Roman numeral analysis

Roman numeral analysis is a type of musical analysis in which chords are represented by Roman numerals (I, II, III, IV, …). In some cases, Roman numerals denote scale degrees themselves. More commonly, however, they represent the chord whose root note is that scale degree. For instance, III denotes either the third scale degree or, more commonly, the chord built on it. Typically, uppercase Roman numerals (such as I, IV, V) are used to represent major chords, while lowercase Roman numerals (such as i, iv, v) are used to represent minor chords (see Major and Minor below for alternative notations). However, some music theorists use upper-case Roman numerals for all chords, regardless of chord quality.[2]

The chord progression vi–ii–V–I in the key of C major. Using lead sheet chord names, these chords could be referred to as A minor, D minor, G major and C major.[1]

In Western classical music in the 2000s, music students and theorists use Roman numeral analysis to analyze the harmony of a composition. In pop, rock, traditional music, and jazz and blues, Roman numerals can be used to notate the chord progression of a song independent of key. For instance, the standard twelve-bar blues progression uses the chords I (first), IV (fourth), V (fifth), sometimes written I⁷, IV⁷, V⁷, since they are often dominant seventh chords. In the key of C major, the first scale degree (tonic) is C, the fourth (subdominant) is F, and the fifth (dominant) is a G. So the I⁷, IV⁷, and V⁷ chords are C⁷, F⁷, and G⁷. On the other hand, in the key of A major, the I⁷, IV⁷, and V⁷ chords would be A⁷, D⁷, and E⁷. Roman numerals thus abstract chord progressions, making them independent of key, so they can easily be transposed.

History

Gottfried Weber's description of the Roman numerals employed on each degree of the major and minor scales, triads at the left and sevenths at the right. Versuch einer geordneten Theorie der Tonsetzkunst, vol. II, p. 45.

Roman numeral analysis is based on the idea that chords can be represented and named by one of their notes, their root (see History of the Root (chord) article for more information). The system came about initially from the work and writings of Rameau’s fundamental bass.

Arabic numerals have been used in the 18th century for the purpose of denoting the fundamental bass, but that aspect will not be considered here. The earliest usage of Roman numerals may be found in the first volume of Johann Kirnberger's Die Kunst des reinen Satzes in 1774.[3] Soon after, Abbé Georg Joseph Vogler occasionally employed Roman numerals in his Grunde der Kuhrpfälzischen Tonschule in 1778.[4] He mentioned them also in his Handbuch zur Harmonielehre of 1802 and employed Roman numeral analysis in several publications from 1806 onwards.[5]

Gottfried Weber's Versuch einer geordneten Theorie der Tonsetzkunst (Theory of Musical Composition) (1817–21) is often credited with popularizing the method. More precisely, he introduced the usage of large capital numerals for major chords, small capitals for minor, superscript ^o for diminished 5ths and dashed 7 for major sevenths – see hereby.[6] Simon Sechter, considered the founder of the Viennese "Theory of the degrees" (Stufentheorie), made only a limited use of Roman numerals, always as capital letters, and often marked the fundamentals with letter notation or with Arabic numbers.[7] Anton Bruckner, who transmitted the theory to Schœnberg and Schenker, apparently did not use Roman numerals in his classes in Vienna.[8]

Common practice numerals

In music theory related to or derived from the common practice period, Roman numerals are frequently used to designate scale degrees as well as the chords built on them.[2] In some contexts, however, arabic numerals with carets are used to designate the scale degrees themselves (e.g. , , , …).

The basic Roman numeral analysis symbols commonly used in pedagogical texts are shown in the table below.[9][10]^(p71)

Symbol	Meaning	Examples
Uppercase Roman numeral	Major triad	I
Lowercase Roman numeral	Minor triad	i
Superscript ⁺	Augmented triad	I⁺
Superscript ^o	Diminished triad	i^o
Superscript number	Added note	V⁷
Two or more numbers(^#-#)	Figured bass notation	V^4–3
Superscript ^# and ^# _#	First inversion	I⁶
Superscript ^# and ^# _#	Second inversion	I⁶ ₄

The Roman numerals for the seven root-position diatonic triads built on the notes of the C major scale are shown below.

In addition, according to Music: In Theory and Practice, "[s]ometimes it is necessary to indicate sharps, flats, or naturals above the bass note."[10]^(p74) The accidentals may be below the superscript and subscript number(s), before the superscript and subscript number(s), or using a slash (/) or plus sign (+) to indicate that the interval is raised (either ♮ in a flat key signature or a ♯ or in a sharp key signature.

Secondary chords are indicated with a slash e.g. V/V.

Modern Schenkerians often prefer the usage of large capital numbers for all degrees in all modes, in conformity with Schenker's own usage.[lower-alpha 1]

Roman numeral analysis by Heinrich Schenker (1906) of the degrees (Stufen) in bars 13–15 of the Allegro assai of J. S. Bach's Sonata in C major for violin solo, BWV 1005.[12]

Inversions

Roman numerals are sometimes complemented by Arabic numerals to denote inversion of the chords. The system is similar to that of Figured bass, the Arabic numerals describing the characteristic interval(s) above the bass note of the chord, the figures 3 and 5 usually being omitted. The first inversion is denoted by the numeral 6 (e.g. I⁶ for the first inversion of the tonic triad), even although a complete figuring should require ⁶
₃; the numerals ⁶
₄ denotes the second inversion (eg I⁶
₄). Inverted seventh chords are similarly denoted by one or two Arabic numerals describing the most characteristic intervals, namely the interval of a second between the 7th and the root: V⁷ is the dominant 7th (e.g. G–B–D–F); V⁶
₅ is its first inversion (B–D–F–G); V⁴
₃ its second inversion (D–F–G–B); and V⁴
₂ or V² its third inversion (F–G–B–D).[10]^(pp79–80)

Jazz and pop numerals

Roman numeral analysis of the standard twelve-bar blues

Play

In music theory, fake books and lead sheets aimed towards jazz and popular music, many tunes and songs are written in a key, and as such for all chords, a letter name and symbols are given for all triads (e.g., C, G⁷, Dm, etc.). In some fake books and lead sheets, all triads may be represented by upper case numerals, followed by a symbol to indicate if it is not a major chord (e.g. "m" for minor or "^ø" for half-diminished or "7" for a seventh chord). An upper case numeral that is not followed by a symbol is understood as a major chord. The use of Roman numerals enables the rhythm section performers to play the song in any key requested by the bandleader or lead singer. The accompaniment performers translate the Roman numerals to the specific chords that would be used in a given key.

In the key of E major, the diatonic chords are:

E^maj7 becomes I^maj7 (or simply I)
F♯m⁷ becomes ii⁷ (or simply ii)
G♯m⁷ becomes iii⁷ (or simply iii)
A^maj7 becomes IV^maj7 (or simply IV)
B⁷ becomes V⁷ (or simply V)
C♯m⁷ becomes vi⁷ (or simply vi)
D♯^ø7 becomes vii^ø7 (or simply vii°)

In popular music and rock music, "borrowing" of chords from the tonic minor of a key into the tonic major and vice versa is commonly done. As such, in these genres, in the key of E major, chords such as D major (or ♭VII), G major (♭III) and C major (♭VI) are commonly used. These chords are all borrowed from the key of E minor. Similarly, in minor keys, chords from the tonic major may also be "borrowed". For example, in E minor, the diatonic chords for the iv and v chord would be A minor and B minor; in practice, many songs in E minor will use IV and V chords (A major and B major), which are "borrowed" from the key of E major.

Diatonic scales

Major scale

The table below shows the Roman numerals for chords built on the major scale.

Scale degree	Tonic	Supertonic	Mediant	Subdominant	Dominant	Submediant	Leading tone
Conventional notation	I	ii	iii	IV	V	vi	vii^o
Alternative notation	I	II	III	IV	V	VI	VII[13]
Chord symbol	I Maj	II min	III min	IV Maj	V Maj (or V⁷)	VI min	VII dim (or VII^o)

In the key of C major, these chords are

Minor scale

The table below shows the Roman numerals for the chords built on the natural minor scale.

Scale degree	Tonic	Supertonic	Mediant	Subdominant	Dominant	Submediant	Subtonic	Leading tone
Conventional notation	i	ii^o	♭III	iv	v	♭VI	♭VII	vii^o
Alternative notation	I	ii	iii	iv	v	vi	vii
Chord symbol	I min	II dim	♭III Aug (or III Maj)	IV min (or IV Maj)	V Maj (or V⁷)	♭VI Maj	♭VII Maj	VII dim (or VII^o)

In the key of C minor (natural minor), these chords are

The seventh scale degree is often raised to a leading tone making the dominant chord a major chord (i.e. V instead of v) and the subtonic chord a diminished chord (vii^o instead of ♭VII). This is called the harmonic minor scale.

Modes

In traditional notation, the triads of the seven modern modes are the following:

No.	Mode	Tonic	Supertonic	Mediant	Subdominant	Dominant	Submediant	Subtonic / Leading tone
1	Ionian (major)	I	ii	iii	IV	V	vi	vii^o
2	Dorian	i	ii	♭III	IV	v	vi^o	♭VII
3	Phrygian	i	♭II	♭III	iv	v^o	♭VI	♭vii
4	Lydian	I	II	iii	♯iv^o	V	vi	vii
5	Mixolydian	I	ii	iii^o	IV	v	vi	♭VII
6	Aeolian (natural minor)	i	ii^o	♭III	iv	v	♭VI	♭VII
7	Locrian	i^o	♭II	♭iii	iv	♭V	♭VI	♭vii

Footnotes

As the symbol for a Stufe, the Roman numeral "I" in C major can signify a major chord, a minor chord, a seventh chord, or indeed many combinations of notes controlled by the root C. The same Roman numeral can also represent the governing harmonic function of an extended passage embracing several or many chords. In this system, therefore, one basic sign applies to all manifestations of a structural harmony, with figured-bass numerals and other symbols indicating inversions and deviations from the basic type. [...] Roman numerals can be used less to indicate local detail and more broadly, and analytically, to denote harmonic function in either the major or the minor mode. This method assumes fluent knowledge of chord quality in both modes, a skill we consider as fundamental as the recognition of key signatures.[11]

References

William G Andrews and Molly Sclater (2000). Materials of Western Music Part 1, p. 227. ISBN 1-55122-034-2.
Roger Sessions (1951). Harmonic Practice. New York: Harcourt, Brace. LCCN 51-8476. p. 7.
Johann Philipp Kirnberger, Die Kunst des reinen Satzes, vol. I. Berlin und Königsberg, Decker und Hartung, 1774, p. 15 and plates to p. 19. It is not entirely clear, however, whether Roman numerals in Kirnberger denote scale degrees or intervals (or both).
David Damschroder, Thinking about Harmony: Historical Perspectives on Analysis. ISBN 978-0-521-88814-1. Cambridge University Press, 2008, p. 6
Floyd K. Grave and Margaret G. Grave, In Praise of Harmony: The Teachings of Abbé Georg Joseph Vogler.
Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst, 3d Edition, Mainz, Schott, 1830–1832, vol. 2, pp. 44–63, §§ 151–158.
Simon Sechter, Die Richtige Folge der Grundharmonien, Leipzig, Breitkopf und Härtel, 3 vols., 1853–1854. Roman numerals are found in all three volumes.
Anton Bruckner, Vorlesungen über Harmonielehre und Kontrapunkt an der Universität Wien, E. Schwanzara ed., Wien, Östrereichischer Bundesverlag, 1950. See also Robert E. Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg, Ann Arbor, UMI Research Press, 1982. ISBN 0-8357-1586-8. pp. 67–84.
Eric Taylor (1989). The AB Guide to Music Theory. Part 1. London: Associated Board of the Royal Schools of Music. pp. 60–61. ISBN 1-85472-446-0.
Bruce Benward; Marilyn Nadine Saker (2003). Music: In Theory and Practice. I (seventh ed.). Boston: McGraw-Hill. ISBN 978-0-07-294262-0.
Edward Aldwell; Carl Schachter; Allen Cadwallader (2011). Harmony and Voice Leading (4th ed.). Schirmer, Cengage Learning. pp. 696–697. ISBN 978-0-495-18975-6.
Heinrich Schenker, Harmonielehre, Stuttgart, Berlin, Cotta, 1906, p. 186, Example 151.
John Mehegan (1989). Tonal and Rhythmic Principles. Jazz Improvisation. 1 (Revised and Enlarged ed.). New York: Watson-Guptill. pp. 9–16. ISBN 0-8230-2559-4.

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[12] As the symbol for a Stufe, the Roman numeral "I" in C major can signify a major chord, a minor chord, a seventh chord, or indeed many combinations of notes controlled by the root C. The same Roman numeral can also represent the governing harmonic function of an extended passage embracing several or many chords. In this system, therefore, one basic sign applies to all manifestations of a structural harmony, with figured-bass numerals and other symbols indicating inversions and deviations from the basic type. [...] Roman numerals can be used less to indicate local detail and more broadly, and analytically, to denote harmonic function in either the major or the minor mode. This method assumes fluent knowledge of chord quality in both modes, a skill we consider as fundamental as the recognition of key signatures.[11]

[1] William G Andrews and Molly Sclater (2000). Materials of Western Music Part 1, p. 227. ISBN 1-55122-034-2.

[roger-2] Roger Sessions (1951). Harmonic Practice. New York: Harcourt, Brace. LCCN 51-8476. p. 7.

[3] Johann Philipp Kirnberger, Die Kunst des reinen Satzes, vol. I. Berlin und Königsberg, Decker und Hartung, 1774, p. 15 and plates to p. 19. It is not entirely clear, however, whether Roman numerals in Kirnberger denote scale degrees or intervals (or both).

[4] David Damschroder, Thinking about Harmony: Historical Perspectives on Analysis. ISBN 978-0-521-88814-1. Cambridge University Press, 2008, p. 6

[5] Floyd K. Grave and Margaret G. Grave, In Praise of Harmony: The Teachings of Abbé Georg Joseph Vogler.

[6] Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst, 3d Edition, Mainz, Schott, 1830–1832, vol. 2, pp. 44–63, §§ 151–158.

[7] Simon Sechter, Die Richtige Folge der Grundharmonien, Leipzig, Breitkopf und Härtel, 3 vols., 1853–1854. Roman numerals are found in all three volumes.

[8] Anton Bruckner, Vorlesungen über Harmonielehre und Kontrapunkt an der Universität Wien, E. Schwanzara ed., Wien, Östrereichischer Bundesverlag, 1950. See also Robert E. Wason, Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg, Ann Arbor, UMI Research Press, 1982. ISBN 0-8357-1586-8. pp. 67–84.

[eric-9] Eric Taylor (1989). The AB Guide to Music Theory. Part 1. London: Associated Board of the Royal Schools of Music. pp. 60–61. ISBN 1-85472-446-0.

[BenwardSaker2003-10] Bruce Benward; Marilyn Nadine Saker (2003). Music: In Theory and Practice. I (seventh ed.). Boston: McGraw-Hill. ISBN 978-0-07-294262-0.

[AldwellEtal2011-11] Edward Aldwell; Carl Schachter; Allen Cadwallader (2011). Harmony and Voice Leading (4th ed.). Schirmer, Cengage Learning. pp. 696–697. ISBN 978-0-495-18975-6.

[13] Heinrich Schenker, Harmonielehre, Stuttgart, Berlin, Cotta, 1906, p. 186, Example 151.

[14] John Mehegan (1989). Tonal and Rhythmic Principles. Jazz Improvisation. 1 (Revised and Enlarged ed.). New York: Watson-Guptill. pp. 9–16. ISBN 0-8230-2559-4.