Functional Harmony
REPRESENTATION
harm — representation for Western functional harmony
DESCRIPTION
The harm representation provides one method
for encoding Western functional harmony. In the harm representation, chords are normally identified
within a key context — such as G minor, or A-flat major. Keys
are normally indicated using the “key” tandem interpretation. If
no key indication is provided, the harmony representation is deemed
both key-independent and mode-independent (neither major or minor).
(Key-independent representations may be useful for comparing harmonic
patterns between groups of works in varying keys.) Changes of key
can be defined at any point in a harm
representation. The defined key context remains in effect until the
occurrence of another key-interpretation or until the key context
is specifically “undefined”: the reserved key-interpretation *?:
can be used to “undefine” the key context.
Chords are labelled according to four attributes: (1) chord root, (2) chord type, (3) inversion, and (4) chord alterations.
Chord roots are identified according to the diatonic scale degree. Scale degrees are indicated through the use of Roman numerals: I for tonic, II for supertonic, III for mediant, etc. The specific roots will vary according to whether the key is major or minor. For example, in the key of C major, the III chord will have E as the root, whereas in the key of C minor, the III chord will have E-flat as the root. By definition, the scale degrees of the minor key are assumed to correspond to the pitches of the harmonic minor scale. Notice that without a major or minor mode distinction, the roots of the III and VI chords are ambiguous — they simply denote mediant and submediant chords respectively.
Of course musical passages may contain chords having altered roots.
Raising or lowering the root is indicated by prepending a minus
sign -
or octothorpe character #
respectively. For example, in
the major key, a chromatic mediant chord based on the lowered
sub-mediant would be encoded as -VI
. In a notated score, the
lowering of the root may be achieved by adding a flat or by adding
a natural — depending upon the prevailing key; however, the
specific accidental used to lower the root is irrelevant to harm. For example, in the key of C minor, a minor
chord having E-natural as a root would be encoded as #iii
.
In the case of triads, there are four possible chord types: major,
minor, diminished, and augmented triads. Upper- and lower-case
numerals are used to indicate whether the third of the chord is
major or minor. For example, the supertonic chord in a major key
would normally be indicated as ii
. In short, major and augmented
triads are indicated through a upper-case Roman numeral, whereas
minor and diminished triads are indicated through a lower-case Roman
numeral. Diminished chords are indicated by the explicit addition
of the small letter o
— for example, the diminished triad
with a root on the leading tone is denoted as viio
. Augmented
chords are indicated by the explicit addition of the plus sign +
— e.g. "III+
for the augmented triad on the mediant degree
(common in the minor key), or V+
for the augmented altered dominant
chord.
In harm, inversions of chords are indicated
using lower-case alphabetic characters: first inversion - b
;
second inversion c
; third inversion - d
; etc. Root position is
implied, so in the absence of a letter designation (a,b,c …)
the token IV
means a IV chord in root position. Figured-bass
notation is not used in harm because it
proves inconsistent in the spelling of extended tertian chords. In
the case of a fully spelled 13th chord in root position, for example,
the figured bass would be 1-2-3-4-5-6-7. However, this same figured
bass would apply to all inversions of the 13th chord, and so it
fails to distinguish any of the possible inversions. In harm, the first inversion of a 13th chord is
signified by the letter b
whereas the hypothetical 6th inversion
of a 13th chord is signified by the letter f
.
Seventh chords are indicated by the addition of the number 7
— as for example in the dominant seventh chord: V7
. Ninth,
eleventh, and thirteenth chords are similarly represented: e.g.
V9
, V11
, V13
. Such extended tertian chords can be encoded in
more detail by indicating whether the interval is major or minor
— signified by use of the upper- or lower-case letter M
.
For example, a dominant minor ninth chord would be represented by
Vm9
whereas a dominant major thirteenth chord would be represented
by VM13
. When the interval is not explicitly indicated as major
or minor, it is assumed that the actual spelling is in accordance
with the prevailing key signature. For example, V9
is equivalent
to Vm9
when the prevailing key is minor.
If it is necessary to specify more precisely the actual intervals
involved in an extended tertian chord, all intervals may be included:
Vm9P11m13
. When more than one interval is given, the intervals
must be specified in ascending order and must include a major or
minor designation. Where intervals are perfect, the upper-case
letter P
is used. Where intervals are augmented, the upper-case
letter A
is used. Where intervals are diminished, the upper-case
letter D
may be used. (It is common practice to represent diminished
intervals using the lower-case letter d
; in harm, however, this would be indistinguishable
from the designation for third inversion.) Thus the half-diminished
seventh chord would normally be represented as viiom7, whereas the
full-diminished seventh chord would be represented as viioD7
.
Doubly-augmented and doubly-diminished intervals can use AA
and
DD
respectively.
Of course it is rare that a musical passage or work remains within
a single key. The use of secondary dominants and modulations requires
that some means be provided for indicating shifting key areas. When
shifts of key are sanctioned, these should normally be encoded using
an “X of Y” approach — e.g. V of V. In the harm representation, such shifts are indicated
via the slash character (/). For example, a dominant seventh chord
on the supertonic degree can be represented as V7/ii. If a passage
modulates to the subdominant and remains there for some time, chord
sequences can be identified as /IV
— e.g. V7/IV
, I/IV
,
vi/IV
, ii/IV
, V/II/IV
, II/IV
, etc. (Notice the use of I/IV
rather than IV
; in long sequences of chords it is preferable to
encode successive chords within the new key area.) There is no limit
to the number of key-area shifts specified in a harmonic token:
e.g. V/V/V/V/V/I
is syntactically legal.
In traditional harmony, a variety of special chords may be encountered
— such as the Neopolitan chord, and the "ethnic sixth"
chords: Italian, French, and German. The Neopolitan chord is a major
triad whose root is the lowered supertonic; it is represented in
harm by the reserved upper-case letter
N
. The Neopolitan chord normally appears as a first inversion
chord, so the Neopolitan sixth chord would be represented as "Nb".
Notice that the Neopolitan sixth chord is equivalent to "-IIb".
The Italian, French, and German augmented sixth chords are represented
as Lt
, Fr
, and Gn
respectively. In addition, the “Tristan
chord” (A4m7m10
above bass pitch) has a special designation as
Tr
.
Occasionally, chords may appear using enharmonically equivalent
spelling. Such chords can be encoded by using the enharmonic prefix
of the tilde character ~
. For example, if a Neopolitan sixth
chord is spelled using the raised tonic rather than the lowered
supertonic, the chord may be encoded: ~Nb
.
In other cases, it may be entirely impossible to identify a chord
in terms of traditional Western functional harmony. Such chords may
be encoded by specifying a set of intervals above the bass pitch
— with the question-mark prefix. For example, in the key of
A major, the chord C4, E4, F#4, G#4, D5 can be represented as:
?-IIIM3A4A5M9
. Notice that this representation reverts to a
descriptive approach and so is no longer truly “functional.”
Chord identifications may be characterized as (1) explicit, (2)
implied, or (3) alternate. Explicit harmonies occur when most or
all of the chordal tones are present. In some cases (such as melodic
lines) the harmonies may be implied rather than explicit. Implicit
harmonies are indicated by placing the chord signfied in parentheses
(). In other circumstances, there will be more than one way of
labelling a given harmony. Alternate harmonies are indicated through
the use of square brackets []
. All other indications are assumed
to be explicit. In the case of bi-tonal works, the user may elect
to pair explicit and alternate encodings, e.g. iii\[v/vi\]
, or
make use of two independent harm spines.
Two or more harm spines may be necessary
in the case of polytonal works.
Barlines are represented using the “common system” for barlines — see barlines.
FILE TYPE
It is recommended that files containing predominantly harm data should be given names with the distinguishing ‘.hrm’ extension.
SIGNIFIERS
The following table summarizes the harm mappings of signifiers and signifieds.
Summary of harm Signifiers
Item | Meaning |
- |
lowered root |
# |
raised root |
+ |
augmented triad |
o |
diminished triad |
I |
chord degree (major) |
i |
chord degree (minor) |
V |
chord degree (major) |
v |
chord degree (minor) |
b |
first inversion chord |
c |
second inversion chord |
d |
third inversion chord |
e |
fourth inversion chord (ninth chords) |
f |
fifth inversion chord (eleventh chords) |
g |
sixth inversion chord (thirteenth chords) |
r |
rest |
7 |
added seventh |
9 |
added ninth |
11 |
added eleventh |
13 |
added thirteenth |
. |
null token |
/ |
secondary function, e.g. V “of” vi |
m |
minor interval |
M |
major interval |
P |
perfect interval |
A |
augmented interval |
D |
diminished interval |
AA |
doubly-augmented interval |
DD |
doubly-diminished interval |
Nb |
Neopolitan sixth chord |
N |
Neopolitan chord in “root position” |
Lt |
Italian augmented sixth chord |
Fr |
French augmented sixth chord |
Gn |
German augmented sixth chord |
Tr |
Tristan chord |
~ |
enharmonically-spelled chord |
() |
implicit harmony |
[] |
alternative functional harmony label |
viiom7 |
half-diminished seventh chord |
viioD7 |
full-diminished seventh chord |
EXAMPLES
A sample document is given below:
**harm
*C:
!! An example.
=1
I
IVb
.
V7/V
=2
I/V
vib/V\[iiib\]
ii/V\[vi\]
viioD7/V
=3
V\[I/V\]
.
IV
Nb
V7d
=4
!! Minore
*c:
i
?-IIIM3A4A5M9
#iiiob
IV
*?:
Lt
=5
ic
Vm9
I
==
*-
PERTINENT COMMANDS
Currently, no special-purpose Humdrum commands produce harm as output, or process harm encoded data input.
TANDEM INTERPRETATIONS
The following tandem interpretations can be used in conjunction with harm:
Tandem interpretations for harm
MIDI channel *Ch1
meter signatures *M6/8
key signatures *k[f#c#]
key *c#:
SEE ALSO
barlines, embel, kern.