[muso] Mathematics

[ClaudiusMaximus]

Listening to In Our Time - Mathematics And Music (downloadable until Wednesday, maybe) made me remember I once promised to start this thread, so here it is.


Note: the maths is hopefully accurate, the rest of it may be made up.


Pythagoras, Intervals, and Rational Numbers

Pythagoras liked triangles.  He noticed that triangles of different sizes made sounds of different pitches when you bashed them.  A triangle of half the size of the first triangle made a sound that was pleasing in combination to the sound made by a triangle that was the size of the first triangle.  And a triangle 2/3 the size of the first triangle was also pleasing (although a little less so), similarly a triangle 3/2 the size of the first triangle created a pleasant effect.

Further investigation revealed that ratios of small integers made pleasant sounds together.

Here are some ratios of small integers, and the corresponding name of the interval in Western-European parlance, and the note-letter if the base note was C:

1::1   unison (C)
2::1   octave (C)
3::2   perfect fifth (G)
4::3   perfect fourth (F)
5::4   major third (E)
6::5   minor third (Eb)
5::3   major sixth (A)
8::5   minor sixth (Ab)

Extending this further gets you into all sorts of trouble as the link above describes:

Say you want to find the ratio for a major second (D).  Then you can start from unison, go up a fifth and down a fourth, which gives you:

(method A)   1::1 * 3::2 / 4::3 = 9::8

Or, you can go up a fourth and down a minor third, which gives you:

(method B)  1::1 * 4::3 / 6::5 = 10::9

Two methods for reaching the same note give different results, which gives big problems if are making music with triangles of fixed sizes, because you'd effectively need an infinite number of triangles to cope depending on your melody (how you reach the note in question), and harmony would be even more problematic.


That's enough typing for one evening, next time I think I'll write a load of nonsense about strings and harmonics and beat frequencies and equal temperament, unless someone has any better ideas or questions or anything.

[butnut]

I don't really have anything to add at the moment, except to say thanks for pointing out the 'In Our Time' programme, and that I love reading about all this. I will post something to this thread when my brain is in better shape.

[The Man With Brass Eyes]

is this the same as Maths rock?

[ClaudiusMaximus]

Much-belated bump - just found this:

Dave Benson - Music: A Mathematical Offering

Also, if you're looking at how to implement some of the ideas, check out:

Miller Puckette - Theory And Techniques Of Electronic Music
which uses Pd for examples.

[NoSleep]

Thanks for those links. You may be interested in Kyle Gann's practical work into tunings & just temperament, also his music, which resembles Harry Partch's (another just temperament user). He makes a good argument against the conventional western equal-tempered note system, and points out that Bach DID NOT use this tuning for The Well Tempered Clavier... he used Werckmeister III. It begs the question as to why we listen to classical music not as it was created to be heard.

This is great stuff...

http://www.kylegann.com/
Just intonation
The history of tuning
700 intervals in an octave, as used in various tunings over the centuries

Terry Riley said western music is played so fast because it's out of tune.

[ClaudiusMaximus]

Terry Riley said western music is played so fast because it's out of tune.

There's a mathematical basis to that argument, if you consider the Fourier Transform (a method for transforming between time-based functions to frequency-based functions).  Without going into the horrible horrible details, the longer the amount of time you consider, the more accurate the frequency information is.  Conversely if you only have a small time period to consider, you can't get very accurate frequency information.

So, if you have something that is out of tune, but only by a small amount, you need a long time period of the same tone to notice it is out of tune, the less out of tune the longer the period required to resolve the frequency detail.  This means if you play faster, it's much harder to notice that it's out of tune (assuming of course that human ears work on a similar basis to the Fourier Transform...).

[NoSleep]

Maybe they don't work the same way as the Fourier Transform, but the reason we don't hear the equal tempered scale as out of tune is probably a cultural one. We're used to it.
If I hear some baroque or other early music played on traditionally tuned instruments, I can hear the adjusted notes in the scales as ''out", even though they are objectively more "in".
When I've experimented with just temperament, I've found that whilst chord voicings sound sweeter, with a tendency for the individuality of the notes to disappear into the sound (much less true in equal-temperament), when you try a melody across these scales, some notes sound "out".

[Gazeuse]

Yes, my ears are fully attuned to equal temperament and all other scales sound decidedly wonky.

Equal temperament is a bit of a bugger when it comes to guitars though...They never sound exactly right. There is an attempt to solve this problem though...

http://www.fretwave.com/

It's two wobbly frets which you fit on the first and second fret of the guitar. It pulls the tuning much tighter and looks pretty good too.

I'm going to get some when I get a really nice guitar.

[NoSleep]

There's a mathematical basis to that argument, if you consider the Fourier Transform (a method for transforming between time-based functions to frequency-based functions).  Without going into the horrible horrible details, the longer the amount of time you consider, the more accurate the frequency information is.  Conversely if you only have a small time period to consider, you can't get very accurate frequency information.

Given this a little more thought. Fourier analysis works in a similar way to an opinion poll, the more data it has access to, the more accurate it's conclusion.
Human perception doesn't 'tune-in' in the same way. We hear what is there to hear, whether we know it's nature or not (in this case, if it's in or out of tune). Understanding it is another matter. I think Terry Riley's quote refers to the sleight of hand approach that musicians/composers in western music have adopted to try and cover up what they understand to be flaws in the system.

Yes, my ears are fully attuned to equal temperament and all other scales sound decidedly wonky.

Equal temperament is a bit of a bugger when it comes to guitars though...They never sound exactly right. There is an attempt to solve this problem though...

http://www.fretwave.com/

It's two wobbly frets which you fit on the first and second fret of the guitar. It pulls the tuning much tighter and looks pretty good too.

I'm going to get some when I get a really nice guitar.

I read almost everything on that link in the end! It certainly solves problems in some situations, whilst creating (slightly fewer) new ones. There'll never be a perfect solution. Guitarists will retune for different sections of a song during a recording, to use the chord voicings they require and to make them all as sweet as possible. But Fretwave is a good compromise for live work.