The Bohlen-Pierce Site
John "Longitude" Harrison and BP 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:

2 · π · log L = log 2

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):

5 · log L + 2 · log s = log 2

this permits to calculate Harrison's entire tuning.

To explore the two equations above, they can be generalized as follows:

A · log L = log N

and

p · log L + q · log s = log N

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:

A = log N / log L   or     L = N1/A

and

s = N1/q - p/(q · A)

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:

A1 = log 2 / log (9/8) = 5.8849 and A2 = log 2 / log (10/9) = 6.5788

Averaging them produces

Aw = 6.23

This is so close to 2 · π = 6.28 that choosing

Aw = 2 · π

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:

Lw = 1.1166 (190.99 cents)    and    sw = 1.0733 (122.54 cents)

But it turns out that Harrison's approach is not limited to the Western scale. Any diatonic Bohlen-Pierce mode (N = 3; p = 4; q = 5) features only one just whole tone: 25:21 (301.8 cents). The resulting constant

ABP = log 3 / log (25/21) = 6.30 ~ 2 · π = 6.28

is even closer to 2 · π than that was the case for the Western scale. The corresponding values for whole and half tone

LBP = 1.191 (302.7 cents)    and    sBP = 1.083 (138.2 cents)

permit a suitable tuning of a BP diatonic scale, as shown in the table below, with the Lambda mode as an example:

 Tone name Just ratio Cents (just) Cents ("Harrison") C 1/1 0 0 D 25/21 301.8 302.7 E 9/7 435.1 440.9 F 7/5 582.5 579.2 G 5/3 884.4 881.9 H 9/5 1017.6 1020.1 J 15/7 1319.4 1322.8 A 7/3 1466.9 1461.0 B 25/9 1768.7 1763.7 C' 3/1 1902 1902

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:

N833 = φ = 1/2 + (5/4)1/2 = 1.618

The scale features p = 4 whole tones and q = 3 half tones. The whole tones are different: L1 = 1.0821 and L2 = 1.0787. Thus we arrive, like in the Western scale, at two different preliminary values for A:

A833/1 = 6.098    and    A833/2 = 6.352

Averaging them results again in a number that is very close to 2 · π:

A833 = 6.255 ~ 2 · π = 6.28

The related whole and half tones are then:

L833 = 1.07959 (132.6 cents)    and    s833 = 1.06001 (100.9 cents)

As the table below shows, this is an exceptionally close approach to the original scale:

 Step Ratio ("just") Cents ("just") Cents ("Harrison") 0 1.0000 0 0 1 1.0590 99.3 100.9 2 1.1459 235.8 233.5 3 1.2361 366.9 366.1 4 1.3090 466.2 466.9 5 1.4120 597.3 599.6 6 1.5279 733.8 732.2 7 1.6180 833.1 833.1

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