~Proceedings ICMCISMCI2014 14-20 September 2014, Athens, Greece
ent. Figure 6 shows the amplitude, phase spectrum of a
circle wave, and the resynthesized circle wave.
DFT (re)synthesized waveform
+Nyquist
Frequency(Hz)
Figure 6. The phase and amplitude spectrum of a circle wave,
and above the resynthesized circle wave.
3.5 N-gon Wave Drum Sound Example
Here is a short example of how to set the parameters to
synthesize a drum like sound with a n-gon wave derived
from a star polygon with Schlafli symbol {420/209}:
n =
q =
1=
(-=
fo =
= 420 (a large n for star polygons)
= n/2 - 1 = 209 (also n/2, more melodic)
n/4 = 105 (or higher or lower, affects the timbre)
= 7/2 = 27/n(n/4) (changes affect the timbre)
= 55Hz (higher or lower values for different pitches)
Figures 7 and 8 show parts of the same n-gon wave. In
figure 7 the circle seems to be filled with the grey color
that is used for the edges of the polygon in the circle.
This is due to the large number of 420 vertices or edges
that are used. The actual n-gon wave is much longer than
seen here.
Figure 8 shows a magnified version of a smaller part of
the same n-gon wave where the edges of the star polygon
can be identified. As can be seen in figure 7, if a star
polygon with a large n (i.e. a large number of vertices or
edges) is used to generate a n-gon wave similar to this
one, the gradient decreases for each edge. At some limit,
i.e. if n would be infinite, it might converge to 0.
Figure 7. A n-gon wave with Schlafli symbol {420/209}, start
phase p = n/2, and number of connected edges 1 = 105. The
wave has a drum like timbre. Because a star polygon with 420
vertices or edges is used, the circle seems to be a grey dot.
4. SCALES
As should appear obvious, every n-gon wave can oscillate with every frequency and they can be used in every
kind of scale, albeit some scales seem to be inherent to
the subject itself.
The scales presented here should be seen as an approach to make the geometrical relations and proportions
of regular polygons and star polygons audible. The n-gon
wave scales described here are derived from regular
polygons and star polygons and a unit circle (i.e. all ngon waves are derived from the same circumcircle radius)
or a unit frequency ratio (i.e. the ratio of the circle wave
wavelength and the n-gon wavelength) of the corresponding circle wave.
Other properties of regular polygons could be chosen as
unit to derive n-gon scales from, for example, a unit incircle, a unit edge or a unit stellation line that connects
the vertices of a star polygon. It is possible to build scales
from the phases of the stellations of a star polygon if they
are used as start phases ().
More than one unit parameter or other geometric properties of regular polygons or star polygons can also be
used for the construction of n-gon wave scales. For instance, unit circle recursion, unit frequency ratio recursion, and the stellations of a star polygon can be combined into a scale.
4.1 Unit Circle Scales
Unit Circle Scales are comprised of n-gon waves with 9
= 0 and T = 7/n derived from n-gons adjacent to one or
more unit circles. A unit circle is used as a centre frequency of a corresponding circlewave of which the other
n-gon wave frequencies of the scale are derived from.
The unit circle wave figures as a form of axis to which
the n-gon waves of a Unit Circle Scale are adjacent to.
Here is a first example. The range of one octave and
one additional higher semitone can be constructed with
two trigon waves and two tetragon waves. To derive the
fundamental frequency of the scale, a trigon with start
phase 0 on the outside adjacent to the unit circle is used
(unit circle exponent Q = 1). For its octave a trigon with
startphse 2/n on the inside adjacent to the unit circle is
used (unit circle exponent a = 0). The ratio of the circle
wave and the trigon waves will then be the fifth on the
outside, and fourth on the inside, with the ratios 3/4 for
the first and 3/2 for the octave trigon wave. The two
tetragon waves are used to build another octave around
the circlewave which then becomes their tritone, the
ratios of the tetragon waves and the circle wave are: X2/2
and 12, respectively.
All n-gon waves with an odd n have different frequencies. The two start phases 9p= 0 and 9p 2=/n do not
change the frequency or the amplitude peaks of the n-gon
waves with an odd n, but the start phases do change the
frequencies and the amplitudes of the even n n-gon
waves.
All n-gon waves with an even n on the inside of a unit
circle with start phase p= 0 and unit circle exponent Q =
0 have the same frequency as the unit circle wave. If the
Figure 8. A magnified image of the n-gon wave shown in figure 8. Parts of the edges of the star polygon in the unit circle are
visible.
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