Roman numeral analysis
In music, Roman numeral analysis involves the use of Roman numerals to represent chords. In this context, Roman numerals (I, II, III, IV, ...) typically denote scale degrees (first, second, third, fourth, ...). When a Roman numeral is used to represent a chord, it is meant to indicate the scale degree corresponding to its root note, which is the note on which the chord is built. For instance, III is the Roman numeral which denotes either the third degree of a scale, or the chord built on that degree. Typically, uppercase Roman numerals (such as I, IV, V) represent the root of major chords while lowercase Roman numerals (such as i, iv, v) represent the root of minor chords (see Major and Minor below for alternative notations).
In the most common day to day use, Roman numerals allow musicians to quickly understand the progression of chords in a piece. For instance, the standard twelve bar blues progression is denoted by the roman numerals I (first), IV (fourth), and V (fifth). In the key of C (where the notes of the scale are C, D, E, F, G, A, B), the first scale degree (Tonic) is C, the fourth (Subdominant) is F, and the fifth (Dominant) is a G. So the I, IV, and V chords are C, F, and G. Similarly, if one were to play the same progression in the key of A (A, B, C♯, D, E, F♯, G♯) the I, IV, and V chords would be A, D, and E. In essence, Roman numerals provide a way to abstract chord progressions, by making them independent of the selected key. This allows chord progressions to be easily transposed to any key.
Roman numeral analysis is the use of Roman numeral symbols in the musical analysis of chords. In music theory related to or derived from the common practice period, arabic numerals with carets are used to designate scale degrees themselves (), whereas in theory related to or derived from jazz or modern popular music uses numbers (1, 2, 3, etc...) to represent scale degrees (See also diatonic function). In both theories, the Roman numeral, number, or careted number, refers to a chord built upon that scale degree. For example, I, , or 1, all refer to the chord upon the first scale step.
Gottfried Weber's Versuch einer geordneten Theorie der Tonsetzkunst (Theory of Musical Composition) (Mainz, B. Schott, 1817-21) is credited with popularizing the analytical method by which a chord is identified by the Roman numeral of the scale-degree number of its root. However, the practice originated in the works of Abbé Georg Joseph Vogler, whose theoretical works as early as 1776 employed Roman numeral analysis (Grave and Grave, 1988) 
Common practice numerals 
|Roman numeral analysis symbols|
|Roman Numeral||Triad quality||Example (tonic)|
|Lowercase o||Diminished triad||io|
|Uppercase +||Augmented triad||I+|
The current system used today to study and analyze tonal music comes about initially from the work and writings of Rameau’s fundamental bass. The dissemination of Rameau’s concepts could only have come about during the significant waning of the study of harmony for the purpose of the basso continuo and its implied improvisational properties in the later 18th century. The use of Roman numerals in describing fundamentals as “scale degrees in relation to a tonic” was brought about, according to one historian, by John Trydell’s Two Essays on the Theory and Practice of Music, published in Dublin in 1766. However, another source says that Trydell used Arabic numerals for this purpose, and Roman numerals were only later substituted by Georg Joseph Vogler. Alternatives include the functional hybrid Nashville number system and macro analysis.
Jazz and pop numerals 
In music theory aimed towards jazz and popular music, all triads are represented by upper case numerals, followed by a symbol to indicate if it is not a major chord (e.g. "-" for minor or "ø" for half-diminished):
- E maj7 becomes I maj7
- F♯ -7 becomes II -7
- G♯ -7 becomes III -7
- A maj7 becomes IV maj7
- B7 becomes V7
- C♯ -7 becomes VI -7
- D♯ø7 becomes VIIø7
|Alternative notation||I||II||III||IV||V||VI||VII|
|Chord symbol||I Maj||II min||III min||IV Maj||V Maj||VI min||VII dim|
|Alternative notation||I||ii||iii||iv||v||vi||vii|
|Chord symbol||I min||II dim||♭III Maj||IV min||V min||♭VI Maj||♭VII Maj||VII dim|
- Jonas, Oswald (1982). Introduction to the Theory of Heinrich Schenker (1934: Das Wesen des musikalischen Kunstwerks: Eine Einführung in Die Lehre Heinrich Schenkers), p.22. Trans. John Rothgeb. ISBN 0-582-28227-6. Shown all uppercase.
- Grave, Floyd Kersey and Margaret G. Grave (1988). In Praise of Harmony: The Teachings of Abbé Georg Joseph Vogler.
- Bruce Benward & Marilyn Nadine Saker (2003), Music: In Theory and Practice, seventh edition, 2 vols. (Boston: McGraw-Hill) Vol. I, p. 71. ISBN 978-0-07-294262-0.
- Dahlhaus, Carl. "Harmony." Grove Online Music Dictionary
- Richard Cohn, "Harmony 6. Practice". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001).
- Gorow, Ron (2002). Hearing and Writing Music: Professional Training for Today's Musician, second edition (Studio City, California: September Publishing, 2002), p. 251. ISBN 0-9629496-7-1.