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Music: A Mathematical Offering (2006) (freecomputerbooks.com)
80 points by ghosthamlet 8 days ago | hide | past | web | favorite | 38 comments






Has it not been updated since 2008? The intro suggests it has been; web searches suggest it has not.

Correct, its not been updated though the book says it would probably have been updated.

I found the link to the new home page of the author:

https://homepages.abdn.ac.uk/d.j.benson/pages/

which links to the same version of the book.


God, how much I love these hand-written html pages. They send a powerful message that the writing there is very serious, interesting, and down to business. Just like a beautiful LaTeX formatting for a paper.

You are using Mozilla/5.0 (X11; Linux x86_64; rv:60.0) Gecko/20100101 Firefox/60.0. You are wearing red.

What?


Inspect the element. It's cute.

It means you have JavaScript enabled.

'This website does not put pieces of poo disingenuously called "cookies" on your computer.'

This made me laugh.


Agreed, no “click through to see the top ten” here.

Music has everything to do with mathematics. It’s all waves just like y = sin(x). Digital music in particular has much in common with calculus in the piecewise manner that discrete samples approximate analog waves. I wrote a program in vanilla JS to demonstrate this idea. It converts XML-formatted music into mathematical terms, and then encodes it into pulse code modulation for playback. Feel free to peek at dietpumpkinmachine.com if you don't mind the sloppy code.

"David J. Benson is a 6th Century Professor in the Department of Mathematical Sciences at the University of Aberdeen." ? Piqued my curiosity!

What a great website.

Music has little to do with math. The math is useful in measuring things and developing the tools but basically, music is about emotional expressivity.

> Music has little to do with math.

As a musician (and audio engineer) I disagree. You don't need to think about math while writing or playing music, just like you don't need to think about physics when riding a bike, but that doesn't mean it has little to do with it.


The kind of math musicians talk about are elementary ratios that concern the relationships between tones and between frequencies, nothing to write home about. And even the latter have little application to composition.

You can be a great musician and even original songwriter and not know any math (or even the names of chords for that matter) at all. And that's even in folk -- it's even less so in electronic music and other genres.


You can absolutely operate (and even operate well) without understanding the formalisms involved in theory, but understanding them opens up a new level of fluency.

I don't need to know how to name common progressions or modes in order to write a song, but if I have my theory down well enough, it's easier to transpose to a different key, or have ideas for chord voicings or variations on the progression that'll enrich the piece (and as far as I can tell, the theory involved here is a form of mathematics).

I don't need to understand the mathematics of waveforms in order to twist a knob and get a different sound and make an aesthetic judgment, but having a model of how waveforms works in my head to correlate with specific sounds helps.

This is what math does. It helps take ad-hoc implicit models -- which people (and now to some extent ML algorithms) can often get by on pretty well -- and move them into explicit formalisms, often providing new power and insight in the process.

The fact that people can build internal models that make them some degree of effective composers/performers implicitly isn't a particular rebuke of the idea that a significant portion of what we call music can be well understood as mathematics.


>You can absolutely operate (and even operate well) without understanding the formalisms involved in theory, but understanding them opens up a new level of fluency.

Sure. My point was that if "you can absolutely operate (and even operate well) without understanding the formalisms involved in theory", (and that's for music theory, the very technical language of music), then imagine how more useless math are in this domain.

Not only you can get by quite well without music theory or math (as you say), but you are 100% set with what you need if you also know music theory alone -- math is entirely redundant [1].

[1] Again, for music (playing, composing, etc) not DSP. If anything you need some simple ratios for coming up with tunings, and that's it.


It's more important if you plan to involve computers in your workflow.

Knowing what a Fourier transform is and how music software applies it can help you avoid problems. Like "why is this song muddy when I add anything to it?" You don't need to know about it to know that two sounds can cancel out, but it helps.

You'll never be worse off for knowing a little about music theory or audio engineering.


>Knowing what a Fourier transform is and how music software applies it can help you avoid problems.

Sure, but that's DSP not music. Whereas the "old wives tale" that music=math has been before the advent of computers.


>> "Sure, but that's DSP not music."

That's why I said "It's more important if you plan to involve computers in your workflow." I didn't make a general statement about all music.

The more electronic your music, the blurrier the line between making music and audio engineering. Most famous electronic musicians, especially EDM artists, are known as much for their audio engineering as for their music.


> The kind of math musicians talk about are elementary ratios that concern the relationships between tones and between frequencies, nothing to write home about.

Perhaps in general this is often true, but Guerino Mazzola might have a thing or two to say about that, having published work applying topos theory to music.[1][2] Admittedly it appears to meet with some controversy and I haven't studied his work myself. My apologies for not having any more accessible links on hand to the subject material.

[1] https://en.wikipedia.org/wiki/Guerino_Mazzola

[2] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...


There is also polyrhythms. I already knew the Chinese remainder theorem when I was taught it in high school, because of music.

When I am riding a bike, I am very much thinking about physics, in order not to fall over and hurt myself. I can say that I have to think about physics regardless of whether I am thinking of it in the terms Newton did, because the physical world exists without the theoretical framework to describe it.

Mathematics on the other hand is purely theoretical. Unless I am engaging in mathematical reasoning, I'm not doing maths.

That said, when I am musicking, I am doing mathematics all the time, at least for things like basic counting, arithmetic and logic. But that's true for so much of the basic activities we do every day that saying that music has a lot to do with maths becomes a pointless truism. Similarly, I may be doing maths when I'm cooking, but no one is going to try to tell me that "cooking is maths". Cooking has little to do with maths, but maths has a lot to do with everything.


There's a difference between physics in abstract, more akin to math, and physics as intuitive assessment of your surroundings.

What is the difference? It seems to me that in order to successfully predict how I should handle my bike when riding it not to fall over, I must be using an abstract model of physical reality that I draw conclusions from.

Approximating isn't the same as an abstract model, which uses glyphs and, well, math.

Glyphs have absolutely nothing to do with it, or do you propose that writing predates abstract thinking? I'd say that abstract thinking is a prerequisite for language. You can't communicate with others unless you can distill the physical world into abstract higher-order concepts. In doing so, you create an abstract model of the world. For example you may point and call that thing a bicycle. In reality, the concept of a bicycle is just a figment of your representational model that happens to roughly coincide with mine.

Maybe an abstract model uses mathematics, but only necessarily in the sense that one could say that I'm using set theory to pick my nose. It's a bit like saying that watching TV is color theory.


Abstract thinking, in the capacity of quick intuitive guessing, does not an abstract model make. You're conflating terms. Running around interacting with the environment requires no more abstraction than an animal is capable of.

> Abstract thinking, in the capacity of quick intuitive guessing, does not an abstract model make. You're conflating terms.

Reasoning in abstract terms requires the capacity to model what you are reasoning about in abstract terms. As far as I'm concerned, that's not conflation. As soon as the world appears to you in terms of higher level concepts, you have a model of it in your head that's abstract; not concrete. You seem to disagree, on the basis that an abstract model necessarily entails using glyphs?

A prediction, e.g. "the bike will tip over if it stops moving" requires that I understand on some level the concept of causality, the concept of a bike, the concept of movement etc. Using my understanding of these concepts, I can predict potential outcomes for my actions. You call it intuitive guessing, and maybe you never needed training wheels, but this is something that most people have to learn and internalize.

In my view, the notion, while riding, that it'll hurt more if you fall off at a high speed is similar in nature to "F = m₁m₂/r²", differing mostly in scope, terminology and subject. Newton's law of universal gravitation is a lot like an "intuitive guess" in that it's a prediction based on limited information that broke down with the introduction of new information.

> Running around interacting with the environment requires no more abstraction than an animal is capable of.

I agree. When did you last see another animal ride a bicycle, though? Even animals that are much quicker to react than us and have mastered within months of their lives movement and spatial awareness to a degree which most humans can only ever hope to ever achieve suck at simple mechanical obstacles. We're amazed when cats, that can jump several body lengths from one thin branch to another to catch a bird before it has thought to fly off, do something as conceptually simple as opening a door using a handle to get where they want to go. A dog that can smell and track down a hare in a field of flowers will amuse us to no end if it shows the slightest hint of recognizing some concept more advanced than a ball.


You don't need math to enjoy or even perform music, or to understand it in a certain way. But there is another way to understand it that is purely mathematical, and it is quite enriching.

The start of the rabbit hole is the pythagorean comma and it has a very real effect on the real-world messy compromises that are involved in tuning instruments to play with each other in multiple keys.

The terrible beauty and ugliness that are one and the same stems from the simple fact that if you go up a fifth, say, C to G, it sounds the purest if the pitch is exactly 1.5 times as high, or 3/2. If you keep going up five notes at a time, eventually you get back to a C, having gone through all the other notes in the process.

But the pure, good sounding way to get from a C to a higher C is to double its frequency. And you never can get to exactly a power of two by repeatedly multiplying by 1.5. So the C you get by going up by octaves and the C you get by going up by fifths come out different.

A consequence of that is it is impossible to play all the different chords in a key as pure, perfectly-in-tune chords without at least one "floater" note (my term) that changes pitch depending on which chord it's part of at the time. See Just Intonation for more information. It's geeky stuff, but has an effect on the way real musicians play certain instruments, such as harpsichord and pedal steel guitar, and it has bothered and fascinated certain musicians ever since the ancient Greeks if not before.


As a jazz musician, you have no idea what you're talking about. Music theory is a subset of set and group theory, and there's no way to analyse music without math.

Music is not meant to be analysed. It is felt. The opera singer is the model for all music. TBH I have often felt jazz and classical music to be cold and mechanical, except for a few songs.

It's not math, per se. But music is certainly about patterns. And some of the ratios in musical patterns certainly have a foundation in math. I think you cannot arrive at beautiful music starting from math. But you probably can arrive at some interesting mathematical patterns starting from music.

The author says as much in the introduction:

"I should close with a disclaimer. Music is not mathematics."


Agreed on what music is intended to do. But being expressive can mean understanding different scales, different rhythms, different harmonic progressions, the capabilities of different instruments, etc.

One thing I did not learn in my childhood music lessons is basic stuff like musical intervals, chord structure, circle of fifths. These are very practical math relationships ... essential to knowing about your options, for both expressive performance and for on-paper composition and arranging.

In later years, I regretted that none of my teachers knew enough about them to help me understand them. They're an indispensable tool in the pro musician's arsenal.


True. Math nerds get excited by the ratios, roots, and groups, but all of that is a rabbit hole - useful for DSP code, but it will do nothing to help you write memorable music.

The book does not assert that to write memorable music you only need mathematics.

Knowing how to count beats and bars can be helpful when writing music, as can knowing how string lengths are related to notes when playing a stringed instrument.



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