Introduction
The octave of the Western tempered scale is made up of twelve “degrees,” each of which is an exact half-step. The frequency ratio of any two notes one degree apart is exactly the same as any other two notes one degree apart.
Historically, though, this wasn't the case. The frequency ratio of a fifth, for example, was considered to be 3:2, and for a fourth, 4:3, etc. These ratios yield very pleasing harmonic results. You can infer the other steps of the scale by stacking two fourths, a fourth and a fifth, etc. However, by doing this, when you got to calculating the seventh step of the scale, you'd wind up with an audible discrepancy. If you calculate steps upward as well as steps downward, you get two different frequency values. The difference between these values is called the syntonic comma.
So, to create the tempered scale, the tunings of each step (other than the tonic and the octave) were adjusted slightly. Mathematically, determining the correct frequencies of each step involves calculating twelfth-roots, which is not trivial arithmetic.
Byzantine theorists, however, broke the octave into 72 degrees instead of twelve. This means that each Western tempered half-step is made up of 6 Byzantine degrees. Byzantine theory allows for the assembly of four basic genera of scales, plus several fractional genera (the latter are used only for brief emphasis of a musical phrase).
In most Byzantine genera, the major scale is made up of two tetrachords (usually identical) separated by 12 degrees (a tempered-scale whole step). This corresponds to the Western disjunct-tetrachord understanding of the major scale.
Full scales
There are three major genera of Byzantine scales. Within these genera, the classic church-modes may be applied. For example, most hymns in Mode IV are in phrygian diatonic.
1. The Diatonic genus
The diatonic scale is made up of the following intervals:
Note that the third note in each tetrachord is lowered by ⅓ of a half-step relative to the tempered scale.
Usually if a melody line is descending, the seventh step of this octave receives a full-flat instead of a ⅓-flat.
This is the most commonly used genus in Byzantine music. In certain ornate and somewhat rare compositions, a diatonic scale comprised of conjunct pentachords (or even hexachords) may be used instead of disjunct tetrachords, which results in a different set of accidentals in one octave than in another.
2. The Chromatic genus
The Chromatic genus is further subdivided into two genera: the “soft” chromatic and the “intense” chromatic:
a. The soft chromatic subgenus
The soft chromatic scale is made up of the following intervals:
You can see that the second step in each tetrachord is lowered by ⅔ of a half-step, and the third step is lowered by ⅓ of a half-step.
b. The intense (or “hard”) chromatic subgenus
The intense chromatic scale is made of the following intervals:
This lowers the second step of each tetrachord by an entire half-step, and raises the third by ⅓ of a half-step.
Many compositions in the soft chromatic genus use a compound scale, the lower tetrachord of which is borrowed from the intense chromatic.
3. The Enharmonic genus
This is the most rarely-used genus. It is identical to the tempered scale:
Partial scales
There are three partial scales (which appear to be borrowed from Ottoman classical music) which may be used for a brief phrase. The modulation into these scales is generally indicated by a particular neume (called a χρόα or “shade”) written over a note. The shift back to the main scale is then indicated by a different neume. For the duration of a “shade,” the melody never leaves the upper or lower boundaries of that shade’s scale-range.
1. The Yoke (Ζυγός)
When the “yoke” neume appears on the fifth step of one of the main genera above (in the key-signature of C, this would be G), that indicates that that G should be considered a part of this scale:
Note that this exclusively allows for a descending figure.
2. The Tilt (Κλιτόν)
The neume for this subscale may appear on the fifth step of one of the main genera above (in the key-signature of C, this would be G). In this case, that G should be considered a part of this scale:
Note that this also exclusively allows for a descending figure.
3. The Sword (Σπαθί)
This is actually two different subscales, depending on which step of the scale the neume is written on.
If the Sword neume appears on the sixth step (A in the case of the key of C), the scale is as follows:
Note that this allows for both ascent and descent while in the “shade” mode.
But if the Sword neume appears on the fourth step (F in our example), the scale is as follows:
Note that the ascent from A to C is the same as the Sword on A, but the descent steps are altogether different, and allow a descent to a lower pitch.
Other possible pitches
In addition to the above, accidentals may be written in the music allowing any number of degrees of sharpness or flatness relative to that step’s natural pitch according to the genus in which it appears.
All of the above scales were standardized by a commission in the 19th century. Prior to that period, other intervals (often with odd numbers of degrees) were substituted based on regional or personal preference.