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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.

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.

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.

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.

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

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

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.

From the introduction to [I]Music: A Mathematical Offering[/I] by David Benson