4. The Hindu-Greek diatonic scale
The relationship between number and tone common to the Vedic Aryans,
Sumerians, Hebrews, and Greeks involves a reference scale of integers
used for defining ratios between the numbers 1 and 2. Usually,
these numbers are products of the prime numbers 2, 3, and 5. Also,
the reference numbers corresponding to tones of a musical scale
should be closed under reciprocation. Details are found in many
musical texts, but McClain is particularly clear on this theory.
For example, some reference scales
found in the literature use the whole number intervals:
[30, 60], [72, 144], [360, 720], and so on. Bypassing many
complications, we choose one of these for our purposes, [72, 144],
which is the smallest reference scale which may be interpreted as
a scale of string lengths on a monochord for the diatonic scale.
McClain calls this range of integers the Christian set.
The heptatonic scale found by the usual Pythagorean method --
in rising order (decreasing string lengths, increasing tones) --
- Lengths: 144, 135, 120, 108, 96, 90, 80, 72
- Tones: D, eb, f, G, A, bb, c, D'.
- Ratios: 15/16, 8/9, 9/10, 8/9, 15/16, 8/9, 9/10
This is the Hindu-Greek Diatonic Scale, also known as the Dorian
Mode of ancient Greece and as Ptolemy's Diatonic Syntonon, and is
shown in Fig. 1. (See McClain, 1976, Chart 1, p. xxi, and Chart 3,
p. 13. See also Levin, 1994, p. 77.)
The pattern of rising intervals -- whole tones (T) and semitones
(S) -- in the Hindu-Greek Diatonic Scale, or Greek Dorian Mode,
is: STT,T,STT. The first note is taken above as D, but could be
any note. This sequence agrees with the Bhairavi Thaat, one of
the ten modes of North India. (See Batish, 1989.)
The Greek Phrygian Mode, also known as the Christian Modus Primus
and as the Modern Dorian Mode, has the pattern TST,T,TST.
(See McClain, 1976, p. 61.) This sequence may be obtained from
the preceding by one rotation, that is, starting the scale one
note higher. It agrees with one of the Modern Minor Modes, and
with the Kafi Thaat of North India.
The Greek Lydian Mode has the pattern TTS,T,TTS.
(See Lauer, in: Godwin, 1989.) It may be obtained from
the preceding by one rotation. It agrees with Modern Major
Mode, and with the Bilaval Thaat of North India.
All these Greek modes are usually presented in Pythagorean ratios
involving powers of the first three primes only: 2, 3, and 5.
Considering the ancient aulos, or Greek flute, as studied from actual
surviving examples, the Early Greek Dorian Mode is found in variants
involving ratios of powers of the first six primes: 2, 3, 5, 7, 11,
and 13. (See Schlesinger, 1939. See also Lauer, in: Godwin, 1989, p. 202.)
The patterns, STT,T,STT etc., are of the form: tetrachord,
whole tone, tetrachord. Using only wholetones and halftones,
there are only three tetrachords which fit into this pattern
and add up to an octave. Therefore, there are nine possible
heptatonic (7-tone) scales made according to this pattern.
Of the nine, only three are symmetric, that is, have the same
tetrachord sequence in both position. And these three are the
Greek Dorian, Phrygian, and Lydian Modes.
According to the rules of Aristoxenos only intervals S and T
are allowed in Greek modes, with S exactly twice, while other
combinations are found in the (older) scales of North India.
Also, S is allowed between the two tetrachords, but there should
not be TTTT in the scale, nor in its extension to two octaves.
See (Mountford, 1920). The seven Greek modes
satisfying these rules are (in order of rotation):
- Mixolydian: STT,S,TTT
- Syntonolydian: TTS,T,TTS (symmetric)
- Phrygian: TST,T,TST (symmetric)
- Dorian: STT,T,STT (symmetric)
- Lydian: TTT,S,TTS
- Ionian: TTS,T,TST
- Aeolian: TST,T,STT
There are no other modes following these rules.