reserve MS for satisfying_equiv MusicStruct;
reserve a,b,c,d,e,f for Element of MS;
reserve MS for satisfying_octave_descendent_constructible
  classical_octave satisfying_octave_constructible classical_fifth
  satisfying_fifth_constructible satisfying_harmonic_closed satisfying_Nat
  satisfying_commutativity satisfying_interval satisfying_equiv satisfying_Real
  non empty MusicStruct,
  fondamentale,frequency for Element of MS;
reserve                              MS for MusicSpace,
        fondamentale, frequency, f1, f2 for Element of MS;
reserve       HPS for Heptatonic_Pythagorean_Score,
        frequency for Element of HPS;
reserve       HPS for Heptatonic_Pythagorean_Score,
        frequency for Element of HPS;

theorem Th92:
  intrval(hepta_4(HPS,frequency), hepta_1(HPS,Octave(HPS,frequency)))
  = (3 qua Real) / 2 &
  intrval(hepta_5(HPS,frequency), hepta_2(HPS,Octave(HPS,frequency)))
  = (3 qua Real) / 2 &
  intrval(hepta_6(HPS,frequency), hepta_3(HPS,Octave(HPS,frequency)))
  <> (3 qua Real) / 2 &
  intrval(hepta_octave(HPS,frequency),
    hepta_4(HPS,Octave(HPS,frequency))) = (3 qua Real) / 2
  proof
    set gamme = heptatonic_pythagorean_scale(HPS,frequency);
A1: hepta_fondamental(HPS,frequency) = 1 * @frequency &
    hepta_1(HPS,frequency) = (9 qua Real) / 8 * @frequency &
    hepta_2(HPS,frequency) = (81 qua Real) / 64 * @frequency &
    hepta_3(HPS,frequency) = (4 qua Real) / 3 * @frequency &
    hepta_4(HPS,frequency) = (3 qua Real) / 2 * @frequency &
    hepta_5(HPS,frequency) = (27 qua Real) / 16 * @frequency &
    hepta_6(HPS,frequency) = (243 qua Real) / 128 * @frequency by Th88;
    set f2 = Octave(HPS,frequency);
    set gamme2 = heptatonic_pythagorean_scale(HPS,f2);
A2: heptatonic_pythagorean_scale_ascending(HPS,frequency).1
        = 2 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).2
      = (9 qua Real) / 4 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).3
      = (81 qua Real) / 32 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).4
      = (8 qua Real) / 3 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).5
      = (3 qua Real) * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).6
      = (27 qua Real) / 8 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).7
      = (243 qua Real) / 64 * @frequency &
      heptatonic_pythagorean_scale_ascending(HPS,frequency).8 = 4 * @frequency
      by Th91;
A3: ex fo be positive Real st frequency = fo &
    Octave(HPS,frequency) = 2 * fo by Def15;
    now
      thus intrval(hepta_4(HPS,frequency),
        hepta_1(HPS,Octave(HPS,frequency)))
        = (((9 qua Real) / 4) * @frequency) / (((3 qua Real) / 2) * @frequency)
          by A2,A1,Def21
       .=(((9 qua Real) / 4) / ((3 qua Real) / 2) * @frequency) / @frequency
          by XCMPLX_1:83
       .= (3 qua Real) / 2 by XCMPLX_1:89;
      thus intrval(hepta_5(HPS,frequency),
        hepta_2(HPS,Octave(HPS,frequency)))
        = ((81 qua Real) / 32 * @frequency) / ((27 qua Real) / 16 * @frequency)
           by A2,A1,Def21
       .=(((81 qua Real) / 32) / ((27 qua Real) / 16) * @frequency) /
           @frequency
        by XCMPLX_1:83
       .= (3 qua Real) / 2 by XCMPLX_1:89;
      intrval(hepta_6(HPS,frequency),hepta_3(HPS,Octave(HPS,frequency)))
        = ((8 qua Real) / 3 * @frequency) / ((243 qua Real) / 128 * @frequency)
        by A2,A1,Def21
       .=(((8 qua Real) / 3) / ((243 qua Real) / 128) * @frequency) /
           @frequency
        by XCMPLX_1:83;
      hence intrval(hepta_6(HPS,frequency),
        hepta_3(HPS,Octave(HPS,frequency))) <> (3 qua Real) / 2 by XCMPLX_1:89;
      thus intrval(hepta_octave(HPS,frequency),
        hepta_4(HPS,Octave(HPS,frequency)))
        = ((3 qua Real) * @frequency) / (2 * @frequency) by A3,A2,Def21
       .= (((3 qua Real) / 2) * @frequency) / @frequency by XCMPLX_1:83
       .= (3 qua Real) / 2 by XCMPLX_1:89;
    end;
    hence thesis;
  end;
