MUS 111 Study Guide B (Chapter 2)
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Both melody and harmony make use of intervals.
A melodic interval is the distance horizontally between any
two notes. A harmonic interval is the distance
vertically between any two notes. Melodic intervals are
measured between two consecutive pitches, one sounding after the
other. Harmonic intervals are measured between two pitches
Intervals are described in terms of both quality
and quantity. Descriptions of quality are:
The term "perfect" originates from the days when early
tuning systems actually used a fifth which was perfectly in tune, or
pure. Today our octave (P8) is in tune, but our 5th is
actually flat by two cents (2/100th of a half step). We still
continue to refer to the fifth as "perfect," even though
it is not.
Descriptions of quantity refer to the numerical
distance one note is from another. Combining qualitative and
quantitative descriptions, we can describe intervals as follows:
m2, M2, m3, M3, P4, A4/d5, P5, m6, M6, m7, M7, P8. Various other
A and d intervals are possible, depending upon how they
Major and Perfect intervals ascending (distance
between first and last note of each measure):
The other intervals, minor (m), diminished (d), and
augmented (A), can be calculated and identified by comparing them
with perfect (P) or major (M) intervals. For example,
"M" reduced by one half step becomes "m."
The chart below, read vertically, may be more helpful
In the left column, starting with a major
interval: (1) increasing the interval by one half-step
produces an augmented interval, (2) reducing the interval by one
half-step produces a
minor interval, (3) reducing the interval by two half-steps
produces a diminished
interval. Reducing a minor interval one half-step produces
a diminished interval.
In the right column, starting with a perfect
interval (e.g., P4, P5): (1) increasing the interval
one half-step produces an augmented interval, (2) reducing the interval
by one half-step
produces a diminished interval.
Minor intervals: these are one half step
smaller than major intervals. The decrease is accomplished
when either the upper note is lowered one half-step, or the lower
note is raised one half step, using the same letter names
(pitch classes). For example, C to E is a major third; C to Eb
is a minor third. (C to D# would be an augmented 2nd.)
Also, C# to E is a minor third. (Db to E would be an
Diminished intervals: these are one half step
smaller than minor or perfect intervals. The decrease is
accomplished by lowering the upper note or by raising the lower
note, but, again, always using the same letter names.
When you derive a d interval from a M interval, be
sure to reduce the M interval by two half steps (same
letter names). It may be easiest to first mentally convert the
M interval to m, then the m to d.
(A diminished 2nd, resulting in a unison, is extremely rare.)
Augmented invertals: these are one half-step
larger than perfect intervals or major intervals, using the same
letter names. (The augmented 3rd and 7th are quite rare,
as they would create a P4 and P8, respectively.)
Terminology for other intervals:
Unison (P1) or perfect prime (PP): two identical
Diatonic and chromatic half steps: in the diatonic
half step, the two tones are spelled with adjacent
letter names, such as F to Gb; a chromatic half step
would involve using the same letter name (pitch class), such as F to
F#. The term diatonic refers to those tones spelled
within the nomenclature of a given key.
Tritone (TT): refers to each of the enharmonic
(sounding the same) intervals of the A4 and d5. Each interval
includes three whole steps, hence, the term tritone.
Inversion of intervals: inverting intervals
simply means flipping them upside down. For example, C to G is
a P5, and G to C is a P4. Each interval has a corresponding
inverted interval. When inverted, seconds become sevenths,
thirds become sixths, fourths become fifths, fifths become fourths,
sixths become thirds, and sevenths become seconds. Also, A2
becomes d7, d4 becomes A5, etc. The sum of the two intervals
should always equal 9. To determine the numerical inversion of
an interval, subtract its number from 9: the inversion of a
third is 9 minus 3 = a sixth, for example. Unisons and octaves
remain the same.
Descending intervals: To spell descending
intervals, you may find it helpful to use the following approach:
First, determine the inversion of the descending interval. If
the objective is to spell the interval a M6 below C, begin by
determining the inversion of the M6: m3 above = Eb.
After the inversion is calculated, the pitch is simply
placed an octave lower: C down to Eb is a descending M6.
In addition to the quantitative factors being
inversely proportionate, the qualitative factors are, as well.
Any M interval becomes m when inverted; any m
interval becomes M when inverted. Likewise, A
becomes d when inverted; d becomes A when
Compound intervals: these are intervals larger
than an octave. An interval is usually described by its
diminutive form (3rd rather than 10th) unless the distinction is
necessary for some reason.
Consonance and dissonance: consonant
refers to a harmonious or pleasant sound; dissonant refers to
a harsh or disagreeable sound. While just which intervals meet
either definition has changed over the years, the following general
categorizations tend to be agreed upon, at least, by some theorists:
Consonant intervals: m3, M3, P4, P5, m6, M6, P8.
Dissonant intervals: m2, M2, TT, m7, M7.
Chord--a group of tones sounding
simultaneously or in close succession. When chord members
sound at the same time, a block chord is produced.
Sounding in succession, broken or arpeggiated chords
Triad--a three-note grouping of pitches
comprised of two stacked thirds. The lowest note is called the
root, above which are the third and fifth.
Since both major and minor thirds exist, four different combinations
Major triad: M3 and P5 above the root;
major third on bottom, minor third on top.
Minor triad: m3 and P5 above the root;
minor third on bottom, major third on top.
Diminished triad: m3 and d5 above the
root; two stacked minor thirds.
Augmented triad: M3 and A5 above the
root; two stacked major thirds.
Triads in a key: Triads can be built above
each note of any major or minor scale. When only scales tones
are used, any note or combination of notes, including triads, is
considered diatonic. When a tone is altered by an
accidental, the chord is considered an altered chord, except
in minor, where the #6 and #7 scale degrees are considered
diatonic. A triad in a key is identified by the scale-step
number of its root, and expressed by a Roman numeral. The
Roman numeral not only designates the scale-step location of the
root, but also the triad construction. For example, uppercase Roman numerals indicate a major triad; lowercase
Roman numerals indicate a minor triad. A lowercase
numeral with a small o symbol indicates a diminished
triad; and uppercase numeral with a + indicates an augmented triad.
Chords larger than a triad: While we will
study extended-chord structures at a later time, it is common
to encounter chords comprised of additional stacked thirds.
The thirteenth chord is the largest possible diatonic chord in that
the next third higher would duplicate the root.
Seventh chords: consist of a triad with an
additional 3rd stacked on top.
The symbol Ø7 represents the
"half-diminished" chord, which consists of a diminished
triad, but with a m7.
The major-minor seventh chord is often called a
"dominant seventh" because the Mm7 structure is only found,
diatonically, above the dominant, or 5th, scale degree.
Diatonic seventh chords in major and minor (harmonic
Inversion of chords: Like intervals,
chords can also be inverted. Chord inversions change
intervallic content and sound without changing the overall spelling.
Lowest Note: Inversion:
(Root position) none
Third (7th chords only)
Analysis symbols for chord inversions:
Inversion in triads:
Root position: The Roman numeral appears
without any other symbol--uppercase for a major chord; lowercase for
a minor chord (with diminished symbol, if applicable).
First inversion: The Roman numeral
indicates the scale degree on which the root of the chord is
built and the quality of the chord (via uppercase / lowercase
symbols). In addition to the Roman numeral is the Arabic number 6
(see illustration below).
This number indicates the distance of the upper note(s) above the bass
note (3rd of the chord). Technically, the full designation of a
first-inversion triad is [6 over 3], indicating a note a 6th and a 3rd
above the bass. In practice, however, the number 3 is omitted as
a shortcut. (Note: The text "subscripts"
the 6 following a Roman numeral (V6), however, some theory
texts use "superscript" (V6). Either way is
acceptable; however, a single Arabic number should be placed
either above or below the baseline of the Roman numeral. In that
the text also superscripts the added "7th" (V7),
I prefer to be consistent and superscript all Arabic numerals
indicating inversion, or the added 7th. For chord symbols
requiring two Arabic numbers, they may be placed evenly with the Roman
Second inversion: Again, the Roman
numeral indicates the scale degree on which the root of the
chord is built and the quality of the chord (via uppercase / lowercase). In order to distinguish second inversion from first
inversion, both Arabic numbers must be present to indicate the
distance of the upper notes above the bass note (5th of the
chord). The designation for a second inversion triad, then, is
[6 over 4] (see illustration above), indicating that one note is a
fourth above the bass and another is a sixth above the bass.
Inversion in seventh chords:
Root position: At this point in our
studies the chord most commonly found with the added 7th is the
dominant chord (V). The Arabic number is simply added to the
Roman numeral if the chord appears in root position: V7.
First inversion: A seventh chord in first
inversion has the 3rd of the chord in the bass. Since the
seventh chord contains four notes, one would expect that there should
be three Arabic numerals used in conjunction with the Roman
numeral. Again, however, a shortcut is used, so each inversion
only contains two Arabic numbers. Even though in first
inversion, a seventh chord consists of pitches a 3rd, 5th and 6th
above the bass, the 3rd is assumed and only the pitches a 5th and 6th
above the bass note (third of the chord) are indicated.
Second inversion: The shortcut symbol
used to indicate a dominant seventh chord in second inversion (fifth
in the bass) is [4 over 3] (see illustration above). This
indicates that the chord consists of pitches a 3rd, 4th and 6th
(implied) above the bass.
Third inversion: The shortcut symbol used
to indicate a dominant seventh chord in third inversion (seventh in
the bass) is [4 over 2] (see illustration above). This indicates
that the chord consists of pitches a 2nd, 4th and 6th (implied) above
No matter what the arrangement or quantity of upper
voices, the chord position is determined in relation to the bass
voice. (Compound intervals are reduced to their simplest
form for identification.)
Figured Bass: Figured bass notation is
a shortcut system for harmonizing notes above a given bass note,
usually in a chorale. Arabic numbers and other symbols are
provided for this purpose. All symbols used are in relation to
the bass note. For example, if no symbol appears
beneath a given pitch, the indication is a triad in root position;
the note is the root of the chord.
If a 6 appears below a pitch, a first-inversion triad
is indicated; the given note is the "third" of the chord.
If a [6 over 4] (see illustration in No. 20) appears
beneath a pitch, a second-inversion triad is indicated; the given note
is the "fifth" of the chord.
If a 7 appears below a pitch, the indication is a
seventh chord in root position; the given note is the root of the
If a [6 over 5] (see illustration in No. 20) appears
beneath a pitch, a first-inversion seventh chord is indicated; the
given note is the "third" of the chord.
If a [4 over 3] (see illustration in No. 20) appears
beneath a pitch, a second-inversion seventh chord is indicated; the
given note is the "fifth" of the chord.
If a [4 over 2] (see illustration in No. 20) appears
beneath a pitch, a third-inversion seventh chord is indicated; the
given note is the "seventh" of the chord.
In addition to numeric indicators, other symbols are
used as well. A sharp, flat or natural standing
alone beneath a pitch refers to the third above the bass note
(root position implied). For example, if the given note is C and
a flat appears beneath it, a C minor chord is indicated.
A number preceded by a flat lowers that particular note one
half step; just the opposite if a sharp appears before a
number. A natural before a number would be used to cancel
the pitch indicated by the key signature. Also a number with a slash
through it means that interval above the bass should be raised one
half step. A dash following a number means that voice
should be held through, even though the bass changes.
While we will eventually be working with figured bass
in relation to the grand staff, the examples below utilize only a
single staff for purposes of illustration.
Staff notation basics:
The single note consists of three parts:
notehead, stem, flag/beam.
The ascending stem is attached to the right side of
The descending stem is attached to the left side of
The middle line on any staff marks the dividing point
where stem direction changes for single notes. For notes on or
above the middle line, stems are down. Stems are up for notes
below the middle line.
When two parts share one staff, stems ascend on notes
for the upper part, and descend on notes for the lower part ...
regardless of their location on the staff.
To indicate two parts performing the same pitch
(unison), use a single notehead with both ascending and descending
stems. For two whole notes in unison, use two slightly
overlapping (touching) whole notes.
Dotted notes: When the note is in a space, the
dot appears in the same space. When the note is on a line, the
dot is normally found in the space above. Exceptions may be
found when various chord tones share the same stem.
Vertical arrangements of notes: All note
sounding simultaneously must be vertically aligned.
Horizontal arrangement of notes: The notes
should appear in the measure in approximate proportion to their time
Accidentals are placed directly before the
affected note and on the same line or space as the note.
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