Frequencies, harmonics, math………confusing

I’m trying to create some formulas for frequencies, let’s say you’re given 440Hz and you’re asked to find the frequency of
a) the major 2nd above it
b) the augmented 4th below it
c) the octave above it

so for a) it would be F = 9f / 8
for b) it would be F = f / sqrt(2)
and for c) it would be F = f^(n-1) , n is not equal to 1 ; where n is how many octaves above you want it.

but the problem is, how about the major 3rd above it? Should it be
a) F = 81f / 64 or
b) F = 5f / 4

If you want to know how I came up with those two different frequencies, in (a) I used the 5ths built up. So if a fifth above is 3*f/2, and 4 fifths above is the major 3rd, except 2 octaves higher, then that would be F = 3*(3*(3*(3*f/2)/2)/2)/2/4 (the 4 is to compensate for the 2 octaves difference) which then becomes 81*f/64. In the second one (F = 5f / 4), I use the harmonics to get it. If the 5th harmonic is the major 3rd above the fundamental, except 2 octaves above, then it would be 5f (fifth harmonic) / 4 (compensate for the two octaves)

Using (a), you get 556.875Hz while you get 550Hz with (b), using f = 440Hz. Now that’s quite a BIG difference.

You might ask why I ran into this problem…
First of all, you should know that there are 2 types of tuning. One is the tuning we use now, and the other is based on the harmonic series. If we use the harmonic series to tune, it would sound good in one key but off in another key.

I actually experimented with the harmonic series and came to know that you CAN’T play the harmonic series accurately on a piano. The harmonic series and conventional tuning stay same for the first few overtones but after that they become different.

for calculation of the perfect 5th above the fundamental there are two formulas.

a) F = 3f / 2 according to the harmonic series
(harmonic series formula is like this – F = nF where n is the harmonic you want, as the fifth is the 3rd harmonic, it will be 3f. But this will be one octave above where you want it, so it becomes 3f / 2)

b) F = 3f / 2 according to the conventional tuning where you divide the original frequency by two (lower the octave) and add the original frequency (F = f / 2 + f)

so both a) and b) yield 660Hz

Another example would be to find the middle of the octave (augmented 4th above the base frequency)

the two formulas:
a) F = f * sqrt(2)
b) F = 2*f / sqrt(2)

using 440Hz, you get 622.16Hz with (a) while you get 622.34795Hz with (b). Not that great a difference, and let’s say, with 20Hz, you get 28.28Hz with (a) while you get 28.284Hz.

You might say that that’s not that big a difference, but we’re looking for the accurate formula. So I’m not sure which formula should be used. I’m going to look into logarithm to solve my problems… but I don’t have the time at the moment.

Note : I have not put much research into this, so don’t take my formulas as the ‘correct’ formula. It’s just to get a rough idea of where the frequencies are. But anyway, I’ll just include what I’ve had time to work out.

Key : F = altered frequency
+n where ‘n’ is a number – moving the note up by a major ‘n’ e.g. 2nd
-n -moving the note down by a major ‘n’ e.g. 3rd
+mn – moving the note up by a minor ‘n’ e.g. minor 6th
-mn – moving the note down by a minor ‘n’ e.g. minor 7th

So F+2 will be moving the note up by a major 2nd and
F-m7 will be moving the note down by a minor 7th

f = original frequency

Frequency Calculations
Major 2nd
F+2 = 9*f / 8
F-2 = 8*f / 9

Minor 2nd
F+m2 = 3*f*sqrt(2) / 4
F-m2 = 4*f / 3*sqrt(2)

Major 3rd
F+3 = 81*f / 64
F-3 = 64*f / 81

Minor 3rd
F+m3 = f * sqrt[sqrt(2)]
F-m3 = f / sqrt[sqrt(2)]

Perfect 4th
F+4 = 4*f / 3
F-4 = 3*f / 4

Augmented 4th/Dim 5th
F+aug4 = f * sqrt(2)
F-aug4 = f / sqrt(2)

Perfect 5th
F+5 = 3*f / 2
F-5 = 2*f / 3

Major 6th
F+6 = 27*f / 16
F-6 = 16*f / 27

Minor 6th
F+m6 = 9*f*sqrt(2) / 8
F-m6 = 8*f / 9*sqrt(2)

Major 7th
F+7 = 243*f / 128
F-7 = 128*f / 243

Minor 7th
F+m7 = 81*sqrt(2)*f / 64
F-m7 = 64*f / 81*sqrt(2)

If you notice, they aren’t accurate, and if you want to do a minor 6th up, you can actually just take the maj 3rd down and transpose it up an octave

Oct+ = 2^(n-1) * f , n is not 1, where n is the amount of octaves

F = nF where n is the harmonic wanted

That’s all for now, I guess, bye!

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