an 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)

**IS THIS THE LOST MUSIC OF THE SPHERES?**

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
"**.**

*"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

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 a ratio of 2 to 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.

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