How Have Composers Used Mathematics in Classical Music?

[Tyrion101]

I was wondering what classical composers, modern, and not used math to write compositions, I am aware of a couple of examples, including the 12 tone system. What has anyone else come up with? I have heard, that Bach's compositions are "mathematically perfect" though, I have no idea what that means yet.

 

[Pythagorean]

Bach used the rules of music theory, which has as set "allowable intervals" and conditions. You could argue that it's mathematical, but its really based on musical theory (which defies acoustic physics and violates the true harmonic series; human sound processing strays slightly from reality by about a "syntonic comma").

 

[PeterJ]

There's lots of examples - Stockhausen, Boulez, Bach, Stravinsky et al. Then there's 'ten green bottles, hanging...'

It's a natural thing really. Music is all numbers. We might say that Debussy was using maths by dividing the octave into six tones. Boulez is one of the most interesting I'd say, with his interest in stochastics. Composers have always set themselves problems and given themselves contraints in order to fuel their ideas, and often these are mathematical.

Paul Simon wrote one song by deciding he had to include all 12 semitones in the melody, and that is kinda mathematical.

 

[Ken Natton]

I am not sure what you are looking for here. Mozart, famously, was a talented mathematician. The fact that shows in his music is pointed to both by its detractors and its enthusiasts.

If you understand the terms ‘just intonation’ and ‘equal temperance’, you will understand how deeply those ideas are wrapped up in the technical details of harmonics. It takes someone with a very mathematical mind to decide to write a set of preludes and fugues in every tonal key so that they can all be played on a keyboard tuned to equal temperance and then call the collection ‘The Well-Tempered Clavier’.

 

[Pythagorean]

The main issue to grasp is that is that in music theory, 'four perfect fifths' and 'two octaves plus a major third' should end up at the same note, but the physics don't quite work out that way (each interval is not rigidly defined, it's relatively defined) so we have to distribute that error somehow.

 

[AlephZero]

The main issue to grasp is that is that in music theory, 'four perfect fifths' and 'two octaves plus a major third' should end up at the same note, but the physics don't quite work out that way (each interval is not rigidly defined, it's relatively defined) so we have to distribute that error somehow.

You only have to "distribute the error" if that's what you choose to do. For centuries western music just took a major third to be 81:64, and treated it as a dissonance (which 81:64 is, compared with 80:64 or 5:4).

Or instead of "distributing the error" you can just pile it all up in one place and then never go there, musically. That system (and various approximations to it) lasted for a few more centuries

Even now, equal temperament is probably less common in practice than it is in theory. Most orchestral instruments only play in an approximation to it (and with different approximations for different instruments!). Even instruments like pianos that could theoretically be tuned in ET are intentionally NOT tuned that way.

Probably the most common source for sounds that ARE accurately tuned in ET are sample libraries for computer-generated music - and that is one reason why it's usually easy to tell them apart from a live recording, if you listen carefully.

There is a scientific basis for psychoacoustics (i.e. the study of how people HEAR sounds, as opposed to what you can measure in a physics lab) but in the final analysis, "music" is just "whatever the people in a particular culture decide to call music". "Western classical music" is only one option.

 

[zoobyshoe]

I have heard, that Bach's compositions are "mathematically perfect" though, I have no idea what that means yet.

It means that Bach frequently set himself the task of solving difficult logical challenges within the framework of music theory and solved them. These aren't really "math," per se, they're more like logic puzzles.

And, I have to say that the compositions where he did more of this than anything else are often the least enjoyable to listen to.

It's like this: an artist can draw a geometric figure whose point is to be intrinsically attractive, or emotionally compelling, or, he can draw a geometric figure whose point is to illustrate the proof of a theorem. The latter are almost invariably aesthetically awkward and unbalanced looking, but they prove an intellectual point, and that point can be amazing. The former are attractive to the eye, and evoke strong emotion, but they can be without any intellectual infrastructure, so to speak.

This:

actually says vastly less than it pretends to say. Whereas this:

actually has mind blowing ramifications.

It's in the nature of art, though, that the former should be the more important. Art is about communicating viscerally. By which I mean, that's what it does best.

 

[H2Bro]

You might look into the fugues Bach composed for King Frederick of Prussia. I've heard they are mind blowingly impressive from a musical theory perspective.

Or, read up on Pythagoras. He originated the idea of harmonics, and the format of mathematical proofs.

 

[Pythagorean]

You only have to "distribute the error" if that's what you choose to do. For centuries western music just took a major third to be 81:64, and treated it as a dissonance (which 81:64 is, compared with 80:64 or 5:4).

Or instead of "distributing the error" you can just pile it all up in one place and then never go there, musically. That system (and various approximations to it) lasted for a few more centuries

Even now, equal temperament is probably less common in practice than it is in theory. Most orchestral instruments only play in an approximation to it (and with different approximations for different instruments!). Even instruments like pianos that could theoretically be tuned in ET are intentionally NOT tuned that way.

Probably the most common source for sounds that ARE accurately tuned in ET are sample libraries for computer-generated music - and that is one reason why it's usually easy to tell them apart from a live recording, if you listen carefully.

There is a scientific basis for psychoacoustics (i.e. the study of how people HEAR sounds, as opposed to what you can measure in a physics lab) but in the final analysis, "music" is just "whatever the people in a particular culture decide to call music". "Western classical music" is only one option.

Sure. I would still consider those cases distributing the error (either to the third or to an unused note).

Western music has some niceties though. I always thought there was something about the I IV V. Especially since the three dominant notes in those chords are some of the first notes in the harmonic series and represent integer ratios of the tonic.

 

[BWV]

This:

actually says vastly less than it pretends to say. Whereas this:

Don't know what that picture is, but in general while art may use math as a tool, it has far more to say than that

Which to a smarter entity than us, would be as trivial as 2+2=4

 

[BWV]

http://gmik.wordpress.com/2008/11/25/complexity-in-geometry-iannis-xenakis/

https://www.youtube.com/watch?v=SZazYFchLRI

 

[Pythagorean]

Sure. I would still consider those cases distributing the error (either to the third or to an unused note).

Western music has some niceties though. I always thought there was something about the I IV V. Especially since the three dominant notes in those chords are some of the first notes in the harmonic series and represent integer ratios of the tonic.

To expand on this (since I misused the word dominant), western music is based on the circle of fifths (which the I-IV-V represents a small series of) and the fifth is the third harmonic (the first two and the fourth being the prime and octaves of the prime). So there's a lot of symmetry arguments for this choice of notes.

The harmonic series is really what brings it all together though, imo. The first ten harmonics:

1. tonic
2. octave (basically the tonic)
3. fifth
4. octave (again)
5. major third
6. fifth (again)
7. minor seventh
8. octave (again)
9. major second
10. major third (again)

and now we can almost completely construct the major scale (everything but that dissonant major seventh, mother of the tritone chord; the chord that a band like Korn bases their music off of). A sixth is an inverted third (much like a fourth is an inverted fifth).

It somehow seems meaningful to me, that if you ring a string, these components ring with the string as it rings as a whole, in halves, in thirds, fourths, etc. It gives the feeling of some objective basis for music psychology (at some point, our psychology reflects being able to measure meaningful differences as they pertain to our survival). Our auditory cortex is definitely "wired" for frequency space, topographically.

For the full harmonic series:

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

 

[BWV]

To expand on this (since I misused the word dominant), western music is based on the circle of fifths (which the I-IV-V represents a small series of) and the fifth is the third harmonic (the first two and the fourth being the prime and octaves of the prime). So there's a lot of symmetry arguments for this choice of notes.

The harmonic series is really what brings it all together though, imo. The first ten harmonics:

1. tonic
2. octave (basically the tonic)
3. fifth
4. octave (again)
5. major third
6. fifth (again)
7. minor seventh
8. octave (again)
9. major second
10. major third (again)

and now we can almost completely construct the major scale (everything but that dissonant major seventh, mother of the tritone chord; the chord that a band like Korn bases their music off of). A sixth is an inverted third (much like a fourth is an inverted fifth).

It somehow seems meaningful to me, that if you ring a string, these components ring with the string as it rings as a whole, in halves, in thirds, fourths, etc. It gives the feeling of some objective basis for music psychology (at some point, our psychology reflects being able to measure meaningful differences as they pertain to our survival). Our auditory cortex is definitely "wired" for frequency space, topographically.

For the full harmonic series:

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

I wonder what early instruments could reliably produce a 10th harmonic. Most "primitive" music around the world is pentatonic (5-note instead of 7-note scales) which does line up better with partials. The first and only time the 6th scale degree occurs is the 13th partial, but it is an out-of-tune minor 6th which would not give the diatonic scale pattern (i.e. the pattern of white keys on a piano). You simply cannot directly derive diatonic scales from the partials for one note, even after adjusting the temperament. One can get the major scale by building major triads on the 1st, 4th and 5th scale degrees. Also there is no real relation between the major seventh and the tritone. In 18th & 19th century music, harmonic and melodic tritones are commonplace and you can find plenty of major sevenths in Bach's harmonies (and the term "tritone chord" does not have any real meaning)

 

[Pythagorean]

by tritone chord, I just mean the diminished chord. From the western seven-note scale, you can construct seven chords: three majors, three minors, and the diminished chord.

And of course the seven-note scale includes the pentatonic in it (removing the major elements of the diminished chord. In C Major, you take away B and F, the tritone interval in the dim chord and you have the pentatonic).

And then for some blues, you just take the pentatonic and add a note between D and E. All these little colors are what gives music unique characteristics. But I think the pentatonic structure is pretty fundamental in all music.

 

[zoobyshoe]

Don't know what that picture is, but in general while art may use math as a tool, it has far more to say than that

I'm pretty sure you didn't understand my post because you're more or less saying the same thing I concluded with, but you're saying it as if contradicting me.

Which to a smarter entity than us, would be as trivial as 2+2=4

I don't understand the point of pointing this out. All kinds of things would be different to an entity smarter than us.

 

[BWV]

just saying the "intellectual infrastructure" or content of even a child's drawing is in a lot of ways far beyond any mathematical concept, as it reflects and and expresses something about the consciousness / personality of the artist. Mathematical objects are logical constructions that may contain more hard information, but are in the end more easily understood. Don't think there are any mysteries remaining in Pythagoras's theorem, but there are in great works of Art.

The last bit was just a paraphrase of the Bertrand Russell observation that all math is a tautology

 

[zoobyshoe]

just saying the "intellectual infrastructure" or content of even a child's drawing is in a lot of ways far beyond any mathematical concept, as it reflects and and expresses something about the consciousness / personality of the artist. Mathematical objects are logical constructions that may contain more hard information, but are in the end more easily understood. Don't think there are any mysteries remaining in Pythagoras's theorem, but there are in great works of Art.

I agree with this completely.

There are two separate and very different things going on in Bach's music and any given piece is more of one than the other. One is his authentically "artistic" ability to express emotion, states of consciousness, personality. This is most apparent in the famous "Toccata and Fugue in d minor". The other is his so-called "mathematical" bent, which really isn't mathematical so much as it's logical. I can't tell you which piece best embodies this, maybe one of the higher number fugues in book II of the WTC or something from Art of Fugue, I'm not sure.

Bach is praised by music theorists for the latter kind of composition and often distained for the former, despite the fact the latter kind are frequently not particularly pleasing to the ear, or emotionally engaging. These compositions are analagous to the diagram of the Pythagorean theorem. However remarkable the mathematical insight of Pythagorean theorem is, it does not make a viable visual composition. It is neither balanced, nor does it move the eye around the page in any natural way. It's pretty awkward and aesthetically unsatisfactory in terms of visual composition, almost a perfect illustration of what you shouldn't do in a good visual composition. Just about all good geometric proofs are the same: they are not aesthetically pleasing as visual events. Their great merit lies in their intellectual infrastructure, the logic they illustrate.

This unfinished fugue by Bach is a perfect example of what I mean by that class of his compositions where he was working out some arcane train of logic in music theory with no apparent regard to the aesthetic effect:

[YouTube]_-Zcs9WF8ik[/YouTube]

In a similar vein, Escher left a lot of "mathematical" work like this:

which solves some kind of challenge he set himself in tessellation, but which, despite the perfect logic behind it, is actually pretty ugly to look at.

This painting is only pretending to be geometrical. It's hand waving at the kind of thing Bach and Escher could authentically do under the guise of being "Sacred" geometry. The geometric figures are actually included for their aesthetic qualities alone, or for their separately superimposed functions as symbols. It's one small step away from a crop circle:

This one is Art Deco, an honestly and frankly decorative geometric style. No pretenses or arcane byzantine logic being worked out:

I think the third image is the most artistic in that it is trying to do what art is best at: communicating something about the mind/emotions of the artist (not that it is a particularly great example of that endeavor. "Most artistic" doesn't mean "it's the best artwork of the four" at all. I just mean this artist let his/her artistic compass swing in the direction art is most naturally want to go).

 

[BWV]

one of the most famous 20th century fugues, perhaps surpassing the complexity of the quadruple fugue in BWV 1080 is Der Mondfleck from Arnold Schoenberg's Pierrot Lunaire where there are two fugues and a canon playing simultaneously, but the rigor of the counterpoint is essentially unperceptable, providing the background texture to the singer obsessing about being unable to wipe a speck of moonlight off his coat. Schoenberg basically throws away the technique, as a less rigorous free counterpoint would have accomplished the same effect

http://tinyurl.com/9umephw

https://www.youtube.com/watch?v=vhwy3mk5jhY