2. Pentadecaphonic system construction

Please consider the frequency interval between A3d= 220 Hz and A4d = 440 Hz,  (from now, the pedix ' d ' will  indicate the dodecaphonic note, whereas the pedix ' p ' will indicate the  pentadecaphonic note); the dodecaphonic system is composed by:

     Note      Freq.[Hz]

A3d          220

A3#d        233

B3d          247

C4d          262

C4#d        277

D4d          294

D4#d        311

E4d           330

F4d           349

F4#d         370

G4d           392

G4#d         415

A4d            440

The interval of four semitones between A3dand C4#dcan be divided into five semitones, obtaining:

       Note          Freq.[Hz]

                   A3p          220       =   A3d  

A3#p        230

B3p          241

B3#p        253

C4p          265

                         C4#p       277        =   C4#d   

The same approach can be done for the intervalsof four semitones between C4#dand F4dand between F4dand A4d,obtaining:

       Note       Freq.[Hz]

                    C4#p       277        =   C4#d

D4p         290

D4#p      304

E4p         318

        F4p         333        

                     F4#p       349         =    F4d

 G4p         366

 G4#p       383

  H4p         401

   H4#p       420

                       A4p         440         =   A4d

The above scale with the pedix ' p' divides the interval between A3 and A4 intofifteen semitones and can be easily shown that it is 'equal tempered'. Bydoubling the frequencies defined, it is possible to build the wholepentadecaphonic system.

    Play the pentadecaphonic  A Chromatic  Scale  (flute tone, from A3 = 220 Hz to A4 = 440 Hz)   Other Scales

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