PYTHAGORAS, TRITONE HARMONICS AND THE SONAR TUNING
Historically, Pythagoras is considered to be the first person who had the profound
insight that – hidden behind the apparent variety and complexity of nature and the
cosmos – there is a pattern and an order that can be expressed by remarkably
simple mathematical ideas. He and his followers, the Pythagoreans, were known
for saying „all things are numbers“ and essentially believed that our world is
rational, i.e. that all important phenomena can be described by ratios of integers.
Their prime example was a connection they found between music and mathematics,
famously carried out on a monochord. They discovered that intervals between
notes can be expressed by ratios of integers: the smaller the numbers, the
more beautiful and consonant the interval. The most pleasant sounding interval
to their ears, a perfect octave, was produced when the ratio of the lengths of
two vibrating strings was exactly 2:1. The next simplest ratios were 3:2 (a perfect
fifth) and 4:3 (a perfect fourth).
But they were wrong. It soon became obvious that the relationship between
numbers and nature was far more complex than they had hoped for and could
not always be explained by simple integer ratios. It must have been quite a shock
to them when they had to admit that even such a basic and fundamental ratio
as the one between the diagonal and the side of a square (√2:1) was found to be
irrational, i.e. not expressible as a ratio of integers  ironically, this can easily be
proved using Pythagoras' own theorem. Another fact soon became apparent that
surely didn't suit their cherished beliefs.: that perfect octaves and perfect fifths
were not compatible, meaning that seven octaves and twelve fifths do not –
as you would expect  produce exactly the same note. Even today, this fact (which
led to the Pythagorean comma and equal temperament) is hard to accept
and even could be considered a flaw of nature. Heaven might be a place where
seven octaves and twelve fifths are the same note – but the real world we live in
is a darker and irrational place where compromises and adjustments have
to be made.
In SONAR, the electric guitars are tuned in tritones C / F# / C / F# / C / F#
where the bottom C is a major third below the standard E. The bass guitar is
also tuned in tritones C / F# / C / F#, where the bottom C is again a major
third below the standard E of a bass guitar.
The tritone is a very interesting interval. In a usual harmonic context, it is
considered to be the most dissonant interval and has even earned the nickname
diabolus in musica. In 12tone equal temperament, the tritone is exactly
half an octave (which corresponds to the irrational ratio of √2:1, just like the
ratio of the diagonal and the side of a square). In many analyses of the works of
20th century composers, the tritone plays an important structural role. Perhaps
the most noted is the axis system, proposed by Ernö Lendvai in his fascinating and
controversial analysis of the use of tonality in the music of Bela Bartok.
A very large proportion of SONAR's music is played using only the first four natural
harmonics of the guitar and bass strings: first octave, fifth, second octave and
major third. In this way, the perfect, rational harmonics of the major scale clash
with the irrational ratios of the tritone tuning to produce a symmetric hexatonic
scale C / C# / E /F# / G / A# (a mode of limited transposition in Messiaen's
terminology resulting from the C major and F# major triads)), which gives SONAR its distinctive and instantly recognizable sound.
Expressed in decimal numbers (rounded to three decimal places), the Sonar tuning has the following ratios:
 Sonar Tuning 
equal temperament 
C  1  1 
C#  1.061 = 3/4*√2  1.059 = 2^{(1/12)} 
E  1.25 = 5/4  1.260 = 2^{(4/12)} 
F#  1.414 = √2  1.414 = 2^{(6/12)}=√2 
G  1.5 = 3/2  1.498 = 2^{(7/12)} 
A#  1.768 = 5/4*√2  1.782 = 2^{(10/12)} 
C  2  2 
In spirit, SONAR is a very Pythagorean band. Numbers are everywhere in our music – not
only in the tuning (an amalgam of rational and irrational worlds) and in the harmonic
material, but also in our approach to odd metres and polyrhythms (another sonic
representation of integer ratios). In the wake of Pythagoras, we also believe that a
composition is most satisfying if – hidden behind the apparent variety of all the sounds
and rhythms – there is a simple idea that organically generates this complexity.
Stephan Thelen, Warth, Switzerland, September 2011
