Music and Math

[bwv 1080]

Astronomy is not math, neither is music

[Karl Henning]

Astronomy is not math, neither is music

The very point.

[millionrainbows]

Astronomy is not math, neither is music

You need to read about the quadrivium, then.

[bwv 1080]

You need to read about the quadrivium, then.

No i dont.  Ancient greeks are not an authority and whatever was in the quadrivelum means nothing

[millionrainbows]

It's easy to see how music is geometric. The 12 notes get divided up into symmetrical groupings, based on the number 12 rather than the tonal way, which is asymmetrical.

Tonality was/is based on sensual constructs: major/minor chords are not symmetrical, and the octave is "divided" at the fifth, which is the way tonality works best, with the IV and V.

Geometrically, the modernists have divided the octave at 6, or the tritone, which is not harmonically derived, and even sounds dissonant, especially when compared to the fifth. All the intervals except 7 (fifth) and 5 (fourth) are symmetrical: 1, 2, 3, 4, and 6  (the rest are inversions).

The fact that there are 12 notes did not always exist, and everything (7-note scales) was based on the sound of the fifth and major third. "12" was arrived at through Pythagoran procedures (not Pythagoras himself, who did not need 12 notes). This "twelveness" itself is an arbitrary, geometric quality.

[millionrainbows]

...arithmetic was pure number, geometry was number in space, music number in time, and astronomy number in space and time. Morris Kline classifies the four elements of the quadrivium as pure (arithmetic), stationary (geometry), moving (astronomy) and applied (music) number

[Archaic Torso of Apollo]

Beethoven was so mathematically incompetent that he never even properly learned how to multiply and divide. (He could add and subtract.)

Apparently this caused a lot of trouble with his personal finances.

He probably didn't care what was in the Quadrivium, either. Although his life probably would have been easier if he had.

[Jo498]

The idea was that the mathematical subjects should be taught/learned in ascending complexity: arithmetic deals only with "one dimension" (numbers"), then plane geometry, then threedimensional geometry, then movement in three dimensions (and celestial mechanics was the only well described regular natural movement until the late middle ages/early modernity). Music theory deals with proportions (of chord length), it was therefore seen as the more complex partner of arithmetic.

"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in its relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantities as such, music the relations between quantities, geometry magnitude at rest, spherics [astronomy] magnitude inherently moving." (Proklos)
https://en.wikipedia.org/wiki/Quadrivium

(It is not completely different from today when one starts with arithmetic and plane geometry in middle school and in college physics point mechanics in three dimensions will usually be taught before elastic continua with the vibrating string as the simplest case.)

[millionrainbows]

well, ever since I saw intervals expressed as ratios, I've been a believer.

[bwv 1080]

A progression which is directly inherited from, and heavily influenced by, the quadrivium of yore. Dismissing the latter as meaningless for today's world is either ignorant or provocative.

Why should I respect the ancient philosophical construct of the quadrivium any more than, say, Aristotelian concepts of science?

[bwv 1080]

Math without astronomy, aplenty --- astronomy without math, hardly if at all. Just saying.

Math is a tool, one is that is essential to astronomy and sometimes useful for some composers.  Go back and read the OP - just realized I started this damn thread a decade ago

[Jo498]

Why should I respect the ancient philosophical construct of the quadrivium any more than, say, Aristotelian concepts of science?

You should not respect it as some "higher truth" but as a historical reality that was important and influential for education for at least 2000 years (from the 5th century BC or if the Pythagorean tradition is plausible even earlier until at least the 17th century when Kepler tried to fit the planetary orbits into "harmonic proportions") and the deeply entrenched idea of the connection (NB not identity) between music and maths is expressed in it quite clearly.
It's like asking "why should one respect the golden ratio when it is just some stuff the ancients came up with" and ignoring how it has dominated parts of aesthetics, architecture etc.

[Karl Henning]

well, ever since I saw intervals expressed as ratios, I've been a believer.

This composer begs your pardon, as he did not come dressed for church.

[millionrainbows]

This composer begs your pardon, as he did not come dressed for church.

Oh, I see. Care to elaborate?

[millionrainbows]

Meanwhile, the "wholesale resurrection of ancient threads" continues...

[Karl Henning]

Meanwhile, the "wholesale resurrection of ancient threads" continues...

I refer you here.

[Monsieur Croche]

Try to restart, reframe and remove the discussion that started in the Mahler thread. 
ISTM that there are two unsupportable positions on the topic:

A) the Quantophiliac, who looks for deterministic mathematical relationships to aesthetics, forgetting that music is art and therefore cannot be fully defined by the sum of any set of empirical observations.

B) the Quantophobe, for whom mystical inspiration is all and ignores the fact that math is often a useful tool in achieving and describing aesthetic objectives.

Certainly composers have taken direct inspriation from mathematics.  Bartok w/ golden mean proportions, Ligeti with strange attractors and other fractal phenomena etc.  Music is not math any more than architecture is math, but as Monk sez, all musicians are subconsiously mathematicians.

I can save you a lot of time.

Mathematics is not an art.
Science is not an art.

Music is not mathematics.
Music is not a science.
Music is a fine art:  music is only "like unto" maths and science in that it is an abstract discipline, that overlap but slight.

Any one involved in any of the three disciplines can be said to have an interest in a type of order.

Bartok did use the ratios of the golden rectangle to determine the number of measures of sections in a piece.  I.e. section 'a' was so many measures long, 'b' so many, etc.  This was about segment durations in linear time, no longer about a rectangle:  he did this in several short pieces, rather like just a few studies.  He  did not apply it all the time, or in any of his larger-scale pieces that I know of. 

If that same ratio of number of measures per section of a piece had been used by a 'bad' composer, the golden rectangle ratio would have done exactly jack-shit to make a bad piece any better.  I.e. a skilled and talented composer can make that set of related duration segments work (they could also make other non-golden rectangle ratio segments work):  it is not the set of ratios itself that has any 'artistic magic.' or intrinsic worth.

It was GMG member North Star who sagely pointed out that maths are used to such a near universal effect in delineating and quantifying so much of our world and universe that mathematicians are going to be prone to believe that there must be a lot of quantifiable maths in music.  For the acoustics department, sure (its a science, after all, that uses a lot of maths:-)  For the 'art' part?

Well, Timothy Leary's description of the brain's very most basic impulse is good here.  For the sake of illustration, let us agree the oceans in this quote are 'art,' and specifically, the art of classical music.
"Imagine your mouth, actual size, adrift on all the oceans of the world.  That mouth is calling out, "Order."

As Karl Henning has already said, a musician's imagination can by catalyzed by any number of things, a mathematical premise included.  A la the example of Bartok and the golden rectangle, the inspiration (math or other) is rarely applied or carried out literally... it was just a premise to springboard an idea.

If there were some tried and true set of maths as applied to music to 'make pieces work,' wouldn't there have been several hundred tomes on the subject by now -- and wouldn't more people, using those tomes as teachers and guides, be writing stunningly good pieces?

[nathanb]

The modern human has a very poor imagination.

Edit: The modern human, on average, is also extremely bad at mathematics, or so says lots and lots of personal experience.

[Monsieur Croche]

The modern human has a very poor imagination.

Edit: The modern human, on average, is also extremely bad at mathematics, or so says lots and lots of personal experience.

"It is said," that "Musicians" are often adept at maths, languages, and they are also quick to learn how to fly an airplane.

Well, "It is said." lol.

[bwv 1080]

I can save you a lot of time.

Mathematics is not an art.
Science is not an art.

Music is not mathematics.
Music is not a science.
Music is a fine art:  music is only "like unto" maths and science in that it is an abstract discipline, that overlap but slight.

Any one involved in any of the three disciplines can be said to have an interest in a type of order.

Bartok did use the ratios of the golden rectangle to determine the number of measures of sections in a piece.  I.e. section 'a' was so many measures long, 'b' so many, etc.  This was about segment durations in linear time, no longer about a rectangle:  he did this in several short pieces, rather like just a few studies.  He  did not apply it all the time, or in any of his larger-scale pieces that I know of. 

If that same ratio of number of measures per section of a piece had been used by a 'bad' composer, the golden rectangle ratio would have done exactly jack-shit to make a bad piece any better.  I.e. a skilled and talented composer can make that set of related duration segments work (they could also make other non-golden rectangle ratio segments work):  it is not the set of ratios itself that has any 'artistic magic.' or intrinsic worth.

It was GMG member North Star who sagely pointed out that maths are used to such a near universal effect in delineating and quantifying so much of our world and universe that mathematicians are going to be prone to believe that there must be a lot of quantifiable maths in music.  For the acoustics department, sure (its a science, after all, that uses a lot of maths:-)  For the 'art' part?

Well, Timothy Leary's description of the brain's very most basic impulse is good here.  For the sake of illustration, let us agree the oceans in this quote are 'art,' and specifically, the art of classical music.
"Imagine your mouth, actual size, adrift on all the oceans of the world.  That mouth is calling out, "Order."

As Karl Henning has already said, a musician's imagination can by catalyzed by any number of things, a mathematical premise included.  A la the example of Bartok and the golden rectangle, the inspiration (math or other) is rarely applied or carried out literally... it was just a premise to springboard an idea.

If there were some tried and true set of maths as applied to music to 'make pieces work,' wouldn't there have been several hundred tomes on the subject by now -- and wouldn't more people, using those tomes as teachers and guides, be writing stunningly good pieces?

Dont see that you saved me any time, just obfuscated what I said back in 2007 without contradicting any of it.  To try to add some real clarity I would add that music theory is not math in the sense that there is no axiomatic foundation such as exists in statistics, computer science or other fields in applied math.  However, I do like the Monk quote