an historical survey of the origins of the circle

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An Historical Survey of the Origins of the Circle:Music and Theory

Jamie L. Henke

Dominant-tonic progressions are considered the primary building blocks ofmusic during the common practice period. As such, they command a substantialamount of theoretical study and analysis; nonetheless, only a limited amount ofresearch has been devoted to the evolution of this phenomenon. EdwardLowinsky states that "the cadence is the cradle of tonality,"1 likewise thecadence is the origin of the circle series. A major concern is whether cadencesfound in music of the fifteenth century can be considered predecessors of the V-I, when composers of the period did not associate the cadence with a conceptof verticality as presently understood. Yet historical perspective allows for theperception of germinal ideas in the fifteenth-century cadence that develop intothe V-I progression found in the Baroque, Classical, and Romantic periods.

Certain contrapuntal practices facilitated the development of the common-practice authentic cadence as is presently understood. During the fifteenthcentury, every composition was to end with a perfect consonance, and thisconsonance was to be approached from the nearest imperfect interval. Contrarymotion was highly emphasized, and the prohibition of parallel perfect intervals,not so important in the thirteenth and fourteenth centuries, became extremelysignificant.

Avoidance of parallel perfect intervals arises from one of two importantelements that occurred in the fifteenth century. First, composers of the periodoften emphasized one axial pitch in a composition, forming a genesis of thepresent day tonal center. In order to maintain this axial pitch, parallel fifths hadto be avoided. This is due to the fact that keys related by the interval of aperfect fifth inherently share six common pitches. Thus fifths can potentiallyreinforce more than one specific note.

Secondly, some composers wrote for four voices rather than the traditionalthree. Fifteenth-century rules dictate not only that a composition must end witha perfect consonance, but also that this perfect consonance must be

approached by an imperfect consonance. For example, if a piece ended on asonority based on C, the pitches used to approach this consonance should be Band D. If a third or fourth voice is added, the only choice that conforms to therestrictions imposed is the pitch G. These three pitches, whether present in afifteenth- or nineteenth-century composition, can be combined to form what isknown presently as a V chord in the key of C. Consequently, the incorporationof a fourth voice precluded any type of cadence other than the V-I similar to thatfound in the common practice period.

Another element that played a part in the development of the V-I cadentialformula is the interval of a perfect fourth. In the thirteenth and fourteenth

centuries, the perfect fourth was considered consonant, but by the fifteenthcentury it was deemed a dissonance. As a result, this causes a leap in the bass


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voice because the pitch G can only occur below the D, any other position wouldform a fourth above the bass. Thus, formation of the cadence became evenmore controlled.

As mentioned above, compositions were to end with a perfect consonance. All

of the aforementioned rules were designed to develop an approach to this goal.If tonality is to be considered goal-directed motion, couldn't the approach andarrival of a perfect consonance be considered goal-directed motion of thefifteenth century? Hence, (1) the shift to four voices, (2) the increase inimportance of the fifth and final, and (3) the prohibition of parallel fifths combineto form catalysts of the common practice cadence. The developmental processof these ideas is illustrated through excerpts chosen from nine composerspracticing their art from 1400 through 1672. Whenever possible, the composerswere chosen in order to trace trends from teacher to pupil, or at least betweencomposers who reportedly had contact with one another. Four of the composersare included in this teacher to pupil relationship: Adrian Willaert, Andrea

Gabrieli, Giovanni Gabrieli, and Heinrich Schtz. Guillaume Dufay, JohannesOckeghem and Jacob Obrecht represent the second category, that ofcomposers reportedly in contact with one another. Only two composers,Giovanni Palestrina and Jan Sweelinck, do not fit within either of these groups.They were chosen to fill historical gaps not covered by the other composers.

Even though the addition of a fourth voice provided much of the impetus forcadential development, seeds of this idea are found in three-voice works aswell. In the following example, Alma redemptoris mater2 by Dufay (1400-1474),foreshadowings, not only of the V-I, but of the ii-V-I cadence as well. Figure 1.aillustrates an ancestor of the V-I cadence, and figures 1.b and 1.c illustratepredecessors of the ii-V-I, first moving to F and then to C.

Figure 1.a


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Figure 1.b

Figure 1.c

Nonetheless, it is in the four-voice works of Dufay that most of the innovativeaspects of fifteenth-century composition are observed. He, Compaignons,

resvelons nous3 manifests a strong axial emphasis of the pitch G. Indeed, mostof the composition is formed by shifts between G and its fifth, D. Since this workis in four voices, a greater variety of pitches are present at the cadences thanthose found in the three-voice works. For example, in measure two, all cadentialpitches are present. A reductive analysis of measures 1-9 is included in figure 2in order to illustrate the predominance of final and fifth, as well as relativelytraditional cadences in this piece.


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It becomes apparent from the reductive analysis in figure 2 that eachcontrapuntal line is woven around the fifth and final of the piece. The result is avertical emphasis of these pitches as well, producing cadences such as those inmeasures 2, 4-5, and 8-9.

Similar cadences are present in the music of Ockeghem, (1425-1497), whocomposed during roughly the same period as Dufay. Several examples occur inthe Kyrie from the Missa L'Homme Arme, in figure 3 [Davison and Apel, 1946,

pp. 76-77]. As with He, Compaignons, resvelons nous, one axial pitch ishighlighted, in this case G. However, the work does not emphasize the fifth andfinal as exhibited in the Dufay example. Notice the leap in the bass in eachcadence. When the upper voices move primarily in conjunct motion, the bassrequires the leap in order to satisfy fifteenth-century compositionalrequirements. Another interesting device employed in this work appears infigure 3.b with the Landini cadential material in the upper voice. The leap of afourth found in figures 3.a, 3.b, and 3.c is applied in figure 3.d where the bassvoice moves from D to F by fourths and then to B in the tenor voice of the nextmeasure. Not all of these tones are chord roots, nonetheless the patternforeshadows the bass motion found in many circle progressions today (see

figure 3).


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Figures 3.a, 3.b, and 3.c

A true precursor of the circle series appears in the Tsaat een meskin by JacobObrecht, who lived from 1451-1505 [Davison and Apel, 1946, pp. 82-83]. Thispiece employs not only circle progressions, but multiple circle progressions inseries as well. During the second half of the fifteenth century, a set of multiplecircle progressions (henceforth referred to as circle series), is novel in and ofitself. Not only did Obrecht extend the cadential formula from dominant-tonic tosupertonic-dominant-tonic, but he employed a circle series as the foundation fora section of music in a manner rarely, if ever, seen before this time. In figures4.a and 4.b, Obrecht utilizes the series D-G-C-F to form the structure forapproximately three measures of music. Immediately following each of thesetwo examples is another circle series, a dominant-tonic that has been extendedinto a subdominant-dominant-tonic on the pitches C, D, and G. This may be thefirst appearance of this cadential pattern; a pattern that would become prevalent

in music of the common practice period.

One more circle series occurs in this piece at the same pitch level as the others,this time spanning seven measures (figure 4.c). The G in the series isprolonged and finally advances to C and F.

Figure 4.a


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Figure 4.b and 4.c

Circle series were employed by other composers contemporary to Obrecht,among them Adrian Willaert, (1490-1562). In the fourth Ricercare of his IXRicercari4, Willaert employs the pitches A-D-G and C, as seen in figure 5.a. It ispossible to interpret this as a five note circle due to the short duration andpassing nature of the Bb, illustrated in the following diagram.

Figure 5.a


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Figure 5.b

A five note circle series occurs in Willaert's Victimae paschali laudes [Davisonand Apel, 1946, pp. 117-118] represented in the preceding figure 5.b. This circleseries progresses from D to Bb in a span of two measures. As with the Obrecht

examples, Willaert employs the circle series as more than a cadential formula,rather it becomes the foundation for this particular phrase.

Not only phrases, but at times entire works are based on circle progressions inthe music of Andrea Gabrieli, (1510-1586), who studied under Adrian Willaert.For example, the entire eighteen measures of the Intonazione settino tono[Davison and Apel, 1946, pp. 146-147] are diagramed below. The score of theIntonazione settino tono follows in

Figure 6.a.

In pieces of a more complicated nature, such as Ricercare del 12 tono [Davisonand Apel, 1946, pp. 147-148], circle progressions appear as a sequence ofsimple cadences, extended cadences and circle series, (figure 6.b). Cadences


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like those found in measure 13-14 occur throughout the composition, andextended forms of the cadence, the supertonic-dominant-tonic, take place inmeasures 28-30, 49-51, and 52-54. In measures 49-51, a supertonic-dominant-tonic is followed by the same cadence type in augmentation.


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Figure 6.b

Also observe that each time the supertonic occurs, it is in first inversion, aforeshadowing of one of the common cadential formulas of the Classical period,that of incorporating the fourth, fifth, and first scale degrees in the bass. This

bass motion may also be an outgrowth of the root motion associated with voice-leading similar to the subdominant-dominant-tonic cadential formula of theeighteenth century, found as far back as Willaert's "Victimae paschali laudes"[Davison and Apel, 1946, pp. 117-118] in figure 6.c.

Figure 6.c

The only circle series in the composition appears in the form of an extendedcircle progression that occurs in measures 7-9 (Figure 6.b). All circles in thispiece seem to fit into a coherent plan determined by Gabrieli at the beginning ofthe composition. For example, in measures 13-14 he presents only the circleprogression G-C; later he extends this by one note back in the circle to D-G-C,forming the supertonic-dominant-tonic cadence, and finally, moves one notefurther back in the circle for the A-D-G-C employed in the circle series inmeasures 7-9. This is important in that the goal of every type of circle in thepiece is the pitch C. Earlier it was stated that during the fifteenth-centurycomposers emphasized one axial pitch; however, even with the development ofthe fifth and final, circle progressions emphasize several different pitches. Forexample, in the Victimae paschali laudes by Willaert, Andrea Gabrieli's teacher,there are circles that end on D, G, C, F, and one that proceeds to Bb.Interestingly enough, even though these pitches combined together form onelarge circle, A-D-G-C-F-Bb, Willaert does not employ these to attain one specificgoal, but rather to emphasize different pitches within the piece. It is AndreaGabrieli who first uses the circle progression to target a specific pitch, possiblytouching on the concept of key and tonic.

Palestrina (1525-1594), employs circle progressions in the same manner as

Gabrieli, presenting the original circle and then extending it. In figure 7.a, thecircle moves from A through D to G; in figure 7.b, this circle is expanded to


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incorporate the C (second measure of the figure). Notice that the two circles arecompleted on different pitches unlike the example by Andrea Gabrieli. Just asAndrea Gabrieli expanded upon ideas found in Willaert, Gabrieli'sdevelopments are enhanced by his student and nephew Giovanni Gabrieli, whoprovides another link in this evolutionary chain.

Figure 7.a and 7.b

Many of the ideas previously mentioned appear in the music of GiovanniGabrieli, among them the use of the circle series for prolongation, already seenin works of Obrecht. Gabrieli (1554-1612) obviously took the lessons of hisuncle seriously, composing circles of greater length. The "Ricercari (5)"5 fromGabrieli's organ compositions contains two extended full circle series. Thenovelty of these circles is that they incorporate not five-note circles, the largest

seen thus far, but rather thirteen- and eleven-note circles within a two measurespan (see figures 8.b and 8.c). Both series originate from the melodic use ofthirds in the piece, occurring for the first time as seen in figure 8.a.

The extended full circle series in figure 8.b breaks when the melodic sequenceof thirds is completed at the arrival of beat three in measure three. In thesecond series Gabrieli breaks the pattern, this time on beat three of the firstmeasure (figure 8.c) in order to avoid the tritone formed by the Eb to A motion inthe circle. Instead, he substitutes a first inversion C minor triad for the Eb. Thispractice of substitution becomes quite common in compositions of this era, andis occasionally employed in music of the common practice period as well.


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Figure 8.a

Figure 8.b

Figure 8.c

Circle series in the music of Jan Sweelinck (1562-1621) may not be as lengthyas those of Giovanni Gabrieli, however, this is more than compensated for byprolific employment of smaller circles such as those present in Fantasia in echo[Davison and Apel, 1946, pp. 209-211].

Two types of circle series occur in this piece, those that proceed quicklyoccupying a short span of time--for example, one measure--and those thatinclude a structural function of prolongation encompassing several measures ormore. An example of the former is found in figures 9.a and 9.b. Note that threemembers of the series are contained within the span of one measure, and thateach of the members, with the exception of the very first, have durations of onebeat.


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Figure 9.a and 9.b

However, in figure 9.c, the same circle series occurs, this time over fivemeasures. Each pitch of the series incorporates the entire measure, except forthe A that receives only two beats in the second measure. Actually, the fermataabove the A would seem to compensate for its smaller durational value incomparison to the other pitches in the series.

Figure 9.c

The circle, E-A-D-G-C, is increased in length by one pitch in figure 9.d thusbecoming E-A-D-G-C-F. Sweelinck treats this series in much the same manner

as the aforementioned one in that it occurs in an expanded form later in thepiece. In this particular occurrence of the circle, the E receives two beats, the Afive, but the rest of the circle takes place within constraints of one beat perpitch.

Figure 9.d

Just as with the circle series in figure 9.c, the series is also expanded later inthe piece, in figure 9.e. In this instance, the material on the pitch E again occurswithin two beats, the A in four beats instead of five, D, G, and C are expandedto two beats each, and F is given one beat as before.


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Figure 9.e

The final link, or culmination, of this evolutionary chain is provided by the musicof Heinrich Schtz, (1585-1672), a pupil of Giovanni Gabrieli. The influence ofGabrieli is quite apparent, especially in the number of consecutive circleprogressions that Schtz employs. Gabrieli's compositional style is alsoobserved in Schtz's avoidance of the tritone. Both composers replace thechord root of the tritone with a chord root that is usually a third away from thereplaced root. This phenomenon appears in music by Monteverdi as well, inworks such as "Chi vuol verder" and "Godi pur del bel sen."

The following figure diagrams a set of circle progressions from the Ricercar (5)by Gabrieli. Note not only the length of this progression, but also thecharacteristic omission of the tritone (figure 8.c, p. 22).

C is a third away from both pitches it could be substituted for; the E forms atritone with the previous Bb, or the Eb that would form a tritone with thefollowing A. Gabrieli concludes the circle on Bb, perhaps to avoid anotherencounter with a possible tritone.

Music by Schtz incorporates all of the aforementioned phenomenon, as well asnew developments. In the first two staves of the "Allelujah" in the "Freuet euchdes Herren ihr Gerechten" from the Symphoniae Sacrae (table 1), Schtzavoids the tritone by the same substitution method found in Gabrieli, in thisinstance replacing the Bb with the D a third away. However, in measure 142Schtz does not avoid the tritone; this may be due to the following intrinsicdifference between the patterns of circle progressions used by the twocomposers.

It appears that during the time of Gabrieli, the function of goal directed motionfrom circle progressions was not yet recognized, at least not to the extent foundin Schtz. At times in the music of Gabrieli the tritone is circumvented, at othersit is not (see table 1). This inconsistency, coupled with the fact that the circle

progressions of Gabrieli do not come to rest on the tonic points to anotherfunction of this use of circle progressions. Circle progressions used by Schtz


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do tend to convey a feeling of arrival, a true sense of goal-directed motion. Inthe previously mentioned set of measures from the "Allelujah," Schtz appearsto include the tritone the second time through the circle in order to establish aprecursor of relative minor, and to prepare a strong arrival in this area atmeasure 143.

A lavish number of consecutive circle progressions are found in "Von Gott willIch nicht lassen" by Schtz. Sixteen groups of circle progressions occur withinthe span of only 206 measures. Some of these extended progressionsincorporate the tritone, while others make use of the third substitution previouslymentioned. Each of the progressions in "Von Gott will Ich nicht lassen" targetsthe pitch G. At this point there can be little doubt that the circle progression isemployed by the composer to facilitate goal-directed motion. Even though thesecircles have the same goal, they move toward this goal in different ways. Sevenof the sixteen sets of circle progressions are interrupted; and in this case arecompleted, not by a substituted third, but by third inclusion. Some avoid the

tritone by breaking the pattern completely at the point at which the tritone wouldoccur. Examples are from "Von Gott will Ich nicht lassen" unless otherwisestated (table 1).


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In studying music of these periods the evolutionary progression of the circleseries becomes readily apparent. With such a profusion of this phenomenon inmusical literature, it would follow logically that a coinciding abundance oftheoretical discussion would also be available. However, this is not the case.Barring the customary circle of keys presented by many theorists, very little ismentioned regarding employment of circle series.

Possibly the earliest reference to the ascending fourth or descending fifth ideacan be found in Musurgia Universalis6 by Athanasius Kircher (1601-1680). Here

he refers to "circulationem harmonicam," or harmonic circles that are ascendingand descending always by fourths or fifths, and that can be employed in such a


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manner as to end where one has begun. It seems that theorists after Kircher donot set aside a portion of their work for this idea. Instead, writers such asHeinichen, Mattheson, C. P. E. Bach, Kirnberger, and Rameau incorporate itinto other aspects of theory in which it plays only a secondary role to the topic ofdiscussion.

Two of the most famous circles are found in the works of Heinichen (1683-1729), and Mattheson (1681-1764). Figure 10.a represents the circle asdeveloped by Heinichen in 1728, and 10.b represents the interpretation byMattheson proposed in 1731.

Heinichen developed his circle through discussions with Kuhnau concerningKircher's method of demonstrating an orderly progression of keys by fourths orfifths. Since Kircher's method proved somewhat awkward by allowing a majorkey to proceed only to another major key, and a minor key to another minor key,Heinichen evolved his circle to permit more freedom.


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Heinichen states that the composer may proceed through the circle in thefollowing schemes:

(1) proceed through the entire circle of keys from either left to right, or right toleft;

(2) move through alternate degrees of the circle of keys, either left to right, orright to left, starting with C major;

(3) same as above only starting with A minor.7

According to Heinichen, the musical circle is useful in composition, inaccompanying from unfigured bass, in playing preludes on keyed instruments,and to facilitate the abolition of the ancient modes. Nonetheless, even thoughHeinichen considers the circle of keys to possess these many functions, heneglects to discuss any one of them in even cursory detail.

When Mattheson announced his circle of keys, he believed it to be animprovement on the circle by Heinichen. However, Mattheson is even morerestrained in his treatment of the circle. The only significant difference betweenthe two circles is that Mattheson simply requires one to traverse the circle inonly one direction to get to any key, rather than both directions, as is necessaryin the Heinichen design. Mattheson states his circle is an improvement onHeinichen's, and Heinichen declares Kircher as his source.

C. P. E. Bach presents a circle of keys in his Essay on the True Art of PlayingKeyboard Instruments (1753-1762)8, but advocates modulation only to closelyrelated keys, and again devotes minimal discussion to the employment ofcircles in composition. J. P. Kirnberger in The Art of Strict Musical Composition(1771-1779)9 presents the circle series as a chain of seventh chords intendedto prevent repose (figure 11).

If one wants to keep the full cadence and yet prevent its effect of repose, theseventh need only be added to the tonic triad, whereby it becomes thedominant of a new key.10

Figure 11.

This could conceivably work with any pattern in the bass, so why doesKirnberger choose a pattern of ascending fourths for this example?


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This question may never be answered because, as mentioned previously, theascending fourth pattern is presented only as a backdrop upon which toillustrate various theoretical constructs, in this case the function of the seventh.

Kirnberger then proceeds to discuss various correct methods of modulation, as

does C. P. E. Bach. Kirnberger states that modulation should result only inmovement to closely related keys, and that each of these keys shouldencompass at least one to two measures. This poses several interestingquestions. If modulation through these keys is supposed to occur over one totwo measure spans, one key at a time, how is this reconciled with the music ofJ. S. Bach in which one can find many instances of full circle series occurringtwice over the span of two measures? Is it possible that the term "key" has adifferent meaning in this particular context? Or is it possible that at this point inhistory only the linear aspects of the circle series were noted, and thus theharmonic elements of the circle series were not considered as modulatory keyrelations?

Few answers to these questions can be found in the theoretical works ofRameau. Rameau discusses a fundamental bass that can proceed byconsonant intervals of a third, fourth, fifth, or sixth, each of which can eitherascend or descend. The only other statement of importance made in referenceto the fundamental bass is that the smaller intervals are preferred over thelarge. Using the smaller intervals for the fundamental bass would include thethird, but whenever Rameau refers to the fundamental bass in his examples, italways progresses by fourths. All of the following examples in figure 12.a-12.dare from Rameau's Traite de L'Harmonie (1722).11

Figure 12.a

In figure 12.a one finds almost two full circle series. If Rameau had included theF in the second half of the first measure both circles would be complete. Inexample 12.b, Rameau again illustrates the fundamental bass as a progressionof alternating descending fifths and ascending fourths, this time including twocomplete full circle series. Figures 12.c and 12.d represent the fundamentalbass as a progression of ascending fourths and fifths as well. It is alsointeresting to observe that each time Rameau presents a fundamental bass infourths, every chord, with the exception of the final one, incorporates theseventh as


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a necessary chord element employed to avoid a cadence. In fact, in figure 12.c,Rameau includes the fundamental bass as a foundation upon which he canshow the preparation and resolution of sevenths and

seconds, and in figure 12.d, refers to the fundamental bass as a "fundamental

bass of sevenths." As with those authors previously discussed, the circle seriesaspects of the Rameau examples are not defined because of their secondarynature to the subject at hand.

Figure 12.b


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Figure 12.c


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Figure 12.d

Some interesting questions arise from the previous illustrations. Does Rameau

really consider the seventh a requirement for this particular form of thefundamental bass, or was it conceivable to have a fundamental bass withoutsevenths? Why does Rameau present so many examples of the fundamentalbass in which the interval of a fourth is employed, when he considers the third,fifth, and sixth perfectly suitable consonances to apply as the fundamentalbass?

Only one theorist appears to have discussed the circle series based upon itsactual behavior in musical literature, especially in regards to the prolongationalelement seen in compositions as early as Obrecht. In his Die Grundstze dermusikalischen Komposition (1853-4)12 , Sechter treats the circle series, not as

an abstract theoretical entity, but rather as a valid theoretical construct withspecific, varied functions in music. Voice-leading, seventh and ninth chords,major and minor, and diatonic and chromatic structures or variations of thecircle series are analyzed in depth. The following figures are presented with texttranslated from the original German by the author.

This order has its good points, in that the fifth of each following chord isprepared, as the octave of the proceeding chord becomes the fifth (referring tofigure 13.a).


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When employing I-IV-V-I one can insert a II chord in between in variousmanners as follows:

I-IV-II-V-I or I-IV-II-V7-I or I-IV-II7-V7-I. One can then substitute a VI chord forthe IV chord.

I-VI-II7-V-I or I-VI7-II-V7-I or I-VI-II7-V7-I

To lengthen the statement, proceed the VI with a III.

I-III-VI-II-V-I or I-III-VI-IV-II7-V-I or I-V-III-VI-II-V-I

In order to extend the statement even farther, one follows the chord of the fourthwith the seventh, the third follows, and so on as seen below.

I-IV-VII-III-VI-II-V-I or I-IV-VII7-III7-VI7-II7-V7-I

All of these sentences can be formed with root position chords, or with theirinversions. Incorporating seventh chords into the statement prolongsrestlessness that is only resolved in the chord of the first step.

Figure 13.b

Sechter also discusses the circle series in terms of goal-directed motion,specifically citing various capabilities in regards to modulation, thus expandingupon the germinal ideas of Heinichen and Mattheson. In figure 13.c, Sechterillustrates the method by which one can modulate from G major to A minorthrough employment of the circle series.


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The example in figure 13.d illustrates Sechter's concept of prolongation withinthe circle series, an idea that has appeared in music since Obrecht, butnonetheless overlooked by most theorists before and after Sechter. Theconcept of prolongation within the circle series implies a hierarchy, similar tothat found in Schenker, in which each member of the circle series maintains its

status until the following member of the series is sounded, regardless of whatmaterial occurs in between.

If chords related by a leap of a fourth or fifth are interrupted by other material,only the two fundamentals are valid, as follows.

Sechter practices his theory within his compositional style. The followingdiscussion is based on a page of the "Pastoral-Fuge" from the Drei Fugen.13Elements of goal-directed motion, modulation and prolongation are all presenthere (see figure 14).

The first goal at the top of the page is the pitch D that is prolonged for threemeasures until the C natural appears to set the stage for the next arrival on G inmeasure 21. The G is then sustained until the C# and D# in measure 23, inconjunction with the B major triad, break the circle and begin a new one thatfirst rests on E in measure 24. The full circle is completed in measures 25-26,however it is then continued so that the arrival of D, the tonic of the piece,coincides with the statement of the fugal theme. Patterns such as these occurthroughout the composition, as well as in other Sechter works.


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Figure 14.

Two things become evident from the previous series of analyses. The first isthat, with the exception of Sechter, the most accurate portrait of the circle seriesis to be gleaned only through study of the actual music; and second, if theprofusion of circle series within this music is to be considered an indication,there exists an abundance of prospective theoretical analysis. This is by no


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means an exhaustive or definitive study on the subject, but rather a catalyst thathopefully facilitates further research and imput on this topic by other individuals.

Figures

1.a Davison, Archibald T, and Apel, Willi, Historical Anthology of Music.Cambridge, Mass.: Harvard University Press, 1946, p. 70, mm. 39-44.

1.b Davison and Apel, p. 70, mm. 20-23.

1.c Davison and Apel, p. 70, mm. 25-28.

2. Author

3. Davison and Apel, p. 77, m. 8-41.

4.a Davison and Apel, p. 82, mm. 8-12.

4.b Davison and Apel, p. 83, mm. 27-38.


5.a Willaert, Adrian, IX ricercari per sonar con tre stromenti. London: Schott andCo. Ltd., New York: Associated Music Publishers, 1933, mm. 94-99.

5.b Davison and Apel, pp. 117-118, mm. 58-60.

6.a Davison and Apel, pp. 146-147.

6.b Davison and Apel, pp. 147-148.


7.a Palestrina, Giovanni, Ricercari sopra li tuoni a quattro. New York: SchottMusic Corp. (Associated Music Publishers Inc.), 1933, pp. 18-19, m. 14.

7.b Palestrina, pp. 18-19, mm. 24-25.

8.a Gabrieli, Giovanni, Composizioni, Vol. II. Italy: G. Ricordi and C. Milano,1958, pp. 18-21, mm. 4-5.

8.b Gabrieli, Giovanni, pp. 18-21, mm. 44-46.

8.c Gabrieli, Giovanni, pp. 18-21, mm. 74-75.

9.a Davison and Apel, pp. 209-211, mm. 5-6.

9.b Davison and Apel, pp. 209-211, mm. 9-10.

9.c Davison and Apel, pp. 209-211, mm. 24-28.


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9.d Davison and Apel, pp. 209-210, mm. 10-13.

9.e Davison and Apel, pp. 209-211, mm. 87-91.

10.a Arnold, F. T., The Art of Accompaniment from a Thorough-Bass. London:

The Holland Press, 1961, p. 267.

10.b Arnold, F. T., p. 276.

11. Kirnberger, J. P., The Art of Strict Musical Composition. Yale UniversityPress, 1982, p. 117.

12.a Rameau, Jean-Phillipe, Treatise on Harmony. Trans and ed. by PhilipGossett, New York: Dover Publications, Inc., 1971, p. 255.

12.b Rameau Jean-Phillipe, p. 257.

12.c Rameau, Jean-Phillipe, p. 261.

12.d Rameau, Jean-Phillipe, p. 305.

13.a Sechter, Simon, Die Grundstze der musikalischen Komposition. Leipzig:Breitkopf and Hrtel, 1853-1854, p. 13.

13.b Sechter, Simon, p. 102.

13.c Sechter, Simon, p. 105.

13.d Sechter, Simon, p. 21, p. 95, p. 199 (author's transcription).

14. Sechter, Simon, Drei Fugen. Wein, Munchen: Ludwig Doblinger (BernhardHerzmansky) K.G., 1972, pp. 2-5, mm. 16-32.

an historical survey of the origins of the circle

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