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Francesco Gasparini's Twenty-One Keys: Do they reflect the use of meantone?

July 11, 2003
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Carl Sloane is a pharmacist by education, a freelance translator by vocation, and an amateur harpsichordist.

In his famous treatise on continuo playing,1 Francesco Gasparini gives a table of twenty-one keys which the student could expect to encounter. The absence of certain keys proves that the contents of the table are determined by the tuning which Gasparini used, and because of his standing, it would be of particular interest if the precise nature of this tuning could be established.

There are immediate indications that Gasparini had in mind some type of regular meantone (RMT)2 with the usual range of Eb to G# and the wolf between G# and Eb. These indications, supported by substantial evidence that meantone was common in Italy well into the eight-eenth century,3 include the complete absence of Ab and the apparent expectation that the table would be universally applicable. However, the matter cannot be automatically considered settled, since the the book was written at a time when meantone was being discarded in favor of temperaments with no wolf. Tempérament ordinaire, which eliminated the wolf by spreading it over several fifths, had been in use in France for some time when it was described by d'Alembert,4 and Werckmeister's tunings,5 which usually solved the problem by reducing the number of tempered fifths, had been published at least by 1691. In addition, there are apparent inconsistencies in the table itself.

The keys are illustrated in the form of figured scales which begin on the tonic, rise to the sixth degree, descend a ninth to the dominant, then leap back to the tonic. They are divided into two groups, those of "great usefulness" (gran giovamento):

G, g, a, A, Bb, b, C, c, d, D, Eb, e, E, F, f, f#, and those used in modulations:

bb, B, c#, eb, F#.

Major keys in group 1 thus run from Eb to E around the circle of fifths, minor keys from F to F#.

Tagliavini6 argues against RMT, pointing out that the absence of Db major is difficult to understand when the more highly inflected Gb major (enharmonic F#) is present. To his rhetorical suggestion that this paradox may be due to the presence of the wolf in the dominant chord of Db, he replies that C# minor should then logically be absent as well. He does not attempt to establish the criterion used to exclude keys from group 1.

The composition of group 1 does not initially seem compatible with the use of RMT: F minor, with its poor tonic chord, and E and Eb majors, with a poor chord on V and IV respectively, would not be expected in this group. On the other hand, there is plenty of contemporary evidence showing that poorly tuned intervals were used regularly in practice (Ref. 3, 156-8, 193), and on this basis, the makeup of group 1 can be logically explained.

In RMT, the most complex major keys in group 1, Eb and E, each have a single note outside the range Eb to G#; the most complex minor keys, F and F#, each have two such notes, at least one of which is on the sixth or seventh degree (see Fig. 1). Accordingly, the keys in group 1 may have been chosen on the understanding that major keys were allowed a maximum of one unavailable note and minor keys a maximum of two, the greater freedom in the minor keys being due to the variable inflection of VI and VII.

It is worth noting that even in group 2, Gasparini's key signatures never have more than three flats or four sharps, thus staying within the same limits as the major keys in group 1. (The section on modulation--(pp. 111-114)--gives key signatures which exceed these limits, but it also illustrates keys not included in the table, so that Gasparini has here presumably sacrificed some degree of rigor.) In addition, the order within each pair of parallel keys in group 1 is obviously determined by the complexity of the key signature, suggesting a certain preoccupation with key complexity and unavailable notes.

 Owing to the manner in which unavailable notes enter as one moves around the circle of fifths, exact location in the scale was probably of secondary importance in group-1 keys. But ultimately position must have become of critical importance. From Fig. 1 it is apparent that the most elementary keys not in the table would have an unavailable note on at least one of I, II or V. Although the presence of the wolf on either I or V (an unavailable note on one member of either of the pairs I/V or II/V) may have been the underlying reason for outright rejection, I think that a more likely working criterion was the spelling of I, II and V, and that the presence of F# major does not imply the inclusion of Gb, any more than the presence of C major implies the inclusion of B#. It seems likely, especially in view of the more lenient treatment of minor keys in group 1, that the presence of the wolf on the dominant of C# minor (or the wrong spelling for II) was considered acceptable. This hypothesis is admittedly not as credible as it would be if Gasparini had not figured V with a major third, because some softening of the effects of the wolf would be expected in certain positions of the chord G#-B-D#, especially of the first inversion, by the presence at the relevant pitch of a partial from the B-natural.7

To the extent that RMT is established, Gasparini's table shows that, contrary to most modern opinion, G# was not retuned to Ab for compositions in F minor.7 In addition, the table gives a valuable clue to the tuning used by Domenico Scarlatti. There is almost nothing in the Venice and Parma codices to suggest that Scarlatti retuned for F minor, in which he wrote extensively (in the Parma codex, it occurs more often than any other minor key), but there is some rather pretty evidence    that he retuned for Ab major and several of the keys in group 2.8 One is therefore strongly tempted to conclude that Scarlatti used the same tuning as Gasparini.

Notes

                        1.                 Francesco Gasparini, L'Armonico Pratico al Cimbalo (1708; reissue, New York: Broude Bros., 1967), 83-6.

                        2.                 "Regular" means only that the eleven good fifths are the same size.

                        3.                 Patrizio Barbieri, Acustica, Accordatura e Temperamento nell'Illuminismo Veneto (Rome: Torre d'Orfeo, 1987), 152-8.

                        4.                 Jean-Le Rond d'Alembert, Elemens de Musique Theorique et Pratique (1752; reissue, New York: Broude Bros., 1966), 48-9.

                        5.                 Andreas Werckmeister, Musicalische Temperatur (1691; reissue, Utrecht: Diapason Press, 1983), 78-9.

                        6.                 Luigi Ferdinando Tagliavini, "L'Armonico Pratico al Cimbalo. Lettura Critica," in  Francesco Gasparini (1661-1727)--Atti del primo convegno internazionale (Comune di Camaiore) (Florence: Olschki, 1981), 133-55, at 149-51.

                        7.                 C. Sloane, "A Further Note on Tempered Minor Chords," Journal of Sound and Vibration 170, 2 (1994): 261-2.

                        8.                  Carl Sloane, "The Case for Meantone in Scarlatti," Continuo, 16, 6 (1992): 1516.

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