Keep in mind that when most of the artworks in which people calim Godlen Mean as a design principle were created, people did not think of "mathematics" in the same way we do today, ie. as a complex conceptual language which can be used to represent various aspects of reality. They thought in terms of proportions, and they used geometry as the primary manner of manipulation. That's why a log progression is called a "geometric" progression, as opposed to linear, which is "arithmetic". Using similar triangles and other tricks was the only way they had to construct such progressions.

Be VERY careful with any claims to the presence of the Golden Mean in music and art. If you look hard enough, you can find anything anywhere. My favorite: did you know that the keyboard is based on the Golden Mean? How so? you might ask. Well, everyboy knows a Fibonacci series rapidly arrives at the Goldon Mean, right? Now the keyboard has:

13 notes to the octave, divided into

8 white keys and

5 black keys, the latter of which are divided into two groups of

3 and

2.

QED.

Needless to say, this argument is totally ignorant of the real and quite well-documented historical development of the keyboard. It also violates the basic musical principle of repetition at the octave, thus the octave really only has 12 notes, with 7 white keys. But hey, for the Golden Mean junkie, these are but trifles.  ;-)

Hello, I've read this and was wondering if this is interesting for this blog.

http://derstandard.at/2000004299463/Da-ist-Musik-in-der-Mathematik

regards Georg

See the work of Princeton's Dimitri Tymoczko for the richest approach I am aware of to establishing a clear relation between music and math.  For example in 

http://dmitri.mycpanel.princeton.edu/geometry-of-music.html

When you ask whether art and music are branches of mathematics, you may very well be limiting consideration to Western art and music alone and leaving Eastern art and music out of account.  When calculating the golden mean, what do you do about Indian musical scales less rigid than the diatonic scale and varying with the need of the composer and/or the performer?  Use of the Golden Ratio presupposes mathematic relations BEFORE the artwork comes into being.  But music with flexible scales may mean that mathematical relations are created AT THE SAME TIME as the artwork.  The same could apply to the latest graffiti art.  If artistic figures are not preplanned, then mathematical relations are problematic and may even vary as the artwork is coming into being.  You would have to speak about mathematical probabilities, but I suspect this goes beyond the scope of your question.   So I agree with Cecilia: sometimes yes, other times no (perhaps the majority of cases). 

Could you mathematize microtones in general-- I.e., tones of diminishing musical intervals--?  If you can assign such tones musical notation on scores, then mathematization is surely possible.  Examples of microtones appear in Spanish flamenco and deep song singing and guitar-playing, and in classical Arabic melodies with flattened tones, i.e., tones that end a half interval flatter then when initiated.  What form of mathematics could apply?  Infinitesimal calculus, without a doubt.  Is mathematical calculation desirable?  Possibly, if it facilitates notation and composition. 

Its always been my sense (as a visual artist who works with graphic music notations) that the relationship between music and math is most clearly drawn on the page. Symmetry in  Bach, for instance, is pretty clear when you have the sheet music in hand. The inscription on the page influences the listener (who can read it), and must have moved the composer as well, when he picked up on the visual regularities. 

There are a lot of books that describe this relationship.

One of this is "The golden ratio" of Mario Livio.

http://www.amazon.com/The-Golden-Ratio-Worlds-Astonishing/dp/0767908163

Ms Qefsere,

I have some thoughts to share.  Lets look at the genesis of disciplines of thought and the archaeology of  knowledge.  When our ancestors tried to communicate, they used some sounds - both vocal and instrumental.  Didn't the hunters use bugle or other sounds to trap their prey?  They also painted on the cave indicating the location of 'things of importance' - like danger, water,  animal, etc. They used tools for various tasks and engaged in crude methods of measuring distances and observing night sky, etc.  In my opinion, this is possible only with a scientific bent of mind.  Being able to produce sounds (vocal and instrumental), being able to paint or draw or even represent (code) and being able to get a sense of time and space, so on definitely, have to make a premise in "logic" which is integral to Mathematics (science?).   If you believe there is logic in music and art, they should necessarily relate to Mathematics.  The question of what is first - egg or chicken?- needs more analysis.  That is, are art and music branches of mathematics is a moot question which needs some empirical answers.

Regarding modern art and music - metrics, scale, etc definitely have mathematical relationships.  Haven't we heard of a saying that " there is a pattern in madness?'" and "chaos theory?".  A Very good question.  

Actually, in a strict sense, the obvious answer is "No". Mathematics is a limited discipline, a complex abstract language for the quantification and the representation of reality. One can use mathematics as a tool to describe certain aspects of music and other arts, but they are not themselves mathematical disciplines. Anyone who has ever tried to teach any subject which requires even a modicum of mathematics to musicians knows that they are more often than not utterly miserable at maths, sometimes at the most basic concepts, such as square root. QED

Years ago I wrote my masters thesis on the Golden Ratio (Phi) and the Fibonacci series. (see D'arcy Thompson). It is indeed thought provoking learn that a proportion which seems to be inherent in biological growth was also identified by the Greeks in their geometry as being 'beautiful' . Phi is derived form pentagonal geometry and pentagonal symmetry is common in nature, perhaps this is the reason. But, most of the so called scholarship on golden mean geometry in art is, in my opinion, vacuous. (Ghyka et al).

Mathematics is not one thing, and 'art' and 'music' are such general terms as to make your larger question unanswerable.

To continue - my conclusion then, which I now believe more strongly, is that impositions of Golden section geometries - or other such analysis, on paintings, is about as useful as phrenology. In *some* cases, it can be shown to have been applied - by evidence in the artist's work. In obvious cases, such as Islamic patterning, it is not a subject of conversation because it is irrefutable. 

This source, about branches of mathematics

http://ls.poly.edu/~jbain/Cat/lectures/01.BranchesofMath.pdf

literally does not include arts and music as a branch of mathematics

Giorgio Griffa: The Golden Ratio And Inexplicable knowledge – Revd Jonathan Evens

The Golden Ratio – also known as the Golden Mean Proportion, the Golden Section, and the Divine Proportion – has found expression, since the Renaissance, in many works of art in the form of Golden Rectangles, pentagrams, spirals and triangles. Within the development of modern art, it featured as part of a quest for an objective, scientific approach to art. That search began with Georges Seurat, who was widely believed to have developed a scientifically based understanding of colour known as Neo-impressionism or Pointillism in conjunction with the mathematician Charles Henry. This style spread across Europe, with the Italian Divisionists becoming strong proponents and paving the way for the development of Futurism...

Severini had ‘turned to a numerically/geometrically based figurative painting, arguing that painting had been discovered as a science at the time of the Renaissance and that this was a progress which could not be negated by a return to the Egyptian or the Romanesque.’ Severini writes in his autobiography The Life of a Painter that ‘many artists liked to discuss geometry and mathematics’ but that he found their discussions insufficient thinking ‘that artists should apply, and would benefit from, strictly observed rules of geometry and mathematics’ which had value ‘beyond their constructive value’ through ‘something strictly innate to artistic creativity.’ ...

http://www.artlyst.com/features/giorgio-griffa-golden-ratio-inexplicable-knowledge-revd-jonathan-evens/

In the discussion https://www.researchgate.net/post/Is_Democracy_the_best_form_of_government_in_a_Federating_State the problem of creating a subject "Mathematical History" was formulated, which then found its continuation in the discussion https://www.researchgate.net/post/What_gives_credibility_to_science_Can_we_really_rely_on_scientific_discoveries. Here the main goal was set - to narrow the gap between the interpretations of the same historical events and the forecast of future historical events. I think it's time to create a subject called "Mathematical Music". Maybe here the main goal would be a better understanding of music through mathematical models and the forecast of future music using these models. Yes, it is sometimes said that talent is the key to success in understanding and composing music. However, is it possible to manage these events using mathematical models, which would contribute to the best achievement of musical results?

Ljubomir, Gennady and all - why do you feel mathematics has to dominate the arts? If you look back on this thread, I posted - 2 years ago or more. As a graduate student I wrote my masters diss on the Golden Section and Fibonacci numbers. Lovely stuff, fascinating. But if you think contemporary art is about beauty (or truth), I'm sorry to say you are out of touch. Nor do any artists I have known in the last 30 years -and I've known hundreds - care about this stuff - well, maybe 1 in 1000. You welcome to have your opinions, but the math most artists are now interested in - if they're interested in any - is about file compression, machine learning or google metadata manipulation. Gennady, if you are interested in mathematical music, look at David Cope's work. ie

https://www.youtube.com/watch?v=2kuY3BrmTfQf

http://artsites.ucsc.edu/faculty/cope/experiments.htm

I love the rule of thirds, fibonacci code or the golden mean. One views all through nature.

1, 1+1=2, 2+1=3, 3+2=5, 5+3=8 etc, etc, etc. Flowers 5 petals, crustacean and ammonites follow the code. The works to of the Renaissance painters can easily be divided by the codes. The human body parts can be divided in the same manner. Once one learns to see the pattern, it becomes easier to draw according to the code.