RE: tetrachords

There seem to be two schools of thought about the melodic structure of
scales; one is based on tetrachords, the other on "maximal evenness".
Let's flesh out what they mean for some common scales.

First, let's consider the modes of the major scale. All of them are
maximally even in 12-, 19- or 26-tone equal temperament; that is, there
is no way to more closely approximate a 7-tone equal-tempered scale in
these tunings than the major scale and its modes. Now look at the modes
themselves:

(C D E F) (G A B C)
(D E F G) (A B C D)
(E F G A) (B C D E)
F (G A B (C) D E F)
(G A B (C) D E F) G or G (A B C (D) E F G)
A (B C D (E) F G A) or (A B C (D) E F G) A
(B C D (E) F G A) B

Each octave species contains two identical tetrachords, shown in
parentheses. In the first three modes, the tetrachords are disjunct, and
in the last four, they are conjunct. So in the first three modes, any
melody played in one tetrachord can be repeated an approximate 3:2 away,
and in the last four, any melody played in one tetrachord can be played
an approximate 4:3 away.

Why is this important? Well, one theory is that the minor third was the
first melodic interval ever used, and the most primitive melodies had
only two notes a minor third apart. See for yourself how easy it is to
sing just about any text in such a melody, and how common it is in
popular sing-song. Later, a third note was added, a whole-tone away so
that the outer two notes formed a perfect fourth, the 4:3 ratio. This
interval is a simple enough ratio so that even without harmony, had some
sort of pretonal strength to it. This three-note scale, or "trichord"
was then repeated either starting on the last note of the previous
trichord, or a whole-tone higher such that the last note of the second
trichord was an octave, or 2:1 ratio, away from the first note of the
first trichord. Thus were formed all the modes of the pentatonic scale:

(C D F) (G A C)
(D F G) (A C D)
F (G A (C) D F)
(G A (C) D F) G or G (A C (D) F G)
(A C (D) F G) A

(These are maximally even pentatonic scales in 7- or 12-tone equal
temperaments).

Then the minor thirds were filled in. In the West, the minor thirds were
divided into a whole tone (already familiar) and a half tone (new).
Ordering these two intervals the same way in both trichords leads to two
identical tetrachords, thus the modes of the major scale above.

In the Arabic world, the minor thirds were often divided instead into
two equal parts, 3/4-tones which were not familiar from the trichords.
This led to the following scales (+ means quarter-tone sharp, - means
quarter-tone flat):

(C D E- F) (G A B- C)
(D E- F G) (A B- C D)
F (G A B- (C) D E- F)
(G A B- (C) D E- F) G or G (A B- C (D) E- F G)
(A B- C (D) E- F G) A

Also appearing was a scale, Mohajira, which put the 3/4-tones at
opposite ends of the tetrachords and the whole-tone in the middle:

(E- F G A-) (B- C D E-)

(This is a maximally even heptatonic scale in 17-, 24-, or 31-tone equal
temperament).

Many scales in the East may have been formed by halving the whole tones,
rather than the minor thirds, in each trichord. Though a clear violation
of the maximal evenness principle, some of the resulting scales are
found outside the West:

(C C# D F) (G G# A C)
(D F F# G) (A C C# D)
F (G G# A (C) C# D F)
(G G# A (C) C# D F) G
G (A C C# (D) F F# G)
(A C C# (D) F F# G) A

Also appearing was a scale (Major Gypsy or Persian) which put the
half-tones at opposite ends of the tetrachords and the minor third in
the middle:

(C# D F F#) (G# A C C#)

In India, musicians may have had a very high sensitivity to intonation,
which forced them to distinguish two types of whole tones. Tuning the
original minor third to 6:5, the whole tone completing the 4:3 would
have to be 10:9, while the one that, along with the two 4:3s, completed
the 2:1, would have to be a 9:8. They appear to have reckoned the former
as 3 srutis, the latter as 4 srutis, and the minor third as 6 srutis,
making a 22-sruti octave. There seems to have been an early attempt at
dividing the minor third into two equal parts, resulting in a scale
known as Gandhaara grama, which did not survive the ancient period and
whose true nature is shrouded in mystery. More successful were the
Western-type approach of dividing the minor third into a half-tone (2
srutis) and a whole-tone (4 srutis). When the two minor thirds were
divided the same way, the resulting "diatonic" scales could have the
following sequences of intervals:

(3 2 4) (3 2 4) 4
(3 4 2) (3 4 2) 4
(4 2 3) (4 2 3) 4
(4 3 2) (4 3 2) 4
(3 2 4) 4 (3 2 4)
(3 4 2) 4 (3 4 2)
(4 2 3) 4 (4 2 3)
(4 3 2) 4 (4 3 2)
4 (3 2 4) (3 2 4)
4 (3 4 2) (3 4 2)
4 (4 2 3) (4 2 3)
4 (4 3 2) (4 3 2)

There are four distinct scale types here, each represented by three
modes. The four scale types are the ones that appear to have been most
important in a period in Indian music history, according to a paper by
Lewis Rowell, John Clough, and others. They show that these scales can
be singled out from all 7-out-of-22 scales by demanding two properties.
One is that there be only one tritone (half-octave interval). The other
is a sort of second-order maximal evenness involving the numbers 7, 12,
and 22. I pointed out to John Clough that a different but equally simple
definition of second-order maximal evenness allows one to obtain the
four scale types without the additional tritone rule. However, I did not
make a big deal out of it, since I believe the acoustical-tetrachordal
derivation to be more relevant.

In modern Indian scales the two minor thirds were, however, not always
divided with the intervals in the same order. Also the Eastern-type
approaches of dividing the whole-tone are used in India. Finally, even
the scales above each have only three octave species with identical
tetrachords. Probably, the majority of Indian scales cannot be
considered to have two identical tetrachords.

If you read my paper on decatonic scales in 22-tone equal temperament,
you will see that one version of the scale has the property that all its
octave species contain two identical pentachords, either conjunct or
disjunct, while another version is maximally even. The pentachordal
scale can be derived from the Indian scales above by treating the 22
srutis as equal and dividing all 4-sruti intervals in half. As far as
melodic appeal goes, I think the tetrachord (trichord, pentachord)
concept wins, as evidenced by comparing the two scales played
melodically. However, the maximally even scale contains 8 consonant
tetrads to the pentachordal scale's 6. Another case of melody and
harmony being at odds for scale construction (something Ivor Darreg
liked to point out).