1991 violin intonation - a historical survey

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Patrizio Barbieri Violin intonation: a historical survey

Although reconstructing the history of intonation for keyboard and fretted instruments (such as the lute, or viola da gamba) has proved a sufficiently vexing problem for musicologists, it is easy to see that when one deals with instruments whose intonation is variable-as in the case of unfretted bowed strings-the problems are multiplied. Most ancient theorists avoided the subject, or confined themselves to reporting uncritically on the traditional divisions of the monochord, without attempting to ascertain whether or not those divisions were respected in practice.

In this article I shall therefore depend mainly upon unpublished sources of practical or experimental origin, including some that concern singing. Indeed, these are the only documents that provide a secure point of reference in so foggy an area. They show that violinists of all schools, at least until the middle of the 18th century, played in just or in mean-tone intonation; moreover, the Italians, especially during Corelli's time, enjoyed playing in quarter-tones.

Just (syntonic) intonation First, I shall describe very schematically the nature ofjust intonation. The Pythagorean scale in use in the Middle Ages was generated from a chain of perfect 5ths; thus, the 12 notes of the octave can be obtained in this way:

Eb°_ Bb,_ F C G'D' A_ E Bo F# ,C#tOG#0 o o 0 0 00 0 0 0o 0

(the bottom zeros between the letters mean that the 5ths are pure, i.e. beatless; the zero exponents remind us that each successive note has been reached by a chain of perfect 5ths). This scheme produced major 3rds enlarged by a syntonic comma compared to the consonant ratio, and minor 3rds narrowed by the same amount, always with reference to the consonant ratio (ex.ia). In the Renais- sance, when such intervals began to be used harmonically, theoreticians tried narrowing one 5th in each four by exactly a comma, as a means of keeping all of the major 3rds and three-quarters of the minor ones consonant (ex.ib). For example, one might narrow the 5ths Bb-F, D-A and F#-C#, obtaining: Eb+1 Bb+F C G Do A E-1- 1 E-1 B1 F C-2G-2

O -1 0 0 0 -1 0 0 0 -1 0

(here the exponents indicate the cumulative alterations of pitch, in syntonic commas, with respect to the Pytha- gorean scale). In this sample solution only the minor 3rds D-F and F#-A remain Pythagorean, being formed by a chain of three perfect 5ths.

Because it poses many practical problems in performance, some scholars believe that just intonation is only a myth;1 however, such an opinion contradicts much historical evidence. Let us examine three of these problems.

Problem i Many authors observe that an interval of a 5th or of a 4th altered by a comma-that is, by a quantity equivalent to about 22 per cent of the semitone in equal temperament-would be too dissonant to be acceptable to the ear. Moreover, in syntonic intonation this dis- sonance cannot be avoided. Suppose that in a melodic succession of the type

CO-FO-A-1 DO-GO-CO

(in which the 3rd F-A is pure and the 5th, or 4th, D-A is altered by a comma), the violinist wished to keep D-A pure: he would thus produce the new succession

CO-F0- A-1 D-1-G-1-C-1

completely formed by beatless consonances, but which fails to return precisely to the initial C. Such cumulative corrections would continually alter the pitch of the composition.

On this point I would suggest that in a public recital a 5th mistuned even by a whole comma would not sound

0 0 0 0 N° 0 0

1° ° °. , I. I *,, Ex.l(a) Pure 5ths form Pythagorean major 3rds (81:64=one syntonic comma larger than the 'just' ratio 5:4) and Pythago- rean minor 3rds (27:24 = one syntonic comma narrower than the just ratio 6:5).

-1 0 0 0 ~~--.:, 0 0

(b) If one of the four 5ths is narrowed by a comma, all the 3rds of the example become just (i.e. in the ratio 5:4 and 6:5, respectively for the major and the minor).

EARLY MUSIC FEBRUARY 1991 69

as horrible as one might at first imagine. Precise measurements recently made by Charles Shackford have established that, in the normal day-to-day performance of a string trio, the 5ths are tempered from -1 to +1.5 commas (i.e. a range of half a semitone).2 More than a century ago Cornu and Mercadier measured mistunings up to two-thirds of a comma, even in the best soloists.3

In any case Giuseppe Tartini unequivocally states that he strictly employed the syntonic intonation, leaving the comma in the 5th D-A, 'where nature had placed it, without thinking of dividing it'.4 This would seem to contradict the notion that the open 5ths of his violin were absolutely pure, and thus would inevitably have produced some Pythagorean intervals.5 However, Benja- min Stillingfleet, referring to such a passage in Tartini's Trattato, affirmed in 1771 that good violinists avoided open strings because they were tuned in perfect 5ths, leading thus to a Pythagorean intonation;6 this is confirmed by other authors, such as Carlo Botta.7 The statements of these two writers agree with an example from Johann Philipp Kirnberger (illus.i), in which he specifies

ber ,ozHart fd)on au4 fufI)l. eo balb bic Siolin, ober jebee (ei3gen infirumcnt nad reinen £Zuinten gea ftimmt ifi, mug in folgenbcn Mo$ ten bad [ltte a fcfon in bcr lpplicao tur gcgriffen rocrben, mtil bat b1ote u l Iod ifi;

- . - - +

1 Johann Philipp Kirnberger, 'Stimmen, Stimmung', Allge- meine Theorie der shonen Kunste, ed. by J. G. Sulzer, iv (Leip- zig, 1786-7), pp.382-4: p.383

that the A in the third bar is not to be played on the open string because-with the violin tuned in perfect 5ths-it would be 'too high'.

In fact the technique of avoiding open strings was not widespread in the 18th century. Indeed, the English amateur Roger North commented (in a manuscript from C.1726):8 Of the first sort [of rules for studying the violin] the chief is the sounding all the notes under the touch, and none with the strings open; for those are an harder sound than when stopt, and not always in tune, which the stop (assisted by the ear) affects with utmost niceness; so that upon instruments so handled, all the semitones, whatever the keys are or however

they change, are in tune to the most scrupolous of the ear. And besides all this, the power of the finger in giving temper and commixture to the notes, hath a superlative effect of sweetness [... ] To perform this [finger-stopping] well is a soveraigne skill, but seems more abstruse than really it is; for among us the old way of using the open strings hath a prepossession, and it is not easy to leave it off. But in time, beginners will take into it, and then common practise will make it familiar.

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2 Galeazzi's syntonic fingering charts, in different keys (1791). The fingerings have been calculated in order to have, starting from each one of the different key-notes, the same succession of major and minor tones: thus, the 5th to be narrowed by one syntonic comma is shifted according to the key. Galeazzi takes into account only the keys having up to four sharps or four flats in the signature

70 EARLY MUSIC FEBRUARY 1991

Eilll ~ _ -I -- II :_.~~- - - - - -U- . - . ..

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I

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(a) Syntonic:

C C: Db D D: Eb E F I I i I I

major tone / tone mir tone /i mor r semitone -

(5/4)

(b) Mean-tone:

C C# Db D Dt Eb E IF 1- I i I I I Ii

\ mean tone < mean tone major semitone /

(c) Pythagorean:

C Db Ct DI Eb Dt El F i. ' I I I I'

major tone major tone / minor semitone/

(5/4+1 comma) 3 Relative size of tones and semitones in the syntonic, mean-tone and Pythagorean systems

Italian violinists very probably carried the technique with them to England, since their intonation was upheld as a model even early in the 18th century.9 Problem 2 In the just scale of C major we have the following succession of major and minor tones:

mo° Do E-1 i B-1 C° major minor major minor major

To transpose the same scale to G, the violinist would have to raise the A by one comma, in order to transform G-A into a major tone. We can easily imagine how com- plicated the task would be if our violinist had to transpose the same melody into all 12 tonalities: he would continually face the dilemma of choosing-and with the speed of lightning-between two notes differing by only a comma (that is about two millimetres, less than one- tenth of an inch, in the central part of the fingerboard). Nonetheless, Francesco Galeazzi-who spent many years in Rome as a violinist at the Teatro Valle-stated in 1791 that the best performers changed the position of the major and minor tones according to the key of the composition.'° He included a fingering chart containing such alterations, by commas (illus.2). Furthermore, he sug- gests that his chart would make

sgangheratamente ridere i poco avveduti Suonatori, quelli specialmente, che sono materiali, e suonano per mera pratica [. . .]. the less precise players laugh heartily, especially the ordinary ones who play merely for the practice [. . .]

Thus, only first-class virtuosi were able to play in this way: ordinary violinists, whatever the tonality in which they played, always put their fingers in the same places. Alexander Malcolm (1721)" and Charles Delezenne (1826-7)12 offer clear historic confirmation of this habit, which caused changes in the character of the piece as it was transposed into different keys. Hubert Le Blanc (1740) states that a few violinists of his day, in order to shift the position of major and minor tones, even had the habit of slightly adjusting the tuning of the open strings according to the key in which they were going to play.3 Problem 3 There is not a unique and 'natural' just intonation system; rather, there are many, according to the 5ths that one chooses to narrow by a comma. Towards the end of the 18th century, the French experimental physicist Jacques-Alexandre Charles demon- strated-even at the Paris Conservatory of Music-that


Giambattista Viotti and Pierre Baillot employed a kind of just intonation in which G-A was made equal to a major tone (the narrowed 5th in the key of C major was thus A-E).14 He called it the Gamme europeenne, in opposition to the ancient Gamme grecque (in which, as we have already seen, the narrowed 5th was D-A): Gamme grecque: CO DO E-1 F° GO A-l B-1 CO Gamme europeenne: CO DO E-1 F° GO A° B-1 CO According to Charles, this habit came about because- complying with the modern tonal feeling-violinists played both the scale of C and its dominant, G major, as strict parallels (and this notwithstanding the fact that the Gamme europeenne retained more Pythagorean 3rds).

In 1751 d'Alembert and de Mairan-in agreement with Rameau's theory-had already observed that a the- oretician of their time, Mr Estve, sans doute pour avoir plus de consonances, fait l'intervalle du sol au la, un ton mineur, au lieu queplusieurs musiciens font cet intervalle un ton majeur, divisant ainsi la gamme en deux tetracordes parfaitement egaux e semblables, ut re mifa, sol la si ut.15 Mr Esteve, undoubtedly in order to obtain a greater number of consonances, makes the interval from G to A a minor tone, while many musicians make this interval a major tone, thus dividing the scale in two perfectly equal and similar tetra- chords: C-D-E-F and G-A-B-C. The evidence of Malcolm'6 and Hesselgren17 also sup- ports the Gamme europeenne.

Mean-tone (mesotonic) intonation Mean-tone can be considered a tempered just intonation-because instead of narrowing by a whole comma one 5th in every group of four-each 5th is narrowed equally by a quarter of a comma:

Eb-4 Bb-4 F -4 C - G - D -, A -/E -4 B -4 F- 4 Ct -4 G

The three 'problems' examined in the preceding sec- tion thus evaporate, since: (i) excessively narrowed 5ths do not occur; (2) large and small major 2nds are replaced by a single tone of intermediate size, i.e. the 'mean tone' (illus.3b); (3) there is no longer a distinction between the various syntonic tunings, such as 'Greek' or 'European'.

Since the quarter-comma temperament became the standard for tuning harpischords and especially organs for a good part of the 18th century, such a solution for violin tuning eliminates all differences of pitch and intonation between keyboard and unfretted bowed strings.

4 [facing] (a) Vibrating string-lengths in mean-tone intonation (total string length from nut to bridge is 13 inches): A. Warren (1725). This author has carried out his calculations according to the equal-tempered system with 31 notes to the octave; string lengths are almost exactly the same as the quarter-comma temperament. (On the left there is added, in modern notation, the first notes of the G-string.)

(b) Syntonic-mean-tone fingering chart: [Francesco Geminiani], The Art of Playing on the Violin (doubtful work, wanting the title page), plate between pp.4-5 (London, British Library, d.47.g-3). The author clearly warns: 'Note also that as Gt and Ab, or A: and Bb, or also D: and Eb, etc, are not the same notes, you must not stop them with the same finger.'

(c) Syntonic-mean-tone fingering chart: Georg Simon Lohlein, Anweisung zum Violin-spielen (Leipzig, 1774), p.35

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5 Reconstruction, with the quarter-comma temperament, of Geminiani's fingering chart: [Du Perron] (1769), p.688 (compare illus.4b)

Although this tuning would be very logical and in full agreement with the keyboard practice of the day, only a few authors suggest it for the violin: 1 Adriano Banchieri (1609), whose position will be examined below; 2 Ambrose Warren (1725), an English organ tuner, who commented:18 'It must be own'd, that all just performers by the voice or on the violin, do always, in all occasions, sing, or stop according to this doctrine' (illus.4a); 3 Jean Dumas (1756), a French mathematician;19 4 an anonymous author (who can be identified with one 'Monsieur du Perron'), whose article in the Journal des seavans (1769) reconstructs Geminiani's well known fingering chart according to the quarter-comma temperament (illus.5).20 (Note that both David Boyden and J. Murray Barbour in 1951-2 concluded that the fingering


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chart in the various English editions of Geminiani's violin method lies closer to mean-tone than to syntonic tuning.)21

One of the consequences of mean-tone intonation was that open-string 5ths had to be tempered. Although this practice was not widespread, a few authors mentioned it: 1 Jean-Philippe Rameau (1737), referring to the Italian- French violin player Jean-Pierre Guignon(Giovanni Pie- tro Ghignone, probably a pupil of Giovanni Battista Somis);22 2 Jean Dumas (1756), who is very positive about its employment both on violin and violoncello;23 3 Jean-Laurent de Bethizy (1764) and the Abbe Rou- ssier (1770);24 4 Antonio Eximeno (1775), a Spanish theorist who had lived in Rome from 1768;25 5 the anonymous author of the New Instructionsfor the Violoncello (c.1795), who warns players that they should tune while taking care 'to make the 5ths rather flat than sharp';26 6 surprisingly, the Italian Luigi Picchianti, who writes in the mid-lgth century that in order to avoid making the open-string major 6th G-E a Pythagorean interval, e indispensabile, che le tre quinte siino un poco calanti, e tali appunto le formano praticamente i suonatori di Violino, ed ipiui senza saperne il perche27 it is necessary that the three open-string 5ths be a little flat, and this is the way violin players tune them in practice, the majority of them without knowing why.

When violins tuned in perfect 5ths had to play with keyboard instruments, many problems of pitch arose: J.-C . Petit (1740) even suggested a harpsichord tuning 'en faveur du violon', i.e. with some diatonic 5ths left pure.28 As late as 1830, in an anonymous Venetian manuscript, the key of E major was classified as

difettoso nei complessi armonici [. .] poiche ammette molte quinte di seguito partendo dal C e percio esiste una differenza rimarcabile fra gli strumenti da tasto mobile e quelli da tasto fisso.29

problematic in ensemble music [.. .] because from C [to E] there is a chain formed by many 5ths, and therefore a notice- able difference [of pitch] between bowed and keyboard instruments.

Indeed, we know that in keyboard instruments the four 5ths C-G-D-A-E were still markedly tempered at that time.

In conclusion, we cannot state that Baroque violinists

played strictly with 'just' or mean-tone intonation, but we can at least be sure that they used a tuning of the syntonic-mean-tone type. In fact we have evidence that: (1) their major 3rds were just, and (2) the sharps were lower than the enharmonically equivalent flats (e.g. D# was lower than Eb).

As far as the first point is concerned, towards the end of the 16th century, Vincenzo Galilei had already noticed that major 3rds were sung at least approximately in just intonation: his statement is fully reliable, because as a reference instrument he had adopted the lute, whose equal-tempered major 3rds were much larger than syntonic ones.30 Even in the mid-lgth century Filippo Fod- era-a 'scientific' violin amateur of Palermo-showed experimentally that on bowed instruments the 3rds were played in just intonation.31

Turning to the second point, we must observe that the well established practice of syntonic-mean-tone tuning-favoured, especially in Italy, by the wide diffu- sion of harpischords and organs with split keys-had in fact made violinists accustomed to differentiating instinctively the precise tuning of enharmonically equivalent notes. In addition to the diagrams in illus.4-5, this is confirmed by a variety of authors: Pier- francesco Tosi (1723);32 Giovanfrancesco Becattelli, a Florentine maestro di cappella (1726);33 Robert Crome (1740-50) (illus.6); l'Abbe le fils (1772) (illus.7); Michel Woldemar, who claims to have been a pupil of Antonio Lolli (1798) (illus.8); Anicot l'ain6 (c.1800o);34 Carlo Ger- vasoni (1812).35 The Venetian Melchiorre Balbi, maestro di cappella at the Basilica del Santo in Padua, in 1829 stated that even the most ordinary violin player, when unaccompanied, customarily preserved a sensibilissima syntonic distinction between two enharmonically equivalent notes.36

DIAL GUE II Fourth String. Third String.

J Jo.j j2 23 3 4 03 3

6 Enharmonic distinction between G#/Ab and D#/Eb: Rob- ert Crome, The Fiddle New Model'd or a Useful Introduction for the Violin (London, 1740-50), p.13


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7 Enharmonic distinction between G7/Ab, D3/Eb and A/Bb: Joseph-Barnabe Saint-Sevin, dit l'Abb le fils, Principes de violon (Paris, [1772]), P.73

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le violon (Paris, [1798]), p.{

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8 Enharmonic distinction between C#/Db, G#/Ab and D#/Eb: Michel Woldemar, Grande methode ou gtude elementairepour le violon (Paris, [1798]), P.7


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Ex.2 Ferdinand III (of Austria), Chi volge ne la mente, opening, showing an enharmonic distinction between D:/Eb (con- tralto part)

|I |J V - I I - "I Chi vol - ge, chi vol - ge ne la men - te, Chi vol - ge, chi vol-ge ne la men - te

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Ex.3 Domenico Mazzocchi's distinction between G/Ab, in his Planctus matris Euryali diatonico-chromatico-enharmonice, (1638): from Athanasius Kircher, Musurgia universalis, i (Rome, 1650), p.661

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Performance of the Enharmonic Diesis

We have seen that, for example, D-D# was a minor semitone and D-Eb a major one. We will now examine the practical circumstances in which a singer or a violinist actually had to play the step Dg-Eb, i.e. the enharmonic quarter-tone (actually closer to one-fifth of a tone).

Leaving aside the experimental microtonal compositions of Vicentino and Colonna-Majone,37 we find a simple example at the beginning of a madrigal of Ferdin- and III of Austria (ex.2). This monarch was a pupil of his Kapellmeister Giovanni Valentini, owner of a harpsichord with split keys.3 Domenico Mazzocchi too-in his famous Planctus matris Euryali diatonico-chromatico-enharmonice (1638)-employed the G#-Ab enharmonic diesis (ex.3). For instrumental consort, we have a complete composition published by Abdias Treu in 1635 (ex.4); but perhaps the best musical example is the motet Derelinquat impius viam suam by Galeazzo Sabbatini (ex.5).39

According to the French physicist Charles Hebert (1733), Antonio Montanari-one of the best Italian violinists of his time-employed enharmonic intervals when playing the Sinfonie of his teacher, Arcangelo Corelli.40 In his manuscript Hebert also writes:

I'jtalien n'est pas si eloigne de l'usage des cordes enharmoniques que le francois; je fonde cette pensee sur la joye qui se repand tout a coup sur le visage de toute une assemblee jtalienne d l'occasion de quelque simphonie ou concert, des que l'on touche quelques unes de ces cordes, c'est un viva qui passe de l'un a l'autre dans un jstant, fasse en presence du Sanctaire, tandis qu'un francois nou- vellement arrive ne pourroit contenir ses grimaces [. .]41 Italians are not so far removed from the use of enharmonic notes as the French; I have formed this opinion looking at the joy that suddenly appears on the face of a whole Italian audi- ence, in a sinfonia or concerto, when one of these notes is played: in an instant all of them exclaim Viva!, even if they are in a church, while a newly arrived Frenchman could not con- ceal his distaste [...].


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Ex.5 Galeazzo Sabbatini, 'diatonico-cromatico-enarmonico' motet Derelinquat impius viam suam, bars 63-92 63

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Ex.6 Jean-Philippe Rameau, Les Indes galantes (1735): the A#/Bb quarter tone in the famous 'earthquake' of Act 2

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But, still according to Hebert, in 1733 the use of such enharmonic intervals was already fading, because the harpsichords with split keys were no longer in fashion. According to de Blainville, Pietro Locatelli also employed such distinct pitches in his first concerts.42

With respect to the French, Hebert's statement explains why Jean-Philippe Rameau was unable to make French audiences accept his enharmonique-chromatique and enharmonique-diatonique genres. In the famous

'earthquake' of Les Indes galantes (Act 2), the musicians would not play the required quarter-tone, and his piece was performed as une musique commune.43 This composition is in the enharmonic-chromatic genre, since the enharmonic diesis follows the chromatic semitone (ex.6).

In contrast to the French, some Italian violinists were still employing quarter-tones in the second half of the 18th century: see Nicola Mestrino (ex.7) and-still later,


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in Pythagorean intonation-Bartolomeo Campagnoli (ex.8).

In Italy microtones were also employed in melodic embellishment. Andrea Angelini Bontempi, who in the first half of the 17th century studied in the strict Roman schools, states that quarter-tones were easily produced by the best singers, just for this purpose (ex.9a). The middle staff of ex.9b shows how Angelini developed an enharmonic diminuzione: it was not written down on the score, he says, because such 'distendimenti e allenta- menti di voce' (raising and lowering of the pitch) belonged to the art of the singer, and not to the science of the composer. Analogous to Bontempi's distendimenti, crescimenti della voce had already been prescribed by the Romans Ottavio Durante (Arie devote, 1608) and Domenico Mazzocchi (Madrigali, 1638). For the latter, half of the enharmonic diesis (indicated by the sign V) represented the 'sollevazione, or (as some say) messa di voce)44

Such effects were also tried on strings. Francisco Valls, maestro di cappella at a series of major Spanish churches between 1688 and 1740, presented in manuscript a piece

Ex.9 Giovanni Andrea Angelini Bontempi, Historia musica (Perugia, 1695), p.158

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for three violins and violoncello, the first few bars of which are shown in ex.loa. He says that it is based on the scale shown in ex.lob, with regard to which he confines himself to explaining that the sign x indicates an elevat- ion in pitch of'two and one-half commas', ie 'half the diesis #. Moreover, the discussion gives him the chance to conclude with a strong measure, affirming that such a piece sera dificultosissima su pratica: aun que hay algunas vozes, y instrumentistas, que son tan desentonados, que naturalmente tocan, 6 cantan enharmonicamente, que no hay oldos que lo pue- den sufrir. will be rather difficult to play, although there may be some singers and instrumentalists so far out of tune, who by nature sing or play enharmonically, that one cannot find an ear that can endure it. According to Michel Woldemar, this enharmonic ornamentation-always with reference to bowed instruments-was peculiar to the Italian school. Woldemar also transcribed an Andante amoroso exactly as Mestrino had performed it in the Paris Concerts Spirituels, c.1786-9 (illus.9). In his performance, beyond any doubt Mestrino had to make clear the individual quarter- tones: the normal glissandi per via di striscio, in which one passes from one note to another without a break, were in fact common enough at that time. In 1736 Gior- dano Riccati confirmed the evidence that such glissandi were performed 'daily by violinists and by singers'.45

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Pythagorean and 'expressive' intonation Toward the middle of the 18th century the modern harmonic-tonal system had also begun to impose its con- straints on tuning: the pull of the tonic on the leading note and of the sixth degree on the minor 7th, for example, had become so strong that these two semitones, which in the syntonic-mean-tone system were major, were now minor. Thus the sharps became higher than the enharmonically equivalent flats (illus.2c). For these reasons, this 'Pythagorean-expressive' intonation might also be termed 'functional'.46

Although a preference for such an inversion of the relation of sharps and flats had already appeared in a manuscript of Christiaan Huygens in the late 17th century,47 only in 1760 was the first codification of such a change recorded. In that year, in fact, the fingering prescribed by Delusse for the flute placed the sharps higher than the flats.48 In the case of singers and violinists, the inversion of the relation between the two semitones was announced ten years later by the Abbe Roussier (1770).49 According to this French theorist, it was brought about by the violinist [Pierre] Vachon and by the cellist [Jean- Pierre] Duport. Thus the syntonic major 3rd (5:4) was enlarged, becoming Pythagorean (81:64).

The statements of Roussier might anyway be viewed with suspicion, coming from a supporter-almost a fanatical one-of Pythagorean intonation 'of the ancient Greeks'. Fortunately Anton Bemetzrieder, a music theorist from Diderot's circle, provides us with independent testimony (1776).50 This author, while confirming that the sharps were played higher than the enharmonically equivalent flats, nevertheless observes: Le Virtuose hausse le 'diese' tantot beaucoup, & tantot peu; il baisse plus ou moins les notes 'bemols'; il joue la meme note 'bemol' differemment, suivant qu'elle est sixte mineure ou tonique; le meme 'diese' differemment, suivant qu'il est sensible, tierce majeure ou tonique: il exerce le meme empire sur les notes naturelles; le 'si' de la seconde corde du violon, qui s'accorde avec le 'mi' de la chanterelle, ne contenteplus l'oreille sensible & exer- cee dans l'accord de sixte, qu'ilfait avec le 're' de la troisieme corde vuide. The virtuoso raises the sharp sometimes more, sometimes less; he plays the same flat note differently according to whether it is the minor 6th or the tonic; the same sharp differently according to whether it is the leading note, major 3rd or tonic. It is the same with the natural notes: the B of the second string on the violin, which is tuned to the E of the first string does not please the sensitive and skilled ear in the chord of the 6th, which it makes with the open D-string. Thus, the intonation described by Bemetzrieder, also

unequivocally of the Pythagorean type, has a strong expressive-functional component. Indeed, alluding ironically to Roussier, who continually returns to the virtues of the ancient Greeks, Bemetzrieder specifies that in the a solo pieces, the virtuoso effects all of these nuances solely by instinct, 'without the aid of Greece'. The controversy between the two theorists very soon brought on one of those querelles, even if in a low key, so dear to the French musical scene in the 18th century. Alexandre-Theophile Vandermonde began by claiming that the first to be aware of the new type of intonation was Bemetzrieder,51 while Louis Dupuy (1781) asserted the priority of Roussier.52 The polemic subsided only in 1781, when Bemetzrieder left Paris permanently for London.

In Italy the first evidence of the new practice is that of Abate Eximeno (1775).53 A little later Galeazzi (1791) noticed with surprise that among the violinists-and contrary to what arose from the syntonic tuning he favoured-the semitone between leading note and tonic was small rather than large.54 In the same year Bar- tolomeo Campagnoli brought out his method for violin, the first to prescribe explicitly a tuning of the Pythago- rean type. However, he preserved the enharmonic distinction between sharps and flats, which are made identical by the temperament only in a few exceptional cases (illus.lo).55

In the 18th century the natural tendency to lower the tuning of the flats was probably one of the factors that favoured the attempts to introduce in practice the seventh harmonic, until then considered the diabolus in musica. To take one example, the minor 7th generated by the seventh harmonic (whose ratio is 7:4) is in fact from the melodic point of view 'compassionable' almost to the point of caricature. Its character is derived from the fact that it lies only a little more than half a semitone from the note to which it must resolve, in contrast to what happens in the case of its syntonic and Pythagorean relatives (illus.nl). This is the new harmonic relation that Tartini proposed to introduce into normal violin practice, indicating it by a special sign to distinguish it from the normal flat (ex.ni). Violinists used the seventh harmonic also to effect the Neapolitan 6th, according to the testimony of the Italian opera composer Francesco Bianchi. With reference to ex.12, in his manuscript he says:56 Dans les instrumens a chevalet mobile le sib du second accord naturellement, est un demiton enharmonique, et on recule le doit pour simple instinct en jouant. En effet l'harmonie de la comme produite par celle de ut et fa peut admettre le sib [ ...] la


1o Bartolomeo Campagnoli, Nuovo metodo della mecanicaprogressivapersuonare il violino (Milano, [1797?], Ricordi). See, at the top of the fingerboard, the position of the temperamento, halfway between the sharp and the enharmonically equivalent flat

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nature veut que ce soit sibb 1/7 ce qui se preuve par l'effet admirable que cette sixthe mineure diminuee produit a l'oreille. On unfretted string instruments, the Bb on the second chord is naturally an enharmonic semitone, and in playing one instinctively draws the finger back. In effect the harmony of A [considered] as a product of [the harmonies of] C and F incorporate the Bb [...] nature wants it to be Bbb 1/7 [=the seventh harmonic], which is obvious from the admirable effect this small minor 6th has on the ear.

Although during the 19th century the favourable response to Pythagorean intonation became almost ritu- alistic, its practical application must have been much influenced by considerations of the expressive aspects of intonation. Bernhard Romberg, in his well-known Methode de violoncelle, a method approved by Cherubini and Spontini, claimed for instance that he played the leading note higher in the minor mode than in the major mode.57

As for the Italian 'taste' in intonation, a valuable source on the experimental side is that of the Roman Fil- ippo Natali (1886), who employed a frequency meter of his own invention. The melodic intonation he recorded, although clearly Pythagorean in orientation, remains difficult to classify given that all the pitches (including the diatonic ones) are adjusted according to their func- tion from moment to moment.58 The meticulous work accomplished a few years earlier by Cornu and Mer- cadier (1869) had yielded results which were much more definite; the two French physicists concluded that the Pythagorean and the syntonic-mean-tone scales, in theory mutually exclusive, were compatible: the former was preferred in melodic contexts, the latter in harmonic ones.59

Moreover, purely emotional and habitual factors must have differentiated the intonation in enharmonically equivalent keys, as for example in F# and Gb major (which in the Pythagorean system, in relation to their respective tonics, are still controlled by the same succession of ratios). In one of their objective tests, Alexandre and Prevost (1862) describe the following curious experiments: Nous avons unefois demande a un musicien dont la delicatesse de l'oreille est bien connue, de jouer un air en ut majeur; puis nous l'avonsprie dejouer le meme air en prenantpour tonique la note qui se trouve sur le piano entre le fa et le sol naturels. L'air exe- cute, nous lui demandames dans quel ton il venait dejouer.-En fa#, repondit-il, sans hesiter.-Nous lui dimes qu'il avaitjoue en solb. Non, reprit-il, c'est trop eclatant, c'est du fa#, et non du solb. Tenez, continua-t-il, voici du solb majeur, et il joua le meme air, sans sortir du ton, mais en introduisant quelques modifications qui donnerent de suite a l'air une certaine douceur, quelque chose qui, di mon oreille, semblait se rapprocher d'un ton mineur.6


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Once we asked a musician whose ear was very sensitive to play a tune in C major; then we asked for the same tune in the key whose tonic lies between F and G natural on the piano. That completed, we asked him in what key he had just played. 'In F#' he said without pausing. We said that he had played in Gb. 'No' he replied 'it is too brilliant, it is F# and not Gb. Wait' he continued 'here is Gb major': and he played the same tune not changing the tonic, but introducing several modifications that gave the piece a certain sweetness, which to my ears seemed to bring it closer to the minor mode.

The orchestra and tempered intonation It is well known that in the late Renaissance and early Baroque band there was a marked disagreement between the main families of instruments: lutes and viols (tuned with equal, 'irregular' or even so-called 'equal-semitone' temperaments), harpsichords and organs (mean-tone tuning), and violins (normally with perfect open-string 5ths). Unlike the majority of theorists, Adriano Banchieri claims that he successfully solved this problem in a mass performed in Verona in 1607.61 His list of instruments includes violins, a consort of viols, two harpsichords, church organ, lutes, chitarroni and sackbuts. Open strings of the strumenti budellati (i.e. with gut strings) were tuned in unison with those of the harpsichord (illus.12): therefore the general tuning of lutes, viols and violins was undoubtedly mean-tone, even if this solution might have caused difficulty in some unisons of the fretted instruments.62

In 18th-century Italian opera, enharmonic modula- tions became more and more fashionable, thus favour- ing the introduction of equal semitones (harpsichords with split keys were becoming obsolete). In any case, it is surprising that, completely forgetting their love for a distinction betwen flats and sharps, the Italians became the strongest supporters of tempered intonation in ensemble music. The Abbe Roussier (1782) writes that at the Paris Opera the harpsichord had already been banished (because of temperament), but Nicola Piccinni, a supporter of tempered vocal intonation, had it restored to the orchestra.63 On this subject see also Mancini (1777), maestro di canto at the Imperial Court in Vienna, and Vandermonde (1780).64

In any case, even in tempered intonation, orchestral violinists succeeded in keeping most of the diatonic intervals pure. Charles Delezenne (1853) showed that even in his lifetime the temperament of the orchestral violin players in Lille was 'mean-semitone': that is, the diatonic pitches were left in syntonic intonation, but tones were split exactly in two when chromatic notes were needed. His monochord is the following:65

(For the convenience of the reader, in this scheme Dele- zenne's logarithmic units have been converted into cents: 1oo cents = 1 equally tempered semitone.)

We find further evidence of this kind of temperament in a public experiment organized by Thomas Salmon before the Royal Society of London (1705).66 The violinist was Gasparo Visconti (Gasparini), a pupil of Corelli, who in those days performed in London with great success: To prove the foregoing propositions, two viols were math- ematically set out, with a particular fret for each string, that every stop might be in perfect exactness: upon these, a Sonata was perform'd by those two most eminent Violists, Mr Freder- ick and Mr Christian Stefkins, servants of her Majesty; where by it appeared that the theory was certain, since all the stops were owned by them, to be perfect. And that they might be proved agreeable to what the best ear and the best hand per- forms in modern practice, the famous Italian Signior Gasper- ini, plaid another Sonata upon the Violin in consort with them, wherein the most compleat harmony was heard. (Salmon's 'mathematical' division was similar to Dele- zenne's 'mean-semitone'.)67 In illus.1o we see that, though his scheme was related to Pythagorean intonation, Bartolomeo Campagnoli had the same rather naive idea about temperament.

Conclusions 1 Until at least the mid-18th century violinists played in a kind of 'just' or mean-tone intonation. In fact we have evidence that (1) their major 3rds were pure and (2) sharps were played lower than the enharmonically equivalent flats (for example, the D# of a B-D# chord was lower than the Eb of a C-Eb chord). Although flats and sharps as a rule had a distinct intonation in everyday performance, the actual employment of the enharmonic step (e.g. D#-Eb of the above-mentioned example) can be found only in some experimental compositions. We know that in the majority of cases open strings were tuned in pure 5ths (only a few authors mention tempered 5ths): thus performers tried to avoid playing on the open strings not only to preserve the homogeneity of the sound, but also in order to avoid Pythagorean intervals.68 2 Towards the middle of the 18th century the sharps began to be tuned higher then the enharmonically equivalent flats: this habit was due to the new 'functional' and 'dynamic' role that semitones had in the modern harmonic-tonal system (e.g. the pull of the tonic on the leading note and of the sixth degree on the minor 7th made-respectively-the sharps to raise and


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the flats to lower). In this new 'Pythagorean-functional' intonation, the pitch of the notes was also conditioned by strong emotional factors and by the character tra- ditionally attributed to the different keys, this last one linked to the mode (major or minor) and alterations in the signature (sharps or flats). In any case, even in the middle of the 19th century the struggle between syntonic and Pythagorean had not completely faded, as a report of Delezenne clearly shows;69 in 1869 Cornu and Mer- cadier checked experimentally that syntonic was preferred in harmonic contexts and Pythagorean in melodic ones.

3 The increasing use of enharmonic transitions also demanded some kind of tempering of the semitones, so that there should be no harmonic distinction between sharps and flats; in large 19th-century orchestras, string players had also to find a compromise with fixed-intonation instruments. Both 'mean-semitone' and our modern equal temperament satisfied this necessity.70 Furthermore, according to Robin Stowell, 'the addition of vibrato is nowadays considered one of the most effec- tive ways of constantly adjusting string intonation to that of an accompanying medium, but there is no evidence of such a practice being employed' until the early 19th century.71

translated by Sandra Mangsen

Patrizio Barbieri is professor of electronics and specializes in research into the tuning of musical instruments. He has published a book and a number of articles on temperament and harmonic theory.

1 See, for example, J. M. Barbour, 'Just Intonation Confuted', ML, xix (1938), pp.48-6o.

2 C. Shackford, 'Some Aspects of Perception-I. Sizes of Harmonic Intervals in Performance', Journal of Music Theory, v (1961), pp.162-202: p.185

3 A. Cornu and E. Mercadier, 'Sur les intervalles musicaux', Comp- tes rendus hebdomadaires des seances de l'Academie des sciences, lxxiv (1872), pp.321-3: p.322

4 Giuseppe Tartini, Trattato di musica (Padua, 1754), pp.99-101 ('dove la natura lo ha posto, senza pensare a dividerlo').

5 See E. R. Jacobi, 'Giuseppe Tartini's "Regola per bene accordare il violino"', Music East and West: Essays in Honor of Walter Kaufmann, ed. T. Noblitt (New York, 1981); P. Barbieri, Acustica accordatura e temperamento nell'Illuminismo veneto (Rome, 1987), p.146*.

6 Benjamin Stillingfleet, Principles and Power of Harmony (London, 1771), p.38

7 Carlo Botta, 'Memoire sur la nature des tons et des sons', Memoires de l'Academie des sciences litterature et beaux arts de Turin, xii (1802), p.2o8

8 Roger North on music: Being a selection from his essays written during the years c.1695-1728, ed. J. Wilson (London, 1959), p.234

9 See below under 'The Baroque band and tempered intonation' Curious remarks on the employment of open strings are made by: Pie-

tro Signoretti, Methode contenant les principes de la musique et du violon (The Hague, 1777) i, pp.8-9. This author presents a diatonic ascending scale, from the bottom g to b", in which all the four open strings are employed; in the same scale, descending, on the contrary he avoids using the open E, A and D strings. These are his explanations: 'Il faut observer que les trois Notes a vide, c'est-a-dire le Re, le La et le Mi, n'ont lieu qu'en montant la Game; car en la descendant, ces trois Notes se font du quatrieme doigt: la raison est parce qu'en montant la Game, on va de corde en corde plus sonore par la proportion de grosseur et de tension. Par consequent ces trois Notes a vide, dont le son a plus d'eclat, se marient mieux avec les suivantes; au contraire en descendant la Game, on passe de corde en corde moins sonore, dont le son est de plus en plus grave. Il en resulteroit que ces trois Notes a vide seroient trop criardes, et feroient un tres-mauvais effet, ce a quoi on remedie en les faisant du quatrieme doigt, dont le son n'est pas si criard, and par consequent se rapproche plus de celui des autres Notes suivantes.'

lO Francesco Galeazzi, Elementi teorico-pratici della musica, con un saggio sopra l'arte di suonare il violino, i (Rome, 1791), pp.100-22 " Alexander Malcolm, A Treatise of Musick (Edinburgh, 1721), p.322

12 Charles E.-J. Delezenne, 'Memoire sur les valeurs numeriques de la gamme, Memoires de la Societe des sciences, de l'agriculture et des arts de Lille, v (1826-7), pp.1-57: pp.38-9

13 Hubert Le Blanc, Defense de la basse de viole (Amsterdam, 1740), PP-54-5, 136

14 Jacques-Alexandre Charles, Cours de physique (Paris, 1802), Paris, Bibliotheque de l'Institut, Ms 2104-Piece 17, pp.274-8. See also Jean- Baptiste Biot, Traite de physique experimentale et mathematique, ii (Paris, 1816), p.72.

15 Jean Le Rond d'Alembert and Dortous de Mairan, Systheme de musique et de temperament de Mr Esteve (Paris, 31 March 1751), Paris, Archives de l'Academie des sciences, Registres des Proces-verbaux, T. 70 and Pochette de seance

O6 Malcolm, A Treatise, p.321. 17 Frederic Hesselgren, Etude sur les intervalles harmoniques (Turin,

1903), p.14 18 Ambrose Warren, The Tonometer (Westminster, 1725), p.15 19 [Jean Dumas], 'Memoire sur l'harmonie temperee et son appli-

cation au clavecin', Memoires de mathematique et de physique rediges a l'Observatoire de Marseille, (1756), pp.84-106: pp.103-4

20 [Du Perron], 'Description dans l'intervalle d'une octave du systeme du partage de la dixseptieme majeure parfaite en quintes egales, et son emploi dans la tablature de quelques instrumens de musique' Journal des scavans (1769), pp.681-94. For the identification of this author see P. Barbieri, 'L'intonazione violinistica, da Corelli al Roman- ticismo' Studi musicali, xix (1990), n.21.

21 D. Boyden, 'Prelleur, Geminiani and just intonation', JAMS, iv (1951), pp.202-19; J. M. Barbour, 'Violin intonfation in the 18th century', JAMS, v (1952), pp.224-34: p.233

22 Jean-Philippe Rameau, Generation harmonique (Paris, 1737), pp.91, 102-3

23 See n.19 above. With reference to the quarter-comma temperament, he says 'que dans l'accord usite de ces Instruments [violin et violoncello], les Quintes sont affoiblies [...] J'invite les amateurs de la Theorie, a reiterer ces experiences, et i observer que dans les cordes minces, dont la grosseur peut etre negligee, l'affoiblissement des Quintes ne laisse pas de se manifester sensiblement.'

24 Jean-Laurent de Bethizy, Exposition de la theorie et de la pratique de la musique (Paris, 2/1764), pp.127-8; Pierre-Joseph Roussier, Memoire sur la musique des anciens (Paris, 1770), p.241

25 Antonio Eximeno, Dubbio (Rome, 1775), p.72 26 New Instructionsfor the violoncello (London, C.1795), p.9 27 Luigi Picchianti, Principj generali e ragionati della musica teorico-

pratica (Florence, 1834), p.101 28 J.-C. Petit, Apologie de l'excellence de la musique: Avec un nouveau

sisteme, et methode demonstrative pour accorder le clavecin et l'orgue (London, 1740), pp.26-32 (French-English text)


29 Trattato teorico pratico del sistema armonico, manuscript by anonymous author (c.183o), Venice, Biblioteca Nazionale Marciana, Ms.It.Cl. iv.1848=11661. The chapter 'De' tuoni' has been published in Barbieri, Acustica, pp.261*-2*.

30 Vincenzo Galilei, Dialogo ... della musica antica et della moderna (Florence, 1581), pp.30-31; Galilei, Discorso ... intorno all'opere di mes- ser Gioseffo Zarlino da Chioggia (Florence, 1589), pp.132-3

31 Filippo Fodera, Su I'arte di suonare il violino (ms. dated Palermo, 1834, Palermo, Biblioteca Comunale, ms. Qq. G. 102), pp.50-51

32 Pierfrancesco Tosi, Opinioni de' cantori antichi, e moderni (Bologna, 1723), p.12

33 Giovanfrancesco Becattelli, 'Lettera critico-musica [...] sopra due difficult& nella facolta musica da un moderno autore praticate (Prato, 15 April 1722)' Supplementi al Giornale de' letterati d'Italia, iii (1726), pp.1-55: pp.31-2

34 Anicot 1'ain&, Principes de violon (Paris, c.18oo), p.4 35 Carlo Gervasoni, Nuova teoria di musica (Parma, 1812), p.36 36 Melchiorre Balbi, 'Appendici' in Antonio Calegari, Trattato del

sistema armonico (Padua, 1829), pp.127-8 37 All transcribed in P. Barbieri, 'I temperamenti ciclici da Vicentino

(1555) a Buliowski (1699): teoria e pratica archicembalistica', L'organo, xxi (1983), pp.129-2o8; Barbieri, 'La "Sambuca Lincea" di Fabio Colon- na e il "Tricembalo" di Scipione Stella', La musica a Napoli durante il Seicento, ed. D. A. D'Alessandro and A. Ziino (Rome, 1987), pp.167-216.

38 See P. Barbieri, 'Cembali enarmonici e organi negli scritti di Atha- nasius Kircher. Con documenti inediti su Galeazzo Sabbatini', Enciclo- pedismo in Roma barocca, ed. M. Casciato, M. G. Ianniello and M. Vitale (Venice, 1986), pp.ill-28: p.117; Barbieri, 'Juan Caramuel Lob- kowitz (1606-1682): Uber die musikalischen Logarithmen und das Problem der musikalischen Temperatur', Musiktheorie, ii (1987), pp.145-68: p.157.

39 Transcribed from Kircher's Musurgia universalis: see Barbieri, 'Cembali enarmonici', pp.119-20. 40 Charles Hebert, Traite de l'harmonie des sons, (Bologna, 1733) (London, British Library, Ms. Add.6137), p.374

41 HWbert, Traite, p.66 42 Charles-Henry de Blainville, Histoire generale critique et philolo-

gique de la musique (Paris, 1767), p.171 43 J.-Ph. Rameau, Demonstration du principe de l'harmonie (Paris,

1750), p.95 44 See M. A. Mabbett, The Italian madrigal, 1620-1655 (PhD Diss., U.

of London, 1989), pp.130-31. 45 Giordano Riccati, letter to F. A. Vallotti, dated Castelfranco

Veneto, 30 April 1736 (Padua, Biblioteca Antoniana, Ms.A.vi.539, fasc.8, ff.6-6c.), published in Barbieri, Acustica, pp.137*-40*.

46 See J. H. Chesnut, 'Mozart's Teaching of Intonation', JAMS, xxx (1977), pp.254-71:p.256

47 C. Huygens, Oeuvres completes, xx (The Hague, 1940), p.74 48 See Pierre Sechet, 'Reflexions' in J.J. Quantz, Essai ... (Berlin,

1752; Paris r/1975), p.1o. 49 Roussier, Memoire sur la musique des anciens, p.214 50 Antoine Bemetzrieder, Traite de musique (Paris, 1776), pp.xxiii-

xxix. The same remarks had been made in 1770 by Jean-Jacques Rou- sseau: see N. Zaslaw, 'The Compleat Orchestral Musician', EM, vii (1979), pp.46-57.

51 See [Jean-Benjamin de Laborde], Memoires sur les proportions musicales, le genre enharmonique des grecs et celui des modernes (Paris, 1781). The recueil contains the 'Observations' by Alexandre-Theophile Vandermonde (pp. 39ff.), followed by the 'Remarques de M. l'Abbe Roussier sur les Observations de M. Vandermonde' (pp.42ff.).

52 Louis Dupuy, review of Laborde's Memoires (see n.51 above), Journal des scavans, (1781), pp.707-13

53 Eximeno, Dubbio, p.53 54 Galeazzi, Elementi, p.122 55 On the subject, see also Boyden, 'Prelleur, Geminiani... ' p.212,

and R. Stowell, Violin Technique and Performance Practice in the Eight- eenth and Early Nineteenth Centuries (Cambridge, 1985), pp.248, 253-4.

56 Francesco Bianchi, De I'attraction harmonique (late 18th century) (Florence, Biblioteca del Conservatorio di musica, Ms.B.2400), Chap- ter VII, art.7

57 Bernhard Heinrich Romberg, Methode de violoncelle (Paris, 1840), pp.20, 127.

58 Filippo Natali, II Diapason differenziale (Rome, 1886), pp.74-83. As far as embellishments are concerned, he noticed that the notes of a turn (gruppetto) underwent a sort of gravitational attraction towards their central note and thus their intonation was much closer than normal to this last note; the same can be said about the appoggiature (intoned more like a messa di voce than a real semitone). 59 Alfred Cornu and Ernest Mercadier, 'Sur les intervalles musicaux', Comptes rendus hebdomadaires des seances de l'Academie des sciences, lxviii (1869), pp.301-8

60 M. Alexandre and P. Prevost, De la theorie mathematique de la musique (Geneva, 1862), p.21

61 Adriano Banchieri, Conclusioni nel suono dell'organo (Bologna, 1609), pp.49-55

62 See P. Barbieri, 'Conflitti di intonazione tra cembalo, liuto e archi nel "concerto" italiano del Seicento' Studi Corelliani, iv (Florence, 1990), pp. 123-53.

63 P.,J. Roussier, Memoire sur la nouvelle harpe de M. Cousineau luth- ier de la Reine (Paris, 1782), pp.36-7

64 Giambattista Mancini, Riflessioni pratiche sul canto figurato, (Milan, 3/1777), p.81; Alexandre-Theophile Vandermonde, Second memoire sur un nouveau systeme d'harmonie (Paris, [1780]), p.7

65 Ch. E.-J. Delezenne, 'Sur la transposition', Memoires de la Societe des sciences, de l'agriculture et des arts de Lille, xxxi (1853), pp.24-90: p.27

66 Thomas Salmon, 'The Theory of Music, Philosophical Trans- actions of the Royal Society, xxiv (1705), pp.2072-2100oo: p.2073

67 For a discussion of Salmon's tuning and other references con- cerning the 'mean-semitone' temperament see M. Lindley, Lutes, Viols and Temperaments (Cambridge, 1984), p.68, and Barbieri, 'Conflitti di intonazione, §1.3.3.

68 Remarks on tuning in pure 5ths can be found in Stowell, Violin Technique, pp.250-53, and in Barbieri, 'L'intonazione violinistica, nn.36, 48.

69 Ch.-E.-J. Delezenne, 'Sur les principes fondamentaux de la musique', Memoires de la Societe des sciences, de l'argiculture et des arts de Lille, xxvi (1848), pp.39-128. See also the separately published version of the same memoir (Lille, 1848), p.31.

70 About equal temperament, suppported by Spohr, see Stowell, Violin Technique, p.253. Anyway, when playing in a string quartet, performers did not like tempered intonation; in this regard, in 1876 the Genoese music historian Cornelio Desimoni mentioned the contrast between the pianoforte's equal temperament and the 'armonia pura del quartetto o simili stromenti ad arco': see M. Tarrini, 'Cornelio Desimoni (1813-1899), Note d'archivio per la storia musicale, v (1987), supplement, p.59.

71 Stowell, Violin Technique p.254. For further evidence on the subject of violin intonation, see P. Barbieri, 'L'intonazione violinistica.

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