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, where asthe 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 A3d and C4#d can 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 intervals of four semitones between C4#d and F4d and between F4d and 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 into fifteen semitones and can be easily shown that it is 'equal tempered'. By doubling the frequencies defined, it is possible to build the whole pentadecaphonic system.

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

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