LucyTuning is a meantone temperament system in which the fifth is 600+300/π ≈ 695.5 cents, 4.5 cents flatter than that of 12-tone equal temperament. Its main advocate is Charles E. H. Lucy, who discovered it among the eighteenth century writings of John Harrison.
An upward step of a whole tone, in LucyTuning, a Large interval (L), in LucyTuning it will be 2(600+300/π)-1200 = 600/π ≈ 190.99 cents. The major third (an interval of two Large intervals) is therefore 1200/π cents, which is an octave divided logarithmically by π or the π-th 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, a small interval (s) is (600-300/π)-1200/π = 600-1500/π cents, or 122.535 cents.
If we call the whole tone L and the diatonic semitone s, for Large and small, the familiar diatonic scale is LLsLLLs in major mode, and a LucyTuned diatonic scale will be one with the above specific values for L and s.
As the system is derived from pi, each new note which is added by steps of fourths and fifths will arrive in a new and unique position in the octave.
The first known writings by Harrison about this system were in John Harrison's book "Concerning Such Mechanism ..... " from the 1770's. A later 1770's Harrison manuscript, "A True and Full Account of the Foundation of Musick ...... ", was found in the Library of Congress in 2002. Modern writings on the subject are in "Pitch, Pi, and Other Musical Paradoxes' by Charles Lucy 1988, and may be found in the British Library.
In Robert Smith's Harmonics of 1749 we find the following description of Harrison's system of tuning:
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.
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 twenty-first century.
Lucy patented the system in 1988, claiming the tuning as well as its use in fretted string instruments as well as electronic instruments.
88-ET can be used as an approximation to LucyTuning. Using 14 steps as the Large interval (L) and 9 steps as the small interval (s), 88-ET gives a close approximation to the diatonic scale LLsLLLs of Lucy Tuning.
Here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series; the difference column measures in cents the distance from an exact fit to these ratios.
|interval name||size (88-ET steps)||size (cents)||just ratio||just (cents)||difference|
|fifth V (3L+s)||51||695.45||3:2||701.96||6.50|
|fourth IV (2L+s)||37||504.55||4:3||498.04||-6.50|
|third III (2L)||28||381.82||5:4||386.31||4.50|
|flat third bIII (L+s)||23||313.64||6:5||315.64||2.00|
|septimal minor third||20||272.73||7:6||266.87||-5.86|
|whole tone, major tone||15||204.55||9:8||203.91||-0.64|
|Large Interval II (L)||14||190.99||-||-||-|
|small interval bII (s)||9||122.54||-||-||-|