Mathematics of Music: Interconnections & Questions

[chaoseverlasting]

Music and mathematics are irrevocably related. One of the things that struck me about both is that they are both products of our mind and exist solely in our heads (this observation, as much as I wish it were, is not mine, but I wholly stand by it), and we find relations between mathematics, music and the world around us, to help us interpret the world, but they are only really there in our heads.

I have recently (about four months ago) taken a healthy interest in music, and about two weeks ago realized the strong interconnections between mathematics and music. It really is astounding. I suppose there are a lot of people on this forum who are similarly interested in both and I'd like to put some questions to them and anyone else who can point me in the right direction.

What I understand of the relation between the two has to do with my limited knowledge of harmonics. The note 'A' is taken to be the base line and its frequency is fixed at 440Hz. Doubling the frequency (the first harmonic) gives the same note, just one octave higher.

There are 12 fundamental notes, and their frequencies are related by the twelfth root of two. For example, if you take the note 'A', whose frequency is set at 440Hz and you multiply it by 2^{\frac{1}{12}}, you get the next note 'A#' and so on until when you multiply the twelfth time, you get the same note 'A' an octave higher. This mathematical arrangement, I believe, is called the 'perfect temperament', as it forms a perfect geometrical progression between the same notes on different octaves.

When we combine these different notes, we get chords. What I want to ask is this, why do we choose particular notes for a chord, and why do we dwell on particular intervals on a scale (such as 1st, 3rd, and 5th note of a scale for a major chord) to give a particular type of chord?

I may have misunderstood some of the relations between the two so please feel free to correct me where ever I've made a mistake.

 

[JasonRox]

are both products of our mind and exist solely in our heads

Whoa!

Care to elaborate on that conclusion?

 

[chaoseverlasting]

The concept of numbers for example, without us giving any reference to it, exists only in our heads.

 

[robertm]

I agree that numbers, just like words, are a human creation, however mathematics is not. In a related thread someone made a very good point: Do not confuse the map with the territory.

The nature of the universe certainly cares about how much how many ect... Numbers are simply the language we have created to describe our world in an unambiguous way.

 

[Pythagorean]

When we combine these different notes, we get chords. What I want to ask is this, why do we choose particular notes for a chord, and why do we dwell on particular intervals on a scale (such as 1st, 3rd, and 5th note of a scale for a major chord) to give a particular type of chord?

Well, let's start with the most popular chord progression in western music (I IV V, the first fourth and fifth fundamental notes)

Let's contain the discussion to one string. When you pluck that one string, you hear one note, but in reality there is a harmonic series (1/2 the string is also wiggling, producing an octave higher note, 1/3 of the string is also wiggling, producing the 4th, and 1/4 of the string is wiggling, producing the 5th)

So as humans, we like the purer tones (that occur in perfect intervals of the string). The 4th and 5th notes of the I-IV-V progression are already in the 1st note when you ring it.

There are higher-order notes that aren't perfect intervals that serve to add color and personality to the chords, but humans tend to prefer the purer tones that come from interval 'waving' of the string.

http://en.wikipedia.org/wiki/Harmonic_series_(music )

 

[Crosson]

This community is very sensitive to perceived insults, but I find it funny that when an acclaimed mathematician makes similar comment it is not interpreted as an insult but rather a jewel of wisdom.

Leopold Kronecker said "God gave us the integers, and we made everything else."

Richard Feynman said "God cares not for our mathematical difficulties, nature integrates empirically."

Ironically, it is more likely that results (theorems) have an independent existence in nature as compared with specific proofs, which almost certainly exist only in human heads. The irony is that in this age of hyper-rigor the emphasis is on iron-clad proofs, not results.

In reference to the original post, I am reminded of Arthur Schopenhauer's view that music is the highest form of art, because it is the only form that does not merely copy ideas. As much as I love the visual arts, I must admit that, as it consists of colors and shapes, paintings can only come so far from the sights that I can see in nature. Music, in contrast, is totally unlike the sounds of nature.

The reason for this difference is that the wave nature of sound has always been a part of human experience throughout our evolution, whereas our experience with light during this same time was entirely in terms of rays traveling in a straight line. When we combine sound waves of various amplitudes, phases, durations, and frequencies, we hear rhythms, beats, chords, timbre and tone; when we do the same thing with light waves (in a modern laboratory) the result is just a fuzzy splotch of light seen as a particular color that is no more striking than any other.

Schopenhauer thus considered music to be a pure expression of the human will, sufficiently untainted and unconstrained by it's own building blocks to allow us to express something that truly "only exists in our heads."

A similar criticism, again coming from the position of abstract purity, can be used against fictional literature in favor of mathematics. In fiction the creative work is done long before the work is complete, since once the boundaries of the story are established all that remains is a relatively mechanical process of generating dialogue, visualizations, setting descriptions, etc. In contrast, any proof in research level mathematics involves considerable creativity at every single step! The poet Voltaire said that "there was more imagination in the head of Archimedes than in that of Homer" and I am also reminded of an anecdote involving a graduate student in mathematics who dropped out to become a poet. Upon hearing this his former advisor, David Hilbert, exclaimed "I knew he would not be creative enough to be a mathematician!" As I am sharing this I realize that I have been influenced by these views to the extent that I no longer regard them as surprising, as their author's intended, but rather as a plain truth.

In conclusion, I promote the view that mathematics is a truly creative activity in comparison with fiction, in particular because fiction merely recycles the same plots, characters, and settings ad nauseam whereas mathematics is constantly generalizing, hypothesizing, and diversifying upwards and beyond what came before. Arguably fiction does not progress at all, while mathematics progresses exponentially. Arguably, the very concept of progress belongs to mathematics, science, and technology while fiction can never stray very far from the mundane stagnation of daily human life. While I am not nearly as critical of visual art as I am of literature, clearly both of these can never escape their nature as consisting of typical components that we have all seen before rather then entirely abstract components that come totally from within ourselves.

 

[chaoseverlasting]

This community is very sensitive to perceived insults, but I find it funny that when an acclaimed mathematician makes similar comment it is not interpreted as an insult but rather a jewel of wisdom.

Leopold Kronecker said "God gave us the integers, and we made everything else."

Richard Feynman said "God cares not for our mathematical difficulties, nature integrates empirically."

Ironically, it is more likely that results (theorems) have an independent existence in nature as compared with specific proofs, which almost certainly exist only in human heads. The irony is that in this age of hyper-rigor the emphasis is on iron-clad proofs, not results.

In reference to the original post, I am reminded of Arthur Schopenhauer's view that music is the highest form of art, because it is the only form that does not merely copy ideas. As much as I love the visual arts, I must admit that, as it consists of colors and shapes, paintings can only come so far from the sights that I can see in nature. Music, in contrast, is totally unlike the sounds of nature.

The reason for this difference is that the wave nature of sound has always been a part of human experience throughout our evolution, whereas our experience with light during this same time was entirely in terms of rays traveling in a straight line. When we combine sound waves of various amplitudes, phases, durations, and frequencies, we hear rhythms, beats, chords, timbre and tone; when we do the same thing with light waves (in a modern laboratory) the result is just a fuzzy splotch of light seen as a particular color that is no more striking than any other.

Schopenhauer thus considered music to be a pure expression of the human will, sufficiently untainted and unconstrained by it's own building blocks to allow us to express something that truly "only exists in our heads."

A similar criticism, again coming from the position of abstract purity, can be used against fictional literature in favor of mathematics. In fiction the creative work is done long before the work is complete, since once the boundaries of the story are established all that remains is a relatively mechanical process of generating dialogue, visualizations, setting descriptions, etc. In contrast, any proof in research level mathematics involves considerable creativity at every single step! The poet Voltaire said that "there was more imagination in the head of Archimedes than in that of Homer" and I am also reminded of an anecdote involving a graduate student in mathematics who dropped out to become a poet. Upon hearing this his former advisor, David Hilbert, exclaimed "I knew he would not be creative enough to be a mathematician!" As I am sharing this I realize that I have been influenced by these views to the extent that I no longer regard them as surprising, as their author's intended, but rather as a plain truth.

In conclusion, I promote the view that mathematics is a truly creative activity in comparison with fiction, in particular because fiction merely recycles the same plots, characters, and settings ad nauseam whereas mathematics is constantly generalizing, hypothesizing, and diversifying upwards and beyond what came before. Arguably fiction does not progress at all, while mathematics progresses exponentially. Arguably, the very concept of progress belongs to mathematics, science, and technology while fiction can never stray very far from the mundane stagnation of daily human life. While I am not nearly as critical of visual art as I am of literature, clearly both of these can never escape their nature as consisting of typical components that we have all seen before rather then entirely abstract components that come totally from within ourselves.

I couldn't follow the whole post, but do you think mathematics is purely in our heads? I do. One of the other posters very correctly said that you should not confuse the map with the territory, and that is absolutely correct. But mathematics is the legend on the map of the territory, where the map is our observation of the phenomena of physics, chemistry or, in fact, any other field where mathematics is applicable, whereas the territory may be significantly different from the map we possesses of it because we do not possesses the understanding of the territory to accurately define the map.

Insofar as this analogy is concerned, mathematics (IMO), is just the legend on the map. As we use mathematics as a legend to define the physical phenomena we understand (or at least try to do so), music, I think is the legend to map of our emotions, and a song a route charted through those emotions. Transferring all properties of mathematics to music, one finds that music readily adheres to all of them, although the terrain mapped is of a different type. That is why I believe that mathematics and music are purely products of our minds and exist solely there.

 

[chaoseverlasting]

Well, let's start with the most popular chord progression in western music (I IV V, the first fourth and fifth fundamental notes)

Let's contain the discussion to one string. When you pluck that one string, you hear one note, but in reality there is a harmonic series (1/2 the string is also wiggling, producing an octave higher note, 1/3 of the string is also wiggling, producing the 4th, and 1/4 of the string is wiggling, producing the 5th)

So as humans, we like the purer tones (that occur in perfect intervals of the string). The 4th and 5th notes of the I-IV-V progression are already in the 1st note when you ring it.

There are higher-order notes that aren't perfect intervals that serve to add color and personality to the chords, but humans tend to prefer the purer tones that come from interval 'waving' of the string.

http://en.wikipedia.org/wiki/Harmonic_series_(music )

When you pluck a string, do all harmonics exist simultaneously?

 

[Pythagorean]

When you pluck a string, do all harmonics exist simultaneously?

I'm not sure what you mean by all harmonics. The harmonics do sound simultaneously with the fundamental note that you struck by intention, yes...

When you pluck the string, the lowest note you hear will the be note of the string itself, the fundamental, the whole string vibrating (if you looked at the link I provided). So there's no harmonic that is lower than the fundamental of the string you plucked.

On the high end, well, the string has mass and volume, so there's a point where the string won't be able to make tight enough vibrations. It's also a matter of how hard you pluck it. I guess one could argue that the harmonics up to the critical high end (where the string can't bend that tight to make those waveforms) are always there, just at amplitudes near to zero.

 

[chaoseverlasting]

So the string vibrates and all harmonics are present up to the highest frequency that the string can accept. If the string were perfect, then all harmonics would be present and we would be aware of harmonics which lie withing the human hearing frequency range.

 

[fuzzyfelt]

This community is very sensitive to perceived insults, but I find it funny that when an acclaimed mathematician makes similar comment it is not interpreted as an insult but rather a jewel of wisdom.

Leopold Kronecker said "God gave us the integers, and we made everything else."

Richard Feynman said "God cares not for our mathematical difficulties, nature integrates empirically."

Ironically, it is more likely that results (theorems) have an independent existence in nature as compared with specific proofs, which almost certainly exist only in human heads. The irony is that in this age of hyper-rigor the emphasis is on iron-clad proofs, not results.

In reference to the original post, I am reminded of Arthur Schopenhauer's view that music is the highest form of art, because it is the only form that does not merely copy ideas. As much as I love the visual arts, I must admit that, as it consists of colors and shapes, paintings can only come so far from the sights that I can see in nature. Music, in contrast, is totally unlike the sounds of nature.

The reason for this difference is that the wave nature of sound has always been a part of human experience throughout our evolution, whereas our experience with light during this same time was entirely in terms of rays traveling in a straight line. When we combine sound waves of various amplitudes, phases, durations, and frequencies, we hear rhythms, beats, chords, timbre and tone; when we do the same thing with light waves (in a modern laboratory) the result is just a fuzzy splotch of light seen as a particular color that is no more striking than any other.

Schopenhauer thus considered music to be a pure expression of the human will, sufficiently untainted and unconstrained by it's own building blocks to allow us to express something that truly "only exists in our heads."

A similar criticism, again coming from the position of abstract purity, can be used against fictional literature in favor of mathematics. In fiction the creative work is done long before the work is complete, since once the boundaries of the story are established all that remains is a relatively mechanical process of generating dialogue, visualizations, setting descriptions, etc. In contrast, any proof in research level mathematics involves considerable creativity at every single step! The poet Voltaire said that "there was more imagination in the head of Archimedes than in that of Homer" and I am also reminded of an anecdote involving a graduate student in mathematics who dropped out to become a poet. Upon hearing this his former advisor, David Hilbert, exclaimed "I knew he would not be creative enough to be a mathematician!" As I am sharing this I realize that I have been influenced by these views to the extent that I no longer regard them as surprising, as their author's intended, but rather as a plain truth.

In conclusion, I promote the view that mathematics is a truly creative activity in comparison with fiction, in particular because fiction merely recycles the same plots, characters, and settings ad nauseam whereas mathematics is constantly generalizing, hypothesizing, and diversifying upwards and beyond what came before. Arguably fiction does not progress at all, while mathematics progresses exponentially. Arguably, the very concept of progress belongs to mathematics, science, and technology while fiction can never stray very far from the mundane stagnation of daily human life. While I am not nearly as critical of visual art as I am of literature, clearly both of these can never escape their nature as consisting of typical components that we have all seen before rather then entirely abstract components that come totally from within ourselves.

I really enoyed reading this, and was hoping you might expand on your thoughts even more, for example, I am wondering how progress is reconciled with abstract purity.

 

[ehrenfest]

I heard a lecture on the relationships between mathematics and music. It was really great. It was given by a mathematician who also played the guitar on the side. The lecture consisted partly of a power point and partly of him actually playing his guitar with a real professional band band they hired just for that purpose.

My favorite part was when he said that the trig identity

\cos A + \sin B = 2 \cos \left((A+B)t/2\right) \cos \left((A-B)t/2\right)

was one of the main reasons why music sounds so good (it explains the phenomenon of beats).

 

[chaoseverlasting]

I heard a lecture on the relationships between mathematics and music. It was really great. It was given by a mathematician who also played the guitar on the side. The lecture consisted partly of a power point and partly of him actually playing his guitar with a real professional band band they hired just for that purpose.

My favorite part was when he said that the trig identity

\cos A + \sin B = 2 \cos \left((A+B)t/2\right) \cos \left((A-B)t/2\right)

was one of the main reasons why music sounds so good (it explains the phenomenon of beats).

Thats the equation of a standing wave created on a guitar string isn't it? And beats are created by the difference in frequencies of the two waves or is it the difference in phases?

 

[ehrenfest]

Thats the equation of a standing wave created on a guitar string isn't it? And beats are created by the difference in frequencies of the two waves or is it the difference in phases?

That is the equation of two guitar strings vibrating at different frequencies A and B. If A and B are close to each other, then A-B will be small so that the net effect will that the "intensity" the sound wave will be cos((A-B)t/2) which will oscillate on a timescale that you can hear.