Hello, Alex
From your excellent website, it'seems that you may be just the mathematical brain needed to make sense of the following. It involves an apparently perplexing finding in the field of music. (Unfortunately for your bank manager, this is for intellectual interest only, and not for hard cash!)
An octave is split into 12 equal intervals, the ratio of each step being the 12th root of two, viz. 1.0594630943593. Thus (1.0594631) ^ 12 = 2
For the purposes of calculating errors, tuning, etc., each semitone is split into 100 cents, the ratio of each step being the 1200th root of two, viz. 1.0005777895065548593. Thus
(1.00057778951) ^ 1200 = 2
Now, 1 cent is approximately an increase of 1 part in about 1730, thus
(1+1/1730.7341) ^ 1200 = 2
The mathematical definition of a frequency ratio (musical interval) calibrated in cents is such that
Ratio = 2 ^ (cents/1200).
Thus, a 2:1 frequency ratio is 1200 cents whilst 600 cents (6 semitones) is the half-way point of root 2.
So, to convert any interval defined as a frequency ratio into cents,
we take the log of the ratio, multiply it by 1200 and divide by the log of 2. For example, for the ratio 3/2 (a natural or "just" fifth)
log (1.5) *1200 / log (2) = 701.955 cents
But we can also use natural logs, where we find that:
1200 / Ln(2) = 1731.23405
Is this is a fluke? If not, why the discrepancy with the above value of 1730.734?
Thank you for taking the time to read this
Regards
Trevor Skeggs
(an interested amateur with the Electronic Organ Constructors Society)
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