow it down a bit. If you want to focus on Bach, definitely check out the book "Goedel, Escher, Bach" by Douglas R. Hofstadter. Any good book store should have it. You'll be exposed to concepts that are common to Bach's music, Goedels theories (particularly his famous Incompleteness Theorem), and Escher drawings. They are then extended to biology, computer sciences, game theory, and there are a lot of fun riddles akin to Lewis Carroll's Alice in Wonderland. It's a huge book, and the subjects covered are not that easy to comprehend, but it doesn't get much more exciting and holistic than that.

[russ]

Well, ive decided that im going to invesigate whether Bach was a mathematician, and all that jazz. If anyone had any info on this more specific request that would be fantastic. Thankyou to all the previous replies .

[xvimbi]

Well, ive decided that im going to invesigate whether Bach was a mathematician, and all that jazz. If anyone had any info on this more specific request that would be fantastic. Thankyou to all the previous replies .

I have no idea what you mean. Do you mean that JSB solved mathematical problems? That he advanced mathematical theory? That he earned his living as a mathematician? Or that one can find mathematical patterns in his music, and - more specifically - that he used mathematical patterns/algorithms to compose his music?

[stormx]

I am a Mathematician, and a beginner piano player.

Although there is relationship between music and math, i beleive it is somewhat overrated...

[russ]

Sorry about the vagueness,
i wanted to investigate how Bach used maths in his music, like the last two things xvimbi said, if there is any mathematical patterns in his music, or anything like that. basically the theoretical side of things - its becoming more urgent now, if anyone knows anything can you reply soon, thanks.

[ted]

I tend to agree with stormx. Leaving aside the question of whether algorithmic composition, still in its infancy, qualifies as using mathematics to write music, I feel the supposed connection is more of a high level analogy than anything else. Both fields are heavily concerned with abstract form, and as Hofstadter asserts in the book rightly mentioned by xvimbi, self-reference and its associated ideas of chaos are intrinsic to abstract form of any sort possessing more than a rudimentary degree of complexity.

As it happens, Bach used a lot of self-reference, as indeed do a large number of musicians and composers, even the thoroughly romantic ones. However, there are more differences than similarities. Music is intimately concerned with subjective beauty and the transmission of shared experience; mathematics is concerned with what is objectively and rigorously true. Contrary to nineteenth century romantic ideals, truth and beauty are not always compatible let alone bedfellows. 

Therefore while a large number of people with a love of abstraction understandably embrace both fields with enjoyment and success, the assertion that the actual mechanism of one can drive the other seems to me to be false. It has been attempted, many times and in many ways in recent decades, to directly apply mathematical rules to creating music. I do not like doing it personally, and so far I haven't much liked the results of forays made in that direction by other people. But each to his own taste.

[trunks]

I used to be a high school mathematics teacher for many years.
The human ear detects a doubling in sound frequency by perceiving it as an octave higher in pitch.

Therefore the human ear is a base-2 logarithmic scale.

[Derek]

There are also those who say studying math will make your music better, or studying music will make your math better. Why then have I excelled the past few years in piano but have been a straggler in math courses, taking many of them twice and getting C's in the rest?   I honestly think the real connection in academics is that both math and piano tend to discipline students. My piano teacher doesn't let me get away with: "Well, I think I'll practice that at home, it'll take me a while to get."  He says: "Well you can also try to do it right now. "  And he's never unpleasant about it, but I think it has sharpened my mind in a way very few educational experiences in the past have.  (still not to the point of being an outstanding student, though) But this proves my point--its a discipline thing rather than an intrinsic similarity between mathematical and musical thought.

[asyncopated]

Hi,

I tried to resist posting here, but to no avail.    So, I just wanted to make a few comments.  Despite my wanting to believe that there is a link, I guess that the connection is tenuous at best.  But if are searching for one, perhaps the following will help.

There are a large number of mathematicians and physicist that are interested in music.  I come from a scientific institution, so I should know.  Apart from the discipline required in both subjects, I suppose we also share a common ground, in a quest for abstract beauty.  Another possible reason is that many scientists, including myself, consider music a kind of a release form science.  Music seems to find a place as an intellectual equivalent, but is sufficiently different, addressing a completely alternative dimension of life (as opposed to the mundane -- solving equations). 

I guess that apart from that there are similarities in terms of philosophical aspects.  Much of science is based on a reductionism.  That is, you try to construct a picture of the world using a few basic rules and elements.  For example, with Newton's laws, you can explain different things, from how cars move to how moon orbits the earth one three simple ideas.  Essentially these ideas it tell you about how things that move. 

One can say the same for a Bach fugue.  The structure of a fugue can be reduced to primary and perhaps secondary subjects (a simple idea).  There are vague rules for how the polyphonic voices can carry the subject, and how one can carry on to develop it.  There are a simple set of rules for how these voices can interact. E.g. Triad and semitone intervals are seldom used and if they are, only to deliberately create a discordant effect.  Fifths and its inversion fourths are encouraged because of harmonious balance. Ending in thirds gives the phrase a character that it is unresolved.  This is because it is sufficiently far away (in terms of what you have to do to get a third from root note) from the tonic.

Essentially the there is a huge amount of beauty and subtlety in a fugue.  (Not to mention double fugues, the Kyrie in Mozart's requiem comes to mind)  How one manages to realize such an idea is beyond me. 

It seems apparent to me that Bach, as with many modern mathematicians was obsessed with symmetry and was able the find than extract beauty from it.  Although bound by a different set of rules, he required it of many of his pieces.  For example the St. Johns Passion is a musical palindrome.  Modern mathematicians live by symmetry -- from the time of Hamilton and perhaps even before. 

Perhaps if I am permitted to stretch things a little, I would say that Bach, although probably never having solved an equation in his life is still an intuitive mathematician.  He solved abstract problems in music rather than math, using basic principles that are pillars which uphold both disciplines -- reduction, abstraction and symmetry. 

al.

[Daevren]

Music is described with the language of music. Not in the language of mathematics.

Of course music is based on physics, which we describe in the language of math. Music itself has its own rules. I don't really see what they have to do with math. Of course music is very structured and has its own objective rules and guidelines but this has nothing to do with math.

You could call complex percussion music where you have to count to 23 or something related to math at most.

[asyncopated]

I am doing this to make a point.  Be forewarned it requires some (advanced?) maths .  This does not in any way affirm or invalidate the beauty of the piece.  It is also probably not very useful, except that if you understood that, you will probably also see where bach's ideas came from.  It's also just one version mathematical analysis and certainly not refined.

Let's talk about bach's first invention, in particular just the first few bars of the treble.  It is a fantastic piece, astounding in compositional value.

Let the subject in the first bar + semiquaver be the subject. S.  Also let the subject be a concatenation (joining) if two part S:=S_1^S_2 the first with S_1 seven semi-quavers and the second 4 quavers. I.e S_1:={c d e f d e c} and S_2:={g c b c}


Also we define the following transformations Mx[ ] - is a mode transformation to where x is an integer from 1 to 7 giving each of the modes.  Such that S==M1.  Ionian mode is the same as the original in this case.  (note we can extend this beyond 1 to 7)

Let Kx[] be a key transformation - transpose K5 is S played in the key of G.

Let I[] be an inversion about the key.  The first few note S1:= {c d e f d e c}
  I[S1]=={c b a g b a c} (descending first).

Lastly, incidental notes are just denoted in curly brackets {}.

The first few bars can therefore be written

S^{d}^M5^{e}^M13[I[S_1]]^{g}^M11[I[S_1]]^{e}^M9[I[S1]]]^{c}^M5[K5[I[S1]]] (notice we have modulated to the key of g)^{a d c d}^M3[K5[I[S1]]] ...

https://www.sheetmusicarchive.net/compositions_b/b2part_1.pdf

Basically, you can write the musical ideas as transformations -- inversions (in key), stretches in time, key changes, mode changes, inversions in time, and see how he develops the subject and or subsets of the subject.  And we haven't even introduced interactions between voices.

Like i say, this is not very useful for musicality -- and certainly not at all musical.  Does it add to the music? I don't know.  Does it add to an understanding of the composition? Probably.  Was bach able to (intuitively) plan and see all these transformations and how they interacted with one another? Definitely.  Was bach good at math?  In one type of math (the type that I am demonstrating), a genius.

al.

[russ]

Right, well, asyncopated, you officially rock. I'm sure il be able to understand that if i read it a bit more thoroughly - and my mums a maths teacher, so im sure between the both of us we'll be able to work it out. Thanks a heap.

I suppose I was opting for the differences between the "Just Scale" (that Galileo (?) I think developed as a rule for stringed instruments) and the "Well Tempered Scale", as in the differeing frequencies altered the sounds of notes. But i suppose your theories could be another aspect of how Bach's music was influenced by maths, or how today we can somewhat mathematically analyze Bach's pieces.

Once again, thankyou very much for your time and effort.

[asyncopated]

Hi Russ,

You're welcome!  The ideas about the differences between the just scale and equal temperament are very interesting as well.  I never found out how bach, figured out (historically) out equal temperament.  If you do learn of the story let me know!

I have not actually seen a formal mathematical analysis (i.e. written in terms of a symbolic structure ) , but most musical analyses treat fugues and the like in a very robust and comprehensive way. 

Here is a suggestion if you want to pursue this.  I would start with rounds, move on to cannons and after that to fugues.  Rounds have the simplest structure in that all voices sing exactly the same notes.  It is also very easy to do a harmonic analysis for this reason. 

Cannons are more complex in that each voice may have simple transformations of the subject, but no incidental notes are permitted.

Fugues are similar to cannons, but to make things more flexible - incidental notes are allowed, figures using subsets of the subject and many more complicated transformations are also allowed.

Also xvimbi also suggested reading "Gödel, Escher, Bach". It is a technical book, and not too easy to read.  If you are up to it, i would recommend the same.

Have fun!

[sznitzeln]

I think Gödel, Escher, Bach is very unscientiffic. He takes a little connection that exists between these 3 things, and magnifies it, and stretches it. It is so overlong. He also tryes to make the connection more significant that it really is. He also makes some statements about human intelligence vs AI , that I think is ad hoc.
But the book is entertaining. So it has some value. But if you want to learn stuff, study music, study mathematics, study art

[Eusebius_dk]

D-sharp * mezzoforte - quaver / (stringendo + A-flat) = 378.62541

[trunks]

Um . . . perhaps this is less decipherable than
A-sharp = Immediately adjacent A * twelfth root of 2 (!!)