Harrison discovered this scale experimenting
with monochords and a viol, and trained a church choir in
Lincolnshire to sing it.
From this information I constructed a number
of computer programs to explore all the possible permutations
and combinations which could generate a practical scale. My
problem was to decide which of the hundreds of intervals which
I calculated by addition and subtraction of multiples of the
Large (L) and small (s) interval should be used. Faced with
pages of possible results, I decided to select intervals by
finding which notes would match if I constructed a circle
of fifths using the ratio of 1.494412.i.e. (1.494412^2)/x.........to.....(1.494412^n)/x.
x being an exponential of 2, the use of which as a divisor
would give a result between 1 and 2 and hence in the first
octave. Although I had originally intended to use MIDI pitch
bend for the scale, I dreamt of an instrument which would
be simple to play, in any key of up to 7 sharps or flats,
for any competent guitarist.
The following morning my feet led me to Denmark
St. where I casually mentioned that I had dreamt about this
very strange guitar, and asked how I could make one. The only
guitar I owned, liked, and was willing to modify was a Westone
Raider II. I agreed a price with Graham Noden of Andy's, and
rushed home to calculate the fret positions. I listed the
note names, frequencies in Hertz, ratios, and fret positions
for 20 frets per octave and a total of 35 frets; and delivered
the Westone, instructions, and a detailed explanation of the
mathematics to determine the fret positions.
On Christmas eve, Graham phoned me to say that
he had started on the guitar, but that the ninth fret seemed
to be in the wrong place, so would I please check my calculations.
I did, and called him back with the position for an extra
fret, between what had previously been the ninth and tenth
When I called to pick up the finished job,
Graham had already left for Christmas. I took home the first
instrument to play the scale a renowned genius had first described
two hundred and eleven years earlier. The tuning had to be
done by ear, as only the open fifth string would be the same
as any other guitar at 110 Hertz. I set it from a tuning fork,
tuned the others from the appropriate frets, and slowly strummed
an open G Major. The intervals sound foreign, but this chord
certainly sings. Now for an open E; the same result. It works.
Harrison must have imagined this moment when he wrote his
book for some future generation to decipher. I spent the next
two hours trying chords and melodies which I had previously
only played on a conventional guitar. It was difficult at
first to remember that sharp and flat notes must conform to
the key of the piece, but with a little practice everything
became playable except for the frets between the ninth and
the octave, where something was definitely wrong. I checked
the measurements. They matched my instructions exactly. Graham
had certainly done his part perfectly. I checked the ratios
against the fret positions; perfect up to the ninth, but between
10 and the octave nothing made sense. Two hours later I realised
I had selected the wrong intervals, because
my program had listed six fret positions on each line and
after nine, I had counted the intervals out of synchronisation
with the fret positions. Back into the computer. Two days
later I had cracked it, and added a few extra frets on my
instruction sheet. Lucy Guitar Mark V has twenty-five frets
to the octave and a total of forty-five. It can be played
in any key of up to 11 flats or 11 sharps.
Initially the layout of the Mark V looks confusing,
for there are six pairs of close frets in each octave, and
above the first octave, it was essential to use mandolin wire
as the adjacent frets are very close. Although the guitar
sounded harmonically perfect I found the average guitarist
was initially intimidated by Mark V.
Details and specifications of LucyTuned guitars
This apprehension resulted in Lucy guitar Mark
VII with nineteen frets to the octave by eliminating the least
used of the frets in pairs. Mark VII is easier to play, but
lacks the tonal versatility of Mark V, for it limits the number
of keys which may be used to up to 7 flats or 9 sharps, or
any conceivable key, if an error of (2L - 3s) = 14.367 cents
between sharps and adjacent flats is tolerable. By compromising
and replacing the pairs of frets on the Mark V by a single
midpoint fret, this error may be reduced to 14.367/2 = 7.2
cents. Any pitch is sharpened or flattened by adding or subtracting
the difference between a Large and a small interval i.e. (L-s)
= 68.451 cents.
Applying the scale to synthesisers, I found
that if the black keypads were all assigned to appropriate
sharps or all to flat pitches, the system sounded consonant.
Mixing sharps and flats caused interesting effects, but contradictory
altered notes tended to sound dissonant.
To produce the most viable use on keyboard
instruments of only twelve keypads per octave, the appropriate
pitches for the black keys need to be programmed, or fast-loaded,
from a selection of choices dependent upon the tonality of
the piece to be played, but for some experimental keyboards
of 31 or 53 keypads per octave the tunings may be fixed, and
modulations between sharp keys and flat keys achieved.
LucyTuning using pitchbend and MIDI
This book is a constantly evolving document,
to also report progress to readers. Many instruments, including
harmonicas, banjos, basses etc. have now been produced, tuned
or modified to this scale. The 19 frets per octave guitar
has served its purpose as a starting point for timid players,
but the limitations and tuning compromises soon became apparent.
The 25 fret guitar has become the most versatile fretted instrument
as all the notes are 'in tune', but this I'm sure will also
evolve as musicians reach further into new tonalities. (as
it has) As hundreds of people are now using these discoveries,
research continues particularly into the connections to physics,
mathematics, topology, and to music other than equal temperament
for composition and performance.
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