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STRUCTURE AND INTERPRETATION OF MUSIC CONCEPTS
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______________________________________________
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==============================================
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CLASS 9: The Tonal System
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_________________________
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=========================
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WHAT IS TONALITY?
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The tonal system organizes notes into a "context" in which they
have
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ROLES and EXPECTED BEHAVIOR. For that purpose, music in the tonal
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system is always related to a SCALE, which is the context object.
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The role and behavior of notes in a tonal piece is determined by
their
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position (relative status) in the scale.
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The central, most important role is "played" by the TONIC note of
a
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scale -- its first note:
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"Tonic is the tone of ultimate rest, the other notes tend
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to move toward the tonic" (L. B. Meyer 1956).
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The definition emphasizes the expected behavior role, that the
tonal
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system attributes to its different notes. The behavior role is
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classified as STABLE or DYNAMIC. The tonic is the most stable
tone
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in the SCALE. Usually it is the last note in a melody or music
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piece. The tonic or the notes of the tonic triad are usually the
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beginning notes.
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A small experiment about the sense of tonality:
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===============================================
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1. Listen to a "typical" tonal music piece and try to perceive
the
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feeling of tonality -- identify the tonic note.
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Suggestion: The first movement of Mozart's "Eine Kleine
Nachtmusik"
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(~5 min),
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or,
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the Minuet (only 2 melodic lines;
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~1 min. without the trio).
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The end should feel like the (a?) ultimate rest of the piece.
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2. The following example includes a small segment, with various
ending
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stable).
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3.
Kleine
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4.
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suggestions.
Rate the ending notes from 1 (most unstable) to 5 (most
Do it fast and intuitively - only 2 seconds for rating between
the examples.
Repeat the process (for a refined rating).
Tonal-ending-1: Different endings.
Listen only to the first 4 bars of the first movement of "Eine
Nachtmusik" .
Try to answer the same questions as above.
Does it feels like we get to the (an?) ultimate rest?
Listen to the first 4 bars, as in (3), and to various ending
suggestions.
Rate the ending notes as in (2).
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scale,
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their
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catch:
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Tonal-ending-2: (3) with different endings.
TONALITY -- A CIRCULAR VIEW:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
* A tonal piece can be conceived as a scene whose meaning,
interpretation, content, and almost anything one would like to
attribute to it, derives from its CONTEXT (its scale, primary
embedded scales, scale-inter-relations).
--> THE SCALE DETERMINES THE PITCH BEHAVIOR.
BUT:
* The context of a piece (scale) is defined by its pitch content
and its interaction with the rhythm dimension.
Pitch content -- simultaneous and sequential combinations, and
interaction.
--> THE PITCH BEHAVIOR DETERMINES THE SCALE.
So -- indeed, any definition of tonality gets into that circular
The tonality of a piece is defined by its pitch content which
is dictated by its tonality.
Examples of pitch behavior, as determined by context (scale):
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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1.
class)
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Therefore,
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notes are
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movement
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2.
(attached
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same
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dictate
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into
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into
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degree.
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context
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dominance
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Consider the behavior of G# and Ab (same music-level pitch
in C major.
G# will be usually part of a transposition to A major (A) or A
minor (Am), so it will function as a leading tone to A.
Ab will probably be part of a modulation to Fm or Eb.
it will go down to G.
Ab and G# are not part of C, so they will point to some other
scale and behave according to there function in that context.
Another possibility: In a CHROMATHIC PASSAGE, the chromatic
not assigned any harmonic function. In that case, an upward
is indicated by an uplifting alteration -- #, and a downward
movement is indicated by a downgrading alteration -- b.
Consider the two chord progressions in the note example
to these class notes).
The first chord in both progressions consists of exactly the
music-level pitch-classes (and music pitches as well).
However, they consist of different tonal-pitch-classes, which
the difference in their behavior.
In the first progression (A), the first chord functions as a
seventh chord on the dominant (V) of F major, and is resolved
the tonic triad, with the Bb note (IV degree) being resolved
the III degree. The IV degree here plays a central role, since
it implies that this is a dominant chord in F major, and being
a DYNAMIC degree, it "requires" a resolution into the III
In the second progression (B), the first chord functions as a
German chord (part of broader harmony repertoire of chords).
It is interpreted as a subdominant chord (IV) within the
of the E or Em scales. Therefore, it is resolved to the
of E (a B major chord).
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A large part of the research in Music Cognition indeed
concentrates
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on issues of the emergence of tonal sensation. Studies in music
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analysis, traditionally, deal with the understanding of the music
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content of a piece, with respect to its scale context.
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Given that circular characterization, all that is left is to dive
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into the components of the tonal system, and leave the philosophy
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of tonal sensation for later discussion.
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The TONAL DOMAIN OF NOTES
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=========================
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The DOMAIN OF NOTES in the western tonal system is generated by
the
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selection of seven Pitch Classes from within the twelve Pitch
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classes that generate the well-tempered domain of notes. These
are
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called the DIATONIC pitch classes.
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The selected diatonic pitch-classes: 0, 2, 4, 5, 7, 9, 11
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** PITCH-CLASS ORDERING: a cyclic, modulo 12, ordering of the
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pitch-classes 0, 1, ..., 11.
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** DIATONIC PITCH-CLASS ORDERING: a cyclic, modulo 7, ordering
of
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the diatonic pitch-classes:
0, 2, 4, 5, 7, 9, 11.
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Names of the diatonic pitch-classes: C, D, E, F, G, A, B,
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respectively.
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Names of the diatonic interval-classes created by the
diatonic
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pitch-classes:
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FIRST, SECOND, THIRD, FOURTH,
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FIFTH, SIX, SEVENTH,
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(OCTAVE),
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respectively.
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This "double-pitch-class" characterization gives rise to a rich
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***TONAL PITCH-CLASS*** terminology, that reflects both the
diatonic
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pitch-class and the 12-tone pitch-class orderings. For example:
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C, C#, Db, D,...., with further possibilities for C##, Dbb, etc.
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The tonal pitch-class domain is no longer totally ordered by a
proper
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< relation based on the music pitch-class, since it includes
values
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that map into the same music pitch-class.
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Clearly, it can be totally ordered by a weak < relation, but it
seems
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the
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class
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spans
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that a lexicographic like relation, defined by considering both
diatonic and the pitch-class ordering is more meaningful.
This is part of the tonal-music arithmetics, introduced later on.
THE SCALES:
===========
The scales of the tonal system derive from the diatonic pitchordering.
* A SCALE-TYPE is defined by a sequence of intervals, that
a whole cycle within the diatonic pitch-class ordering.
The sum of the intervals within a scale type is the OCTAVE
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when it
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the
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with the
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semitones:
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harmonic.
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*
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diatonic
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interval.
The two main scale types are the MAJOR and the MINOR scales.
- The MAJOR SCALE-TYPE (henceforth, we omit the word "type"
can be understood from the context) is defined by starting
sequence with the 0 pitch class.
The major scale is defined by the sequence of semitones:
2, 2, 1, 2, 2, 2, 1.
- The MINOR SCALE-TYPE is defined by starting the sequence
9 pitch class.
The minor scale-type is defined by the sequence of
2, 1, 2, 2, 1, 2, 2.
The minor scale-type has several variants: melodic,
MELODIC SCALE-TYPE: 2, 1, 2, 2, 2, 2, 1.
HARMONIC SCALE-TYPE: 2, 1, 2, 2, 1, 3, 1.
- During the history of western tonal music, further
"modes" have been used.
A SCALE is an "instance" of a scale type:
A sequence of tonal pitch-classes, obtained by projecting the
scale-type on the tonal-pitch-class domain (using the
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pitch-class ordering). Therefore, a scale is an ORDERED
SEQUENCE
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of tonal pitch-classes, that spans a whole cycle in the
diatonic
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pitch-class ordering -- an octave.
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The pitch-classes in a scale are called: DEGREES.
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A scale is named "after" its first pitch-class:
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C-Major (CM for short), C-minor (Cm), ....
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* Naming the scale degrees:
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I
-- Tonic.
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II -- Super tonic.
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III -- Mediant.
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IV -- Subdominant.
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V
-- Dominant.
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VI -- Submediant.
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VII -206
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Dominance ordering of the scale elements:
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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The degrees of a scale are ordered by their DOMINANCE status:
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TONIC (I)
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MEDIANT (III)
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DOMINNAT (V)
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SUBMDIANT (VI)
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SUBDOMINANT (IV)
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SUPER TONIC (II)
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VII
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That is, the scale is totally ordered by a dominance ordering
(which is
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different from its pitch-class ordering).
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Note that the names of the scale degrees refer to their relative
interval
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with either the tonic or the dominant-- probably, reflecting the
dominant
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status of the latter.
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Stable/dynamic classification of scale elements and intervals:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1. Scale elements:
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The dominance position implies a stable/dynamic
characterization.
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The tonic element is the most stable.
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The leading element (VII-th degree) is the most dynamic. Its
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occurrence requires a RESOLUTION -- move towards a more stable
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element.
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The stable/dynamic characterization follows the dominance
ordering
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position. Tonic -- most stable, VII -- least stable.
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2. Scale intervals:
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The scale intervals are classified as CONSONANT/DISSONANT.
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The intervals of the tonic triad and their inversions
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(complimentary intervals) are CONSONANT. The other intervals
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are DISSONANT.
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* Consonant intervals are stable -- can stand for themselves.
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* Dissonant intervals are dynamic -- require resolution to a
stable
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interval.
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The tonal domain of notes
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~~~~~~~~~~~~~~~~~~~~~~~~~
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The need for this, supposedly "redundant" domain of pitch-classes
arise
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from the diatonic role that pitch-class play with respect to a
scale context.
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For example, an F# might function as a VII in G Major, or as an
uplifted
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IV in C major, while Gb can function as a downgraded II in F
major.
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The number of pitch-classes in the western tonal music is larger
than the
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number of pitch-classes to which they are mapped on the physical
dimension.
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This rich domain is used to reflect the diatonic role of the
pitch-classes.
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* Tonal pitch-class domain (TPC):
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TPC = Diatonic-value x alteration value =
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= {C, D, E, F, G, A, B} x {bb, b, &, #, ##}
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* Tonal pitch domain (TP):
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TP = Octave x TPC = Int x TPC.
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The Octave domain is taken as the set of integers. The exact
meaning
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of the tonal pitch-class values and the tonal pitch values is
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determined by the mapping of the values on the tonal pitch
dimension
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to the physical pitch dimension.
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HARMONY
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=======
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Tonal harmony deals with inter-relationships among notes, that
arise
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from the interval-structures created among them.
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The interval-structures are characterized by chord-types and
their
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properties.
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This is a broad view of harmony, that concentrates on the
interval
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structures, and is not restricted to simultaneity of note
occurrences.
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The Triad:
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~~~~~~~~~~
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The interval structures of tonal harmony are generated by
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TRIAD chords.
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A triad consists of a sequence of two THIRD intervals. The
various
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kinds of triads arise from constructing them along a scale type.
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The triads are also commonly referenced with respect to the
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corresponding scale degrees.
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- A TRIAD-TYPE is a sequence of two third intervals.
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- A TRIAD is an instance of a triad-type, obtained by
projecting
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a triad-type in the pitch-class ordering.
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- Main triad-types:
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* MAJOR triad-type: 4, 3 semitone intervals.
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* MINOR triad-type: 3, 4 semitone intervals.
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* DIMINISHED triad-type: 3, 3 semitone intervals.
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* AUGMENTED triad-type: 4, 4 semitone intervals.
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THE TRIADS BUILT ON SCALE DEGREES:
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- MAJOR SCALE:
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Tonic (I), Dominant (V), and subdominant (IV) triads -301
Instances of the major triad-type:
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I
- C, E, G.
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IV - F, A, C.
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V
- G, B, D.
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II, III, VI triads -- Instances of the minor triad-type:
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II - D, F, A.
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III - E, G, B.
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VI - A, C, E.
VII triad -- instance of the diminished triad-type: B, D, F.
- MINOR SCALE:
Tonic (I), Dominant (V), and subdominant (IV) triads -Instances of the minor triad-type:
I
- A, C, E.
IV - D, F, A.
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- E, G, B.
III, VI, VII triads -- Instances of the major triad-type:
III - C, E, G.
VI
- F, A, C.
VII - G, B, D.
II triad -- instance of the diminished triad-type: B, D, F.
- HARMONIC MINOR SCALE:
Tonic (I), subdominant (IV) triads -- Instances of the minor
triad-type:
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- A, C, E.
IV - D, F, A.
V, VII triads -- Instances of the major triad-type:
V
- E, G#, B.
VI - F, A, C.
III triad -- instance of the augmented triad-type: C, E, G#.
II, VII triads -- instances of the diminished triad-type:
II - B, D, F.
VII - G#, B, D.
- MELODIC MINOR SCALE:
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Tonic (I), II triads -- Instances of the minor triad-type:
I
- A, C, E.
II - B, D, F#.
IV, V triads -- Instances of the major triad-type:
IV - D, F#, A.
V
- E, G#, B.
III triad -- instance of the augmented triad-type: C, E, G#.
VI, VII triads -- instances of the diminished triad-type:
VI - F#, A, C.
VII - G#, B, D.
Harmony plays a major role in melody structuring, and in the
construction of its accompaniment. Music analysis distinguishes
between SMALL RANGE to LARGE RANGE harmony.
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* Small range harmony -353
study of the chord structure, with respect to the scale
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context (key).
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* Large range harmony -356
study of key (scale context) relationships.
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The scale context assigns ROLES/FUNCTIONS to the chords, based on
their
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consonance/dissonance or stable/dynamic characterization.
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These roles determine various chord successions.
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For example:
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The CADENCE part in a major scale is based on the
progression
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of
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I -> IV -> V -> I.
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Finer chord progression rules refine the progressions, using
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characterization of chord functions. In the cadence that
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might imply the progression
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I -> II -> V -> I
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that arises from similar harmonic scale function of the II
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and the IV degrees. Both degrees function as subdominant
degrees
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in the cadence.
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The chord progressions provide the basis for the chord structure
of
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the piece. Each part in a progression can include a whole
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segment/piece that carries the same chord function. For example,
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the above cadence progressions, can be extended into:
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I -> VI -> IV -> II -> V -> I.
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This way, a basic cadence can be extended into a complex chord
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progression, triads can be elaborated into richer chords (e.g.,
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in Jazz, chords are extended with additional notes -382
like 7, 9, 11, 13 chords).
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In general, the way chords are combined into chord progressions
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is dictated by the chord functions and by the pitch-class
dominance
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in the context scale. These properties yield "rules"
(characterizations)
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of typical voice leading for creating chord progressions.
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STUDIES IN THE TONAL SYSTEM:
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============================
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Analytical research in the tonal system study, not only the
content of
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music pieces, but also the fundamental assumptions that underlie
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the tonal system.
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For example, there are studies into the essential SELECTIONS
done by
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the tonal system, and even the 12 note selection within an
octave.
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Such studies try to understand the advantages/limitations of
different
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selections of pitch-classes, on which music systems can be built
(Dahlia
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Cohen).
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An intensive direction in the study of tonal music basics is in
the
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direction of MUSIC PERCEPTION. Such studies try to understand
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when/why/how does the sensation of a tonal context arise. They
also
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try to find characteristics of various tonal contexts.
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For example:
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Experiments done by Krumhansl suggest that the perception of key
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(scale) depends on the frequency of occurrences and duration of
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the pitch classes. One product of these experiments is a table of
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pitch class rating in a given scale. These experiments might
imply
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methods for creating tonal melodies (Krumhansl, Cognitive
Foundation
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of Musical Pitch, Oxford University Press, 1990).
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