MUS 111 Study Guide B (Chapter 2)

Basics II

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  1. 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 sounding simultaneously.

  1. Intervals are described in terms of both quality and quantity.  Descriptions of quality are:
    Perfect (P)
    Major (M)
    minor (m)
    Augmented (A)
    diminished (d)
    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 are notated.

  1. Major and Perfect intervals ascending (distance between first and last note of each measure):

  1. 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 ...

A   A
M   P
m   d
d    

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.

  1. 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 augmented 2nd.)

  1. 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.)

  1. 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.)

  1. Terminology for other intervals:

    Unison (P1) or perfect prime (PP):  two identical pitches.

    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.

  1. 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.

  1. 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 inverted.

  1. 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.

  1. 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.

  1. 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 are produced.

  1. 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 are possible:

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.

  1. 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.

  1. 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.

  1. 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.

  1. Diatonic seventh chords in major and minor (harmonic form):

  1. 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:
      1                    (Root position) none
      3                    First
      5                    Second
      7                    Third (7th chords only)

  1. 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 numeral:

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 the bass.

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.)

  1. 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 chord.

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.

  1. 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 notehead.

The descending stem is attached to the left side of the notehead.

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 values.

Accidentals are placed directly before the affected note and on the same line or space as the note.  

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