Originally Posted by Dataf1ow
Although it is this simple to derive octaves, it's not so simple to derive equal tempered pitches the same way. The western octave is NOT divided equally, so that 1.25 multiplication will put you somewhere between B and Bb (or A#).
Well kind of. The western scale is divided equally across twelve frequencies. That's why it's called an even tempered scale. It's just a logarithmic scale instead of a linear one.
It's slightly more complicated to calculate the difference between notes than octaves. But it's by no means hard. Just use a scientific calculator.
The basic formula for the frequencies of the notes of the equal tempered scale is given by
fn = f0 * (a)n
f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz.
n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away.
a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...
The wavelength of the sound for the notes is found from
Wn = c/fn
where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature."
Source: [Only registered and activated users can see links. Click here to register]
So it's just a matter of deciding which value you want to determine and then running the numbers.
Originally Posted by Dataf1ow
Although using whole number ratios to define pitch relations is the basis of an entire tuning system, called jsut intonation, you might be interested in reading more. The 1.25 example would be a ratio of 5/4, and give you the 'Just Third' to G.
Even better, if you are simply wanting to adjust the pitch from one note in the scale to another, you can just calculate these using an online tool like this one: [Only registered and activated users can see links. Click here to register]
Once you have the two frequency values, it's simple as f1/f2 to get the ratio of f1 to f2 or f2/f1 to get the ratio of f2 to f1.
eg: A = 440 / C = 523.3 = 0.84081788648959
So if you want to reduce the speed to make C into A just reduce the playback speed by 0.840. and you're set. However, if there are other notes present in the sample, these will be detuned and likely inharmonic to other sound sources. To ensure that every note retained it's harmonic relationship to the western scale you'd have to run the calculation for each note.
There are a lot of granular synthesis tools that are less labour intensive and usually yield more interesting results.
Originally Posted by 5995
Thanks for the reply! It seems you can easily play things either twice as fast or half as fast, but anything else will end up sounding somewhat wrong. It's funny how much maths there is in music..
Not really when you consider that music is one of the earliest expressions of our understanding of mathematics.