The Theories of Tartini

ASTUDY OF OF THE THEORIES TARTINI GIUSEPPE Alejandro Enrique Planchart Caracas, Venezuela Before beginning a systematic study of the theories of Giuseppe Tartini (1692-1770), i t is necessary to examine briefly his position and his goals a s a theorist and to give a summary of the contents of the works to be discussed. Of a l l the theorists of his time no one was more singlemindedly concerned with throwing light upon the most basic principles of harmony; in consequence, anything that in his opinion did not serve this purpose was not brought into consideration [T,cf. l e t t e r to Count Decio Agostino Trento, without page number, a t the beginning of the book]. Thus we find little o r nothing said about practical rules for chord progression o r about the structural o r affective u s e s of harmony. To him, the basic principles of harmony a r e a n exact science, derived from nature to be s u r e , but from a nature rigidly ruled by mathematical proportions in a Cartesian sense. In his work, mathematical relation and proportion take priority, and he derides those who think that the principles of harmony may depend upon feeling o r artistry. He is quite aware of the difficulties into which other theorists have been led in attempting to explain consonant minor harmony and dissonance, and he t r i e s throughout his work to relate the three apparently different foundations of major, minor, and dissonant harmony to an a priori principle which will account for all three. This universal principle, he believes, is to be found in the relationship of the circumference of a circle to i t s diameter projected a s the side of a circumscribed square (Figure 1 . ) Figure 1. This leads him to devote much space in his work to long and abs t r u s e mathematical and geometric proofs in which he seeks to demons t r a t e the validity of this a priori principle and the necessary inclusion of the principles of major, minor, and dissonant harmony in this univ e r s a l principle to thus produce a complete and closed system. To most of his contemporaries these proofs were incomprehen- THEORIES O F GIUSEPPE TARTINI sible, and those who understood them found e r r o r s in T a r t i n i ' s calculations and attacked his system violently. Even T a r t i n i 1 s follower, Benjamin Stillingfleet says: "One cannot without some impression of compassion, s e e him wandering in the perplexing labyrinths of a b s t r a c t ideas, almost without guide, o r a t best with one which i t is most likely would mislead him [§, 16]!' And l a t e r on: "What I have already said will be sufficient for my not entering into a detail on this long chapter Ch. 111 a s such a detail would be extremely tedious to some, unintelligible to others and would appear (at least) strange to the only men who a r e qualified to form any judgement on this matter, I mean the mathematicians [6, 17]!' This s e e m s to explain why there has never been a n attempt to account fully f o r this part of T a r t i n i 1 s work, to show where the e r r o r s a r e and in what way they affect his conclusions. Such a n attempt is made in this essay. Tartini's theoretical works a r e : the Trattato di Musica [TI that appeared in 1754, De' Principj delllArmonia [PI that appeared in 1767, and a Risposta di G. T.. alla crittica del di lui Trattato [ a ] that appeared in 1767. His other works a r e two small t r a c t s on ornamentation [A] and violin playing [L,]. He has been credited with the authorto ship of an anonymous ~ i s ~ o s t a l J . J . Rousseau that appeared in 1769. This essay is a comparative study of the Trattato, the Principj o and the Risposta of 1767; although it would have b m i r a b l e t pare these works with the anonymous Risposta, no copy of this work has been available to me. The Trattato is a long and abtruse work which contains the most comprehensive account of T a r t i n i ' s theoretical thought. After a n introduction, where Tartini exposes the m a t h e m a t i c a l p r e m i s e s Tor the demonstrations in the second and third chapters, the work begins with an account of several phenomena which point to the possibility of a physical basis for harmony. Of particular importance here is his a c count ofthe difference-tones whichare the most secure physical foundation for his system; in the Principj he claims to have discovered this phenomenon in 17 14 [P, 361. In the second and third chapters we find the bulk of his mathematical calculations and geometric proofs; he t r i e s to prove the harmonic nature of the circle and to derive f r o m the r e l a tion ofthe circumference to the diameter the major and minor systems, the diatonic and chromatic dissonances, and even a n enharmonic system. The l a s t two chapters a r e devoted to the derivation of the scale, the establishment of the basic b a s s progression, and a discussion of several intervals found in the music of his day. The work concludes by answering some possible arguments that could be raised against his system. The Principj is a much c l e a r e r work than the Trattato; the mathematical proofs of the e a r l i e r work a r e for the most part absent here. There is a general re-exposition of the principles of the Trattato, but limited to the diatonic system. The work is divided into four parts. The f i r s t deals with the physical foundations of harmony, and contains accounts s i m i l a r to those in Chapter I of the Trattato. The second [x, .. . 1. Risposta d i un anonimo Venice: Antonio Decastro, 1769. a 1 celebre signor J. ..J . . . Rousseau TABLE I TARTINI'S THIRD SOUNDS for the fifth: for the fourth: I - for the m a j o r third: for the minor third: f o r the m a j o r tone: f o r the minor tone: for the major semitone: for the minor semitone: 51" and his mistaken calculations lead him astray. The third t r e a t s what Tartini calls "the musical foundation of harmonyu and consists of the derivation of the scale and the establishment of the basic b a s s progression. There is an excellent account of Rameauys theories bu Joan Ferrls [8]. Tartini observes that when a monochord string is plucked. Rameau s t a t e s that musical sound is not one but three 15.101. the a u r a l impression is that of a single sound [T. and i t will Rameau be evident to anyone who compares the works of the two theorists that. 101. [~eneva:Henri Albert Gosse et Jean Gosse. . In this sense the sounds of the monochord string a r e "harmonic monads" and the t e r m aliquot p a r t s does not 121. differencetones). le traiix! de th6orie musicale de M W i n i .THEORIES O F GIUSEPPE TARTINI deals with the systematic foundation of harmony and is concerned with the derivation of intervals. . He firmly believes that he has found the answer to a l l the problems of harmonic theory and has discovered a new s e t of properties for the geometric figures with which his proofs deal. a unity that divides itself into multiplicity only to return again to unity a s i t s basic principle [T. In the O b s e m t i o n s sur les principes de ltha. 121. . which a r e in the harmonic proportion of 1:1/3:1/5 [T. Likewise. 17631. 4 By an aliquot part of any quantity. . 1871. an integral divisor. occasiondes par quelqu&s gcrits modernes sur ce sujet. Tartini's system is a closely woven one. F r o m this he deduces that the essence of harmony is unity. the basic theoretical conclusions of both men a r e quite s i m i l a r 3 P [z. but since the sounds produced by the pipes a r e in harmonic proportion. line.. The Risposta is a small polemic work in which Tartini answers the objectibns raised to his Trattato by J e a n Adam ~ e r r e ? It contains v e r y little that has not been dealt with m o r e extensively in the other two books. i t produces three distinct sounds: the fundamental. 3 .rmonie. The most important proposition in the Ris osta is a geometric explanation for the formula of the "third sound (i. In the Trattato. In the Risposta he mentions that he has developed a new science which however he will not yet publish since the few glimpses of i t that appeared in the Trattato were so ill-received 13. e. This is denied by Shirlaw [a. et particuli&rement par. He finds further support for this in the fact that instruments such a s the tromba marina produce no sound unl e s s the string is divided into a n aliquot part4 of the total length (unity) of the string [T. such a part as will measure the whole wlthout a remainder. he notices that when a n organ key (with a mixture stop) is pressed many pipes of different pitch sound. 2 . The attack on T a r t i n i ' s system by the French theorists led Fdtis to consider T a r t i n i ' s theories a s directly opposed to the theories of viii. the twelfth. . The fourth relates the material ofthe three previous chapters into a unified whole. 11-12]. in spite of T a r t i n i ' s polemic attitude towards Rameau and the attack on Tartini's system by Rameauts followers. This work was not available to me.131. and the seventeenth. Unfortunately i t s author is "quite ignorant of geometry and algebra [B. That is. 2931. or surface is meant . properly describe them [T. 2881. even though i t is one string. they ignore their findings in setting up harmonic s y s t e m s based solely on 1. will be heard [ T . generally lower than either of the two sounds being played. a s octave doublings. Both claim to have heardthe sounds 112 and 114 produced by a plucked string.ALEJANDRO ENRIQUE PLANCHART When Tartini s t a t e s that the three sounds of the monochord string a r e harmonic monads. 31. He opens the Principi with a discussion of the specific a s s e r t i o n s of Rameau [& 1941 and dlAlembert [_P. their s y s t e m s invalidate themselves by not making use of their empirical findings [? 31. In the conclusion of the Trattato. 171. artificial [T. 43ffl with respect to this question. p. 34): for the octave and the unison no third sound is given since Tartini claims that these intervals produce none. He admits that he has been unsuccessful in his attempts to determine the presence of the sounds 112 and 114 in the vibration of the plucked string because. the third sound is the r e a l physical fundamental b a s s of any given interval and of any given p a i r of melodic lines. Rameau's emphasis on multiplicity would not contradict-directly T a r t i n i ' s assumption of unity a s the basic harmonic principle. 1/ 4. 13-14]. a third sound. a t best. To him. I. 131. In his s e a r c h f o r a m o r e comprehensive physical foundation for a harmonic system he t u r n s to his discovery of the difference-tones. which he calls the "phenomenon of the third sound [E. they a r e swallowed up by the fundamental sound of the s t r i n g [_P. and even 1/6 in addition to the three sounds he has already reported a s being produced by a plucked string. 171. Tartini admits that this may be true but that it would not invalidate his theories since these additional sounds a r e a l s o in harmonic proportion [T_. Whether o r not multiplicity should be considered beyond the three sounds (1. 113. Any interval will produce a third sound. Any other b a s s would be absurd o r . sometimes with an octave displacement [T. 41. Tartini admits that other theorists claim to h e a r the sounds 1/ 2. Tartini points out that although they make this claim. he determines their existence through a s e r i e s of experiments in which two instruments play s e v e r a l intervals. 113. and 115 [P. This account is accompanied in the Trattato by a l i s t of the third sounds for the different intervals (Table I. Tartini observes that any combination of intervals out of the . the successive third sounds produced by their combination constitute the t r u e fundamental b a s s of the melodies. The third sounds of an interval and i t s inversion a r e the same. he does not place himself in direct opposition to Rameau since he considers multiplicity a function of unity and r e g a r d s the division of unity into multiplicity and the resolution of multiplicity into unity a s p a r t s of a complete cycle [T.14ffl. 511' As reported in the Trattato. If two violins play the following: it will be noticed that the third sound glides from that of one interval to that of the other. 115) which a r e audibly produced by the monochord string i s a matter of extensive discussion in both the Trattato and the Principj. He claims that if Rameau and dlAlembert have indeed heard these sounds. If the intervals a r e played with just intonation and loud enough. 1'701. never ceasing to sound even though an incommensurable number of intervening intervals have occurred [X. The complex 6:3:2. 181. In the Principj. the resultant (third sound) is 112. However. and the complex formed by E u l e r l s law is 6:3:2:1. the third sound. This formula enables Tartini to determine the third sounds f o r the octave and the unison. 5-61. The integral divisors of 6 a r e 3. by his formula. T a r t i n i a s s e r t s that the product of the values that r e p r e s e n t the simplest arithmetic expression of a n interval will give the c o r r e c t third sound. still mistakenly. e . 18-19]. f o r m a harmonic whole [E. he mistakenly identifies a s 112. 25ffl. is. which is not in harmonic proportion. the product. he attempts to justify both 1 and 112 a s third sounds. which is included in the senario. 170-11. When a n interval is expressed in a m o r e complex form. 51. the octave a s 2:4. In the case of the fifth. T a r t i n i overlooks this contradiction. The interval of the fifth. that is. an octave above the fundamental rT. As a consequence of the identification of the third sound of the harmonic s e r i e s a s constant in 112. the product of these t e r m s is 6. and 1. and a l l the harmonic divisions in between. in the conclusion of the Trattato he mentions that other theorists have identified the third sound a s 1. he gives a mathematical formula for the deduction of the third sound of any interval. This goes against the basic p r e m i s e of the system. The third sound of the octave 1/2:1/4. The formula for the third sound and the operation it r e p r e s e n t s a r e s i m i l a r to a formula proposed by Leonhard Euler. the product of the t e r m s of the ratio must be divided by the common factor in o r d e r to a r r i v e a t the c o r r e c t third sound [_P. the interval. 2. while 1 is the arithmetic one k. i. which cons i d e r s harmony a s a projection of unity. Tartini fails to notice that he h a s invalidated his a s s e r t i o n that the third sound of the senario is constant in 1. 112. together with a l l i t s integral divisors. This. 221. He explains that they a r e inaudible because they a r e in unison with one of the generating tones [_P. given 1/3:1/4. trying to defend the Trattato a t a l l costs. Tartini r e v i s e s his view and identifies the third sound. the resultant is 112 and s o on [T. thus 6 will r e p r e s e n t the third sound of the interval 3:2. can be expressed arithmetically a s 3:2. in addition. will f o r m a complex s i m i l a r to the harmonic complex of T a r t i n i l s s y s t e m of the third sound [L 39ffl. Tartini believes that he h a s found a novel property of harmonic ratios. o r the fundamental of the s e r i e s . It is important to note that T a r t i n i d o e s not consider the third sound a s a foreign element added to the generating interval. 1/ 2: 1/ 3. does not upset his a s s e r t i o n of the third sound a s the fundamental b a s s and physical root of any interval o r harmonic s e r i e s [T_. If 6 is expressed a s 1 of the harmonic s e r i e s . 3 will be 112 and 2 will be 113 [_P. T a r t i n i a r g u e s that E u l e r l s formula fails to produce a harmonic proportion. he says. which in the Trattato. Given I / 2: 113.6-71. a s constant in 1 o r in unison with the fundamental of the harmonic s e r i e s [_P. In the Risposta. E u l e r ' s formula is based on the multiplication of the t e r m s of the ratio a t which two s t r i n g s vibrate when producing a n interval. 61. . 3 x 2 = 6. The vibration of two s t r i n g s producing the interval of a fifth a r e in the ratio 3:2. This is done by means of an involved mathematical proof that attempts to demonstrate that 112 is the geometric expression of the third sound.THEORIES OF GIUSEPPE TARTINI senario up to the sounding of the whole senario produces a single third sound. Figure 2 . Example 4. . Example 2 . Example 3. Example 5. a s in the major triad. This makes the second interval of any harmonic s e r i e s the interval that determines the harmonic nature of the s e r i e s [_P. Tartini points out that. if the third sounds were more audible. in the case of the minor third. derived from the arithmetic division of the string. Returning to the phenomenon of the monochord string. As soon a s he attempts to explain the minor system. In consequence. Tartini concludes that the law of the third sound is the law of harmonic proportion and has i t s b a s i s in unity. third. e. the complex a r r i v e d a t by means of E u l e r ' s formula is not even consonant. g. 151. So f a r . 8-91. Any attempt to construct a s e r i e s within the senario using a l a r g e r difference will never reach the second interval which is necessary to determine the nature of the s e r i e s [P. 111. Minor harmony. had been considered by other theorists a s "borrowed" from arithmetic science and essentially foreign to harmony. 2).681. the ratios contained within the senario should be considered a s necessarily basic [_P. Moreover. the third sound of an interval which has 1 a s one of the t e r m s of i t s ratio cannot produce a proportion with that interval because it will be identical with one of the sounds of the generating interval. trouble a r i s e s . Tartini admits the "imperfection" of minor harmony a s compared to major major harmony has but he denies that minor harmony "priority of nature" over minor may s t e m from a different principle and considers it a s a n intrinsically inseparable consequence of major harmony [T. while the one a r r i v e d a t by T a r t i n i t s formula is. Given a string of determined length. the nature of the third sound consists in three properties: first.THEORIES O F GIUSEPPE TARTINI produced by T a r t i n i ' s system. the sounds that he has heard in the monochord string. the s y s t e m by difference 2 produces the essence of the senario in 1:1/3:1/5.671. second. 6:5. the third sound will s t a r t producing harmonic proportion with the generating sounds f r o m the second interval on in any ordered s e r i e s [_P. 121. The phenomenon of the monochord string is then 231. one by difference 1 (superparticular) a s 1:1/ 2: 1/ 3 and another by difference 2 (superpartiens) a s 1:1/3:1/5. Tartini has worked only with the harmonic division of the string which produces the major system. 91. The argument in both t r e a t i s e s is essentially the same. 681. any interval which has the unit a s one of i t s t e r m s is only potentially harmonic. The resultant is a dissonant combination which causes Tartini to comment that. but two (Ex. In the Trattato he gives the third sounds f o r the minor triad. he divides the string harmoni- - - . 151. F r o m this. while the system of the third sound by difference 1 produces the whole of the senario. in consequence. i t s simultaneous union with the two generating sounds and a l l the harmonic divisions in between gives r i s e to the harmonic s y s t e m of which the third sound is the fundamental bass [_P. i t is intrinsically physical harmonic unity a s a principle. i s [E. it produces harmonic proportion with the generating sounds. This amounts to a n admission of m o r e than one principle for any successful harmonic system. and Tartini finds this assumption absurd and fundamentally opposed to the idea of system itself [T. However. 60. which a r e not one. only music with m a j o r triads would be possible [T. To Tartini. only a partial one while that of the third sound is the general one [E. Within the senario t h e r e a r e only two possible s e r i e s . and to this end Tartini considers it in connection with the straight line represented in the square. I n t h e D & o n s t r a t i ~ a [ ] Rameau J. 2. he continues t o d e r i v e minor harmony from t h e p r o p o r t i o n 6:5:4 and maintains t h a t while major harmony is a d i r e c t product of nature. In T a r t i n i t s opinion. Rameau relates minor harmony to the co-vibration of multiples of the fundamental string [3. The straight line of the monochord string. A ~ . a fundamental C would generate F . 38). a s Shirlaw points out. He overlooks the factthat the system of advances consists of fractions such a s 516 which a r e not aliquot parts of the string and could not be produced by such instruments a s the tromba marina. produce another s e r i e s of string lengths which produce . 25ffl. p. when it vibrates to produce the harmonic proportion 1:1/3:1/5. is actually made of intervals belonging to different harmonic s e r i e s and a t best it proves only that minor harmony is some s o r t of inverted major h a r mony E. minor harmony i s " i n d i c a t e d by n a t u r e " [B. S t i l l . 3 (p. 2215 ~ a r t i n iconsiders Rameau's point when he superimposes the systems of fractions and of advances upon a constant fundamental sound (Ex. 69. in such a way that they would be inseparable and form a single principle [T_. in o r d e r to be valid. it i s necessary to unite the two categories. physical and mathematical. Tartini is aware that m e r e physical proofs a r e incomplete and s e t s out to s e a r c h for a mathematical foundation for his system. He calls this the system of advances and considers it the physical proof that minor harmony and the subdominant a r e a n inseparable consequence of major harmony [T. Each of the infinite number of radii that enter into i t s construction can 211.C . These divisions. . mistakenly. must be evident in the observed physical facts. 295-61. produces some s o r t of circular figure. 2361. 26-71. 38). Rameau had s t a t e d . and the a i r m a s s e s s e t in motion by the string take a spheric shape.ALEJANDRO ENRIQUE PLANCHART cally (Fig. when taken to B. admits t h a t t h i s observation was erroneous. However. 2. to be a harmonic unity. Each of the segments taken from A to the point of division produces one of the sounds of the s e r i e s shown in Ex. "where the point is to establish a system. He s e e m s not to notice that great-C is not the fundamental of that s e r i e s and that the successive third sounds a r e the fundamentals of each interval.the s e r i e s of sounds shown in EX. mathematical principles. The resultant third sounds (F and ~b in particular) a r e the s a m e a s the sounds of Rameau's sympathetic strings [_P. 5 ) . 201:' The unity which he believes to have found in the physical foundation of harmony is best expressed by the circle which is in itself one. His s e r i e s . t h e longer s t r i n g s v i b r a t e i n segments corresponding t o t h e unison of the e x c i t i n g s t r i n g . This is not enough. this premise has a definite physical basis. This would imply a s e r i e s of "undertones" i n i n v e r s e p r o p o r t i o n t o t h e s e r i e s of "overtones!' Accordingly. 4. These 5. He calls this s e r i e s the system of fractions and it is nothing other than the senario. The circle must be proven s e r v e a s unity [T. t h a t a v i b r a t i n g s t r i n g w i l l cause t o v i b r a t e o t h e r s t r i n g s which a r e i t s r i u l t i p l e s i n l e n g t h a t t h e i r r e s p e c t i v e fundamentals. T a r t i n i attempts to show that xy will be the harmonic mean and xz the arithmetic mean of the ratio Ax:xB into [z. through this and other such approximations. is a discrete geometric proportion since both 12 x 6 and 9 x 8 equal 72. The l i n e t o which T a r t i n i r e f e r s is t h e diagonal of a square which i s always i r r a t i o n a l since it involves a length equal t o w . 2 is the geometric n mean between 1 and 4. minor. F r o m this premise. the irrational geometric mean that l i e s between the harmonic and arithmetic means can be exp r e s s e d discretely i n t e r m s of rational integers. of which 9 is the arithmetic mean and 8 is the h a r monic mean. I consequence. the geometric mean. Here it is necessary to introduce some of T a r t i n i t s arithmetic p r e m i s e s . Taking the c i r c l e AB (Fig. By this Tartini means a proportion in which the product of the harmonic and the arithmetic means is equal to the product of the extremes. In the Trattato p r e m e s s o to the Trattato he s t a t e s that. p. and chromaticism. F o r example. 211. This is s not a valid premise. Tartini admits this when he s a y s that we cannot have a notion of this mean a s geometric because i t cannot be expressed by integers but only by a line [T. The c i r c u l a r line presupposes the existence of a straight line in the radius.. given the duple r a t i o 60:120. In this way. 1. p. which. but Tartini.THEORIES O F GIUSEPPE TARTINI phenomena can be best expressed by the c i r c l e which is the most p e r fect of a l l c u r v e s [_T. placed tangent to the circumference. Tartini operates with the second one. attempts to warp geometry to suit his system. F o r example. reduced to i t s simplest t e r m s is 10:7:5. In T a r t i n i ' s proportion 120:84:60. becomes the side of a square which c i r c u m s c r i b e s the c i r c l e ( s e e Fig. upon completion of the c i r c u m ference produces the diameter [?_. 11. 6. Another t e r m which we encounter in T a r t i n i ' s theory is the d 2 c r e t e geometric proportion. the proportion 12:9:8:6. 3. the two r a t i o s that f o r m the proportion 1:2:4. . and to prove that the limit of consonant harmony is to be found in the senario. which Tartini establishes a s 84 [T. this does not hold. besides. the a r i t h metic mean will be 90. 42) with the diameter d i v i d e d r a t i o n ally a t any point. The nature of the geometric mean is that the square of the mean is equal to the product of the e x t r e m e s o r that the differences between the e x t r e m e s and the mean a r e i n the s a m e ratio a s the two successive ratios which f o r m the proportion. The f o r m e r would b e m6-1 the l a t t e r i m (7). and dissonant harmony. The diameter. 32). and the counter-harmonic mean will be 100. the square of 7 is 49 while the product ofthe e x t r e m e s is 50. 1-21? Thus he makes a distinction between what he .t e r m s the complete irrational geometric mean and the incomplete r a i to n a l 21. This proportion. l e t u s say x. T a r t i n i t s f i r s t goal is to prove the harmonic nature of the circle. and the differences between 1 and 2 and 2 and 4 a r e i n the ratio 1:2 o r 2:4. Tartini attempts to demonstrate the h a r monic nature of the c i r c l e and to derive f r o m i t the basic principles of major. 211. T h e r e is a mean missing. the harmonic mean will be 80. It is obvious that the differences between the mean and the e x t r e m e s a r e not in the s a m e ratio since the mean does not produce the s a m e ratio with both ext r e m e s . the square of 2 is equal to the product of 1 x 4. .Figure 3. This s e e m s to indicate that the length of the perpendicular dropped from any point on the diameter to the circumference of a circle is the harmonic mean of the ratio of the line segments into which the diameter is divided by the point on the diameter [T. is ~ 35:37. a.THEORIES O F GIUSEPPE TARTINI which the diameter AB is divided.the theorem of pythagoras? Ax2 + A Z (74). Substitu -t i n g 3 f o r Ax and 7 f o r xB we have 3 x 7 = 2 1 which i s t h e square of 5 8. The sum of t h e squares of t h e s i d e s equal t o t h e square of t h e hypotenuse. that the proportion 15 :2 1:25: 35 is a numerical expression for the lengths of the lines in the proportion Ax:xy:xz:xB [T. Tartini's e r r o r in this proof l i e s in the expression ofthe discrete geometric proportion of the ratio 3:7. Ax:xB = 3:7 = = then: Ax2 - X B ~ 9 49 21 (see footnote 7) 25 (because it i s r2). we F2 z2 7. 22. 241.391. which. If. G2 G2 = = Expanding the ratio 3:7 (by multiplying by 5) we a r r i v e a t the dis3rete geometric proportion 15:21:25:35. However. l a t e r in the Trattato. The second proof of the harmonic nature of the c i r g e is through the same figure with the chord Ay and the hypotenuse Az added to it (Fig. Tartini says that Ay will be the harmonic mean and Az the counterharmonic mean of the ratio Ax :xz. then: Ax:xB = 5:9 - -- Ax2 X B ~ = = = 25 81 45 49 (because it is r2) -2 XY z2 5 = By = Ay2 (70) and + Ax2 = . p. of a r i g h t triangle i s . If we reduce the discrete geometric proportion to i t s original form. Tartini s e e m s to have been vaguely aware that something was out of o r d e r in his expression of the ratio 3:7 a s the proportion 15:2 1:25:35 since he attempts to demonstrate. of which 21 is the harmonic mean and 25 the arithmetic mean [T. If AX:^ i s expressed a s a discrete geometric proportion. then 15:21:25:35 becomes 3:21/5:5:7. He multiplies a l l the t e r m s by five except the harmonic mean. z 2 : z 2 = 70:74.>ence t h e square of i s equal t o Ax times xB. reduced to the simplest t e r m s . with the diameter dividedrationally a t x. Given the circle AB. a If. i s t h e perpendicular dropped from t h e v e r t e x of t h e r i g h t t r i a n g l e AyB. 42). proposition 111. 4. 2115 does not represent the segment xy which i s actually for which 2 I V 3 i s a close enough approximation to make the e r r o r l e s s evident. T a r t i n i 1 s e r r o r l i e s in the expression of his t e r m s a s a proportion. Tartini constructs the following proportions using the s q u a r e s of the remaining segments: . p. 56-7. Given the c i r c l e and the s q u a r e (Fig. The universal system. 5. Tartini attempts to prove the necessity of this limit a s follows. This again s e e m s to indicate the harmonic nature of the circle. u. the only s y s t e m that can be derived is that of consonant harmony and a s such a particular system. contains both the arithmetic and harmonic means. the diameter can be divided harmonically g d infih m . The Principj contains no calculations of this s o r t . and further. that the nature of the square s e e m s to be both arithmetic and counter-harmonic [_T. must be derived f r o m the chords. IV]. sines. 22ff. of which 35 is the harmonic mean and 37 the counterharmonic mean. Tartini never revised his calculations. Tartini s a y s that the diameter (arithmetic per s ~ should be 491. Tartini claims. proposition 1 1 . which a r e derived f r o m the diameter divided harmonically 531. he s t a t e s "the c i r c l e is a n infinite number of harmonic means [&.ALEJANDRO ENRIQUE PLANCHART have 30:35:37:42. but in o r d e r to produce a closed s y s t e m Tartini is forced to find a limit to the divisions of the diameter. As Zarlino does. 42) with the diameter s e t equal to 120 and divided harmonically through the simplest s i x divisions. Having a r r i v e d a t the conclusion that the c i r c l e is intrinsically harmonic while the s q u a r e is intrinsically arithmetic and counter) harmonic. the s q u a r e s of the chords and of the hypotenuses in the figure (which according to T a r t i n i a r e the harmonic and counter-harmonic means of the proportions considered in the preceding proof) will be: (m) (m [x. the arithmetic a t the point where the sine touches the diameter. a s in the previous proof. but in the Risposta. But f r o m a diameter divided harmonidivided harmonically cally. he s e t s this [T. He is aware o f t h i s when he a t tempts to justify his view by saying that the sine (xy). H ~ r ethe and Az t e r m s 35 / 6 and 371 6 do not r e p r e s e n t the lines Ax but only approximate them. Theoretically. published the s a m e y e a r a s the Principj. limit a t the f i r s t s i x divisions of the diameter (the s e n a r i ~ ) proposition VI]. and protracted sines a s well. the harmonic a t the proposition point where the sine touches the circumference [T. Omitting the s q u a r e s of Aa' and of Aa" since both a r e the same. 22ff. a s geometric mean. 11 Again here. If we reduce the proportion 30:35:37:42 to i t s original f o r m we have 5:35/6:37/6:7. 141:' This placed him in opposition to a l l the mathematicians who had demonstrated that the c i r c l e is the locus of the geometric mean of the infinite number of r a t i o s into which the diameter can be divided. complements. In both the Principi and the Risposta. a discrete geometric triple ( a s 1:3) [T. 84 i s . This exclusion of the minor sixth brought criticism from S e r r e . a discrete geometric duple ( a s 1:2). 581. T a r t i n i i s u s i n g t h e f a l s e premise t h a t a mean of t h e r a t i o 120:60. a disc r e t e geometric proportion of the ratio 2:5. F i r s t . a s we saw in the p r e vious proof. and 80 a r e the means of the duple ratio 120:60 (which a r e the values assigned to the diameter and the radius of the given circle)? He has managed to base the entire calculation on a circle with the diameter divided in the basic ratios of the genario and has obtained what he considers a unified expression f o r a l l the means of the duple ratio. Second. All accepted consonant intervals a r e included in the senario except the minor sixth which occurs between 115 and 118. Tartini adds the extremes: and extracts the square root of each total: Tartini observes that 100. the squares of the chords. the third. 84. The limitation of the harmonic system to the senario c r e a t e s another problem. A s we have seen.THEORIES OF GIUSEPPE TARTINI lower extreme harmonic mean counter-harmonic mean upper extreme 4000 Ab12 4800 Ab1I2 5200 6000 2700 A C ' ~3600 Ac1I2 4500 5400 The f i r s t proportion is a discrete geometric sesquialtera ( a s 2:3). the t r u e geometric mean of the ratio 120:60 is not 84 6 ( ) b u t m . T His calculations fail because two of his premises a r e incorrect. the second. a r e not the harmonic mean of the ratios produced by the sines and the segments of the diameter that form a right triangle with the sines and the chord. 90. Tartini r e g a r d s this a s proof that the integral extension and limit of the harmonic s y s t e m a r e in the s e n a r i ~ [ I 601. Tartini answers that the minor sixth i s incapable of reduction to a harmonic proportion other than the continuous 9. which i n f a c t it i s not. the last. . 000. 000 for the radius. the oc24. and to have a system i t is necessary to have a proportion. Because of this. produces the harmonic triple 1:1/2:1/3 [T. but the octave expressed a s 2:4 canbe divided arithmetically 2:3:4 and the fifth. He repeats the calculation two other times. 33ffl. in the construction of the circle. The octave and the fifth a r e then united in the f i r s t harmonic proportion of the system. However. and 6. 2 All this is exemplified. an absurdity since the harmonic system is essentially diatonic [E.461. The harmonic triple. inverted. 2. 000. the triple proportion 1:1/2: 113 1 . that a n interval produced entirely by arithmetic division cannot be included in the harmonic s y s t e m [E.283. form the minor sixth (Ex. the radius being unity which. 481. The octave itself does not form any system since it is only a ratio. the proportion is indivisible. In this form. he adds an attempt to derive the minor sixth from the senario. 14 for the diameter. the explanation is m o r e involved. The s m a l l e r ratios of each proportion (3:4 and 5:6).507 for the circumference[T. expressed a s 4:6 can be divided arithmetically a s 4:5:6.000. To Tartini it is obvious Example 6. 241. which. 221. The basic interval of any system is the octave since i t is the f i r s t interval of any superparticular s e r i e s that s t a r t s f r o m unity. he undertakes the derivation of the s y s t e m from the r e lationships contained within the figures of the circle and the square. Once Tartini has established the universal principle of his system in the circle. in the case of the harmonic system. this ratio is the fifth (1/2:1/3).000 for the diameter. 2. the l a s t time using such values a s 1. The derivation of the intervals of the senario in the Trattato p r e sents no difficulty since they can be derived from the diameter divided harmonically. the c i r c l e then represents harmonic a s well a s arithmetic unity. This produces the arithmetic s e r i e s 1:2:3. where Tartini has avoided proofs concerning the circle. and 22 f o r the circumference. when put one above the other ( a s musical intervals). 27ffl. He s t a t e s that the five intervals of the senario must not be derived f r o m the division of an already existent ratio.000. upon completion of the circle. has been doubled in the diameter and (approximately) tripled in the circumference. In the Principi.185.THEORIES O F GIUSEPPE TARTINI harmonic proportion 1/5:1/6:1/7:1/8. Tartini attempts to approximate this proportion through long calculations using Archimedes1 measurements of 7 for the radius. which is composed of a n octave and a fifth can be expressed arithmetically a s 1:2:3. . In the inversion. which requires the use of the non-diatonic seventh aliquot part. 6). according to Tartini. He finds the system based on the three sounds (1:113: 115) of the monochord s t r i n g unsatisfactory since the [x. namely. The tave is considered the a priori ratio of the system second ratio of any system is then the ratio that determines the nature of the system. 73ffl. 116. The derivation of minor harmony follows in bothtreatises a s i m i l a r course which consists in deriving i t from the advances produced by the harmonic division of a line (the diameter) a s was explained e a r l i e r in this essay. after the explanation of the senario. eel: (See Ex. . 1/4. Ad'. . 1.1/4. 621. 114. 46).'l1 Two fourths. which is essentially arithmetic. dissonances a r e derived from the circle in the following manner: Given the circle AB with the diameter divided up to the f i r s t s i x divisions and having the sines and the chords drawn (Fig. 113.361 essentially arithmetic. the octave (1: 1/21. 7b). c c ' . r e t : (See Ex. 7c). Be': (See Ex. 1 . 1. two major thirds. 58ffl. 7a)under the third (Ex. Tartini deduces the systems given b g o w _ ~ Bequals the pitch C. F r o m the s a m e example. ddl. 112. Acl. formed by the two fifths c-g and g-d' which produce a continuous geometric sesquialtera [T. cannot be obtained unless the fraction 1/7 i s included [_P. etc. he draws the following law: "Generally. 1 112. 8). 7c). 10. the terms. non s i pongano insieme questi t r e termini. . in the Trattato. . will each produce a continuous geometric proportion and. and under F r o m these both the successive formations of the senario (Ex. 113. El. a dissonant combination. 7a). any chord will be dissonant " " that contains two similar intervals of different species except (more by custom than by reason) the octave. 1/6.- .ALEJANDRO ENRIQUE PLANCHART basic ratio of the harmonic system. p. the minor third. . -- . a s such. K 1 . 1/5. and the last interval of the senarip. The dissonant one will be that which does not [belong] and cannot possibly belong to either of the two mentioned . se 1 s i seranno nell'accordo due i n t e r v a l l i simili d i specie diversa eccetua t a (piu per uso che per ragione) l a ottava [T. Che nelle p e r t i i n t e e a l i della sestupla armonica. ~ F r o m the squares of A . Tartini's next concern is to determine which interval is the dissonant one when two similar intervals appear in the same chord and he says: "Of two similar intervals of different species. 621. 1/4.1/2. to the circle by his r e m a r k that the circle must be considered in conjunction with the a projection of the diameter of the circle which is square [T. the dissonance of the ninth occurs. Tartini gives the following rule: "Of the p a r t s that form the s e n a r i ~ 1. Tartini then places the f i r s t (Ex. must be obtained by dividing harmonically the ratio 1:1/3. Tartini explains the relation of such a system of division. Che in genere qualunque accord0 m s i c a l e sara dissonante. 1/2. However. He goes then to the system of the third sound and finds that by the superparticular s e r i e s all the intervals of the senario can be obtained without deriving any interval from another and that the octave finds i t s proper place a t the beginning of the s e r i e s [E 59-60].E ' . F r o m the squares of a a l . 6. Ab'. In the Trattato. : 115. F r o m the squares of Ba'. bbt.must not be put to ether [by themselves] although they a r e contained within the senarip!' O The explanation of the rule i s that 1: 1/ 2 : 1/ 4 form a continuous geometric proportion and a s such they a r e the potential principle for dissonant harmony [T. in the second position.' . that one will be consonant which intrinsically belongs to the harmonic [major] o r arithmetic [minor] system. . Tartini observes that. In the Trattatg. 741. benche contenuti nella sestupla [_T. - - 8 KB. If the f i r s t three aliquotparts with odd denominators a r e added to the senario. he states. l / 11 etc. che i n niun modo appaxt iene. eleventh (fourth). augmented twelfth (augmented fifth). with the bass moving likewise by the same interval 99ffI a s in Ex. F i r s t he considers the possibility of a system in which the dissonant tones a r e a r r i v e d a t f r o m the ratios outside the senario which have odd numbers a s denominators. .761. .. Che de due i n t e r v a l l i s i m i l i d i s p e c i e d i v e r s a sar& il consonante q u e l l o che intrinsecamente a p p a r t i e n e a 1 sistema armonico. yet he includes it by definition since i t contains two fourths and is t h e r e fore dissonant by the law given previously [T.901. Example 10. In the Principi. 10. This would explain the special treatment of the minor seventh (117) which is not always prepared. Tartini observes that the thirteenth is not present in this system. e .THEORIES O F GIUSEPPE TARTINI systems1112 In the same manner. Tartini deals with the remaining notes of Ex. the dissonant note and the dissonant interval should be p r e pared by a s i m i l a r consonant interval. producing the resolution shown in Ex. 82ffl. This Tartini finds inconclusive since the odd-numbered aliquot p a r t s produce a l l the out-of-tune tones which cannot be considered diatonic and which a r e a l 12. 9. The resolution of dissonances should proceed downwards by s t e p o r half step [_T. 82 and 1301. 901. In the case of the second he concludes that the dissonant note is not the upper note of the interval and this he considers a reason against calling the interval a "dissonance of the second" E. 117. He proposes no name for this combination. and fourteenth (seventh). we have a system in which the dissonances produced by the ratios with odd numbers a s denominators outside the senario would have a consonant basis in the three basic sounds of the monochord string [P. Any dissonance. 119. i. arriving a t the following l i s t of dissonances: ninth.. These a r e the tones produced naturally by the tromba marina and other s i m i l a r instruments. Sarb il dissonante quello. n8 pub appartenere a'due s u d d e t t i s i s t e m i [T. i s curious that Tartini even resolves the augmented twelfth in this mann e r [_T. Tartini is concerned with the diatonic dissonances. 7c). Example 9. should be prepared by a melodic unison. since it stands in this system a s some s o r t of mean between the consonant partials 1:1/ 3: 115 and the openly dissonant ones 1/9:1/11:1/13 [E. It [x. 751. 771. 6 :\a '. m Example .3:iL Example 13.Example 12. E l =l Example 14. Example 17. IEa EsE3 Example 15. [x. their meaning for the word must have been that of successive c o n s o u 143. 61ffl a s in Ex. The union of both major and minor formulae forms a discrete geometric sesquialtera C-G with i t s harmonic mean E and i t s arithmetic mean E~ E. However. 8. 5. this accounts for major harmony only. 18. 951. 2. 13). 107. the duple. which is in turn derived from the senario. 3. T. 49ffl. e. This discrete geometric duple can be expressed mathematically a s 6:8:9: 12 and musically a s in Ex. g is the harmonic mean. Minor harmony is derived from the three lower notes of the arithmetic sextuple [B. The union of both harmonies is accomplished through the use of the "organic" formula. 6 [T. This organic formula can be found in three different positions. the 8 added on top a s a duplication of the bass-tone [E. 11) a r e considered dissonant [z. Inthe music of his time. F r o m the union of successive and simultaneous harmonies. This successive consonant harmony he deharmony rives from the basic ratio of the system. 1. 66ffl. both natures and is called a mixed cadence by Tartini (Ex. 1. 5 with 28. 1051. which is the basic principle of melody. 4. It is from this form of the that Tartini derives his "organic" formula: 1. Tartini a r r i v e s a t the basic b a s s progression for the establishment of a key [T.741 (Ex. 101-21. 88. and the third. The second and l e s s perfect is the arithmetic cadence from the arithmetic mean to the extreme (Ex. can be expressed in i t s essence in the three sounds of the monochord string. . with i t s harmonic and arithmetic means. The third and least perfect combines both means and. The senario a s we have seen. 17). and f the arithmetic mean 70ffl. Of the three positions. 3. second. He returns again to the s y s t e m of the third sound and constructs upon the fundamental b a s s a diato-ic scale. The tones c and c 1 a r e the extremes of the duple. and here Tartini recognizes the theory of inversion although the word The f i r s t position is 1. 14). The f i r s t and more perfect is the harmonic cadence from the harmonic mean to the extreme (Ex. 16). scale which a r e not included in the senarlp (Ex. Tartini concludes that since the Greeks did not have simultaneous harmony. 50). 12 (p. This results in the s a m e dissonances a s the ones derived in the Trattato except for the augmented twelfth which does not appear h e r e since i t cannot be considered a diatonic dissonance.THEORIES O F GIUSEPPE TARTINI tered by skilled players to fit the diatonic system E. but i t may be expressed in yet another way by taking only the three upper tones (Ex. successive consonant harmony is r e p r e sented by the three cadences that can be derived from the duple. After a discussion of the u s e of the word "harmony" by Greek theorists. 15). Those intervals between the fundamental b a s s and the notes in the Example 11. 2. T. 6. the inversion is never mentioned. in s o doing. T a r tini derives the scale. By o r d e r ing the three cadences according to their degrees of perfection and enclosing them with the extremes of the duple. 3. 1121. E a Example 20. Example 2 2 . . Example 21. l E . Example 2 3 . Example 26. a Example 25.Example 19. c. This fundamental b a s s is completely harmonic in the sense that i t is formed totally of major harmonies. Tartini examines the natural scale produced by such instruments a s the tromba marina. 7b) (see p. Chapter IV. The f i r s t fundamental b a s s for the scale consists in underlining it with the progression of the ordered cadences (Ex. a = However. 21). Thus the distance between the . He is quite distressed by the fact that most temperaments s e e m to him quite arbitrary. and finds that the ratios from one tone to another a r e not those used in musical practice. 110. horns. from the union of the two kinds of consonant harmony: simultaneous and successive. and the three resultant chords a r e the b a s i s for the formation of the scale [T.THEORES O F GIUSE P P E TARTINI he considers the f i r s t the most perfect in most cases. Tartini observes that such a scale needs to be tempered since three major thirds will not reach the octave and four minor thirds will exceed it. The second fundamental b a s s is directly derived from the concept Tartini considers the of the discrete geometric duple C. P. The resultant scale is the Ptolemaic syntonic 771. F. Luigi Antonio Sabbatini (1739-1809). Since the note C is duplicated. . scale (svntonic ditoniaion) h the T r a t t a b . trumpets etc. 1061. 13. 71ff] a s in Ex. G. The relative minor is derived from the l a s t note of Ex. 110-111. 98. This leads to his derivation of the scale. both harmonies (major and minor) because. when placed in i t s proper . However.111. he adm!ts that the second position is m o r e important in the minor triad thah in the major triad [T. and G. there a r e only three notes that form the essence of successive consonant harmony: C. 95ffl. The whole work was published a s V a l l o t t i ' s (under t h e t i t l e T r a t t a t o d e l l a modern8 musica) i n 1950. The corresponding figuration for the scale would be a s shown in Ex. in both t r e a t i s e s he warns against the confusion ofthe melody with the successive consonant harmony ofthe cadences which results in the misuse of the scale a s a b a s s melody [T. u . The essence of successive consonant harmony is the discrete geometric duple expressed a s in Ex. o r basic principle of melody. I n t h e Libro second^. The 4th and 7th notes a r e "out of tune" [T. 46) which is E ~ and which Tartini considers to contain inherently . T a r t i n i r e f e r s t o t h e Padre Francescantonio V a l l o t t i (16971780) who was o r g a n i s t a t t h e Franciscan seminary i n Padua a t t h e time. However. V a l l o t t i f i n i s h e d only t h e f i r s t book. we f i n d S a b b a t i n i l s e x p o s i t i o n of V a l l o t t i l s t h e o r i e s on temperament. [x. place a s arithmetic mean between C and G i t f o r m s a major third with G and a minor third with C.first notes of relative scales is proven to be a minor third [T. The o t h e r t h r e e were w r i t t e n by V a l l o t t i ' s d i s c i p l e and successor. 22 1071. 20. F. Upon every one of these three notes a chord is built according to the organic formula. but offers no solution of his own 1001. Tartini gives three fundamental basses for the scale and consequently three ways of adding figures to the scale when used a s bass. he mentions that Vallotti's temperament13 is the safest. - E. u. V a l l o t t i had s t a r t e d w r i t i n g i n 1735 a t r e a t i s e i n f o u r books. 19 (p. 52). To derive the enharmonic tetrachord the seventh aliquot part is interpolated in the scale f r o m a to d f which includes the bb (Ex. It appears after Tartini gives a brief account of the Greek tetrachords and adds. 1321. This produces a fundamental bass that is a retrograde of itself. . 27). the arithmetic mean of the discrete geometric duple C. when the scale is descending. G.1261.d f [x. in the f i r s t one given. 1311. 30) [T.1271. 26).which is the tonality produced by the (given) senario [T. which a r e presently those tones on which the relative minors of the tones underlined by the two cadences would be based IT. a reasonable man and diligent collector of old things.ALEJANDRO ENRIQUE PLANCHART scale a s two similar tetrachords and the fundamental bass underlines this factor (Ex. "1 have read a l l this in Zarlino. a completecircle [T. o r . 1101. 14. 10 non la so 1211. uomo ragionevole. Thus. 24. there is a tritone relationship between the harmonies of the sixth and seventh steps (Ex. To this he answers that the seventh partial. 23). Through the use of G#. The Tartini considers the use of the scale in the b a s s unwise. the relation becomes intolerable since there is nothing to soften it. T a r t i n i l s solution is quite s u r p r i s ing since here he r e s o r t s to the seventh aliquot part which he calls the "consonant seventhU(Ex. 7c) (see p. This tetrachord takes place in A minor. The derivation of the chromatic and enharmonic genera is contained entirely within the Tratatto. What may have been the reason for dividing the scale in s o many t e t r a chords. Tartini relates chromaticism to the minor mode. 55). i n T a r t i n i f s words. Tartini a r r i v e s a t the following chromatic tetrachord (Ex.. The minor t r i a d s a r e on D and A. Tutto cib ho letto nel Zarlino. p. This is inherent in the harmonic nature which tends to a s cend and not to descend D. 46). through a strong cadence. Another characteristic of this b a s s which distinguishes it from the f i r s t is the appearance of minor triads. The corresponding figuration is a s shown in Ex. This fundamental b a s s is harmonic in the sense that it consists of two harmonic cadences (Ex. F. Quia sia stata la cagione di divider la scala in tanti Tetracordi. . but i t underlines. c [TI 109ffI. 25). e diligente raccolittore delle cose antiche. This relation is considered by Tartini a s tolerable when the scale is ascending since it is softened by the passing of the scale from a l e s s perfect mean (arithmetic) to a more perfect one (harmonic). b a s s motion should be determined by the successive consonant harmony while the realm of the scale is in the upper voices. However. [x. which includes the notes G# and B~ with the help of which the chromatic genus can be constructed 122ffl (see Ex. . relative minor of C major. Tartini admits that some theorists will complain that this new sound is not prepared and not resolved by descending. 28. having the same third sound a s the f i r s t six and being moreover the true harmonic mean of the fourth a . The third fundamental b a s s is produced by the observation that.. 29). I do not know!'14 The possibility of interpolating chromatic tones in the diatonic scale in o r d e r to provide for melodic chromaticism finds i t s basic principle in Ex. . Example 3 3 . I Example 31. Example 34. Example 32. Example 29. Example 30.Example 28. Example 3 5 . . All these intervals a r e derived f r o m the interpolated diatonic scale of Ex. 7c) (p. The combination of the duple and the sesquialtera f o r m s the b a s i s for triple m e t e r 1141. even i n the generation of the third sound. 10 (p. This ratio provides duple meter. p. However. 1281. the duple. Tartini concludes by pointing out the possibilities of such harmonic m a t e r i a l s in dramatic music [_T. particularly i n triple m e t e r . The only other chord that could be considered dissonant cannot be dissonant in the inversion in which i t is written since the D# cannot be distinguished. 1291. 49) [T. 46). a stylization [x. The augmentedfifth (in contrast to the augmented twelfth) is treated a s a non-essential combination since the tone that produces i t is not prepared and it a s cends (Tartini u s e s the word discordanza for it) a s in Ex. and resolves i t a s shown in Ex. T a r t i n i considers the augmented twelfth a chromatic dissonance. 1591. . i n contrast with the minor fourteenth.THEORIES O F GIUSEPPE TARTINI is consonant [T. which is considered a diatonic dissonance. and the diminished fourth. The augmented sixth is a l s o treated a s a consonance [T. inverted a s the augmented sixth. a r e derived by Tartini f r o m the basic ratio of the system. 159ffl. 1581. the augmented s e c ond. Rhythm and m e t e r . but a stylization of the syllabic values which he illustrates a s shown i n Ex. He s t a t e s that the diminished seventh is treated a s a dissonance although it is not always prepared. 35) IT. d-a and a-dl and gives a n example of a s m a l l piece composed using the scale (Ex. is not a s t r i c t application of syllabic values to the pitches. inverted a s the augmented fifth. 34. [x. f r o m the t r u e harmonic note ~ b (Ex. 27. The f i r s t of the two settings is a s t r i c t application of the syllable values that r e s u l t s i n confused m e t e r . He claims that the minor seventh before cadences need not be prepared since the difference between the consonant seventh and the t r u e minor seventh is s o s m a l l that the minor seventh is thought of a s a n almost consonant interval E. 32. 36). 28 [T. This scale is based on Ex. Tartini claims that i n Ex. His hArmonization of the scale using the seventh partial (Ex. rhythm is produced in long and s h o r t accents (long and s h o r t note-values) which correspond to the long and s h o r t syllables of prosody [T. 1151. In the process of applying m e t e r to the successive consonant harmony and to the melodic voices. In accordance with the derivation. 33). 31 Tartini considers three other intervals and t h e i r inversions: the diminished third. 52) b e longs therefore to the enharmonic genus. The diminished fourth is treated a s a consonance (Ex. while the second. Tartini observes that the s c a l e consists of two equivalent tetrachords. The minor seventh. 157-581. 35 the only dissonances a r e the 4-3 suspensions a t the cadences. The use of such intervals can be demonstrated i n the scale shown in Ex. since the G o c c u r s i n the chromatic tetrachord. 38. Tartini classifies the augmented second and the diminished fourth a s chromatic intervals and the diminished third a s a n enharmonic interval [T. like harmony. is r e garded a s a n enharmonic dissonance and given the resolution shown i n 1311. Ex. 1301. inverted a s the diminished seventh. The funda' mental f o r this chord should then be a s shovl?l i n Ex. 37. 162ffl. Tartini observes that musical rhythm. . . Example 4 1 Example 42. Example 40.Example 39. T 143ffl. 1371 (Ex. in the Rismsta. Since the Greeks did not have simultaneous harmony. and subdominant D. 40). this i s dropped in the Princiui. he considers a s harmonic modes. he finds the organization of the modes unsatisfactory since the harmonic and arithmetic divisions of the octave do not fall within the same octave (Ex. Tartini believed that the Greeks had an octave-species based on the arithmetic and the harmonic divisions of the octave thus forming two tetrachords (Ex. the consonant intervals were the leaps of the fourth. 1431. and e-e' [T. 39). consists in a statement of the different divisions of the tetrachord [T. expressed a s C. T a r tini sees a basis for the three clefs 1201 and the basic tonal realm of tonic. dominant. and about Greek theory. while any interval smaller than the fourth was considered dissonant [T. published the same year a s the Principj. a minor key. 49ff. fifth. in which the fifth is at the bottom. Tartini mentions in passing that harmony seems to have a definite influence in determining the long and short accents in music. Tartini considers the ecclesiastical modes a s combinations of fourths and fifths. In the Tratthto he uses hexachord nomenclature to refer to any given note. Tartini had little understanding of the organization of the modes and his new modal system i s nothing but the major and minor scales that form the common tonal realm of the music of his time. for him. Tartini adds another s e r i e s of modes based on the arithmetic and harmonic divisions of the octaves A-a. 117ffl. Tartini attacks S e r r e ' s use of . In the discrete geometric duple. Because of this. 98ffl. 41). c. The authentic modes. To these six modes. Tartini's notions about history. A major key. which a r e derived from the discrete geometric duple (Ex. Greek theory. 2. However. 1391. 1211 and an explanation of the adaptation of the tetrachords to the strings of the kithara L_P. 531. 147ffl. Tartini then proposes what he considers an improved and "natural" organization of the modes based on the harmonic and arithmetic divisions of the octave c-c'. Example 38. d-d'. and octave. F. because of i t s harmonic nature will tend to modulate towards the dominant. However. a r e scattered throughout the two treatises. Tartini warns against "leaps" in modulation such a s going from a key in flats to a key in sharps and requires that the principal key be made clear a t the opening and closing of any piece [T. while the plagal ones with the fourth on the bottom a r e considered arithmetic modes. f-f \ and g-g'. Tartini treats modulation very briefly. he considers his diatonic ' system a s essentially the same a s the Greek [P. 146ff. and he notices the association of downbeats with consonance [T_. . G. [z.THEORIES O F GIUSEPPE TARTINI of the syllabic values is clear rhythmically and metrically [T. because of its arithmetic nature will tend to modulate towards the subdominant. 42). Tartini feels that a study of the affective value of the intervals could begin with this and continue to account for those intervals that a r e better suited to each voice. P a r i s : Chez Giuzeppe Tartini. and minor harmony to melancholy. 1779. Therefore. 1351. etc. The two best means by which to induce an affection in a h e a r e r . a s practiced in his time. L. . .. m. sics di Monsieur l e S e r r e di Ginevra. London: Bremmer. [x. [z. llauteur. Del principi dell'armonia musicale contenuta nel diatonic0 genere. can most effectively communicate a n affection [T. He doubts that music. 1767. according to Tartini.ALEJANDRO ENRIQUE PLANCHART the syllable gi which. of the different voices would obscure any attempt to communicate any given passion clearly. some of the old Italian d r a m a s which consisted almost entirely of recitative could most clearly communicate a n affection [T. would voices require another treatise written by one who. hexachord solmisation [R. Giuseppe Tartini alla Signora Maddalena Lombardini inserviente a d una importante Lezione Der i Suonatori d i Violino. according to Tartini. P. Tartini is quite skeptical. even if used constantly. He claims that a harmonic cadence c a r r i e s the characteristics of the major harmony while a n arithmetic cadence has the characteristics of minor harmony.. and sweetness on the other. Greek music could do s o since it was monophonic and since particular passions had their own ranges. philosopher. [1782]. He observes that while octaves and fifths sound quite satisfactory in the bass. 149ffl. 139ff. dynamic qualities. Venice: Colombani. languor. Burney. motion. with perhaps a single instrument. like the Greeks. a r e melodic ornaments and harmony itself 148-91.. Tartini concludes. . Bibliography P. Lettera del defonto Sig. A. Translated into English by Dr. Venice: Antonio Decastro. Traitk des agrkments de la musique. and poet. IV: 75ff. 1767. shows S e r r e l s ignorance of "the masterful Italian solfeggio:' that is. can express any precise affection. A single voice. Such considerations. speed. The effect will be a very general one leaning toward one affection o r another 1411. composd p a r l e cdlebre e t traduit par l e sigr. they a r e quite repugnant in the upper 153-41. 1770. Padua: Stamperia del Seminario. Denis. Regarding the affective power of music. etc. Tartini r e m a r k s on the relation of major harmony to f i r e and joy on one hand. Also available in English in the Journal of Research in Music Education. However. 70-711. would be a musician. Tartini's opinions on the affections a r e found in the fifth chapter of the T r a t t a t ~ . Risposta di Giuseppe Tartini alla crittica del di lui Trattato di mu. in most music of his day which employed many voices this would hardly be possible because the different pitches. R. dissertazione di Giuseppe Tartini. 2. Ddmonstration du principe de l'harmonie Paris: servant de base 3 tout l l a r t musical thedrique e t pratique. "The Evolution of Rameau's Harmonic Theories:' Journal of Music Theory. Rameau. Benjamin. P a r i s : Imprimerie Royale. 1739. Shirlaw. . Jean Philippe. 5.. Padre Francescantonio [and Sabbatini. Padua: I1 messaggiero di S. Matthew. Vallotti. F. . London: J . Leonhard. Petropoli [St. 1754.. 4. Antonio. [1939]. Joan. 1950. 1835-44. Luigi Antonio]. Durand. 3. Petersburg? 1: Typ. Biographie universelle des musiciens et bibliographie gdndsale de la musique. 7. Hughs. Euler. Fktis. The Theory of Harmony. F e r r i s . Brussels: Leroux. Volume VIII. Stillingfleet.Nouvelles rdflexions s u r l e principe sonore [bound with the Code de musique pratique]. and H. Trattato di musica second0 la Vera scienza delllarmonia. . 6. 8. 1750.THEORIES OF GIUSE PPE TARTINI T. Padua: Stamperia del Seminario. III(1959):231-56.. The Principles and Power of Harmonx. 1760.. 1771. London: Novello and Co. Trattato della moderna rnusica.. Academiae. J . 1. Tentamen novae theoriae musicae ex certissimis harmoniae principis dilucide expositae.