[John Chalmers, Divisions of the Tetrachord]

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

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[Joe Monzo]

Shown below is a 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

Below is a rendering of the PIS into modern staff-notation, using 4-note groups of eighth-notes to represent each tetrachord, with tetrachord names shown. The Greater Perfect System is shown with upward stems, and Lesser Perfect System with downward stems. Ties from the last note of one tetrachord to the first note of another indicate conjunct tetrachords.

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

Below are two Tonescape^{®} lattices of the PIS, one showing the ratios and one showing the letter nominals. Both lattices include prime-factor 2, since the PIS was always described in terms of absolute ratios and not with octave-equivalence assumed. The orientation of the axes has 2 on the horizontal and 3 on the vertical.

Below is an ASCII-text version of the same diagram, showing 2,3-monzo, ratio, letter nominal, and absolute cents value from 1/1 for each note:

lattice diagram: PIS (Perfect Immutable System) - diatonic genus pythagorean tuning [8 -5> -- 256:243 | Bb | +0090 | | | -- [7 -4> 128:81 F +0792 | [6 -4> -- 64:81 | F | -0408 | | | -- [5 -3> ----- 32:27 | C | +0294 | | | [4 -3> -- [4 -2> 16:27 16:9 C G -0906 +0996 | [3 -2> --- 8:9 | G | -0204 | | | -- [2 -1> ---- 4:3 | D | +0498 | | | [1 -1> -- [1 0> 2:3 2:1 D A -0702 +1200 | [0 0> ------ 1:1 | A | 0000 | | | [-1 0> -- [-1 1> 1:2 3:2 A E -1200 +0702 | [-2 1> --- 3:4 | E | -0498 | | | -- [-3 2> 9:8 B +0204 | [-4 2> 9:16 B -0996

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.

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The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)