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#2069371 - 04/23/13 02:16 PM Mathematics behind Music
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Hello, I've written a small article in my website explaining as example, why C and C# sound dissonant while C and G sound good.

I hope you find it interesting and usefull. It was funny for me to build the graphs smile If you are interested in seeing the graph of a specific combination of notes just tell me, I've made a script who allow to combine any notes just by entering it's number. I don't know if there are any practical application to this but at least it's interesting I think smile

http://www.miguelmotapinto.com/math.php

If you could give some feedback it would be great! smile

(And if you notice some problem in the webpage, feedback is appreciated as well, because the site was launched yesterday...)

Miguel
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http://www.miguelmotapinto.com/
http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2069383 - 04/23/13 02:35 PM Re: Mathematics behind Music [Re: mpmusic]
JohnSprung Offline
1000 Post Club Member

Registered: 08/02/11
Posts: 1543
Loc: Reseda, California

You should have a subscript n on the f to the left of the =.

You might want to explain more thoroughly how two waves are added, and the generation of beats at the difference of the frequencies.

As for what sounds good or bad, perhaps the thing to do is go through a wide range of beat frequencies. Start with a unison, then go slightly out of tune to a beat of, say, 0.2 Hz, and so forth to a minor second, a fifth, an octave.... Audio corresponding to the wave forms would help, too.
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Yamaha CP33
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#2069410 - 04/23/13 03:39 PM Re: Mathematics behind Music [Re: JohnSprung]
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Originally Posted By: JohnSprung

You should have a subscript n on the f to the left of the =.

You might want to explain more thoroughly how two waves are added, and the generation of beats at the difference of the frequencies.

As for what sounds good or bad, perhaps the thing to do is go through a wide range of beat frequencies. Start with a unison, then go slightly out of tune to a beat of, say, 0.2 Hz, and so forth to a minor second, a fifth, an octave.... Audio corresponding to the wave forms would help, too.


Thank you! smile I will try to do an animation with the first note fixed and increasing the frequency of the second note. And I will try to make a video out of that and put the sound of the combination smile


Edited by mpmusic (04/23/13 03:42 PM)
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http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2069481 - 04/23/13 05:21 PM Re: Mathematics behind Music [Re: mpmusic]
JohnSprung Offline
1000 Post Club Member

Registered: 08/02/11
Posts: 1543
Loc: Reseda, California

It turns out that there's something along those lines already on Wikipedia:

http://en.wikipedia.org/wiki/Beat_frequency

They only have two brief examples, so there's room for you to do better.
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Yamaha CP33
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#2069540 - 04/23/13 07:04 PM Re: Mathematics behind Music [Re: mpmusic]
PaperClip Offline
500 Post Club Member

Registered: 12/21/09
Posts: 522
Loc: Amsterdam, Holland
Nice graphs of the vibrations. You should do more graphs, like octave, fifth and of course the fourth and sixth, seventh. I think comparing the different oscillations would explain more to non math people. Maybe also graphs of both tones, so graph tone C + graph tone D is graph tone (played together(C&D)), displayed in two pictures. But it's just an idea.
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Playing since May 02 2009

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#2069755 - 04/24/13 04:18 AM Re: Mathematics behind Music [Re: PaperClip]
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Originally Posted By: PaperClip
Nice graphs of the vibrations. You should do more graphs, like octave, fifth and of course the fourth and sixth, seventh. I think comparing the different oscillations would explain more to non math people. Maybe also graphs of both tones, so graph tone C + graph tone D is graph tone (played together(C&D)), displayed in two pictures. But it's just an idea.


Thank you smile That's a good idea, display the two separate graphs in one figure and the combination in the other. I will do that in the video I'm preparing with lots of figures and animations.
_________________________
http://www.miguelmotapinto.com/
http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2069806 - 04/24/13 08:14 AM Re: Mathematics behind Music [Re: mpmusic]
CarlosCC Offline
1000 Post Club Member

Registered: 12/06/09
Posts: 1415
Loc: Lisbon, Portugal
Nice page Miguel,
Interesting article about notes and frequencies. Maybe you should put the same examples with the "Romance" notation (Dó, Ré, Mi, Fá, Sol, Lá, Si).
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#2069900 - 04/24/13 10:56 AM Re: Mathematics behind Music [Re: mpmusic]
zrtf90 Offline
2000 Post Club Member

Registered: 02/29/12
Posts: 2437
Loc: Ireland (ex England)
I don't see from the two graphs why the C and C# are dissonant but the C & G are not. The former looks like a round and pleasant beat and the second looks like sharp, jarring knife edge.

Consonance and dissonance might be better explained by the relationship of one note to another forming patterns. When there is a close integral relation between intervals as shown in the harmonic series (2:1 for an octave, 3:2 for a fifth, etc.), the pattern created by the two notes recurs very quickly and is very easily recognised by the brain. When the ratio is expressed in high numbers as, perhaps, the tritone, around 31:22, it's a long time before the pattern repeats so the ear has difficulty picking out or recognising the pattern and regards that as unpleasant dissonance.
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#2070051 - 04/24/13 04:01 PM Re: Mathematics behind Music [Re: zrtf90]
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Originally Posted By: zrtf90
I don't see from the two graphs why the C and C# are dissonant but the C & G are not. The former looks like a round and pleasant beat and the second looks like sharp, jarring knife edge.

Consonance and dissonance might be better explained by the relationship of one note to another forming patterns. When there is a close integral relation between intervals as shown in the harmonic series (2:1 for an octave, 3:2 for a fifth, etc.), the pattern created by the two notes recurs very quickly and is very easily recognised by the brain. When the ratio is expressed in high numbers as, perhaps, the tritone, around 31:22, it's a long time before the pattern repeats so the ear has difficulty picking out or recognising the pattern and regards that as unpleasant dissonance.



Yes I've read about that somewhere but I think it's very abstract, I'm trying to give some concrete, visual examples that anyone can understand even without knowing math.

There are some pdf about this theme like this one: http://www.math.wustl.edu/~wright/Math109/00Book.pdf
_________________________
http://www.miguelmotapinto.com/
http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2073319 - 04/29/13 07:13 AM Re: Mathematics behind Music [Re: mpmusic]
PaperClip Offline
500 Post Club Member

Registered: 12/21/09
Posts: 522
Loc: Amsterdam, Holland
Hi,

Maybe you could also add a graph of a secunde and a graph of a trill to see the differences. A trill doesn't sound so dissonant as a secunde. I wonder why. Maybe a graph could explain that to non math people. That would be so cool. laugh
_________________________
Chris

Playing since May 02 2009

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#2073350 - 04/29/13 08:44 AM Re: Mathematics behind Music [Re: mpmusic]
B&W Offline
Junior Member

Registered: 04/27/13
Posts: 8
Loc: Australia NSW
An audio book I really enjoyed is How Music Works. This kind of stuff is covered very well along with heaps of other interesting things.
http://www.hachettebookgroup.com/titles/john-powell/how-music-works/9780316098304/
or
http://www.audible.com/pd/ref=sr_1_2?asin=B004GMJ81U&qid=1367239391&sr=1-2
_________________________
Sometimes you know not what you want until you know what you've got is not what you want - anonymous

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#2073372 - 04/29/13 09:26 AM Re: Mathematics behind Music [Re: PaperClip]
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Originally Posted By: PaperClip
Hi,

Maybe you could also add a graph of a secunde and a graph of a trill to see the differences. A trill doesn't sound so dissonant as a secunde. I wonder why. Maybe a graph could explain that to non math people. That would be so cool. laugh



That's an interesting point! I will try to do an animated graph representing a trill smile
_________________________
http://www.miguelmotapinto.com/
http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2073374 - 04/29/13 09:30 AM Re: Mathematics behind Music [Re: mpmusic]
KeysAngler Offline
Full Member

Registered: 01/14/10
Posts: 260
Loc: The Fabulous Florida Keys
When the peaks of one note fits exactly between the peaks of a different note then they are consonant

The degree to which they do not symmetrically fit is the amount of dissonance

when the waves move together in such a way one pattern is perfectly contained inside the other there is consonance

On the other hand, when you play a more dissonant interval, you get clashing waves that sound like they are trying to cancel each other. There are degrees of dissonance. There are intervals that are incredibly tense sounding (diminished fifth, flat ninth) to others that are not so much, even considered kind of consonant (thirds, sixths). These degrees of consonance and dissonance and the way we approach them are what makes music interesting


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#2073397 - 04/29/13 10:17 AM Re: Mathematics behind Music [Re: mpmusic]
PianoStudent88 Offline
3000 Post Club Member

Registered: 06/16/11
Posts: 3183
Loc: Maine
Originally Posted By: mpmusic
There are some pdf about this theme like this one: http://www.math.wustl.edu/~wright/Math109/00Book.pdf

That PDF has been published: Mathematics and Music by David Wright. For a textbook, it's incredibly reasonably priced.

The author is teaching a course on it this spring. It's listed only in his "current courses" though, so if you're interested grab the links to the materials now because the page collecting the links will be changed next fall (or maybe in the summer, if Washington University St. Louis has a summer term).

If anyone's interested in delving into the book or the course materials, start another thread and I'd love to do so. (My background: Ph.D. in math, four years as a math professor, adore music and math.)
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#2073418 - 04/29/13 11:10 AM Re: Mathematics behind Music [Re: KeysAngler]
PianoStudent88 Offline
3000 Post Club Member

Registered: 06/16/11
Posts: 3183
Loc: Maine
Originally Posted By: KeysAngler
When the peaks of one note fits exactly between the peaks of a different note then they are consonant

The degree to which they do not symmetrically fit is the amount of dissonance

when the waves move together in such a way one pattern is perfectly contained inside the other there is consonance

That's what we're all taught, but apparently the true answer is not so simple.

From the introduction to Music: A Mathematical Offering by David Benson:
Quote:
What is it about intervals such as an octave and a perfect fifth that makes them more consonant than other intervals? Is this cultural, or inherent in the nature of things? Does it have to be this way, or is it imaginable that we could find a perfect octave dissonant and an octave plus a little bit consonant?

The answers to these questions are not obvious, and the literature on the subject is littered with misconceptions. One appealing and popular, but incorrect, explanation is due to Galileo Galilei, and has to do with periodicity. The argument goes that if we draw two sine waves an exact octave apart, one has exactly twice the frequency of the other, so their sum will still have a regularly repeating pattern whereas a frequency ratio slightly different from this will have a constantly changing pattern, so that the ear is 'kept in perpetual torment'.

Unfortunately, it is easy to demonstrate that this explanation cannot be correct. For pure sine waves, the ear detects nothing special about a pair of signals exactly an octave apart, and a mistuned octave does not sound unpleasant. Interval recognition among trained musicians is a factor being deliberately ignored here. On the other hand, a pair of pure sine waves whose frequencies differ only slightly give rise to an unpleasant sound. Moreover, it is possible to synthesize musical sounding tones for which the exact octave sounds unpleasant, while an interval of slightly more than an octave sounds pleasant. This is done by stretching the spectrum from what would be produced by a natural instrument. These experiments are described in Chapter 4.

The origin of the consonance of the octave turns out to be the instruments we play. Stringed and wind instruments naturally produce a sound that consists of exact integer multiples of a fundamental frequency. If our instruments were different, our musical scale would no longer be appropriate. For example, in the Indonesian gamelan, the instruments are all percussive. Percussive instruments do not produce an exact integer multiple of a fundamental, for reasons explained in Chapter 3. So the western scale is inappropriate, and indeed is not used, for gamelan music.

Even on the piano, when tuners talk about stretched octaves, I believe they are referring to the same phenomenon where what sounds consonant is related to the specific overtones produced by two notes on a given physical instrument, and not just to the pure frequencies of the fundamental of each note.
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Ebaug(maj7)

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#2073471 - 04/29/13 01:05 PM Re: Mathematics behind Music [Re: PianoStudent88]
mpmusic Offline
Full Member

Registered: 08/28/12
Posts: 101
Loc: Portugal
Originally Posted By: PianoStudent88

From the introduction to Music: A Mathematical Offering by David Benson


Looks very interesting the book!
There are some pages available on google books: http://books.google.pt/books?id=Ko1NsIq4...p;q&f=false
_________________________
http://www.miguelmotapinto.com/
http://youtube.com/user/miguelmotapinto

"Music is the key to understand the universe..."

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#2073995 - 04/30/13 08:20 AM Re: Mathematics behind Music [Re: PianoStudent88]
KeysAngler Offline
Full Member

Registered: 01/14/10
Posts: 260
Loc: The Fabulous Florida Keys
[quote=PianoStudent88That's what we're all taught, but apparently the true answer is not so simple.
[/quote]

It'll have to suffice for my purposes. It's been a looooooooooooooong time since math class.

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