A system of Just intonation which extends to intervals with frequency ratios involving the number seven, principally 7:4, 7:5, 7:6, 9:7 and their inversions; these are known as ‘septimal intervals’. The ratio 7:4 corresponds to the interval between the fourth and seventh harmonic partials (i.e. between the third and sixth overtones) of a note, and is somewhat smaller than an equal-tempered minor 7th or augmented 6th; in terms of just intonation, it is 27 cents smaller than a minor 7th (9:5), or 7 cents smaller than an augmented 6th (225:128). In his Tentamen novae theoriae musicae (St Petersburg, 1739), Leonhard Euler put forward the view that the ratio 7:4 gave rise to a natural ‘harmonic’ form of minor 7th; adherents of this view, who included Fétis, Helmholtz and Harry Partch, therefore referred to the interval as the ‘harmonic 7th’. On the other hand, a series of writers beginning with Nicola Vicentino (see Barbour) have viewed the same interval not as a minor 7th but as an augmented 6th (see also Regener).
In his Harmonie universelle (Paris, 1636–7/R), Mersenne attributed consonant qualities to septimal intervals. Kirnberger, in Die Kunst des reinen Satzes (1771–9), invented a notational symbol for flattening a note by the difference between a minor 7th and the interval with frequency ratio 7:4. Euler, and later Helmholtz, saw in the configuration of the fourth, fifth, sixth and seventh harmonic partials an ideal realization of the chord of the dominant 7th. At the same time, however, many theorists – including Zarlino, Rameau and Schenker – have rejected the use of septimal intervals in music.
Some 20th-century composers, although satisfied to work within the system of equal temperament, nevertheless used pitch structures intended to evoke higher harmonics such as the seventh; examples are Skryabin’s ‘mystic chord’, Messiaen’s ‘chord of resonance’ and Bartók’s ‘acoustic scale’. Others, including Partch, Stockhausen and La Monte Young, have employed septimal intervals in their works, seeing in them a means of escaping the constraints of equal temperament and thereby expanding the tonal system.
See also Microtone.
J.M. Barbour: Equal Temperament: its History from Ramis (1482) to Rameau (1737) (diss., Cornell U., 1932)
F.-J. Fétis: Esquisse de l’histoire de l’harmonie (Paris, 1840; Eng. trans. by M.I. Arlin, diss., Indiana U., 1971)
H. Helmholtz: Die Lehre von den Tonempfindungen (Brunswick, 1863; Eng. trans., 1875/R)
H. Partch: Genesis of a Music (New York, 1974)
E. Regener: ‘The Number Seven in the Theory of Intonation’, JMT, xix (1975), 140–54
M.J. Hewitt: The Tonal Phoenix (Bonn, 1999)
MICHAEL JOHN HEWITT