Applications of mathematics to music

[shyvera]

A moment ago, I posted something on the application of group theory to music in the thread real life applications of group theory. Here is an elaboration of that post, and another application of maths to music.

 

Circle of fifths

I mentioned in my previous post that the theory of intervals is based on the cyclic group of order 12. Here is another way of looking at musical intervals. It's a bit more complicated and less intuitive to the non-mathematical musician as it involves more mathematics – which suits people like me much better.

 

Consider the set of all musical notes distinguished by their absolute pitch (or frequency measured in Hz). Define a relation ~ on this set by X~Y iff the notes X and Y differ by a whole number of octaves. Then it is immediately clear that ~ is an equivalence relation and there are exactly 12 equivalence classes, one for each of the 12 notes in the chromatic scale. Denote the equivalence class containing the note X by [X].

 

Now let us take the equivalence class [C] (containing the note middle C) as our reference. All the other equivalence classes [X] are related to [C] by the number of semitones (modulo 12) between X and C. The class [X] can therefore be defined as the interval C-X. (Of course we can choose any other class than [C] as our reference, e.g. the class [A], but as middle C is a much used in music (especially by pianists) as a reference note, we may as well choose [C] as our canonical reference.) Intervals (as equivalence classes) can be added by adding up the number of semitones (modulo 12) between the notes in each interval relative to C. For example, [E] + [F] = [A]. Note that this is relative to our reference note C; for different reference notes, the results will be different. However, given a fixed reference note, such an operation is always well defined. The set of all such intervals under this addition operation is then a cyclic group of order 12.

 

Even temperament

Temperament is a method of tuning a musical instrument by adjusting the ratios of the pitches of notes in a scale relative to a fixed note, called the tonic. Various termperaments are possible, but I am only concerned here with what is called even temperament, in which all semitone intervals have a fixed ratio, and the ratio of one note to the one exactly one octave below is 2.

 

Let r be the ratio of the frequency of one note to that of the note exactly one semitone below. Then if we start with a fixed note, of frequency f, and ascend the chromatic scale, the frequencies of these notes are f, fr, fr², …. The frequency of the note one octave above is 2f and that note is 12 semitones above the note we started with. This gives us

 

[math]2f\,=\,fr^{12}[/math]

 

or

 

[math]r\,=\,\sqrt[12]2\,\approx\,1.059[/math]

 

Thus in even temperament the frequency of each note is always approximately 1.059 times that of the note one semitone below. If we take the frequency of middle C as 256 Hz, then the frequency of concert A (9 semitones above middle C) is 256 × (2^1/12)⁹ or approximately 430.54 Hz. In some concerts, however, the frequency of concert A is set to 440 Hz. In this case, the frequency of middle C is 440 / (2^1/12)⁹ or approximately 261.63 Hz. Whether it is middle C that is tuned to 256 Hz or concert A that is tuned to 440 Hz depends on the performance in question.

[khaled]

Modulo Arithmetic would be fine to mess with that, but all of that on top of assuming a formula for some values ...

 

In the end, it will end by generating random set of notes that none would like to hear, not to mention that even

a random given formula would generate noise on sound output ...

 

If you want to map Music from Mathematics, that's a not easy thing, you have to understand Music and Harmony

then the work of writing a music sheet, which requires knowledge at the Neural Studies and the Artificial Intelligence,

and then to create a model that maps this system in Mathematics ...

 

I once had such project, but it was paused until I get free time ...

[shyvera]

I agree. Mathematics can be very useful as a tool for analysing theories of music, but if you want to compose decent music, you still have to understand the rules of harmony, modulation, etc.

 

By the way, I composed a short symphony a few years ago; I call it my Short Symphony in E Minor. It's in three movements, the second of which leads without a break into the finale. I've made a MIDI version of it; see what you think.

 

http://people.delphi...Symphony2_1.mid

http://people.delphi...mphony2_2-3.mid