Bauhaus-Universität Weimar

Titel:
Acoustic methods of work in relation to systematic comparative musicology
Person:
Kornerup, Thorvald
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit38535/23/
Acoustic Methods of Work. 
23 
4. Out of the calculated golden system I have formed some other Series for 
instance 31-toned see Scheme 20: 
Double super 
division. 
Great Third-Series IV: 
4 cisis 6 eses 
10 e 
16 ges 
26 bes 
Tritonos — V : 
9 feses 
15 fis 
24 beses 
Great Sixth- — VI: 
9 disis 
14 geses 
23 a 
Group B, Geometrical Constructions of golden Tones. 
The golden tones and intervals can also be constructed geometrically as shown in 
Scheme 8, which indicates: 
1) above, part of a regular Pentagon, the angle of which =108° is divided into 3 
parts by means of 2 chords; if the chord be equal to a Tetrachord— the interval C-F, 
(the Fourth f), this interval will be super-divided in »great double super-division« 
I C--D......Es--F = Series I 1 
m { 192.43 ; 118.93 + 192.13 = 503.79 cents J 
2) below: a square placed inside a semicircle, whereby the diameter is super-divided 
into a »small double super division« 
in cents 
corresponding with: 
Cis .... D--E ..... F 
D .....Es--F ..... Ges 
B ..... C - D ..... Es 
118.93 + 192.43 jjl 18.93 = 430.29 
C ..... D--F ..... G 
192.4 + 311.4 H- 192.4 =696.2 
3) further: within the square we can form a »great double super division« 
. j D - Dis .... Es- E ) 
m 1 73.50 -B 45.43 + 73.50 = 192.43 cents j 
Scheme 9 shows the golden system constructed by means of the Pentagram with 
the same chord C .. F, so that the side is = C .. Es in the Pentagon. 
Fourth-Series I: C, Deses, Cis, Des, D, Es, F and further As, Des’. 
C (Es) es 
Further: — = Cosinus 36°, the relation between small and great Third. 
• • G 0 
Scheme 10 shows »double super-division« in 4 Series; the upper and lower edges 
of a square are divided: 
1 from c upwards into ases, direct, 
\ — c’ downwards into eis, retrogade.
        

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