Ad Quadratum Construction and Study of the Regular Polyhedra

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In either case, consonance is lost. The resolution was eventually found in the Tempered
scale popularized by J.S. Bach in the early 18 th century. The Ancients knew of it 39 but
they also knew the importance of finding one’s limits and always kept their melodies
within proper bounds so as to not have to tamper with either intervals or octaves. The
problem was solved by spreading the comma across the octave by dividing the octave
scale in 12 equal tones.
Pythagorean tuning had another problem and that was the discordance of the third 81
.
64
Remember that only the fourth and the fifth, together with the octave and unisson were
considered as concordances by the Pythagoreans.
To remove this discordance, just intonation, which replaced the Pythagorean third 81
64
by the natural third 5
4
, was suggested. It was not adopted however until the
Renaissance since the third was avoided in ancient and early medieval music as was
pointed out earlier.
We shall return to Pythagorean tuning and its relation with the adquadratum construction
but before we shall address the design of the just intonation scale.
Just Intonation:
As the Pythagorean scale was shown to originate out of the play of numbers 1, 2, and 3
through the prime (c=1), the octave (c’=2) and the fifth (g= 3
), the just intonation scale
2
can be constructed through multiplication of division out of numbers 1, 2, and 3 equally,
the natural third 5
4
appearing as a consequence of these multiplications and divisions
by 2 and 3, as shown on fig. 80. This is interesting in the sense that the just intonation is
usually presented as the result of combination of the natural fifth and the natural third.
(e.g., N 3 2
m
5 4
n
).
Since we can show that only 1, 2, and 3 are involved, it brings Just intonation within the
scope of the adquadratum construction.
Acoustically, the operation involves the half-cut and the third cut, which can be done
aurally or visually on a monochord. The first line of fig. 80 gives the intervals to the
prime; it constitutes the c-major scale.
39
In fact McClain (op. cit. p. 5) claims that Plato’s Republic embodies, from a musician’s perspective, a treatise on
equal temperament.