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Extract from Pitch, Pi, and Other Musical Paradoxes (A Practical Guide to Natural Microtonality), by Charles E. H. Lucy copyright 1986-2001 LucyScaleDevelopments ISBN 0-9512879-0-7

Chapter One. (First published in Music Teacher magazine, January 1988, London)



After twenty five years of playing, I realised that I was still unable to tune any guitar so that it sounded in tune for both an open G major and an open E major. I tried tuning forks, pitch pipes, electronic tuners and harmonics at the seventh, fifth and twelfth frets. I sampled the best guitars I could find, but none would "sing" for both chords. At the risk of appearing tone deaf, I confessed my incompetence to a few trusted musical friends, secretly hoping that someone would initiate me into the secret of tuning guitars, so that they would "sing" for all chords. A few admitted that they had the same problem, but none would reveal the secret.

I had read how the position of the frets was calculated from the twelfth root of two, so that each of the twelve semitones in an octave had equal intervals of 100 cents. It is the equality of these intervals, which allows us to easily modulate or transpose into any of the twelve keys.

Without realising it, I had begun a quest for the lost music of the spheres. I still secretly doubted my musical ear, but rationalised my search by professing an interest in microtonal music, and claimed to be searching for the next step in the evolution of music. First stop library, where I found Helmholtz's book On The Sensations Of Tone. He introduced me to thousands of alternative tunings. The only universal 'truth', on which all tunings agreed was that halving the length of a string doubles the frequency, and produces a ratio of 2.00000:1 which is known as an octave. This octave was subdivided in three basic ways;

1) By a geometric progression, with any number of equal intervals.Eg. 12 (as on conventional guitars) [100 cents per semitone or interval]; 31 as advocated by Huyghens (1629-1695) [38.71 cents per interval], or 53 by Mercator and Bosanquet (1876 Treatise) [22.64 cents];

2) By low whole number ratios. Eg. (Just Intonation) 3:2 for the Vth; 5:4 for the major third etc.

3) By cumulative fifths. Eg. Pythagorean Tuning 3:2 for the Vth; but 81:64 for the major third.

There are also hybrids of the other three, Eg. Meantone Temperaments

If the only point of agreement is that the octave ratio should be exactly two; how do we explain the phenomenon of stretched octaves, used by some piano tuners?

The Pythagorean system , instead of arriving at the octave after 12 steps i.e. (3^12)/(2^18) becomes 531441/262144 = 2.0272865 instead of 2.0000 as we assume when tuning a guitar by fifths at the seventh fret. This difference or error is known as Pythagoras' lemma. Every system seems to be an imperfect compromise, which is probably why mathematicians and musicians have devoted millions of hours to searching for the perfect scale.

Initially, the idea of 53 notes on a geometric progression seemed to be a sensible solution, and I read that Bosanquet's harmonium on this scale had been in the Kensington Science Museum since the 1880's. I went to find it, but it was in storage. Instead I found Mr. Chew. I told him of my quest and that I had a hunch that the solution was in some way connected with the music of the spheres and the Greek letter " Pi ".

Pi "That's what Harrison thought."

I enquired further and discovered that John Harrison (1693-1776), an horologist, had discovered longitude and won a £20,000 prize from Parliament after the personal intervention of George III. I was directed to the Clockmakers' Library in the Guildhall, and there found a treasured copy of A Description concerning such Mechanism as will afford a nice, or true Mensuration of Time; together with Some Account of the Attempts for the Discovery of the Longitude by the Moon; and also An Account of the Discovery of the Scale of Musick, only to be refused permission to photocopy any of it, due to its antiquity. I eventually acquired photocopies of the relevant pages from another source. Harrison had written in such an obscure style that I suspected that he intended to hide its secrets from all but the most diligent enquirer.

The essence of what Harrison said is as follows:

"The natural scale of music is associated with the ratio of the diameter of a circle to its circumference." (i.e. pi = 3.14159265358979323846 etc.)

"This scale is based on two intervals;"

1) The Larger note as he calls it; This is the ratio of the 2*pi root of 2, or in BASIC computer terms 2^(1/(2*pi)), which equals a ratio of 1.116633 or 190.9858 cents, approximately 1.91 frets on a conventional guitar. (L).

2) The lesser note, which is half the difference between five Larger notes (5L) and an octave. i.e.(2/(2^(1/(2*pi)))^5)^(1/2), giving a ratio of 1.073344 or 122.5354 cents, an interval of approx. 1.23 frets. (s).

The equivalent of the fifth (i.e. seventh fret on guitar) is composed of three Large (3L) plus one small note (s) i.e. (3L+s) = (190.986*3) + (122.535) = 695.493 cents or ratio of 1.494412.

The equivalent of the fourth (IV) (fifth fret) is 2L+s = 504.507 cents.

To convert between ratio and cents Enter a number in either field, then click outside the text box.


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)/^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 frets.

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 my mistake.

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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Related Pages
Chapter 2 Fourths and Fiths: Circles, Spirals, or Cylinders?
Chapter 3 Scales available in LucyTuning
Chapter 4 Pitch and Colour Connections
Chapter 5 Scalemaking