April 27, 2007
Understanding the Western Tempered Scale (Part 14 of 15):
A Real Live African Tuning from Hugh Tracey (by way of N. Scott Robinson)

By now, it is the stuff of legend. Hugh Tracey, the Brit who fell in love with Africa, its people, and its music, travelling across the continent exploring its music and its scales... But where can we find those scales? And what can we do with them?

N. Scott Robinson has provided me with information from an article published by Hugh Tracey in the African Musical Society Journal. It provides several African scales in Hz (units Hertz). Dr. Tracey travelled with micro-tonal tuning forks (i.e., adjacent tuning forks in the set separated by less than a semi-tone, or half step). As Paul Tracey related to me, his father would find the best musicians in a village and ask them to produce a well-tuned instrument, and then he would measure the pitches of the notes of the instrument. For example, the Ruja scale is given as:

#	Frequency (Hz)
1	208
2	228
3	260
4	284
5	316
6	344
7	392
8	416

We can see that the #8 note is exactly 1 octave above the #1 note (i.e., twice the frequency). Good! Any group of people who don't work with octaves might make scales that are not comprehensible to us.

It turns out that 208 Hz is basically an A flat, which is 207.65 Hz. We could figure out that A flat is this frequency either by looking it up in a table, or by scaling down from A 440 to A 220 (i.e., an octave lower than A 440), and then dividing by that magic factor of 2^(1/12) = 1.059463.

My first step in understanding Dr. Tracey's noted scale is to retune this scale to exactly A flat, which is accomplished by scaling all frequencies down by a factor of 207.65/208 = 0.99832. The intervals will all be preserved in such a transformation, though not in Hz.

It is unclear how important this is in this case, because Dr. Tracey's tuning forks are only accurate to 4 Hz, and our recalibration to exact A flat results in a change of less than 1 Hz, but there will be other cases where this is important. I also show the nearest note in the western scale, the frequency of that note, and the "delta", or how far off the African note is from the western note. (Delta is a Greek letter that usually represents a small change.)

#	Frequency (Hz)	Nearest Western Note	Frequency of Western Note (Hz)	Delta (Hz)
1	207.65	A flat	207.65	0
2	227.62	below B flat	233.08	-5.46
3	259.56	below C	261.63	-2.07
4	283.52	below D	293.66	-10.14
5	315.47	below E	329.63	-14.16
6	343.42	below F	349.23	-5.81
7	391.34	a bit below G	392.00	-0.66
8	415.30	A flat	415.30	0

The problem is that a delta of, say, 5.46 Hz at the bottom of the scale is a bigger fraction of a note than a delta of 5.46 Hz is at the top of the scale. If you have an electronic tuner, it will show you how far off you are from the desired note, and it will measure that delta in terms of "cents."There are 100 cents in every half step, no matter what frequency the note is, or where in the scale you are.

So this is our goal: to get the delta in terms of "cents." That way, we can present a scale in terms of the western intervals like "major 3rd," and something like "-20 cents." This means we are 20% of the way down from the major 3rd towards the minor 3rd.

This is something that musicians will have some intuition about, while "-5.46 Hz" is not very meaningful. Of course, a note with a delta of 0 Hz has a delta of 0 cents (it is right on the western note). A delta of -0.66 Hz is not significant, because Dr. Tracey's measurement was to the nearest 4 Hz. But, what do we do with -10.14 Hz? How do we convert that to cents?

Dear reader, wait till next week, and we will see.

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