Dear Costas Drossos

The relationship between mathematics and music are very old . In the Greek world in the century. VI BC, the Pythagoreans emphasized the role played by the number and proportion in understanding the universe . They felt that the song contains a hidden arithmetic and that harmony is the proportion that unites the contrary principles in the constitution of any being . The Pythagoreans distinguished two types of harmony, sensitive harmony that is felt by musical instruments and harmony that is intelligible in terms of linking numbers

Music existed before the development of mathematics , because the combination of sounds , though largely dominated by mathematical relationships , based on our psychoacoustic perception, ie , our physiological perception of sound . The formation of sound and music is a physical process . The sound is a physical phenomenon and as such is part of the study of physics. Music is the art of combining sounds ( and silences ) . Therefore, to understand deeply music is necessary to know physics.

Music can be used to illustrate some mathematical concepts . Figures time ( duration) of notes, for example of compass are fractions of the type 1/2 , 1/ 4, 1/ 8, etc.

Obviously we can not cover all the possible dimensions of the relationship between mathematics and music as they are manifold and extremely complex .

What surprise me, reading the book:[Francis_James_Budden]_The_Fascination_of_Groups Chapter 23. Groups and music, that considering say a do major scale as cyclic group, then the subgroups of this give the usual accords of Sol^6, fa major, etc. Thus what pleases our ears are just the subgroups! Actually this situation synthesises: music, mathematics, and the physiology of the ears!

Musical notes are related to numbers...

Mathematics of Music

Hi Costas,

If you ask me,... rhythm,... e.g. 1/2, 3/4, 4/4, 3/8, etcetera. This determines how notes will be written and played.

I don't say that a melody is subordinate to rhythm, but when you look at the history of music, musical archaeology if you like, then rhythm is the first element in music popping up in cultures with a limited pallette of musical instruments. sometimes even voices are the only 'instruments' utilized in some cultures close to nature, rhythm is the principal factor in their songs.

Can you live with that Costas?

frank

The personal timing and drive -- which cannot be described precisely -- are very important in music. They may have their own math values, in a way, but I think it is the deviation from the notation (exact math values which may not necessarily be learnt but perceived somehow, e.g. through rhythm) that makes music and musicians unique.

I know that there are educational methods in the US which approach math through music. I am not sure what level of education they are targeting and how efficient they are but this is something peculiar to understand - how music can help translate math, what audiences can be reached (musical vs. all vs. arbitrary) and perhaps namely some relations in the process of education and learning.

Dear Kamal, Sribas, Hristina, Frank, thank you for your answer. Seeing Kamal's answer I remember tha Pythagoras divide mathematics in:

Arithmetics or numbers in calm

Music or numbers in motion

geometry or magnitudes in calm and

Astronomy or magnitudes in motion.

Costas and Kamal,

Implicit in what you have written about the relation between mathematics and music is the fundamental notion of the structure of a set induced by some relation.

To see this idea in the context of music and mathematics, let X be a set of sounds that can be made with a muscal instrument. Then the relationships between the sounds is defined by a musical composition. And then a composed set of sounds (music) induces a structure on the set of instrumental sounds.

For example, one can find a cyclic group in the set of sounds for a major scale. Do you agree with the idea of structured sets of sounds? Can you give an example of such a set of structured sounds?

Dear Costas Drossos

The relationship between mathematics and music are very old . In the Greek world in the century. VI BC, the Pythagoreans emphasized the role played by the number and proportion in understanding the universe . They felt that the song contains a hidden arithmetic and that harmony is the proportion that unites the contrary principles in the constitution of any being . The Pythagoreans distinguished two types of harmony, sensitive harmony that is felt by musical instruments and harmony that is intelligible in terms of linking numbers

Music existed before the development of mathematics , because the combination of sounds , though largely dominated by mathematical relationships , based on our psychoacoustic perception, ie , our physiological perception of sound . The formation of sound and music is a physical process . The sound is a physical phenomenon and as such is part of the study of physics. Music is the art of combining sounds ( and silences ) . Therefore, to understand deeply music is necessary to know physics.

Music can be used to illustrate some mathematical concepts . Figures time ( duration) of notes, for example of compass are fractions of the type 1/2 , 1/ 4, 1/ 8, etc.

Obviously we can not cover all the possible dimensions of the relationship between mathematics and music as they are manifold and extremely complex .

Indeed for music playing mathematician it is an activity of a mathematical mind with right tools at hand. Music – generating notes and pitches with a best possible combinations, compositions, and norm ( or magnitude ) calculations with changing periodicity and frequencies.

As mathematicians, we use our mathematical ability to play a structured activity of a structured system through compositions, additions, magnitude formations, changing periodicity and frequency (timing) of notes and pitches of controlled elasticity, to fit to the best of our test.

I basically say, it is a spatio-temporal activity of group theory, norm analysis (probably norm of functional analysis, because we use not just a length of a traditional vector, but a magnitude of a varying pitch on a certain interval) and physics.

I have found nice interpretation, Music Tech & Math Music! Take a look into a Magic Chart!

http://music-technology.blogspot.com/

Please check books on pitch class set theory and there are many unsolved problems in mathematics to music with respect to designing the tones and tondals

@Kamal : Dear Kamal you are joking! However you do not know that there is really a mathematical Surgery theory:

http://en.wikipedia.org/wiki/Surgery_theory

@James Peters: Dear James, I agree with you. An example of a structured set of sounds is any accord, say Do major for example. Furthermore group theory plays a role into that when sounds travel there is some invariant organised sound that reach the hear. This is a kind of "Geometry of sounds".

There is an interesting paper: The mathematics of Musicians:

Dear Costas,

What a beautiful question and very interesting answers too. I just want to step aside into a bit of philosophising...

Both music and math are a medium through which we attempt to connect to an unknown. Both have sound (!) sets of fundamentals that help us to build structures that elevate the mind towards infinity. James peters was seeking examples of structured set of sounds. In addition to what you have stated I would like to mention 'raga' - in classical Hindustani and Carnatic music traditions 'raga' is a composition of sounds nuanced in a particular style, ascending and descending with a theme that usually connects to expressions of nature or expressions of the heart, for example there are 'ragas' that sync with the break of the dawn or the pitter-patter of rainfall or the rolls and claps of thunder; there are ragas that denote the moods - love, pride, surrender, sorrow, etc.

Constructing an 'accord' or 'raga' is a mathematical process of the mind, even if the musician consciously does not think so. The mathematician does so by applying appropriate mathematical structures in the process of which he seeks a predictable but as yet unknown. The musician does it unconsciously while seeking the unknown in his heart/soul. Both of them start with known fundamentals and some where in their journey manage to loose their self in their pursuit of the abstract. They fall in love and then become one with the beloved....perhaps this is why many mathematicians play instruments. It helps them connect.

Have a great day!

Dear Srinivasan, you posted a great post!

Dear Costas,

The paper by Vicente Liern is excellent! He covers a lot of ground and presents some beautiful results.

Thank you James!

Yordan,

Perhaps you will find Section 3.1 (starting on page 19) of the attached paper interesting.

Pythagoras found the relation of musical intervals with ratios of integers, using the interval of the fifth to create further intervals, implicitly introducing a theory of overtones.

http://eprints.ma.man.ac.uk/1548/01/covered/MIMS_ep2010_103.pdf

The musical interval between two notes that were produced by two different hammers,

depended only on the weights of the hammers, and in particular the consonant musical

intervals (which, in Ancient Greek music, was the intervals of the octave, the fifth, and

fourth), corresponded with weights to fractions, 2/1, 3/2, and 4/3 respectively (p. 21). See Figure 7, page 25.

Pythagoras conducted a series of experiments, as explained in [4], using different

instruments to confirm the relationship between musical intervals and fractions.

Yordan,

Many thanks for your comments. In my earlier post about Pythagorus, I forgot to mention that there is connection between Pythagorus's approach to music and fractals.

See the edited volume by J. Fauvel, R. Flood, R. Wilson, Music and Mathematics. From Pythagorus to Fractals, Oxford University Press, 2006:

http://www.amazon.com/Music-Mathematics-From-Pythagoras-Fractals/dp/0199298939

I asked to have the opinion on mathematics and music from some experts: Jean Maruani, Marja Rantanen and Roland Lefebvre. I attached their excellent paper.

Nishad,

Yes, definitely, viewing music as the mathematics of the senses is reasonable. See Prof. Drossos' comments from earlier posts to see how what you are suggesting can be realised.

Music is closely linked to the way our sensory-motor system control our body. It is why dancing and music goes hand in hand. Try watching a movie without sound. The movement are not perceive as well but most importantly we lack the emotional narrative of the movie. Our sensory-motor system has a lot of built-in timing for the activation of the body along thousand of automated movement control. This information is probably highly organized and might be involved in perceiving music, in dancing on music, and producing the movements for playing music.

Music is related to body movements and it is related to emotions. There are all kind of sounds out there but only music emotionally moves us. There is a huge industry of musical sound but there is no industry of other type of sounds. You would not record street sound and listen to it for hours. What emotionally touch us in general? Natural beauty such as mountain, the power of tunder on the ocean, etc but primary expresssion of human emotions: a crying baby, a crying adult, the love in the facial expression, the beauty and attractivement of another person, etc. Maybe music is such emotional expression. It is a very old idea, not confirmed scientifically yet, that the first human languages were mimetic dansing and singing. Humans are among all mammals, the one who are most able to coordinate their actions as a group. What is best for unite a group than a common participation to a dance , a song. We engage into a common living fiction. We get a sense that what we feel inside is felt by everyone and this create a sense of participation into a bigger living group. It is not an illusion. The illusion is to think that what we feel and what we think is personal. In reality,all came from the group. Living the particiption reality in collective dancing and singing makes our connection to the group explicit and any expression of emotions through movements and musical sound allows can be reacted by the whole group,

Singing bird, singing whales do the same.

Michale Tilson Thomas said nice words about classic music! He speaks about "Mood Trajectory"!

Here is a link to the famous shower scene in the movie "Psycho".

"Psycho"

https://www.youtube.com/watch?v=8VP5jEAP3K4

1. Watch it without sound,

2. Close your eyes and listen to it.

3. Watch it with sound.

All the emotions are in the music and in the screams almost nothing in the visual.

Thank you Costas for this question. I read the paper of Jean Maruani, Marja Rantanen and Roland Lefebvre, "SCIENCE AND MUSIC: FROM THE MUSIC OF THE DEPTHS TO THE MUSIC OF THE SPHERES" which you attached. As they said that connections between science and music (and philosophy) go back to remote periods of history, when they did not exist as such but were part of magic rituals.

I want to thank you all for your interesting posts for this importnant question!

What is the link between the most hard core rational domain ''mathematics'' and the most emotional of the art ''music''?

First both are an artform. The excitement of the mathematician is not simply the proven theorem but the creative journey, a journey which is a navigation in the dark towards a bright clarity guide along the way by a sense of aesthetic.

Second what is central to the mathematical artform is finding structures and patterns.

Leibniz said "Music is the sensation of counting without being aware you were counting." It is I think a mistake to equate ''music'' with ''musical sound''. It is confounding the communication signal with the meaning of the message. The message has meaning for a listener. The signal evoque the message in the listener but it does not mean that what is evoked could be understood completely by analysing the message. These sentences are a message but their meanings will be totally different for different readers. The musical signal is meaningfull , emotionally moving, aesthetically pleasing when the patterns in this signals are interpreted by the part of the sensory-motor system responsible both for body movement and emotional expressions. When listening to music, we are like listening to a fascinating narrative , we are in the expectation of what is coming next with our own sub-conscious expectation and problem to be resolved and are constantly confronted by some unexpected resolution by the composer and musician and stay captivated by the narrative as long as we are partially conforted into our expectation but not totally so and get surprised. We kind of enter a mathematical world of patterns with no single solution and goes onto a kind of mathematical journey at the same time as an emotional journey.

Mozart's first biographer, in collaboration with Mozart's wife, related a congruent account of how Mozart composed:

''Mozart wrote everything with a facility and rapidity, which perhaps at first sight could appear as carelessness or haste; and while writing he never came to the klavier. His imagination presented the whole work, when it came to him, clearly and vividly. …. In the quiet repose of the night, when no obstacle hindered his soul, the power of his imagination became incandescent with the most animated activity, and unfolded all the wealth of tone which nature had placed in his spirit …. Only the person who heard Mozart at such times knows the depth and the whole range of his musical genius: free and independent of all concern his spirit could soar in daring flight to the highest regions of art.''

http://en.wikipedia.org/wiki/Mozart's_compositional_method

‘’For Pythagoras music was mostly a math science which its priority was numbers and

the harmonic relationship between them. The universal movement and that of human soul are

based on the same harmonic relationship. Music was also the image of the universal harmony

which transferred to planets and people character .’’

‘’There are at least four aspects of mathematics which can be related to dance: spatial

exploration, rhythm, structure and symbolisation (Tytherleight, Watson, 1987). Mathematics and music share a concern with numbers and patterns of change. In

music and dance –movement these patterns are called rhythm. And this is their basic common

element. (Eisenhower Southwest Consortium,1998).’’

‘’students who

attended class with music, movement and theatre scored in maths 77% while students who

didn’t attend the same class scored 55%.’’

‘’Recent researches (Nasiou, Pollatou, Gerodimos, Goudas, 2007) proved that music

procession can get positive results to basket ball skills learning.’’

‘’The most important in the research

was that logic-mathematical met and identified with bodily- kinaesthetic. The knowledge

rose inside movement and rhythm . Rhythm and math consist common elements’’

Source:

http://www.emuni.si/Files/Denis/Conferences/EMUNI_HE-R/Proceedings/Papers/62.pdf

Thank you Louis for the paper!

Only recently I observed this question and maybe people already mentioned it and yet, can’t hurt to repeat it.

As far as I know, Pythagoras was the first (at least in the West) who not just analyzed the relation between Math and Music, but who actually established the relation 2:1 between frequencies of two sounds as too close to be distinguished when played together, while 3:2 was sufficiently distinct to be heard as different sounds. The result is that 2:1 defines the octave (because of the 7 distinct basic notes with the 8th actually repeating the first) and the 3:2 resulting in the other 7 that are used until these very days to build melodies (sounds in series) or accords (sounds in parallel).

Other, more complex ratios proved to be too dissonant to lead to any real “music."

Nice Post Christofer! We thank you.

@Costas, not all responses are available on your thread! Anyway, I have found a fine paper regarding the issue, here at RG database! Hope it is not duplicate of some previous answers!

Article Mathematics and Music (Mathematical World, Vol. 28) by David Wright

I always do mathematics while listening music. It keeps me fresh.

Math is an art of thinking. music is art of sounds. painting is art of colours. literature is art of words, etc.

art is an action of giving an (aesthetic) form to these elements. art in math is to form thoughts to solve problems.