The Synergetic Temperament System

Kenneth Hemmerick

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Octaval Subdivision

The synergetic temperament system differs in conceptualization from other systems of temperament. First, it uses a 3-dimensional wave formation to describe the design and behavior of the propagation of sound. Second, it does not use the octaval relationship as the fundamental interval for the structuring of frequencies.

Any sub-division of the octave necessitates a root derivative of 2 which, in every case, is always an irrational number. Due to this mathematical phenomenon, the intervalic relationships structured in context of the octave are also irrational. For example if a tone is desired seven octaves above a given tone, the given tone's frequency is increased by a factor of 1:2 to the seventh power or (2)7.

When the same procedure is followed, except that the interval of a fifth is substituted in order to obtain the same seventh octave tone, the frequency of the former seventh octave tone is overstepped by a small margin. (2:3)12 or (1.5)12 or 129.74663 which is greater than (2)7 or 128 by a factor of the difference of the two intervals. This phenomenon occurs with all intervalic sequential ordering. For an illustration of this behavior see Fig. 11. In this chart, all intervalic sequential ordering should reach an octaval relationship.

Fig. 11

Interval Name Frequency Ratio Ratio in decimal form Closest # of intervals to reach octave Intervalic factorial value True octaval factorial
Unison 1:1 1      
Minor Second 15:16 1.0666... 12 2.1694 2
Major Second 8:9 1.125 6 2.0272 2
Minor Third 5:6 1.2 4 2.0736 2
Major Third 4:5 1.25 3 1.9531 2
Diminished Fourth 25.32 1.28 3 2.097 2
Perfect Fourth 3:4 1.333... 12 31.5693 32
Augmented Fourth 32:45 1.40625 2 1.97753 2
Diminished Fifth 45:64 1.4222... 2 2.0227 2
Perfect Fifth 2:3 1.5 12 129.74663 128
Augmented Fifth 16:25 1.5625 3 3.81496 4
Minor Sixth 5:8 1.6 3 3.81496 4
Major Sixth 3:5 1.666... 4 7.7160 8
Minor Seventh 9:16 1.777... 6 31.56932 32
Major Seventh 8:15 1.875 12 1888.066 2048
Octave 1:2 2 1 2 2
The octave is the most perfect consonance, so perfect that it gives the impression of duplicating the original tone, a phenomenon for which no convincing explanation has ever been found, Its singularity becomes apparent if the acoustical frequencies are compared with the series of color frequencies (spectrum), which does not show any duplication. The fundamental importance appears also from the fact that it is the only interval common to practically all scales ever evolved, regardless of the number of pitch of the intermediate steps. 5

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© 2006 Kenneth Hemmerick