# Representing Ratio's in Algebraic Formulae

• October 1st 2010, 08:50 AM
BIOS
Representing Ratio's in Algebraic Formulae
Okay so i'm looking to create algebraic formula's specific ratio's. This relates to interval relationships in a musical scale for example:

An octave has ratio of 2:1 which is represented by the following formula:
$
\displaystyle \frac {f2}{f1}=2^n$

where n = the number of octaves.

So if i wanted to find the value of the frequency 10 octaves above 20 hertz i would say:

$\displaystyle \frac {f2}{20}=2^{10}$

$\displaystyle f2=(20)(1,024)$

$\displaystyle f2=20,480$

So i'm wondering how you would formulate this for other frequency ratios. i.e:

3:2, 5:4 etc.

Any help would be great.

Cheers

BIOS
• October 1st 2010, 09:08 AM
SpringFan25
any fraction can be related to a ratio as follows:

$\frac{a}{b} = a:b-a$
• October 1st 2010, 09:14 AM
BIOS
What does $a:b$ represent mathematically there in terms of an operation?

Say i wanted to find the frequency a fifth above 20 hertz. And a fifth has a relationship represented by the ration 3:2. How would i formulate that as i have done above with the octave?

For a fifth:

$\displaystyle \frac {f2}{20}=?$
• October 1st 2010, 09:23 AM
SpringFan25
Please be careful using terms like 'fifth' on this forum as you are likely to be misinterpreted. try 'musical fifth'.

You want the following

f2 : 20 = 3:2

if you have had practice on ratios, you can rearrange as follows
0.5 * f2:10 = 1.5:1
0.5 * f2 = 1.5*10
f2 = 1.5 * 20

Otherwise, you can convert to fractions and solve like any other equation:
Step1:
convert them both to fractions using the relationship
$a:b = \frac{a}{a+b}$

Step 2
$\frac{f2}{f2+20} = \frac{3}{5}$

$\frac{f2 + 20}{f2} = \frac{5}{3}$

$1 + \frac{20}{f2} = \frac{5}{3}$

$\frac{20}{f2} = \frac{2}{3}$

$\frac{f2}{20} = \frac{3}{2}$

$f2 = 20 * \frac{3}{2}$

rearrange and solve for f2.
• October 1st 2010, 09:27 AM
BIOS
Sorry mate. Should have explained. A fifth is the term for an interval above a particular frequency that is a ratio of 3:2 with the original frequency.

Interval ratio - Wikipedia, the free encyclopedia

So just looking to create formula's for each interval ratio.
• October 1st 2010, 09:31 AM
BIOS
Is there a way to do it logarithmically as with the original formula? So i find two fifths above, three fifths above etc...
• October 1st 2010, 09:35 AM
SpringFan25
if each step increases the frequency by a factor of 1.5, then

$\frac{f_s}{f_t} = f_t * 1.5^{s-t}$

Or, starting from a frequency of $20$, and going up "t" fifths

$f_{new} = 20 * 1.5^{t}$
• October 1st 2010, 10:09 AM
BIOS
Sweet man. I understand where to go from there. Thanks for all the help!