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Encyclopedia of Microtonal Music Theory

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Ptolemy (Claudius Ptolemaios)

[Joe Monzo]

Ptolemy's different genera, like all those in ancient times, were bounded by the fixed framework notes which delimited each tetrachord. In the reference tetrachord (meson), these notes are namely mese (ratio 1/1) and hypate (a 4/3 ratio below mese).

Below is a table showing the pitches of the moveable notes ocurring in all shades of Ptolemy's genera. Also shown are some of the xenharmonic-bridges posited by Joe Monzo, which connect notes of the same name across Ptolemy's different genera.

Ptolemy's genera - moveable notes

parhypate / trite

 genus      shade           monzo           ratio        cents     Semitones
                        3   5  7 11 23

diatonic    even      [ 2,  0  0 -1  0 >    18 / 11    852.5920594    8.53
diatonic    tense     [ 0, -1  0  0  0 >     8 / 5     813.6862861    8.14
diatonic    ditonic   [-4,  0  0  0  0 >   128 / 81    792.1799965    7.92
diatonic    tonic     [-2,  0  1  0  0 >    14 / 9     764.9159047    7.65
diatonic    relaxed   [ 2, -1  1  0  0 >    63 / 40    786.4221943    7.86
chromatic   tense     [ 0,  0 -1  1  0 >    11 / 7     782.4920359    7.82
chromatic   relaxed   [-2,  0  1  0  0 >    14 / 9     764.9159047    7.65
enharmonic            [-1, -1  0  0  1 >    23 / 15    740.0056325    7.40

some xenharmonic-bridges:
                      [-6,  1 -1  0  0 >  5120 / 5103    5.757802203  0.06
                      [-4,  0  1 -1  0 >   896 / 891     9.687960643  0.1
                      [ 2, -1  2 -1  0 >   441 / 440     3.930158439  0.04
                      [ 4,  0 -1 -1  0 >    81 / 77     87.67615463   0.88
                      [ 3,  1  0 -1 -1 >   270 / 253   112.5864268    1.13


lichanos / paranete

diatonic    even      [ 2, -1  0  0  0 >     9 / 5    1017.596288    10.18
diatonic    tense     [ 2, -1  0  0  0 >     9 / 5    1017.596288    10.18
diatonic    ditonic   [-2,  0  0  0  0 >    16 / 9    996.0899983     9.96
diatonic    tonic     [-2,  0  0  0  0 >    16 / 9    996.0899983     9.96
diatonic    relaxed   [ 0,  0  1  0  0 >     7 / 4    968.8259065     9.69
chromatic   tense     [ 1,  0 -1  0  0 >    12 / 7    933.1290944     9.33
chromatic   relaxed   [-1,  1  0  0  0 >     5 / 3    884.358713      8.84
enharmonic            [ 0, -1  0  0  0 >     8 / 5    813.6862861     8.14

one xenharmonic-bridge:
                      [ 2,  0  0  0  0 >     9 / 8    203.9100017     2.04
		
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