Of the Temperament of Those Musical Instruments, in Which the Tones, Keys, or Frets, are Fixed, as in the Harpsichord, Organ, Guitar, &c.

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This page contains excerpts of the article by Tiberius Cavallo (1749-1809) in the Philosophical Transactions of the Royal Society of London, volume 78 (1788) p.238-54. Cavallo was an Italian 'natural philosopher', meaning scientist, who made his home in London after 1771. His chief interest here was in the use of equal temperament, evidently a novelty, and the article is mainly useful for whatever light it sheds on the unequal tuning in use in 18th century England. Later, Cavallo was consulted by Broadwood about the design of pianos, though it is not clear how useful his 'scientific' advice was in the end.

In the article, Cavallo discusses temperament in general, and naturally begins with the pure intervals of the scale and the deduction that a keyboard instrument cannot contain every pure interval.

Other instruments, in which the notes are not fixed, as the violin, violoncello, &c. are perfect, because the performer stops the string upon them in different places, even for sounding the notes of the same name. Thus a skilful performer, in order to sound A, will stop the string a little farther from the bridge when he plays in the key of C, viz. when C is considered as the key-note, than (...) in the key of D.

The choice between different pitches for A follows from its dual purpose as major sixth above C or perfect fifth above D.

At present, the harpsichords and organs are commonly tuned so, that some concords are very agreeable to the ear, whilst others are quite intolerable; or, in other words, when the performer plays in certain keys, the harmony is very pleasing, in others (...) just tolerable, and in some other keys (...) quite disagreeable.

These remarks already rule out any truly circulating or near-equal temperament.

The best keys to be played in are (...) C, F, Eb, Bb, G, D in the major mood, and (...) C, D, A, B (...) minor. Next to those come the less agreeable keys of A, Ab, E (...) major; besides those, the rest are disagreeable in greater or less degree, so that out of 12 keys, which, on account of the two moods, become 24, there are hardly fourteen that can be used; and for that reason most of the modern compositions in music are written in those keys.

This is the first clue to what the 'common' tuning could have been. However, it is evident that Cavallo has missed out or misstated some of the minor keys; it is impossible that G minor should be omitted from among the 'best' minors. Also, given the 'disagreeable' nature of F# major, it may be that B minor is a misprint. However, the fact that there are fewer 'good' minor than major keys is understandable, since minor tonality requires a greater harmonic range.

If we count the keys Cavallo mentions, the total number of more or less 'agreeable' keys reaches 13. Perhaps one more minor key ought to be added; at any rate, Cavallo's enumeration is not completely reliable.

The designation of both Ab and E major as 'less agreeable' points to a compromise pitch for G#/Ab, within a framework of modified meantone. If C-E were pure, then both E-G# and Ab-C could be made practically Pythagorean; any slight widening of C-E could go to improve one or both of the other thirds. The fact that B major and Db major are 'disagreeable' strongly suggests that the fifths C#-G#-D# were wide, probably making B-D# and C#-E# wider than Pythagorean. Other fifths, for example Eb-Bb, could be slightly wide without making Eb major significantly less agreeable.

It is perhaps surprising to see A major called 'less agreeable'. At any rate it probably indicates that C# was sharpened relative to the regular meantone framework of the central keys.

As long as the perfomer is to play in certain keys only, it is much better to have them tuned in an advantageous manner, than to let those be tuned in a less perfect manner, for the sake of others, which he does not intend to use. Hence the great harpsichord players generally have their instruments tuned in a peculiar [i.e. idiosyncratic] manner, viz. so as to give the most advantageous effect to those concords which they more frequently use (...). And hence (...) the harpsichords and organs are always tuned different from each other, unless they be tuned by the same person with equal attention, and without any particular instructions.

Cavallo here endorses unequal tuning, if one does not want to use all possible keys. You might ask: if, in fact, instruments are always tuned differently, what is the use of Cavallo's list of better and worse keys? We suppose that the players and tuners had, despite their differences, some tastes and practices in common, which is likely since no famous player could always be sure of having an instrument tuned in his peculiar manner.

This practice [i.e. unequal tuning] cannot conveniently be laid aside, viz. when the instrument is to be tuned for solo playing; and for a certain style of music (...). But the case is different when the instrument is to serve for accompanying other instruments in every sort of music, or (...) good singers; for then the disagreement becomes very audible; and for this purpose the harpsichord or organ ought to be tuned according to [equal] temperament (...)

The 'style of music' mentioned here probably means the choice of keys and range of modulation.

When the compositions of old masters are performed (...) with the organ or harpsichord tuned in the common manner, the effect is frequently very disagreeable. This is particularly the case with (...) songs of Handel, Galluppi, [Leonardo] Leo [1694-1744], Pergolese, and others, who wrote in a great variety of keys, and very frequently for those, for which the common way of tuning is not at all calculated.

This remark strongly suggests that the Italian school of the early 18th century (to which Handel belonged) expected a more nearly equal keyboard tuning than was commonly used in late 18th century England. Cavallo clearly would recommend equal temperament for the accompaniments of such composers.

The next part of the article describes Cavallo's setting of equal temperament with the (questionable) help of a monochord.

After a great deal of trouble (...) I at last [rendered] the divisions exact within at least the 300th part of an inch (...) I had a large harpsichord, with a single unison (...) tuned very accurately by help of this monochord. [In] whatever key the performer played, the harmony was perfectly equal throughout, and the effect was the same as if one played in the key of E natural on a harpsichord tuned in the usual manner.

A likely interpretation is that the key on a 'normally' tuned harpsichord that most resembled the effect of equal temperament was E major. A major would be more 'agreeable', and B major less. But if one takes Cavallo as saying literally that E-G# was the same as in equal temperament, it is not clear how to proceed in interpreting his previous remarks on differences between keys. The correspondence between ET and E major could not be exact.

Cavallo summarises:

(...) to serve for solo playing, and for a particular sort of music, it is proper to tune in the usual manner, viz. so as to give the greatest effect to those concords which occur frequently (...); for accompanying other instruments or human voices, and especially when modulations and transpositions are to be practised, then it must be tuned according to the temperament of equal harmony (...).

We may or may not agree with Cavallo's judgement, but it is clear that later 18th century England was a place where very noticeably unequal tuning with strong differences between keys was the norm. This can scarcely have been anything other than modified meantone where the central keys had pure or nearly pure thirds.


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