let HPS be Heptatonic_Pythagorean_Score; :: thesis: for frequency being Element of HPS holds (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 5 = (81 / 64) * (@ frequency)
let frequency be Element of HPS; :: thesis: (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 5 = (81 / 64) * (@ frequency)
set MS = HPS;
set q = Fourth (HPS,frequency);
reconsider n3 = 4 as Nat ;
(spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . n3 is Element of HPS ;
then reconsider r32 = (27 / 16) * (@ frequency) as Element of HPS by Th85;
A1: (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 5 = (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . (n3 + 1)
.= Fifth_reduct (HPS,frequency,((spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . n3)) by Def19
.= Fifth_reduct (HPS,frequency,r32) by Th85 ;
consider r, s being positive Real such that
A2: ( r = r32 & s = (3 / 2) * r & Fifth (HPS,r32) = s ) by Th54;
A3: ex fr being positive Real st
( frequency = fr & Octave (HPS,frequency) = 2 * fr ) by Def15;
A4: ( not @ frequency <= (81 / 32) * (@ frequency) or not (81 / 32) * (@ frequency) <= 2 * (@ frequency) ) by XREAL_1:68;
A5: ex r being positive Real st
( Fifth (HPS,r32) = r & Octave_descendent (HPS,(Fifth (HPS,r32))) = r / 2 ) by Th51;
not Fifth (HPS,r32) is_Between frequency, Octave (HPS,frequency) by A4, A2, A3;
hence (spiral_of_fifths (HPS,frequency,(Fourth (HPS,frequency)))) . 5 = (81 / 64) * (@ frequency) by A1, A2, A5, Def18; :: thesis: verum