This is what John 'Longitude' Harrsion wrote in the final pages his book Concerning Such Mechanism (csm....) 
to mathematically describe this tuning system.  CEHL (March 1999).
John "Longitude" Harrison (1693-1776) stamp
I will now give some account how the real scale of Musick is indeed
generated from the proportion which the diameter and radius of a
circle bear respectively to the circumference; but as towards which,
this great, or secret discovery, it is as in the first place, to be
understood, that it was after I had made several strict experiments
of divers, or diverse divisions of the octave, and they as from or by 
such necessary, or proper preparations, or aparatusses to the
purpose, as from my other business I was enabled to make: yea, I may
boldly say as thence, from far more correct, or natural
qualifications to the purpose, than any before me were able to make
or have, (nay, and still are --- as yet the same), and that as so at
last, I found to my great surprize, or admiration (viz. as from the
same strictness of trial of the result of the properties of a circle, 
as here above specified, and as with such, the same apparatusses to
the purpose) the real foundation of the matter to exist, or be, as
thence, by the hand of providence established: and the which (as in
brief) I shall explain as followeth.

Let the ratio of the octave, or, as even here, as well as below to
the purpose, the octave itself, be represented by the logarithm of 2
(viz. .30103): and let that same number be also taken or supposed as
the circumference of a circle ----------
And then, (as in the margin) (.30103 x  2 = .60206 + .09582 = .69788) 
let the space of two octaves and a sharp 3rd be taken, or be as
chiefly, or rather as primarily to the purpose notified, viz. when
(as according to my discovery) the said sharp 3rd is in its most
strictly musical proportion, and that is as when, with respect to the 
octave the same is taken as the diameter of the circle (viz. here, as 
.09582): For the proportion which the circumference of the circle
bears to the diameter (and as true enough to this purpose, as well as 
to others) is about as 3.1416 to 1: So, as 3.1416 is to 1, so is
.30103 to .09582.
And then as five larger notes (but not with tones major and minor, as 
hath been imagined, and that from of old) and as together with two of 
the lesser notes (as all along foolishly styled half notes major)
are, or must be, exactly contained in the octave: so therefore, as in 
taking half the diameter for the larger note, viz. .04791, as I from
strict, or proper experience, found it to be -- as an interval of
melody, right truly pleasant (although, as barely in itself, as well
as the lesser note, nothing to do with harmony), and that four 5ths,
thence as below to be generated (viz. of each containing .17447),
(margin note .17447 x 4 = .69788) and as when, as I am proof sure, to 
be then in their most strictly musical proportion, will, as according 
to nature, be equal to the two octaves and sharp 3rd, and at the same 
time, as already intimated, each one of the four 5ths will also be as 
without any infringement in any case (viz. as with respect to the
product of nature) so generated by subtracting five times the radius
from the circumference, where will be left such a quantity or space,
as the two lesser notes must, with equal shares, take up; and that
will be .06148, so the half of which, viz. .03074 must be the lesser
note; and the lesser subtracted from greater will leave .01717,
properly to be called a flat or a sharp (or the difference of the
notes), and not non-sensically the half-note minor; the lesser note
having withal the same authority to be called a whole note as what
the other has; but they may respectively or properly be styled tone
major and tone minor, viz. without meaning the fictitious nonsense as 
of old: and (as well understood) a 5th must contain three of the
larger notes and one of the lesser (viz. as in the case or cases here 
But as not withstanding, that from what is here above, are indeed the 
real steps or intervals of tune, or of natural melody, exactly
pointed out, or are to be thence truly generated (viz. accordingly as 
they out, or are to be as thence truly generated (viz. accordingly as 
they are taken by the voice or by voices); be also the real
consequence thereof, be also the real consonances, or chords of
natural harmony, truly limited or described; nay as so, in both
respects (viz. as touching both melody and harmony) I found to my
great surprize, to be confirmed upon strict instrumental musick, as I 
have shewn above. But still (and as has just been intimated) that
though from what is shewn above, the true steps of melody, as also
the true consonances of natural harmony, are as touching them all, or 
each of them, exactly to be defined, yet as from thence, no ratios at 
all can be said to be (that of the octave to be excepted), so the
said chords etc. must be denominated as they have all along been: and 
in the logarithm way, as here to the purpose the best way, as the
ratio of any chord is to be had by subtracting the logarithm of the
lesser number from that of the greater, so therefore, and as only
proper, viz. as in what is here, as first above -- may differ from
such ratios, so each chord, or interval, must to its properness or
sweetness of relish, in tunes or lessons of musick, be said to have
respectively such and such flatness or sharpness of latitude: as the
5ths to have, .00162 flat latitude, the 4th (its complement to the
octave) as much sharp; the flat 3rd to have .00109 flat latitude, the 
flat 6th as much sharp; and here I may notify, that the thirds will
bear their flat latitudes better than the 6th will bear their sharp;
nay the 5th will bear its flat latitude of .00162 as well or better
than the sharp 6th its sharp latitude of .00053: But to bear have I
said, as touching them all! whenas, as when in that, their exactly
right degrees, they are only as so rendered perfect! I speak from
strictly due experience (viz. from such as no man before me could
ever make, nay and are as still the same): and therefore, as each
interval respectively so results from the properties of a circle, as
I have shewn, they cannot each one, or any one, as by proof from
hence, be said to have a defect of any part or parts of a foolishly
feigned nonsensical comma; no for this, as here otherwise shewn, is
certainly the true essence of all that can be said of the matter,
whatever nonsense any book, as heretofore in the world may consist
of. Now whether my style of writing in this affair, be right proper
to the purpose or not, I thought it must be better than that the
contents of this book should be in danger of sleeping in oblivion;
yea, notwithstanding what I had -- as verbally communicated to the


LucyTuning homepage
The Old Hundredth in 12tET - midi
The Old Hundredth in Harrison's tuning - midi