In order to understand the operations with intervals it is useful
preliminarily to know the sense of their names. It is natural to
suppose that names for major intervals are derived from major scale.
Really, quantities of semitones in all major intervals correspond
their values in major scale. But there is a exeption for natural minor
scale: its 2th includes 2 semitones but minor 2th includes 1
semitone. Owing to it major 7th and minor 2th give together the
octave interval.
Thus the rule =93maj+min=3Doctave=94 seems somewhat far-fetched.
Do a mode exist whose intervals correspond minor ones? It is phrygian
mode (Ph.M) EFGABCDE with steps S-T-T-T-S-T-T. In major scale the
steps are T-T-T-S-T-T-S. The difference is that in Ph.M semitone steps
are first in each tetrachord and in major scale-last. Because of it
all intervals in Ph.M except perfect 4th and 5th are less than same
named in major scale. Here following takes place: up to 3th step
intervals are more in major scale, 3th step is more in Ph.M and as
result 4th intervals are same in both . The 4th steps are whole tones
in both and correspondingly 5th intervals are same in both scales. The
next interval in Ph.M is semitone and its all following intervals are
minor. One can see that each major interval greater than same name
minor interval only on one semitone. It is wonderful in all this, that
same in major-minor scales and Ph.M 4,5 perfect intervals are most
consonant intervals as well( the only exception among church mode is
locrian mode which has imperfect 5th ).
If one accepts that major intervals are taken from major scale and
minor intervals from Ph.M he can easy check the known rules maj
+min=3Doctave, perf+perf=3Doctave, aug+dim=3Doctave (or prove them
logical).=
The natural minor scale with steps T-S-T-T-S-T-T differs from Ph.M
only concerning the order in 2 first steps. It is interesting to
mention that mood of Ph.M is evaluated sometimes as more sad in
comparison to minor scale. For example on the site
http://brebru.com/musicroom/theory/introduction/modes.html#MajorandMinorScal=
es
is written:
"This mode begins on E. Play all the white key up to the next E. It
has a very sad sound and is heard in lots of Spanish, Hebrew, and
Gypsy music. It is the only mode that begins with a half-step. "
Best Regards
Yuri Vilenkin
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