The modes used in tuning and performance of a stringed instruments like lyre or kithara were called "tonoi" (from Greek "tonos" - to stretch or tension).
Lyre had five, seven or eight strings. The top and bottom strings were tuned to notes which sounded an octave (diapason) apart. The tunings of the intermediate strings varied, using different combinations of large and small intervals ('tones' and 'semitones').
Strings with lengths in the ratio 2/1 produced the interval of an octave (diapason).
Those in the proportion 3/2 produced the interval of the fifth, diapente.
Strings of equal tension with length in the proportion 4/3 produced the interval of a fourth, diatessaron.
All of these intervals are present between strings with relative lengths 2, 3 and 4. Thus the most harmonious of intervals are contained in the number progression 1 : 2 : 3 : 4. This reinforced the concept of spacial and musical harmony being related and the belief that the harmony of the entire universe was inherent in the mystical power of numbers.
The basic musical scale unit of ancient Greece was "tetrachord" (literally meaning "four strings"). It was used to define series of four notes, divided by three intervals. The notes were arranged in descending order. Interval between upper and lower notes was equal to perfect fourth (diatessaron).
Three genera (kinds) of tetrachords were distinguished: diatonic (tone-tone-semitone), chromatic (minor third-semitone-semitone), and enharmonic (major third-quartertone-quartertone).
The octave was built up from two tetrachords and one whole tone between them. The intervals of diatessaron (4/3) and diapente (3/2) when combined (multiplied) give the interval of a diapason (2/1). The interval between them (divide) is 9/8 which is the interval recognized as a whole tone.
Conjunct tetrachords shared a note.
Disjunct tetrachords were separated by a disjunctive tone of 9/8 (a Pythagorean major second).
The scales were constructed from conjunct or disjunct tetrachords. Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves.
For example, the Greater Perfect System (diatonic scale) was derived from two disjunct diatonic tetrachords (EDCB and AGFE) and two more conjunct diatonic tetrachords (AGFE is conjuncted to EDCB, and DCBA is conjuncted to AGFE).
Diatonic scale has semitone intervals between C and B and also F and E. All the rest of the intervals are whole tones.
Harmoniai were defined as scale patterns in the greater perfect system
A different scale pattern in terms of the position of tone and semitone intervals may thus be produced by traversing the scale from different starting points ("shifting the tonic").
Using this method of tonic shoft the following seven harmoniai can be derived:
|E D CB A G FE||Dorian|
|D CB A G FE D||Phrygian|
|CB A G FE D C||Lydian|
|B A G FE D CB||Mixolydian|
|A G FE D CB A||Hypodorian|
|G FE D CB A G F||Hypophrygian|
|FE D CB A G F||Hypolydian|
|(unspaced letters show semitone intervals)|