By Dr. Charles Heiden
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July 17, 2009
From primer to college
textbook, a violin student is apt to
encounter the theory of music framed from a
keyboard bias. This is understandable, even
desirable, perhaps. On the piano keyboard,
the structure of Western music is displayed
in a black and white format relevant for all
musicians. The keyboard approach begins from
the Perfect octave [P8 – first in the
overtone series] spanned in the pianist’s
hand between first and fifth finger, and
“filled”
scale-wise by the player with the pitches of
the major mode. But students who study an
instrument of the violin family need a
perspective adjusted to their experience –
two neighboring strings tuned at the Perfect
fifth [P5 – derived from the second
overtone] and “filled” by a major tetrachord
that relates to the fingerboard distances
between four fingers..
Playing instruments tuned in P5ths, string
players can experience for themselves the
fecundity of this interval. For by
projecting or stacking P5ths, ALL twelve
pitch classes [assuming modern equal
temperament] are generated! The following
exercise demonstrates this 12-stack
[starting, for violas and cellos, on open C,
then joined by violins from open G. [When
the ascent threatens to go “stratospheric”,
the player drops down to the lowest
available pitch of that class-name, as
indicated below.]
Continued Below
Theory
books are prone to present the major scale
as TTSTTTS [T = tone, or whole step. S =
semitone, or half-step].2 This pattern – 2,
then 3 T’s -- corresponds to the geometry of
the black keys of the piano. But for the
string player, the major scale structure is
better formulated 221 (2) 221 [1 = a
semitone or half-step. 2 = 2 semitones or
whole-step]. Thus, the major scale is a
matter of playing the major tetrachord
twice, with 2 semitones as the link between
the units. In a heterogeneous string class,
the beginner learns to mount the major
tetrachord on the open D, next on the open
A, and then combines the two tetrachords to
make the D-major scale.3 Below and above
this D-string centering, the scales mounted
on G- or A-strings begin an education about
key signatures and about interlocking key
structure [What served as the lower
tetrachord of the D-scale becomes the upper
tetrachord of the G-scale. What served as
the upper tetrachord of the D-scale becomes
the lower tetrachord of the A-scale, etc.
around the circle of 5ths].This notion of
interlocking tetrachords [the relatedness of
keys] – will it become, perhaps years later,
the foundation for understanding modulation
and key-architecture in a concerto by
Mozart?
The chart above shows how sharps and flats
become necessary in order to construct the
major scale from a given starting pitch
according to the 221 (2) 221 formula: the
sharps or flats are employed so that the
“landmarks-for-the- ear” [those 1’s or
semitones in the formula] will occur at the
end of each major tetrachord. At the
beginning of each staff in our notational
system, these sharps or flats needed for the
desired key are listed in a zig-zag format
called the “key signature”. Derived from the
stack of 5ths, the two patterns [7 sharps, 7
flats] can be learned mechanically: for
sharps, starting from F#, down 4, up 5 –
zig-zagging in order to stay on the staff
[beware at A!]; for flats, starting at Bb,
it’s the opposite direction, up 4, down 5.
For the signatures of specific keys, apply
the “rules” [they all trace back to the
stack of 5ths]. The last flat in a signature
is scale step 4 – count down to 1. The
second-last flat is 1, the key itself. The
last sharp is 7, the leading tone – count up
to 8 [=1]. To write a given signature in
sharps, write the proper zig-zag, stopping
at the pitch just below the desired keynote.
For example, the B-major signature is F# -
C# - G# - D# - A#. Or, for an example in
flats, consider Gb. Write the proper zig-zag
until reaching the desired keynote, Gb –
then go one more step [the key must be the
2nd-last flat] Thus, Bb – Eb – Ab – Db – Gb
– Cb.
Continued Below
Minor
mode, in a string-friendly presentation of
theory, also can be approached in terms of
P5ths on the key circle, but now the
tetrachord finger-patterns that “fill” the
5ths will be minor plus Phrygian.4
Above, the minor scale starting from d is
the relative minor of the major scale on F.
Human blood relatives share common family
traits. F major/d minor [“doh-ti-la” or
8-7-6 expresses the relation] share the same
key signature, the same set of pitches. But
d-minor starts on d not F, and in other ways
behaves differently. Why this upstart
individuality? To a considerable degree, it
is because the two keys have different
dominants. The V of F is C-E-G. The V of d
is A-C#-E [yes, C-sharp! There must be a
leading tone – that’s what makes the
dominant chord dominate!] Notice d-minor’s
upstart individuality! Not only does this
mode don new clothing [C#] not found in F-
major, but it is this precise rebellion that
denies C [V of F], the old family identity!
The
analogy of family relationship helps to
explain the three forms of the minor scale.
Natural minor? That’s according to the
family crest, the key-signature of F major.
Harmonic minor? That’s with scale-step 7
raised [as an accidental] so that the V
harmony will have the leading-tone
[semitone-to-the-tonic] that it needs in
order to do its job. Melodic minor? That’s
to smooth out the “bump” [2+] in the
harmonic form. For in fig. 5, if C is raised
to C#, then to play melodically from Bb to
C# is to play a skip of 3-semitones, the
enharmonic equivalent of a minor 3rd.
Therefore, as an accidental, raise
scale-step 6 [B] to smooth this gap. The
need arises on the ascent; when descending
-- moving away from the tonic -- no leading
tone is required, and therefore no “bump”
occurs. Descending melodic minor reverts to
the natural, key-signature form. In
practice, composers often mix the natural,
harmonic, and melodic forms of minor.
Intervals is another fundamental area of
theory from which the string player can
benefit greatly if the presentation is
translated into string-friendly terms based
on the P5 and four fingers. Below, the
“Chart of Fingerboard Equivalents” enables a
lesson in fingerboard geography. This is a
chart of relationships that can start at any
pitch. When the player has reached the
tritone [4+/5o], 1-4, then the next moves
will repeatedly cross to the next higher
string [always retaining the guiding finger
as foundation]… P5, m6, M6. etc. In this
way, the player learns, for instance, that
the P4 on one string, and the P8 to the next
string is the same 1-4 hand-span!
This figure [learn to write it] is a
convenient way to remember the 2 classes of
intervals: those above come in 3 sizes, those below
come in 4 sizes. The intervals of the major
scale [bold type on the chart] are either P
or M; regard these intervals as “standard”.
Then for the upper group [3 sizes], 1S
[semitone] smaller than standard is
diminished [ o ], while 1S larger is
augmented [+ ]. For the lower group [4
sizes], 1S smaller than “standard” is m, and
1S smaller than that is diminished [ o ].
While 1S larger than “standard” is augmented
[ + ].
Continued Below
ROUTINES FOR
PRACTICE/DRILL
Route #1 going left to right
The high road:
short (Major scale – bold italics), or long
(chromatic scale), starting string, then to
next higher.
The low road: starting string, then to the next lower string.
Route # 2 “Navaho rug zig-zag”-- i.e.,
crenelated.
Upper road: P1, up to P5, right to m6, down to m2, right to M2,
etc..
Lower road: P1, down to P5, right to 4+/5o, etc.
The
basics of music theory -- reformulated in
string-friendly terms and applied to every
playing task -- need not be
compartmentalized in college level courses.
From the beginning, the basics ought to
contribute to a string student’s progress.
Pragmatic and applied, a string-friendly
presentation can turn dull, intellectual
theory into jubilation. “Allelujah,” the
string student can say – with pride! “Those
88 keys of the piano – every pitch in the
musical universe – all is generated from the
P5 of violin family tuning, by the
tetrachord patterns made by my 4 fingers,
and the geography that they explore on the
fingerboard!”
Charles Heiden, 2009
Bibliography
1. See “Harmonics,” The new Harvard
dictionary of music, Don Michael Randel,
ed., 1986
2. Ibid., article “Scale”.
3. For example, see Michael Allen, et alia,
Essential Elements 2000 Plus for Strings: A
comprehensive String Method, Hal Leonard
Corp, 2001, Violin Book 1: p 6, D String
Notes; p.10, A String Notes; p.11,Down the D Scale; p.26, G String Notes;
p.27, G Major Scale. Absent here is any
recognition that it is the
same 221 tetrachord pattern of whole and
half steps that is being mounted from these
open strings, or that
learning the new G major scale involves
applying a pattern of structure that the
student has already experienced.
4. Charles Heiden, Franz Wohlfahrt, Op. 38 /
Easiest Elementary Method / for Violin /
Revised for D-major Beginners, Heiden Music
Publications, 2008. See “The Three
Tetrachords,” Bk. 1, p.17. And for
string-friendly concepts about minor mode
patterns, see Bk. 2, pp. 22 -24 in the same
C-to-D major rapprochement.
5. Ibid.Bk. 2, p.34
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