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#2069371  04/23/13 02:16 PM
Mathematics behind Music

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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 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 http://www.miguelmotapinto.com/math.phpIf you could give some feedback it would be great! (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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#2069383  04/23/13 02:35 PM
Re: Mathematics behind Music
[Re: mpmusic]

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Registered: 08/02/11
Posts: 3740
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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#2069481  04/23/13 05:21 PM
Re: Mathematics behind Music
[Re: mpmusic]

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Registered: 08/02/11
Posts: 3740
Loc: Reseda, California

It turns out that there's something along those lines already on Wikipedia: http://en.wikipedia.org/wiki/Beat_frequencyThey only have two brief examples, so there's room for you to do better.
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#2069540  04/23/13 07:04 PM
Re: Mathematics behind Music
[Re: mpmusic]

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Registered: 12/21/09
Posts: 525
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]

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Registered: 08/28/12
Posts: 101
Loc: Portugal

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 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.

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#2069900  04/24/13 10:56 AM
Re: Mathematics behind Music
[Re: mpmusic]

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Registered: 02/29/12
Posts: 3287
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]

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Registered: 08/28/12
Posts: 101
Loc: Portugal

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

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#2073319  04/29/13 07:13 AM
Re: Mathematics behind Music
[Re: mpmusic]

500 Post Club Member
Registered: 12/21/09
Posts: 525
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.
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Playing since May 02 2009

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#2073350  04/29/13 08:44 AM
Re: Mathematics behind Music
[Re: mpmusic]

Junior Member
Registered: 04/27/13
Posts: 8
Loc: Australia NSW

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#2073374  04/29/13 09:30 AM
Re: Mathematics behind Music
[Re: mpmusic]

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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]

3000 Post Club Member
Registered: 06/16/11
Posts: 3847
Loc: Maine

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]

3000 Post Club Member
Registered: 06/16/11
Posts: 3847
Loc: Maine

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: 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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