Perfect Immutable System (PIS)

The amalgamation of the Greater Perfect System and the Lesser Perfect System into one "complete" system.

See Systema

[from John Chalmers, Divisions of the Tetrachord]

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Shown below is my very general schematic illustrating the Greek names for the PIS in the diatonic genus, in descending order, along with approximate letter-names to help the Western reader to comprehend.

              -
             |  A  nete hyperbolaion   
 tetrachord  |
hyperbolaion |  G  paranete hyperbolaion   
             |
             |  F  trite hyperbolaion
              - E  nete diezeugmenon   
             |                                                        -
 tetrachord  |  D  paranete diezeugmenon        D  nete synemmenon     |
diezeugmenon |                                                         | tetrachord
             |  C  trite diezeugmenon           C  paranete synemmenon | synemmenon
             |  B  paramese                                            |
              -                  -              Bb trite synemmenon    |
                                |   A  mese                           -
                     tetrachord | 
                       meson    |   G  lichanos meson
                                |
                                |   F  parhypate meson
                                 -  E  hypate meson
                                |
                     tetrachord |   D  lichanos hypaton
                      hypaton   |
                                |   C  parhypate hypaton
                                |   B  hypate hypaton
                                 -
                                    A  Proslambanomenos

Note that the intervallic structure of all tetrachords is identical, and that the complete system includes both of the notes which we would call "B" and "Bb". (See mutation for more on this.)

The tuning of the diatonic genus was almost always given by theorists as Pythagorean, i.e., 3-limit, as shown in the following lattice-diagram:

Bb -- F --- C --- G --- D --- A --- E --- B

-5   -4    -3    -2    -1     0     1     2

              exponent of 3

This conception of the gamut of musical pitch resources lasted until the late 1400s in Europe, when Ramos and other theorists began writing about scales in which some of the pitches were retuned to 5-limit ratios.

Note that the two inner notes in each tetrachord could be retuned to other pitches in the chromatic and enharmonic genera, and even in different shades of the diatonic genus.

see also:

my Tutorial on ancient Greek tetrachord-theory

[from Joe Monzo, JustMusic: A New Harmony]

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(to download a zip file of the entire Dictionary, click here)

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

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