A theory question.

[littleluke]

On a keyboard, A4 (A to the right of middle C) is 440 Hz. This means that A5 is 880 Hz. Therefore the frequency range of this octave is 440 Hz, divided into 12 steps - Semitones.

 

I'd previously thought this would be divided into 12 equal steps of 36.7 Hz, but now I think it's some sort of ratio whereby

 

A5:A4 =2:1

 

So how do I go about working out the frequency of the notes?

 

A bit geeky maybe but there you go!

[GN-Nick]

From my article on guitarnoise "Gin and Diatonic"

 

Remember Mr. "Square of the Hypoteneuse" Pythagoras? He discovered the ratio for these harmonics, amongst other things, a couple of thousand years before Spielberg was born.

 

Octave = 2/1

Fifth = 3/2

Fourth =4/3

 

Pythagoras, who was a mathematician, came up with all of the combinations of these ratios to get numeric values for the scale.

 

For example:

 

A perfect fifth (3/2) plus a perfect fourth (4/3)

= 3*4 = 12/6 or 2/1= an octave

2*3

 

If you look at all the combinations of these ratios, you get the aforementioned and dreaded Pythagorean diatonic scale. In the line below big "C" is the root note and little "c" is the octave.

 

C=1; D=9/8; E=81/64; F= 4/3; G=3/2; A=27/16; B=243/128; c=2/1.

 

This is an exact ratio scale.

 

Pythagoras thought "Wow, harmony in music, math as harmony" until some wise guy came along and told him this system had problems. For instance, twelve pure fifths is not the same as seven octaves, but it should be.

 

This is called the "The cycle of Fifths" problem - not to be confused with recycling fifths, or moonshining.

 

Here is the "The cycle of Fifths" problem in a nutshell:

 

7 octaves = (2/1)^ 7 = 128

12 fifths = (3/2)^ 12 = 129.74

 

Let's just walk through that in a slightly different way:

 

You take a note, we'll call it the "root" and play the note a fifth above it.

Now take the note you just arrived at and do the same. Repeat this a total of 12 times.

 

Where do you end up? Interestingly, (or not as the case may be), you end up at a note amazingly similar to your root note, 7 octaves higher.

 

(As an aside, and this one may take a second or two to understand, but if you flatten this series of fifths into a single octave, you get a 12-step scale. Haven't we seen that before?)

 

But wait, there's more.

 

If you take that note from above, the one that is 12 fifths from the root, and compare it to the original root note, taken up 7 octaves higher, well they should be exactly the same frequency, right?

 

Wrong.

 

The cycle of fifths note has a frequency of 3568.02 Hz

The octave-based note has a frequency of 3520 Hz

 

The difference comes from the fifth being off by .02 cents compared to the octave. Multiply this by 12 and you get an error of 24 cents or roughly a quarter semi-tone or about 48.02 Hz.

 

To get around this dilemma, you flatten the pitch of the fifth by 2 cents. That is a Tempered Fifth.

*******************

 

Here is a link to note frequencies:

 

http://www.phy.mtu.edu/~suits/notefreqs.html

[littleluke]

Nick,

 

Is that a politician's answer?

 

It's most interesting, but just a little too advanced for me at the moment.

 

I just want to know how many Hz from A to A#, and what mathematical process was used to get there. From that I can figure out all the notes above and below.

 

I'm going to re-read your post now to see if I missed it first time (and second and third...!) as I can be a bit slow at times :-)

 

Michael

[littleluke]

No, Sorry Nick

 

Maybe I'm a bit slower than I thought, but I can't de-cypher your answer :-)

[GN-Nick]

If you just want the frequency, go to the link at the bottom of my post.

[boseengineer]

You want a recipe?

To go a half step up multiply with

 

1.05946309435930

 

so A# above 440 is about 466 Hz (up 26 Hz)

A# from 880 is 932 Hz (up 52 Hz)

 

The ratios are constant but not the absolute differences

[littleluke]

 

Originally posted by boseengineer You want a recipe? To go a half step up multiply with 1.05946309435930

 

 

Right I've just gone an octave from A4 multiplying by 1.05946309435930 and it works, thanks for that. But can you tell me how that number is derived?

 

Thanks.

 

Michael

[littleluke]

O.K. I've got it.

 

It's the 12 root of 2 :-)