Lucy tuning
From Academic Kids

LucyTuning is a form of meantone temperament, derived from pi, in which the fifth is of size 600+300/π (= approximately 695.5) cents. Its main advocate is Charles Lucy, who discovered it in the eighteenth century writings of John Harrison. The LucyTuned perfect fifth is 0.0384 cents sharper than the fifth of 88 tone equal temperament, and a mere 0.01015 cents flatter than 3/10comma meantone, and therefore is audibly indistinguishable from either as the just noticeable difference for pitch is five cents for single pitches. The difference is clearly audible, in the beating, if more than one pitch is sounded at the same time.
A major tone is two fifths up and an octave down, so in LucyTuning it will be 2(600+300/π)1200 = 600/π cents. The major third therefore is two tones, or 1200/π cents, which is an octave divided logarithmically by π or the πth root of two. This works out as 381.972 cents, 4.342 cents flatter than a just major third. A diatonic semitone is the interval between a major third and a fourth, which in LucyTuning will be (600300/π)1200/π = 6001500/π cents, or 122.535 cents. Any interval can equally well be expressed in terms of octaves and fifths or whole tones and diatonic semitones. If we call the whole tone L and the diatonic semitone s, the familiar diatonic scale is LLsLLLs, and a LucyTuned diatonic scale will be one with the above specific values for L and s.
In Robert Smith's Harmonics of 1749 we find the following description of Harrison's system of tuning:
 He told me he took a thin ruler equal in length to the smallest string of his base viol. and divided it as a monochord, by taking the interval of the major IIId, to that of the VIIIth, as the diameter of a circle, to its circumference. Then by the divisions on the ruler applied to that string, he adjusted the frets upon the neck of the viol. and found the harmony of the consonances so extremely fine that after a very small and gradual lengthening of the other strings at the nut, by reason of their greater stiffness he acquiesced in that manner the placing of the frets.
While Smith himself interpreted this somehow to mean that Harrison's major thirds were a comma flat, it does seem to say that the proportion of third to octave is 1:π, which only seems to make sense if it is interpreted so that this proportion is logarithmic, or in other words, that Harrison's third is the 1200/π third of LucyTuning.
Charles Lucy on Lucy tuning
 Many musicians believe that extreme precision is significant in musical tuning, as different beat frequencies are heard, which are characteristic of each different tuning.
 I appreciate that I could appear to be beating angels on pinheads to death. Yet......... There is a valid reason for my demands about precision.
 I am thinking beyond only the cent values, and preparing for the day when the precision of our tuning technology improves, the production, analysis and discussion of beat frequencies will then become both practical and significant.
 From this research into tuning, a harmonic and scale structure has been mapped by a unique system of ScaleCoding, which can be used for the analysis, synthesis and organization of musical scales. Using LucyTuning is seems that simultaneously heard notes which are closer on the spiral of fourths and fifths sound more consonant than those which are separated by more steps along the spiral.
Contemporary writings show that there was considerable animosity between Smith and Harrison about their different concepts of music tuning.
Harrison clearly states in his writings that he believed that the most harmonious intervals were at positions other than at integer frequency ratios. He expressed contemptuous regard for just intonation. Competition between these two opposing paradigms continues into the twentyfirst century.
See also: Musical tuning, meantone temperament, diatonic scale
External links
 Tuning system derived from π and the writings of John 'Longitude' Harrison (http://www.lucytune.com)
 Lucy Scale Developments (http://www.harmonics.com/lucy/)
 LucyTuned Lullabies (http://www.lullabies.co.uk/)