theorem Th68:
  for r1,r2,r3,r4,r5,r6 being positive Real st
  pentatonic_pythagorean_scale(MS,frequency).1 = r1 &
  pentatonic_pythagorean_scale(MS,frequency).2 = r2 &
  pentatonic_pythagorean_scale(MS,frequency).3 = r3 &
  pentatonic_pythagorean_scale(MS,frequency).4 = r4 &
  pentatonic_pythagorean_scale(MS,frequency).5 = r5 &
  pentatonic_pythagorean_scale(MS,frequency).6 = r6 holds
  r2 / r1 = (9 qua Real) / 8 & r3 / r2 = (9 qua Real) / 8 &
  r4 / r3 = (32 qua Real) / 27 & r5 / r4 = (9 qua Real) / 8 &
  r6 / r5 = (32 qua Real) / 27
  proof
    let r1,r2,r3,r4,r5,r6 be positive Real;
    assume
A1: pentatonic_pythagorean_scale(MS,frequency).1 = r1 &
    pentatonic_pythagorean_scale(MS,frequency).2 = r2 &
    pentatonic_pythagorean_scale(MS,frequency).3 = r3 &
    pentatonic_pythagorean_scale(MS,frequency).4 = r4 &
    pentatonic_pythagorean_scale(MS,frequency).5 = r5 &
    pentatonic_pythagorean_scale(MS,frequency).6 = r6;
    set gamme = pentatonic_pythagorean_scale(MS,frequency);
A2: gamme.1 = frequency &
    gamme.2 = spiral_of_fifths(MS,frequency,frequency).2 &
    gamme.3 = spiral_of_fifths(MS,frequency,frequency).4 &
    gamme.4 = spiral_of_fifths(MS,frequency,frequency).1 &
    gamme.5 = spiral_of_fifths(MS,frequency,frequency).3 &
    gamme.6 = Octave(MS,frequency) by Def20;
    gamme is pentatonic by Th67;
    then consider frequency be Element of MS,
    r1,r2,r3,r4,r5,r6 be positive Real such that
A3: gamme.1 = frequency & gamme.1 = r1 &
    gamme.2 = r2 & gamme.3 = r3 & gamme.4 = r4 & gamme.5 = r5 & gamme.6 = r6 &
    r1 < r2 < r3 & r3 < r4 < r5 & r5 < r6 & gamme.6 = Octave(MS,frequency);
    @(spiral_of_fifths(MS,frequency,frequency).2) /
    @frequency = (3 * 3 qua Real) / (2 * 2 * 2) &
    @(spiral_of_fifths(MS,frequency,frequency).4) /
    @(spiral_of_fifths(MS,frequency,frequency).2)
      = (3 * 3 qua Real) / (2 * 2 * 2) &
        @(spiral_of_fifths(MS,frequency,frequency).1) /
        @(spiral_of_fifths(MS,frequency,frequency).4) = ((32 qua Real) / 27) &
        @(spiral_of_fifths(MS,frequency,frequency).3) /
        @(spiral_of_fifths(MS,frequency,frequency).1) = ((9 qua Real) / 8) &
        @Octave(MS,frequency) /
        @(spiral_of_fifths(MS,frequency,frequency).3)
      = ((32 qua Real) / 27) by Th62,Th63,Th64,Th65,Th66;
    hence thesis by A1,A2,A3;
  end;
