Contour is the shape or format of an object and can be either bi- or multidimensional. In

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1 Contour Algoriths Review arcos da Silva Sapaio Universidade Federal da Bahia (UFBA) Pedro Kroger Universidade Federal da Bahia (UFBA) Abstract: In this paper, we present soe probles of two usic Contour Relations Theory operations algoriths: the Refineent of Contour Reduction Algorith, which was developed by Rob Schultz, and the Equivalence Contour Class Prie For algorith, which was developed by Elizabeth arvin and Paul Laprade. We also propose two alternative algoriths to solve these probles. Keywords: usic Contour Theory, Reduction Algorith, Algorith, Equivalent contour classes. I. Introduction Contour is the shape or forat of an object and can be either bi- or ultidiensional. In usic, a contour is an abstraction of eleents such as pitch, chord density, duration, tibre, or intensity. A elodic contour, for instance, is a ap of pitches in tie. usical contour is "a set of points in one sequential diension ordered by any other sequential diension" []. The study of contour is iportant because, as is the case with otifs and pitch class sets, it can contribute to usical coherence. The usical Contour Relations Theory provides concepts and operations with which to establish contour identity and siilarity easures for analytical and copositional purposes. This theory has supported the analysis of works by coposers such as W. A. ozart [], Luigi Dallapicolla [6], Arnold Schoenberg [, ], Anton Webern [], Olivier essiaen [], Elliott Carter, Pierre Boulez, Iannis enakis, Alois Haba, ilton Babbitt, Harrison Birtwistle, and Igor Stravinsky [], as well as video gae soundtracks [8]. In this paper, we present soe probles of two Contour Theory operations algoriths: the Refineent of Contour Reduction Algorith [] and the Equivalence Contour Class Prie For algorith [6, 7]. We also propose two alternative algoriths for solving the. II. Contour Theory basic concepts and operations A coplete presentation of Contour Theory concepts and operations is outside the scope or this paper. However, an understanding of contour space, contour segent, coparison atrix, class equivalence, prie for, and contour reduction is necessary to follow this paper s preise. See Beard [], Bor [], and Sapaio [] for further inforation concerning usic Contour Theory. 7

2 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Figure : Fugue subject, The Well-Tepered Clavier by J.S. Bach Table : Coparison atrix of the contour < 5 5 > Contour space is a usical space coposed of the contour eleents (or contour points [CPs] ). The contour space of pitches, for instance, is coposed of all pitches represented as integers fro low to high and nubered fro to n [9], where n denotes cardinality. For instance, the contour space of pitches in the elody shown in Figure is S = {,,,,, 5, 6}, representing the pitches Eb, F, G, Ab, B, C, and D. A contour segent, or, ore siply, a contour, is an ordered set of contour points. For instance, the contour of the first five notes in Figure is < 5 5 >. Noralization (or translation) is an operation that siplifies the contour representation by re-enuerating fro to n. The otive < 5 5 > noralization is < >. The relation of any pair of CPs is ternary: one CP is lower, equal to, or higher than the other. The contour can be analyzed and represented in a linear or cobinatorial way. The linear contour regards only the relations between adjacent CPs; the cobinatorial contour considers relations between adjacent and non-adjacent CPs. For instance, in the contour, the relation between 5 and is lower ( ), whereas it is higher (+) between and. The relations aong the contour points are represented in a coparison atrix (cf. Table ). Two contours are equivalent if they share the sae coparison atrix. The set of equivalent contours fors a contour class. This class is represented by an equivalent class prie for. According to arvin and Laprade citearvin987, the retrograded, inverted, and retrogradedinverted versions of a contour belong to its equivalence class. The prie for of this class can be calculated with a siple algorith (cf. Section IV). Any contour can be reduced to a prie for through the Contour Reduction Algorith []. This algorith, like the Schenkerian reductive techniques, sets up hierarchical levels of pitch salience. High and low peaks are selected: "passing notes" and "inner voices" are pruned. That is, the algorith provides a criterion for associating nonadjacent notes by picking out those that are presuably ost obvious to the ear []. Robert orris [] proposed 5 basic contour prie fors to which any contour could be reduced. Rob Schultz [] proposed a refineent to this algorith because soe contours such as See Lewin [5] for further inforation regarding usical spaces. A contour point is also known as a contour pitch, or c-pitch. Do not confuse this with prie contours, as proposed by Robert orris []. 7

3 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. A < > are not reducible to one of the basic pries by the orris algorith. The equivalence classes and prie contours are useful for checking the identity of the contours for analytical and copositional purposes. For instance, the contours A < > and B < > belong to the sae class, as represented by their prie for < >. The B contour is obtained by inverting and retrograding the A contour. As another exaple of this, the contours C < > and D < > have a coon prie: P < >, as obtained by the Refined Contour Reduction Algorith. III. Refined Contour Reduction Algorith (by Schultz) The Refined Contour Reduction Algorith [] (cf. Algorith ) reoves interediary CPs and preserves salient ones with the support of an auxiliary internal structure called the ax-/in-list. This algorith has a preliinary coponent (Steps to 5), as well as a recurrent one (Steps 6 to 7). CONTOUR-REDUCTION-ALGORITH(C), where C is a contour. Let variable N: Step : Set N to. Step : Flag all axia in C upwards; call the resulting set the ax-list. Step : Flag all inia in C downwards; call the resulting set the in-list. Step : If all c-pitches are flagged, go to step 6. Step : Delete all non-flagged c-pitches in C. Step 5: N is increented by (i.e., N becoes N + ). Step 6: Flag all axia in the ax-list upward. For any string of equal and adjacent axia in the ax-list, flag all of the, unless: () one c-pitch in the string is the first or last c-pitch of C, then flag only it; or () both the first and last c-pitches of C are in the string, then flag (only) both the first and last c-pitches of C. Step 7: Flag all inia in in-list downward. For any string of equal and adjacent inia in the in-list, flag all of the, unless: () one c-pitch in the string is the first or last c-pitch of C, then flag only it; or () both the first and last c-pitches of C are in the string, then flag (only) both the first and last c-pitches of C. Step 8: For any string of equal and adjacent axia in the ax-list in which no inia intervene, reove the flag fro all but (any) one c-pitch in the string. Step 9: For any string of equal and adjacent inia in the in-list in which no axia intervene, reove the flag fro all but (any) one c-pitch in the strings. Step : If all c-pitches are flagged, and no ore than one c-pitch repetition in the ax-list and in-list (cobined) exists, not including the first and last c-pitches of C, proceed directly to step 7. Step : If ore than one c-pitch repetition in the ax-list and/or in-list (cobined) exists, not including the first and last c-pitches of C, reove the flags on all repeated c-pitches except those closest to the first and last c- pitches of C. Step : If both flagged c-pitches reaining fro step are ebers of the ax-list, flag any one (and only one) forer eber of the in-list whose flag was reoved in step ; if both c-pitches are ebers of the in-list, flag any one (and only one) forer eber of the ax-list whose flag was reoved in step. Step : Delete all non-flagged c-pitches in C. Step : If N =, N is increented by (i.e., N becoes N + ). Step 5: If N =, N is increented by (i.e., N becoes N + ). Step 6: Go to step 6. Step 7: End. N is the "depth" of the original contour C. 7

4 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Algorith Refined Contour Reduction Algorith (by Schultz) A axiu point exists if, in a triad of CPs, the iddle CP is equal to or greater than its neighbors. By default, the first and last CPs in a sequence are also axia. The axia-list is calculated in a linear anner with Algorith (in pseudocode). This algorith returns a ax-list fro a given sequence of CPs. The inia and in-list are calculated in an analogous way. For instance, the axia- and inia-list of the contour S < > are = [,, ], and = [,, ]. Algorith axia list algorith A-LIST(S), where S is a sequence of CPs, represented as S = {S, S,..., S n, S n } and n is its cardinality. Let ax-list Add S to ax-list for i fro to n do if (S i+ S i ) (S i+ S i+ ) then Add S i to ax-list end if end for Add S n to ax-list return ax-list In the Reduction Algorith, the CPs are recurrently flagged as axia and inia, and the CPs that are not flagged are reoved. Each loop iteration increases the algorith depth value that represents the coplexity of the contour reduction. The loop finishes when all CPs are flagged. In the first part, the contour is used as the sequence of CPs for ax- and in-list calculus. In the second part, the ax- and in-lists theselves are used for the calculus. As an illustration, consider the reduction of the contour A 5 < > following Algorith. Step. N =. Step. axia flag: {A, A, A, A 5, A 7, A 9, A, A } (Figure a). Step. inia flag: {A, A, A, A 6, A 8, A, A } (Figure b). Step. Conditional: all flagged, jup to Step 6. Step 6. First part: Flag axia fro ax-list: {A, A, A 5, A 7, A, A } (Figure c). Second part: The repeated adjacent axia sequence ({A 5, A 7 }) does not involve either the first or the last CP. Thus, both are flagged. Step 7. First part: Flag inia fro in-list: {A, A, A, A 6, A 8, A, A } (Figure c). Second part: The adjacent inia sequence ({A, A, A 6, A 8, A }) does not involve either the first or the last CP. Thus, both are flagged. Step 8. The repeated adjacent axia sequence ({A 5, A 7 }) contains an intervening inia (A 6 ). Thus, the flag is kept. Step 9. The repeated adjacent inia sequence ({A, A, A 6, A 8, A }) contains a slice with two intervening axia ({A, A 6, A 8 }) and two other slices without it ({A, A } and {A 8, A }). The slices without the intervening axia have the flag reoved. The resulting in-list is {A, A, A 6, A 8, A } (Figure d). 5 In this analysis, we are representing the axia and inia using sall triangles above and below the CPs, with the letters (for axia) and (for inia) and the contour as a sequence A = {A, A,..., A, A }. 75

5 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Step. There are CPs not flagged ({A, A, A 9, A }), and the cobined axia and inia repeat ({A, A 5, A 6, A 7 }). Go to Step. Step. Reove the flags fro the cobined CPs, keeping the CPs near the beginning and end of the contour (Figure e). Step. The CPs flagged in Step ({A, A 7 }) belong to the ax- and in-lists and reain unchanged. Step. Reove non-flagged CPs ({A, A, A 5, A 6, A 9, A }) (Figure f). Step. N =. aintain the CPs as being unchanged. Step 5. N =. Increent of units: N =. Step 6. Return to Step 6. The second iteration begins with the CPs, axia and inia flags illustrated in the figure a 6 Step 6. First part: Flag axia fro ax-list: ({A, A, A, A }) (Figure b). Second part: The repeated adjacent axia sequence ({A, A }) does not involve either the first or the last CP. Thus, all axia are flagged. Step 7. First part: Flag inia fro in-list: ({A, A, A 8, A }). Second part: The repeated adjacent inia sequence ({A, A 8 }) does not involve either the first or the last CP. Thus, all inia are flagged. Step 8. The adjacent repeated axia sequence ({A, A }) has two inia intervene ({A, A 8 }). Step 9. The adjacent repeated inia sequence ({A, A 8 }) has not yet had the axia intervene. The flag of the one repeated inia (A 8 ) is reoved (Figure c). Step. There are CPs that are not flagged ({A 7, A 8 }). Go to Step. Step. There is no cobined CP repetition. Step. There is no cobined CP repetition. Step. Reove non-flagged CPs ({A 7, A 8 }) (Figure d). Step. N =. Increent of unit: N = Step 5. N =. aintain the contour as being unchanged. Step 6. Return to Step 6. The third iteration begins with the CPs, axia and inia flags illustrated in the figure : {A, A, A, A } and {A, A, A }. Step 6. First part: Flag axia fro ax-list: ({A, A, A, A }) (Figure ). Second part: The repeated adjacent axia sequence ({A, A }) does not involve either the first or the last CP. Thus, all axia are flagged. Step 7. First part: Flag inia fro in-list: ({A, A, A }) (Figure ). Second part: There are no repeated adjacent inia sequences. Step 8. The repeated adjacent axia sequence contains an intervening inia. Thus, the flag is kept. Step 9. There are no adjacent repeated inia sequences. Step. All CPs are flagged, and there are no cobined axia and inia repetitions. Jup to Step 7. Step 7. The contour A has a depth of, is reduced to < >, and is noralized to < >. i. Reduction algorith review There are two probles in this algorith:. Cobined adjacent axia and inia repetition is in Step. 6 For siplification reasons, we keep the original sequence index. 76

6 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Contour < > Contour < > (a) Step (b) Step Contour < > Contour < > (c) Steps 6 8 (d) Steps 9 and Contour < > Contour < > (e) Steps and (f) Steps 6 Figure : Reduction algorith flags in the first iteration 77

7 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Contour < > Contour < > (a) Initial state (b) Steps 6 8 Contour < > Contour < > (c) Steps 9 (d) Steps 6 Figure : Reduction algorith flags in the second iteration Contour < > Figure : Reduction algorith flags in the third iteration 78

8 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No...5 Contour < > Figure 5: Cobined adjacent axia and inia repetitions..5 Contour < > Figure 6: Contour < > and flagged axia and inia.. There is indefiniteness in the repeated axia and inia sequences of Steps 8 and 9. Step contains two conditions with two possible results: continue with Step or jup to the algorith s end at Step 7. In the second condition, the expression "no ore than one" leads to an error because prie contours cannot have cobined axia and inia repetitions, and this expression allows for at least one. Thus, the algorith ust not end while these repetitions are in the contour. For instance, the contour A < > (Figure 5) has all CPs flagged and only one ax-in cobined repetition of CPs and. Thus, in Step, the algorith ends, and the repetition is not reoved. To fix this proble, this step should be rewritten as "and there are no repeated cobined CPs in ters of the axia and inia." The expression "string of equal and adjacent axia" in Step 8 is not precise concerning the length of the string. For instance, the contour A < > (Figure 6) has three repeated axia. There are two possible interpretations of "string" in this exaple: the unique sequence with three repeated axia {A, A, A }, or each of the two sequences of repeated axia {A, A } and {A, A }. Considering the unique sequence {A, A, A } as the string, there is an intervening inia 79

9 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No.. Contour < >. Contour < > (a) with inia between the axia (b) without inia between the axia..5. Contour < > (c) with and without inia between the axia Figure 7: Contour with and without inia between axia (A ), and we should ove to Step 9. Considering each pair of repeated axia as a string, there is no intervening inia in the first pair, and we should reove the flag fro one of these axia. In the first case, with a unique string, the algorith ends with the unchanged contour < >, and with the repeated axia. In the second case, with two strings, one of the axia is reoved, and the algorith finishes with the correct prie contour < >. There are three particular situations for the equal and adjacent axia in axia-list:. with inia between the axia (Figure 7a).. without inia between the axia (Figure 7b).. with slices with and without inia between axia in the sae string (Figure 7c). In the first situation, all the axia flags ust be kept; in the second, only the first axia ust keep it flag; and in the third situation, in the slice without inia between the axia, these axia ust be unflag, and, in the slices with inia between axia, the axia adjacent to the intervening inia ust be kept. In this third situation, there are cases where two equal adjacent 8

10 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. axia are between two inia. In these cases, the second axia ust be unflag (See C 6 in the figure 7c). There is a siilar proble in the analogous Step 9, related to axia between equal adjacent inia. We propose a new version of the algorith with these questions reviewed (Algorith ). CONTOUR-REDUCTION-ALGORITH-REVIEW(C), where C is a contour. Let variable N. First part Step : Set N to. Step : Flag all axia in C upwards, and call the resulting sequence the ax-list; flag all inia in C downwards, and call the resulting sequence the in-list. All odification in a axor in-list will result in a new ax-/in- list. Step : If all CPs are flagged, go to Step. Step : Delete all non-flagged CPs in C, and increent N by (i.e., N becoes N + ). Second part Step : Flag all axia in the ax-list upward, and flag all inia in the in-list downward. Step 5: If there are no adjacent repeated axia in ax-list and inia in in-list, go to Step. Step 6: For any string of equal and adjacent axia in the ax-list: if First and last CP are present in the string then Flag the. else if First or last CP are present in the string then Flag it. else Flag all axia in the string. end if Step 7: For any string of equal and adjacent inia in the ax-list: if First and last CP are present in the string then Flag the. else if First or last CP are present in the string then Flag it. else Flag all inia in the string. end if Step 8: For any string of equal and adjacent axia in ax-list: if There is no iniu between the axia then Flag the first axia and unflag the others. else if There are inia between all the axia then Flag all the axia. else Flag the axia that are adjacent to the inia and unflag the others. if There is two adjacent axia between two inia then Unflag the second axia. end if end if Step 9: For any string of two equal and adjacent inia in the in-list: 8

11 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. if There is no axiu between the inia then Flag the first inia and unflag the others. else if There are axia between all the inia then Flag all the inia. else Flag the inia that are adjacent to the axia and unflag the others. if There is two adjacent inia between two axia then Unflag the second inia. end if end if Step : If there is no CP repetition in the ax-list and in-list (cobined), not including the first and last CPs of C, go to Step. Step : Reove the flags on all repeated CPs except those closest to the first and last CP of C. Step : If both flagged CPs reaining fro Step are ebers of the ax-list, flag any one (but only one) forer eber of the in-list whose flag was reoved in Step. Step : If both flagged CP reaining fro Step are ebers of the in-list, flag any one (but only one) forer eber of the ax-list whose flag was reoved in Step. Step : If all CP are flagged, go to Step 7. Step 5: Delete all non-flagged CP in C and, if N =, increent N by (i.e., N becoes N + ); otherwise, increent N by (i.e., N becoes N + ). Step 6: Go to Step. Step 7: End. N is the "depth" of the original contour C. Algorith Reviewed Contour Reduction Algorith IV. Equivalence Contour Class Prie For The equivalence contour class prie for is obtained with the algorith proposed by arvin and Laprade [7] (Algorith ). Algorith Equivalence Contour Class Prie For Algorith EQUIVALENCE-CONTOUR-CLASS-PRIE-FOR(cseg), where cseg is contour: Step : If necessary, translate the cseg so its content consists of integers fro to (n ), Step : If (n ) inus the last c-pitch is less than the first c-pitch, invert the cseg, Step : If the last c-pitch is less than the first c-pitch, retrograde the cseg. The ain feature of this prie for is the copactness in the contour beginning. This feature is inspired by the Post-Tonal Theory s set class prie for. The Contour Interval Succession 7 is a useful tool for how copact the beginning of the contour is. For instance, the contour A < > belongs to the sae class of its retrograde, inverted, and retrograded/inverted versions: R(A) < >, I(A) < > and RI(A) < >. Table contains the interval successions of these contours. The RI (A) version starts with the lowest positive value of the Contour Interval Succession. arvin and Laprade s algorith has soe conditions to invert and/or retrograde the contour to find the prie for of its equivalent class. 7 A sequence with the differences (or contour intervals) between adjacent CPs []. 8

12 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Table : Contour Interval Succession of contour A < >, R(A), I(A) and RI(A) Contour Contour Interval Sucession A < > [,,, ] R(A) < > [,,, ] I(A) < > [,,, ] RI(A) < > [,,, ] i. Algorith review This algorith, however, fails in 8 of 5 equivalent classes according to the arvin and Laprade table [7] (cf. prie for fails in Table ). Each of these classes has two prie fors. For instance, both contours A < > and RI(A) < > belong to the cseg-class 5-, but according to the arvin/laprade algorith, each one has a particular prie for. They are not changed by the algorith steps. Let the processing of the two contours A and RI(A) occur while coparing each step: Step : Both contour A and RI(A) are noralized. aintain the contour as being unchanged. Step : Both contours have the sae cardinality (N = 5), and the sae last CP (). Therefore, the condition (5 ) < is false. aintain both contours as being unchanged. Step : Both contours have the sae first () and last CP (). Thus, the condition < is false. aintain both contours as being unchanged. Hence, this algorith fails with soe classes. To fix this proble, we propose a new algorith that is tested using the arvin/laprade table (Algorith 5). Algorith 5 Equivalence Contour Class Prie for Algorith Reviewed EQUIVALENCE-CONTOUR-CLASS-PRIE-FOR-REVIEW(cseg), where cseg is the contour: Let the array arr: Step : Noralize cseg. Step : Get the result of inversion (I), retrogression (R), and retrogression-inversion (RI) and add the to arr. Step : Sort the array arr. Step : The prie for is the first contour of the array arr. V. Conclusions In this paper, we revealed probles with the Refined Contour Reduction and the Equivalence Class Prie For algoriths that lead the to fail with soe contour inputs. We deonstrated how and why these algoriths fail, as well as how to fix the issues, and we proposed alternative algoriths. The probles with the Refined Contour Reduction Algorith raised here do not allow for an output of the correct reduction of soe of the contours with cobined axia-inia like < >, and with strings of repeated axia (or inia) with intervening inia (or axia), such as < >. However, a review of the string definition and a sall change in the algorith s tenth step fixed these probles. The proble with the Equivalent Class Prie For Algorith led to it returning two prie fors for the sae class in 8 of the 5 arvin/laprade classes, as shown in the table showing the sae [7]. The alternative algorith proposed in this paper fixed this proble. 8

13 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. Table : Equivalent Class with Two Prie Fors Cseg class FP correta FP incorreta 5- < > < > 5-8 < > < > 5-5 < > < > 5-7 < > < > 6- < 5 > < 5 > 6-9 < 5 > < 5 > 6- < 5 > < 5 > 6-5 < 5 > < 5 > 6- < 5 > < 5 > 6-7 < 5 > < 5 > 6-5 < 5 > < 5 > 6-59 < 5 > < 5 > 6-5 < 5 > < 5 > 6-9 < 5 > < 5 > 6-5 < 5 > < 5 > 6-7 < 5 > < 5 > 6- < 5 > < 5 > 6-9 < 5 > < 5 > 6- < 5 > < 5 > 6-9 < 5 > < 5 > 6-78 < 5 > < 5 > 6-8 < 5 > < 5 > 6-8 < 5 > < 5 > 6-8 < 5 > < 5 > 6-8 < 5 > < 5 > 6-86 < 5 > < 5 > 6-88 < 5 > < 5 > 6-9 < 5 > < 5 > 8

14 usat: Brazilian Journal of usic and atheatics Deceber 6 Vol. I, No. References [] BEARD, R. D. (). Contour odeling by ultiple linear regression of the nineteen piano sonatas by ozart. Phd Dissertation, Florida State University. [] BOR,. (9). Contour reduction algoriths: a theory of pitch and duration hierarchies for post-tonal usic. Phd Dissertation, University of British Colubia. [] CLIFFORD, R. J. (995). Contour as a structural eleent in selected pre-serial works by Anton Webern. Phd Dissertation, University of Wisconsin-adison. [] FRIEDANN,. L. (985). A ethodology for the Discussion of Contour: Its Application to Schoenberg s usic. Journal of usic Theory, vol. 9, no. : pp. -8. [5] LEWIN, D. (7). Generalized usical intervals and transforations. Oxford University Press. [6] ARVIN, E. W. (988). A generalized theory of usical contour: its application to elodic and rhythic analysis of non-tonal usic and its perceptual and pedagogical iplications. Phd Dissertation, University of Rochester. [7] ARVIN, E. W., and LAPRADE, P. A. (987). Relating usical contours: Extensions of a Theory for Contour. Journal of usic Theory, vol., no. : pp [8] ORAES, T., and SAPAIO,. S. (5). Relações de contornos entre eleentos sonoros e visuais do jogo Super ario Bros. In: Proceedings of SBGae 5, pp [9] ORRIS, R. D. (987). Coposition with pitch-classes: A theory of copositional design. Yale University Press. [] ORRIS, R. D. (99). New Directions in the Theory and Analysis of usical Contour. usic Theory Spectru, vol. 5, no. : pp [] SAPAIO,. S. (). A Teoria de Relações de Contornos usicais: inconsistências, soluções e ferraentas. Tese de Doutorado, Universidade Federal da Bahia. [] SCHULTZ, R. D. (8). elodic Contour and Nonretrogradable Structure in the Birdsong of Olivier essiaen. usic Theory Spectru, vol., no. : pp [] SCHULTZ, R. D. (9). A diachronic-transforational theory of usical contour relations. Phd Dissertation, University of Washington. 85