John Harrison (© Wikipedia)
In his book "Concerning Such Mechanism" (as made available by Charles Lucy) the famous inventor of nautical clocks John "Longitude" Harrison (1693  1776) describes the mathematical approach he took to create his unique tuning of a diatonic scale. In today's terms, he claims that the logarithm of a whole tone (ratio L) should relate to the logarithm of the octave (ratio 2) like the radius of a circle to its circumference:
Combined with the fact that in a diatonic scale the octave is made up by five whole tone steps (L) and two half tone steps (s):
this permits to calculate Harrison's entire tuning.
To explore the two equations above, they can be generalized as follows:
and
where A is a constant, N is the frame interval of a scale, and p an q are the numbers of whole tone and half tone steps, respectively, in that scale. A slight transformation makes the equations more useful for the calculation of scale parameters:
and
In any Western diatonic mode (N = 2; p = 5; q = 2) there are two different just whole tones: 9:8 and 10:9. This leads to two preliminary values for A:
Averaging them produces
This is so close to 2 · π = 6.28 that choosing
for a Western diatonic mode (Harrison and Lucy) appears to be well justified. The correlated values for the whole tone and the half tone are then:
But it turns out that Harrison's approach is not limited to the Western scale. Any diatonic BohlenPierce mode (N = 3; p = 4; q = 5) features only one just whole tone: 25:21 (301.8 cents). The resulting constant
is even closer to 2 · π than that was the case for the Western scale. The corresponding values for whole and half tone
permit a suitable tuning of a BP diatonic scale, as shown in the table below, with the Lambda mode as an example:












































Thus Harrison's approach works for BP as well. And it is not limited to scales employing rational numbers as a frame interval, as a look at the 833 Cents Scale confirms. In this scale the frame interval is the Golden Ratio, an irrational number:
The scale features p = 4 whole tones and q = 3 half tones. The whole tones are different: L_{1} = 1.0821 and L_{2} = 1.0787. Thus we arrive, like in the Western scale, at two different preliminary values for A:
Averaging them results again in a number that is very close to 2 · π:
The related whole and half tones are then:
As the table below shows, this is an exceptionally close approach to the original scale:



("Harrison") 
































John Harrison, with high probability, would have disliked this scale. But it proves the viability of his basic idea.