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.
|
Note |
Freq.[Hz] |
|
F2 |
83 |
|
F2# |
87 |
|
G2 |
91 |
|
G2# |
96 |
|
H2 |
100 |
|
H2# |
105 |
|
A2 |
110 |
|
A2# |
115 |
|
B2 |
121 |
|
B2# |
126 |
|
C3 |
132 |
|
C3# |
139 |
|
D3 |
145 |
|
D3# |
152 |
|
E3 |
159 |
|
F3 |
167 |
|
F3# |
175 |
|
G3 |
183 |
|
G3# |
192 |
|
H3 |
201 |
|
H3# |
210 |
|
A3 |
220 |
|
A3# |
230 |
|
B3 |
241 |
|
B3# |
253 |
|
C4 |
265 |
|
C4# |
277 |
|
D4 |
290 |
|
D4# |
304 |
|
E4 |
318 |
|
F4 |
333 |
|
F4# |
349 |
|
G4 |
366 |
|
G4# |
383 |
|
H4 |
401 |
|
H4# |
420 |
|
A4 |
440 |
|
A4# |
461 |
|
B4 |
483 |
|
B4# |
505 |
|
C5 |
529 |
|
C5# |
554 |
|
D5 |
581 |
|
D5# |
608 |
|
E5 |
637 |
|
F5 |
667 |
|
F5# |
698 |
|
G5 |
731 |
|
G5# |
766 |
|
H5 |
802 |
|
H5# |
840 |
|
A5 |
880 |
|
A5# |
922 |
|
B5 |
965 |
|
B5# |
1011 |
|
C6 |
1059 |
|
C6# |
1109 |
|
D6 |
1161 |
|
D6# |
1216 |
|
E6 |
1274 |
Play the pentadecaphonic A Chromatic Scale (flute tone, from A3 = 220 Hz to A4 = 440 Hz) Other Scales
Copyright INTEAS - Reproduction forbidden - 07-Sep-2001