May 4, 2007
Understanding the Western Tempered Scale (Part 15 of 15):
How To Get From Delta Frequency To Cents

Last week, we were stuck with a traditional African tuning which Hugh Tracey himself determined in Hz (Hertz) by using his set of micro-tonal tuning forks (i.e., a much finer grid than a half step, the smallest interval in the western scale). This week we finish the job.

If I were a better physicist, I could probably explain this equation. I've never seen it before, and I just sort of wrote it down and know that it must be true, so I'll at least verify it for you.

Scientists always explain the different terms in an equation:

• If you don't know about LOG (the mathematical sign for logarithm) then you won't understand this equation, but you can still plug the numbers into your calculator to get the answers.
• "freq0" is the reference frequency (i.e., the frequency of the western note we are referencing, in Hz).
• "freq1" is the frequency of the note that we want to know how many cents flat or sharp it is from the reference note (i.e., the frequency of the African note).
• "cents" is the number of cents sharp or flat. If it comes out as a positive number, freq1 is sharp from freq0. If it comes out as a negative number, the freq1 is flat from freq0.

Here's the way to understand the equation: If freq1 is twice freq0, we know it is an octave higher, right? But log(freq1/freq0) is then equal to log(2), so the outcome is 1200 cents, which is 100 cents times 12 half steps, which is an octave. You can do the same sort of checking using the frequency ratio for a half step (2^(1/12)) and you should get 100.

Now, applying this to last week's example:

#	Frequency (Hz)	Nearest Western Note	Frequency of Western Note (Hz)	Cents
1	207.65	A flat	207.65	0
2	227.62	below B flat	233.08	-41
3	259.56	below C	261.63	-14
4	283.52	below D	293.66	-61
5	315.47	below E	329.63	-76
6	343.42	below F	349.23	-29
7	391.34	a bit below G	392.00	-3
8	415.30	A flat	415.30	0

The regular major scale in A flat is:

1=A flat, 2=B flat, 3=C, 4=D flat, 5=E flat, 6=F, 7=G, 8=A flat.

This African scale has: 1 (perfect by definition), a 2 that is 40 cents low (i.e., almost exactly between the A natural and the B flat), a 3 that is just 14 cents flat of the western major 3rd, a 4 that is 61 cents flat (ie, it is closer to the C than the D flat), a 5 that is 76 cents flat, a 6 that is 29 cents flat, and a 7th that is almost right on.

Now, in performing these calculations, I realize that Hugh Tracey actually needed something finer than the 4 Hz tuning fork set he possessed, especially at the lower end of the scale. Let's assume Dr. Tracey's ear was excellent and he didn't make mistakes (probably true), and he would always assign an African note to the nearest tuning fork frequency. Then he would typically have up to a 2 Hz error in the frequency of any African note (i.e., if the African note were halfway between two tuning forks).

However, we are looking at intervals, which are differences of notes, so the intervals could be off by up to 4 Hz. Now, at the lower end of this scale (207 Hz), 4 Hz corresponds to 33 cents, or a third of a half step (if you follow me). At the upper end of the scale, his error was as large as 16 cents. On average, his error would be half of these values.

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