Lesson 6a - Roman Numerals in Harmonic Analysis

Thus far, we have focused on the building blocks for constructing music such as pitches, intervals, chords, and melodic interaction, but we have yet to look at how these elements combine to create harmony. To study harmony, we need a tool that describes chords and sonorities based on their function rather than their components.

To this point, we have used leadsheet notation to convey the essential components of a chord: root, bass note, quality, inversion, and any additional pitches. But a single leadsheet chord does not provide context about its role in the music.

Cmin⁷/E-flat

From this, we know that the chord has the pitches C, E-flat, G, and B-flat with the E-flat as the lowest voice, but we do not know how it relates to the chords around it. Is it stable, and does it sound final? Or is it unstable and pulling toward a different chord?

When music theorists want to discuss diatonic (key-based) function, we label chords using a Roman numeral system based on the chord’s position within a particular key.

Goals for this topic

Using the examples below, determine:

• what are the diatonic qualities of all triads and seventh chords in major and minor
  • If we consider these examples to be “diatonic”, does every quality occur naturally or do some require accidentals?
  • What is the relationship between chords in major and minor?
• how Roman numerals are created and labeled
  • What information does each part (e.g. Roman numeral, lower-case vs. upper-case, inversion figures, etc.) of the Roman numeral convey?
  • Are the inversion figures identical to the figured bass discussed in the class reading from this unit? If not, how do they differ?
  • How do leadsheet notation and Roman numerals differ in the information that they convey?

Note that the ABC notation used in these examples has limitations on what kind of text can be entered. In the Roman numeral system, everything other than the Roman numeral itself should be written as superscript.

Conclusions

You may already be familiar with the basics of using Roman numerals for labeling diatonic chords, but it is helpful to have a complete breakdown of the system into its fundamental components.

• Root - The number (e.g. one, two, three, etc.) associated with a Roman numeral denotes the scale degree on which the chord is built.
  • If there is no accidental in front of the Roman numeral, the chord is built off the diatonic scale degree.
    • The only exception to this rule are the vii^o and vii^o7 chords in minor, which are both built off of ti (raised seventh scale degree) but will not have a “#” placed in front of the Roman numeral. We will discuss the reasoning for this in detail over the next couple of units, but to have its standard dominant function in diatonic harmony, the vii^o chord must contain ti. Because of this requirement, we skip using the normal root alteration in our Roman numeral analysis as a shorthand.
    • The naturally-occurring VII chord is used much less and has a more limited scope in common practice.
  • If there is an accidental in front of the Roman numeral, this affects only the root of the chord by raising or lowering it.
• Chordal third - The case (i.e. upper-case versus lower-case) of the Roman numeral denotes the quality of the chordal third.
  • All upper-case Roman numerals have a major third. This includes major triads, augmented triads, major major (major 7) chords, and major minor (dominant 7) chords.
  • All lower-case Roman numerals have a minor third. This includes all minor triads, diminished triads, minor minor chords, diminished minor chords, and diminished diminished chords.

• Chordal fifth - The chordal fifth is altered by adding either a ^o (the diminished symbol) for lowered fifths or a ⁺ for raised fifths.

• Chordal sevenths - To show a chordal seventh is present, you only need to use an inversion figure that implies a seventh chord (i.e. ^{7, 6/5, 4/3,} or ^4/2).
  • A seventh chord inversion figure with no alteration implies a minor seventh above the root. It may be helpful to think of this as the “default” chordal seventh.
    • Having a minor interval as default is different than chordal thirds and fifths, because both of those chord members have “defaults” of a major interval.
    • We do this for two reasons:
      • We cannot raise a M7 above the root; it just becomes the root.
      • A d7 would be lowered twice if we used a M7 as default, and we would have to devise a new system for showing this.
  • A ^M is added before the inversion figure for all major seventh chords, even though they do occur diatonically. This allows us to differentiate chord qualities clearly once secondary functions and mode mixture are introduced later in the course. Non-diatonic harmonies are common enough that the Roman numeral system must be able to clearly distinguish them while retaining its clarity.
  • The diminished symbol, ^o, is added before the inversion figure for fully diminished seventh chords. This also shows that the chordal fifth is lowered, the same as in a triad.
    • If you think of a m7 above the root as the default chordal seventh for inversion figures, then adding ^o lowers the chordal seventh by a half step.
    • As you know, half diminished seventh chords (diminished minor chords) have a lowered fifth above the root but a minor seventh, rather than the diminished seventh of a fully diminished seventh chord. For half diminished chords, we replace the ^o with a ^ø to show that the chordal fifth is lowered but the chordal seventh is a m7 above the root.

Remember that when you add an inversion to a seventh chord, you do not need a ⁷ anymore. The inversion implies the seventh.

By having each part of a Roman numeral describe an isolated chord tone, we are able to accurately describe any chord that can occur in our tonal harmony system, regardless of whether it has a standard function.

Chordal member	Default implied pitch	To raise by semitone from default	To lower by semitone from default
root	diatonic scale degree	sharp symbol in front of Roman numeral*	flat symbol in front of Roman numeral*
third	based on case of Roman numeral	upper case (M3)	lower case (m3)
fifth	P5 above root	+ after Rom num	o after Rom num
seventh	m7 above root	M before inversion figure	o before inversion figure**

*For clarity’s sake, we always use a sharp or flat symbol to show that we are raising or lowering the root, even if you are actually adding a natural. When analyzing pieces that change keys–especially to distant keys–this makes it much easier to look for similar patterns, regardless of whether the key signature uses sharps or flats.

**Because the diminished symbol ^o implies the interval of a d5 AND a d7, you must use the half-diminished symbol if you wish to alter the fifth but leave the chordal seventh as a m7 above the root.

Add and sub

There are two more possible additions for Roman numerals, but these are advanced techniques are will not be necessary until Unit 19. For completeness, we will discuss them here, but if you would rather skip this for now, please do so.

The word ^add is used when a tone is a functional part of the chord but does not belong to a standard triad or seventh chord. A good example of this would be a triad that has a functional 9th above the root. For example, in C major, if a C major triad were to have a D as necessary to the function of that chord, it would be labeled I^{add 9}. This clearly identifies the triad plus the additional ninth, but it also omits the chordal seventh. If it were a major seventh chord, you would not need use the ^add function, because a 9 would imply everything below it. So a I^M9 is assumed to have 9th, 7th, 5th, and 3rd above the root.

The word ^sub is used when a tone replaces another chord tone. The replaced chord tone is always the closest chord tone below the subbed chord tone. For example, if the tonic triad in C major had a 4th above the root instead of the chordal third, you would label that as I^sub4. This implies that the 4th replaces the chord tone directly below it, so this would be a triad with C, F, and G.