[Bell Historians] Converting frequencies to notes

s.ivin at n... s.ivin at n...
Fri Jul 12 17:08:08 BST 2002

> > Db could indicate nearer to D and C# nearer to C
> Well, it depends what tuning system you are using, and in what order 
> the notes are tuned, but it is quite possible, or even likely, that 
> Db is nearer to C and C# is nearer to D. It just depends on the 
> direction you come to these notes in the cycle of fifths.

Also, we are talking two different things here. Using pitch names (letters) without
any reference to a standard (e.g. A=440 Hz) is pretty well meaningless from a
'scientific' or rigorous point of view, but clearly serves a purpose in musical
notation - which can has a lot more worms in it - e.g. C flat, or even C double-flat!
I ask you! I suppose it is comparable to using Andante etc to describe beats per
minute, for ambiguity.

But I'm not sure why we need a Spreadsheet when a cheap calculator can be used
to swap between Hz & deviations in cents from a standard frequency:

Cents = Log(Freq1 / Freq2) * (1200 / Log(2))
(but make sure the log base is the same in both cases)

So in the case which Michael Wilby questioned (to find the difference between
A440 and C256:

440 / 256 = 1.71875;
log(1.71875) = 0.2352....
1200 / Log(2) = 3986.3137... (logs to base 10)
Cents = +937.6... so the higher frequency (440) is 938 cents above the lower (256)
which amounts to 9 semitones and 37 cents, whereas in Equal Temperament A is
9 semitones exactly above C. Hence the C256 can be defined as 37 cents flat of
A440, or A440 as (100-38) sharp of C256.

The reverse is equally straightforward:

Hz = Inverse log (AntiLog?) (Cents / (1200/log(2))) * the ref. Freq.

Find the Hz corresponding to C (based on A440)

(the nearest) C is 3 semitones above A, or 300 cents;

300 / 3986.3137... = 0.07525...
Inv log(0.07525..) =1.1892...
440 * 1.1892... 523.25.... which is an octave higher than 261.625... to
put it in the same octave as the above example.

(Alternatively in this case we could just multiply 440 by the 12th root of 2,
three times!)

Andrew Higson, like most others who have a lot of pitching and recording to do,
moves easily between the two notations, but I think that the Hz or cps notation
is the most compact and completely without ambiguity, and is well suited to
recording data on bells, where there is widespread divergence from any pitch
standard, necessary for old bells and not uncommon on new ones. As an example,
one might want a bell of say 58.5" diameter, which with a regular profile would
naturally give a note between C 60.5" and C# (or Db) 57".

Stephen Ivin

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