JOSEPH S CH I L L I N G E R COURSE S P O N D E N CE C O R RE With: Dr. Jerome Gross Subject: Music Lesson CLXX. TWO-PART MELODIZATION This technique consists of writing two correlated melodies (two-part oou.nterpoint) to a given chord­ progression. The counterpoint itself must satisfy all the requirements pertaining to harmonic intervals. Each of the melodic parts (to be designated as Mr and M 11, or as CPr and CPrr ) must satisfy the requirements pertaining to melodization. The sequence in which two-part melodization should be performed is as follows: (1) the writing of�; • (2) the writi2�g of M with the least number of attacks per H; (3) the writing of M with the most number of attacks perH. It is not essential which melody is designated as Mr and which as M rr . Considering the natural physical scale of frequencies as increasing in the upward direction of musical pitch, we shall evolve t he melody with the least number of attacks ii.Dmediately above harmony, and the • - 0 2• • melody with the most number of attacks above the first melody. Such schemes will be considered fundamental and could be later rearranged. Thus we arrive at the two possible settings: (1) and (2) Octave-convertibility (exchange of the positions of MI and M11) is possible only when the harmonic intervals of both melodic parts are chosen with consideration of su�h a convertibility. This mainly concerns the necessity of supporting certain higher functions (such as 11) by the immediately preceding function (such as 9). All forms of quadrant rotation ( G), @, @ and @)) are acceptable on one condition: Mr and Mrr always remain above the chord progression (Ir). As melodization of harmony by means of one part produced different types of melody in relation to the different types of harmonic progressions, the same possibilit ies still exist for the two-part melodization. • It is to be remembered that some types o f melody in one-part melodization were the outcome of new techniques. For instance, the technique of modulating symmetric melody above all forms of symmetric harmony, or the technique of diatonic melody evolved fro m a quantitative scale above 0 ,,u • all forms of chromatic harmony. All such new techniques shall be applied now to the two-part melodization. This, naturally, will result in the new types of counterpoint. Ir The distribution of attacks of�' M 11 and is a matter of co nsiderable complexity and will be discussed later. For the present, we shall distribute the attacks for all three parts (M1, MII and r) uniformly and by means of multiples. Some elementm forms of the gistribution o f attacks. a 2a a a a a 2a H H a 3a H H MI. 9a 3a 12a 3a Mrr 3a 9a H H a 4a 2a 6a 2a 8a 2a 6a 3a 8a 4a a 4a H a 4a 2a 4a 2a a 2a 8a 3a 6a 4a Sa H H H 2a 4a H H 5a 3a H 6a 4a H H H H H H • • • 3a 12a 4a 12a 3a 15a 4a 16a H H H H H H H H Here the quantities of attacks in are designated per chord. Each original setting of two simultaneous melodies accompanied by a chord-progression offers seven forms of exposit_?.on,. (5) L (6) • • 0 4. ' L Melodization of DiatonicHarmony by means of Two-Part Diatonic Counterpoint. (Type I and II) The melody with least number of attacks and appearing immediately above harmony m u st conform with the principles of diatonic melodization. It is desirable not to include higher functions (9, 11) into this melody (we shall call it Mrr), for the reason that the latter cou ld be spared for the use in melody with the most number of attacks (we shall call it Mr ). Thus the high functions of M1 will be supported by MII• pcales of both melodies must have common source of derivation. This common source is the diatonic scale of harmony. Any derivative scales of the original d c an be employed. Harmony can be devised in four or five parts. Four-part harmony is preferable as the textu re of a duet accompanied by five parts is somewhat heavy. None of the melodies must produce consecutive octaves with any of'. the harmoo:m .parts. should be written as counterpoint to M11 and as melodization of the chord-progression. M1 Identical as well as non-identical scales (which derive thro ugh permutation of the pitch-units of d0) can be used in Mr, Mrr and r. Under such conditions any d0 produces 35 possibilities of modal relations between the abovementioned three components. 0 5. L As we are employing se ven-unit scales, '7 c3 = ° 3! 7t = 5040 6 •24 ('1-3) I = 5040 144 = 35 The number of two-part melodizations which is possible to evolve to one chord-progression (written in one definite d) l.S: - 5040 2•120 - 5040 240 21 Examples of Diatonic Two-Part Melodization Figure I . (please see pages 6 and 7) Chromatization of the Diatonic Two-Part Melodization. In order to produce a greater contrast between M1 and MII either one can be subjected to chromatic variation. If desirable, both melodies can be used in their chromatic version. Chromatic variation is achieved by means of passing or auxiliary chromatic tones. Example of Chromatic V ariation. Figure II, Var. I �d II. By mean s of combining the two variations of Fig. II, we can obtain a new version, where chromatic sections alternate with the diatonic ones. r1gure II, Var, II�. (please see page 8) 0 (4.) . u Figure I. . , t�� r ,. 'f;i . • I� - -eJ 1 ' .. r,I ....... - r - .,.It i: • - • F s l ,, I • If -- .. -- ,6 •� I "• A -.. -. --- f ! ' • * • ¥ -- - • - * I """ • - f $ $ - I � I .� -- I - I I r - ., � :t -- I -1 f ( $> , -- I , $ � . "" • � , • I -""' • � � .... .. � ,J , � . , ,... f t: , j . � • • ' � ·, 11 , � - I • � -- , "" - -- � � ,, L .. I "'I 1r-, • �- ,, pi 6,.I ,_ = '� -s9- I� I• L i,- � ,� -e, � ,_ • C No. t . Loose Leaf KIN ltAND tl:>9$ \\·••>'· N. 0 17. Lesson CLXXIII. Composition of Attack-Groups for I 4 the Two-Part Melodization Mr The quantity of attacks of Mr r H either constant or variable. can be A constant form of the attack-group talces place when every individualH has a definite corresponding number of attacks in Mr and Mrr , v1hich remains the same for every consecutive H. Mr Mrr = H • A canst. L A constant A does not necessitate an even a(Mr ) distribution in a(Mrr) · An even distribution may be considered merely as a special case • E xamples of an even distribution of A: 4a 6a 6a 8a 8a 9a 2a 2a ·3a 2a 4a a a a a a 12a 12 3a 3a 4a a a a E xamples of uneven distribution of A : Mr 2a+3a 4a+2a 4a+2a 4a+6a MII a+a a+a 2a+a 2a+2a a a a a H Mr 4a+2a+3a+6a 6a+3a+6a+4a+2a+9a M rr 2a+a+a+2a 3a+a+2a+2a+a+3a a a H • 0 18. A variable form of the attack-group takes place when A emphasizes a group of· chords, and when each consecutive H has a specified number of attacks for a definite quantity of chords. For example : MI - 2a+a Let A , = II _ a+a a • and let A 3 Mr Mrr = ----H � = A ' + 2 + A3 MI = 4a+3a M rI - 2a+a H a and let A 2 _ 4a+6a+3a _ -------2a+2a+a a then : 4a+3a 2a+a 2a+a a+a a A 4a+6a+3a 2a+2a+a a a Ha All other considerations concerning the distribution and quantities of attacks are identical with one-part melodization (see : "Composition of the Attack­ Groups of Melody" in the branch of Melodization o f Harmony) . Example of Correlated fotta�k-Grou£s in Two-Part Melodization Figµre VI. Mr Mrr -� - 2a+3a • a+a a + H, + 3a+4a �+a a H� + + 4a+3a+2a a+ a+ a a H� I::. . .Q.E. = s 6J2 , S (9) canst.; L l3 XIII; P ' transformation : , '-- T" - 3 12t l.Il 4 • time. � 0 19. Figure VI. � - , ... .. ..- ..... .... .�.�-- - ;.,-;• a •� � 'loo • • .. , • A A r, ... l' I • - �= • � - • , ' .,>-- ,• -f ' ; I� , � ., p 4 - � 'I .. . ., - � II ' k . ,.. �- .... ' • � � I • • • -• • �= � I �: I I - !!!!!!!!!!! � i- •• ' • .' • I J• .. �- •' ' i • • L.i.� --I. C. • f I -- J � a �-... � . - II :, :,: I �f �·. II', � f:r:�... � � - ' " . - -· ... J f ·- r. � I �: "' . • No. 1. l.oose Leaf KIN t59ll tl° w�y. ff. 0 20. Composition of, Durat�ons for the Attack­ Groups of Two-Part Melodizatio n • Selection of durations and duration-groups satisfying the attack-groups com posed for two-part melodization can be based either on theSeries of the - E volution of Rhyth m Families (in which case there is no i nterference between the attacks of the attack-group and the attacks -of the duration-group) or on a direct compos • ition of duratio�-g;:ouFs (which may or may not produce • an interference between the attacks of the attack-group a nd the attack of the duration-group) which would be super­ imposed upo·n the attack-gro ups. When the respective attack-groups are represented S eries, and the number by the durations selected fromStyleof individual attacks in the attack-sub-groups does not correspond to the number of attac ks in the duration-groups, it is necessary to split the respective duration-units. This consideration concerns the first technique only (i.e. , the matching of attack- groups by the series of durations). Musical example of Figure VI is a translation of its corresponding attack-group into j series, where three 1, 1 a nd 1 . One 3 4 2 exception to the series was made at the cadence , where a 4 series binomial, i.e., musical quarter was split in to 4 3+1. The numerical representation of this example of types of split-unit groups were used : melodization appears as follows: 0 21 .. 1/2t+l/2t+l/2ttl/2�tt t + + + + 2 t 3t + H, + l/3t+l/3t+l/3t+l/2t+l/2t+l/2t+l/2t + 2 t 3t t 1L4t+l/4t+l/4t+�/4t+l/3�+1/3t+lf3t+l/2t+l/2t + t + t t 3 t The ab undance of split units and split-unit groups in this instance is due to the abundance of attacks over each H and to relatively low value of tha series. With a series of higher value, the splitting of units would be • greatly reduced • We shall translate now the same example into i 9 series : MI = t+3t+t+3t+t + t+2t+t+t+2t+t+t M II - 4t +5t +5t H ' + 4t - _..::::.;;_.......,;�-9t � 9t ----- + t+t+t+t+t+t+t+t+t +3t +2t + 4t 9t -Figure H3 VII. (please see next page) 0 22. ...··-·--..- ' ' al"'• • -� :.' ' � I .. - � f-'.� �.... I II L, • ' ' ,� �-.. .. .. I '. I� r ' ,,. • I� 1..- n I� I.... I - • � � , - ' •� I�• .., . "' • � �( - ..I . I• I J_· _, .. • . • • 2 • • . . .., � d I I . I -- .. � � .I �: • • ' I ., - , I I . - -.-·IO I . • a J,J:-� T • - "--. I• • • ' . � -• . i,, {;, +: � • - • J. - ...... .- ■.. s ' I • •• II - •- ff ,.., ' ] - . . ..,., . ..:...__-,,,,- . • • • I' � - • • ..__ ,� I= • I t I. I • -� - - • • I.A ... - - .. ( - ■ . .......... . -[ • ' - • :: • � • No. t. Loose Lear 0 C Now we shall take a case where the attack and the duration-groups are composed independently. Let r5+4 represent the quantities of attacks of M 1 to each attack of Mrr, and let ever.y 2 attacks of M 11 correspond to one attack of � . Then the distribution of attacks for all three parts takes the following appearance: a (M 1) - 4a+a aCllrr) a(H '') 3a+2a 2a+3a + + a+ a H 2 a+a H, a+ a + + a a a • a+4a + Ha a+ a a H'f Let us superimpose the following duration-group : T L Then : = r4+3 = $= f§ = Hence, T • = 16t •2 Let T" = f; 1 ( 20 ) 2 (10) 32t = at, then : NT" - l6t; lOa = 32 = 4 8 Each a(M1) corresponds to an individual term of T; each a (Mrr) corresponds to the sum of the respective durat ions of Mr; each a (�) corresponds to the sum of 2 durat ions of M II . The final temporal scheme of this two-part melodization takes the following form : Mr M II + + 3t+t+2t+t+t +t H, 7t 8t 3t+t+2t+t+t 4t +4t Ha 8t + t+t+2t+t+3t +4t + 4t 8t + t+t+2t+t+3t + t+7t 8t H2 + + d 0 � C 24 . Figµre VII.I,. • . l [ ( • ! ] I ( - .. . .,) I • e --. 1 ' �· • jjiil .. p - • , �I - � f ' ., ' ' .. 2- I$ -' ◄ ,. .. - ' .. • ... T I - � ( I • #" - $ � � I • 7 � II 1r SI - - I • I6 � - I,,I • - • • - � I I No. t. Loose Lear KIN 1Sli$ aAJIID e·w,y, N . 0 25. Direct Composition of Durations for the C C 4 Two-Part Melodization Direct composition of durations becomes particularly valuable, when a prop9Ftiona�e distribution of durations for a constant number of attacks between the component parts (Mr, Mrr and �) is desired. Distributive involution of three synchronized powers solves this problem. As it follows from the Theory of Rhythm, the cube of a binomial produces an eight-term polynomial, the square of a b inomial produces a quadrinomial and the first -power group remains a binomial. of attacks of the two adjacent parts two. Thus, the quantity Mr Mrr is and MII � Cubing of a trinomial gives a twenty-seven-term polynomial, the synchronized square producing nine and the first-power group -- three terms. The quantity of attacks between the two adjacent parts remains three. Thus, the number of terms of the original polynomial equals the • quantity of attacks between the adjacent parts . We shall devise now a correlated proportionate system of duration-groups. The distributive cube will serve as T for Mr , the synchronized distributive square as T for Mrr and the synchronized first-power group as T for Ir. We shall operate from the trinomial of the series. This secures the following attack- group correlation: 0 26. a (M1) a ( ia:11 9a -) = ;·(a 5) 3a The entire temporal scheme assumes a the following form: T (Mr) = [(8t+4t+�t) + (4t+2t+2t) + (4t+2t+2t) ] + T (M 11) = (16t ) + + at . + 8t 32tH , T (Ir ) + [(4t+2t+2t) + (2t+t+t)_+ (2t+t+t) ] + + 4t_ ) + + 4t + (8t 16t H2 2t+2t) + (2t+t+t) + (2t+t+t) ] + t 4 [( + + 4t ) + 4t + •(at 16t H3 L Fig_yr� ,ItC. (please see page 27) In addition to this technique, coefficients of duration can be used for correlation of durations in the two-part melodization. Example : Mr Mrr • = == (3t+t+2t+2t)+(3t+t+2t+2t)+(3�+t+2t+2t)+(3t+t+2t+2t) (6t+2t+4t+4t) + (6t+2t+4t+4t) • 0 27. Figure IX. • f ��. J �A :� ,11�, � • I -bl lllliiiiij - • ( [ ! = � ( - = - � ] I • � ] I, • - ..:e- � 1'1 . 1 • ' � ,: ,� lr- •- • - i i7 - �� ' '-"' 6 -· � I � ( � ! s I � � - f_L _L - .. .•• . '.. $l • • � � -- ! � - , - f: '� • No. 1. Loose Lear 0 28. Lesson CLXXIV. Compopition of_pontinuity in Two-Part Melodization • The seven forms of expositions previously classified can be now incorporated into continuity of two-part melodization. The applied meaning of these seven forms can be expressed as follows : (1) MI (2) M11 (3) � -- Solo --- melody : theme A; Solo melody: theme B ,· Solo harmony: theme C; (4) � -- Solo melody with harmonic accompaniment (theme A accompanie d) ; (5) ;; -- Solo melody with harmonic accompaniment (theme B accompanie d) ; (6) Theme A Duet -of two melodies ( The me B ) • (7) Duet of two melodies with harmonic accompaniment Theme A Theme B Theme C The above seven forms serve as thematic elements of a composition, in which they appear in an organized sequence producing a comple te musical whole. Themes A, B and C must be conside red as component parts of the whole in which the y e xpress the ir 0 29. L· particular characteristics. These characteristics which distinguish A from B and C are : (1) High mobility of A (maximum quantity of attacks) ; (2) Medium mobility of B (medium quanti ty of attacks) ; (3) Low mobility of C (minimum quantity of attacks) combined with maximum density (four or five parts) . The planning of continuity must be based on a definite pattern of the variati on of density combined with the vari ation of the quanti ty of attacks. scale of densi ty can be arranged from The low to high as follows : A Iv. • B A, c, (1) , B' C' C A B, B • (2 ) B, B' c, C C L - - More or less extreme points of any such scale produce contrasts. (1 ) -B + A C (2) A + For instance: A B+ A + C + B + C + B B + A + B + ,· C A A C C C + B + C + A + B + B + B + A + B c c A Durations corresponding to one individual attack of the component of lowest mobility (mostly H4 ) become time-units of the continuity . Such units (we shall call them T) can be arranged in any form of rhythmic distribution. Correlati on of the thematic duration-groups 0 30. ( T • s with their coefficie.nts) with the different forms of density constitutes a composition. Assuming that there are three forms o f density and three forms of mobility, we obtain the following combined thematic forms (Low, Medium, High): Low Low Density Mobllity I Low Medium Medium High Medium Low High Medium Low High High Low �- g� g Thus, for instance: Density = High = Ir' . Mobility Low ' Medium Medium 3 2 = 9. _M Densi:tY _ Low = II ' Mobility - Low Density _ High = MII Mobility - Medium - r etc. We shall now devise a composition which will combine �he gradual and the sudden variations of mobility and of density. It i s desirable to have such a scheme of two-part melodization which is cyclic -. and recapitulating, i.e. , one permitting a correct transition from the end to the beginning for all three components .. For the present, we shall not resort to any additional techniques (such as inversions, expansions etc.) , as the complete synthesis will be accomplished in the branch of •Composition • Let Figure VIII serve as the fundamental scheme of two-part me lodization, as this material is 0 31. cyclic and recapitulating. Let us adopt the following scheme of density and mobility: Density = Low + Low + Medium + �i gh + �igh + Medium +High Mobility Low High High Medium Low High High The sequence of thematic elements and their combinations, corresponding to the seven forms of �xpositions and satisfying the above scheme of thematic forms may be selected as follows: r E + Mr 1 E M E + r M 2 � 3 � 't + �E • J,.. + Mr Mrr We shall make T correspond to H and establish the following sequence for the T • s: T = r 5+3 • � :;: 7T 15H. The 7T of � produce no interference in relation to the 7E of W. There is an interference between � V and 8 . :rr-' , however, as � == H 8 ( 7) . 7 (8 ) ' r' = 7 • 8 = 56 TE. As 7 TE corresponds to 15 H, there will be 7 TE• 8 = 56 TE and 15 H•8 = 120H. Thus the complete composition after synchronization evolves into the following form: � ' �• = 56 TE 120H; 0 32. As in Figure VIII T11 = TH, the entire composition consumes 120 measures, which is 15 times the duration of the original scheme of melodization. • Here is the final layout of the composition : Figure• X. + Mrr ( if""" H7 + H,9 + H , ) T., E44 + ff'(H2 ) T£ ES" + M� M I (H3 + H�)T6 E6 + Mr + M rr (Hf + H6 + H..,) T,, E 7 ] + [ Mr r (H, + H, + H 2 )'rg E i + !rt + Mr + Mr M rr (H3 + H ., + Es-) T"E ,. , ] + Ir+ + Mr + � (H"i + H., + H0 ) ¼EAS" + wt (H7 ). T.;t E. E�E>+ (H 3 + H") T9 E '¾ + ' (H5) T,0E 1 0 + � (H& + H7 + Hg) • [ MII ( � + 81 (H 1 + H�) T_;E � :\+ � (H3 ) T� E�" + t/ + H e) T��.t_,+ 0 33. M + r JP Mr � (H, + H� + H 3) T3s- E 3,s-] + Mrr Ts-� Es-.t + IF, (lJa + H , + H 2 ) T33 E.,3 + �(H 3) T.sil E.r-"4 + Mr ./'+ MII TS.> ES;, F • 0 34. • ... t L.esson CLXXy, X• • I • � .-/} - • i,' •Figure - ! - -s .- lrf - • " • , . • :'l C �- ' z [ • - If � - .ii; ,,, I - - !!!!! - • • i.-.. .. -- - · � I .. -tr • � • . ' - � - - ., ' I � . ;II r -- $ [--fl: • -1 ---.; '• --- ! I - - � ii No. t . l.ooae Leaf KtN 1,110 169:) 8 way 'N. 0 L Ii�. -• - .... . • • �· -:,. � . .. ,: 35. • J ,I � I • • � I, * � .. , ' ' •I l:5- -0 -. • I � • - t ( Iii I i!!! I, 'I L , . I I J �_ -s � .. ,. ,I • 116 • • • I I• l ' l!! , II , - � � - - ' i.• 'I ' ,. , $ * - � J No. t. Loose Lear KIN RAIID 0 36. u - ! • �· i,,, I • �- I iiiiiiiiiiil - -• I I 2 • ... ! ( I I .. • -9" ·- � .. .,. I �- -- i.,,II �· ' J * � I J � - .,.... $". - [ • - , , - It (!I) � • � ,I � � . , - $ , • - I ( ii! � • ,--- - i !.• .,, , • -••• * If' No. 1. Loose Leaf KIN 1a110 169$ 11·w,y. II, Y. 0 37• • L • � .... s.,. 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Loose Lear "'" 15$1 8'way. N. Y. 0 - , � 'J �. • - ; ' 41 . • I[, I ' ' � ' �.) • - - I� e '- te • "-j • - ' ' -'_. ' ' $ $; :2 ,. • - �I e • - :. L - - I "$ � -:,. • - � . I It c>, _ " -, - T � � I i � • • 'II� -· � • - ' I I iii . .. , ,. No. t. Loose Leaf KIN tr.1'& s·••J. N. Y. 0 42. .I I - u-... � 1.-� - ::2 ·� , � I :I - .. .. " r � . •� ... � () \....../' - - � , ... , �' � - I I � :!!!!!! - ( _.. - � I t - I- 9 I I $. -I ( � I I• - • .. � "2Sg � II � • ' J• • -,. I -· • � J "fli -- I • � - No. t . toose Lear KIN f t&D:1 l way. N. Y. 0 I� ·- -• � ! • I 43 • • ( . � i,, II � •• :a,- '--� � ] . .. Ji. .. If • • ,� -- ., . • • • • ,, r-.J ... - � ��· .. """ • ·� � - • T ..., .. - -· � ;i 4 - - - t I � I • • • , ... • ' - -- * "li- - - 2!ii • it' - .g I • � - � � ,_ A � ]f 9·' L �. " '. I - - :i. , • - .. i r No. 1. Loose Leaf KIN 111,� 8way. N. Y. 0 J O S E P H S C H I L L I N G E R C O R R E S P O N D E N C E With: Dr. Jerome Gross C O U R S E pubject : Music Lesson CLXXVI. ' TWO-PART HARMONIZATION The principle of writing a harmonic accompaniment to the duet of two contrapuntal parts consists of assigning harmonic consonances as chordal functions . Every combination of two pitch-units producing a simultaneous consonance becomes a pair of cho.rdal functions. This premise concerns all types of counterpoint and all types of harmonization. Pitch-units produc ing dissonances are p erceived throug h the auditory association as auxiliary and p assing tones . Justification of the consonance as a pair of chordal functions gives meaning to the harm onic acc ompaniment. Diatonic Harmonization of the Diatoni,c Two-Part Counterpoint, Under the conditions imposed by Special Harmony, • two-part counterpoint , which can be harmonized by the latter, must be constructed from seven-unit scales of the first grotip, not containing identical intonatioris. As all three components must belong to one key, u according to the definition o f diatonic, the only types of counterpoint which can be diato nically harmonized are types I and II . 0 2. It is important for the composer to realize the modal versatility of relations which exist between the three component s. As M1 may be written in any of the seven modes ( do , d , , d 3 , d�, dq , %-, d0 ) of one scale, and so may M11 and the i:r-t, th e total number of modal variations for one scale is : 73 = 343. This, of course, includes all the identical as well as non-ident ical combinations. Practically, however, this qua ntity must be somewhat limited, if we want t o preserve th e consonant • relation between the P.A. • s o f M1 and M1 1 • It is important to remember that the number of seven-unit scales not containing identical units is 36. Therefore the total manifold of relations of Mr : in the diatonic counterpoint of types I and II is: 343•36 M1 1 : � = 12,348. Any given combination can be modified into a new system of intonations, i.e. , into a new scale, by mere readjustment of the accidentals. All th e above quapt ities, nat ura lly, do not include the attack-relations which have to be est ablished for the harmonization. r M are fixed groups, the As the attacks of MII only relation th at is necessary to establish concerns � . The most refined form of harmonization results from I assigning each harmonic consonance to one H. If counter- 0 3 ., \_ point contains many delayed resolutions of one dissonance, then the number of attacks of MI is quite great and the changes of H are not as frequent. On the other hand, direct resolutions produce frequent chord changes. The assignment of two suc cessive harmonic consonances to one H, amplifies the number of chords satisfying such a set, but at the same time neutralizes somewhat the character of P . This technique, however, permits a greater variety of ' attack-relations between the three components. We shall now proceed with the two-part diatonic harmonization. Let us harmonize counterpoint type I I , where = a . In s uch a case all the harmonic intervals are consonances . of attacks: Therefore we can have the following mat ching Mr = a Mr = 3a Mr = 2a = 2a Mrr = a Mir = 3a etc. Mir �= a = -; � = 'a r E x�mples of Diatonic Harw9nization of �he Two-Part Coun�erpoint Mr = a . Mrr Figure �. (please see pages 4 and 5) E xamples of Diatonic Harmonization of the Two-Part 3a _ - --Counterpoint 4C I -Figure I I II. (please see pages 5 and 6) E xamples of Dia�onicHarmonizati9n of the Two-Part Counterpoint MI - 4a M11 and 6a a Fig,ure III. (please see pages 6 and 7) • 0 4. Figure ,I, u lt\EME: d✓,,,, '·.) ... I, ',. th5 , ... - • , . � 0 ::c 9 • -! \..I ARl'AOK\2.ATION ( 1 ) ,. -· •.1 • • - � .. . ' ., � ,"- of, ., I - - -- ' � ,- � l� • . � � ! I� ·� - r ,, l'l , '�5"$= , � -- - I, , ' . -s � . � � , � I. 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A chromatic variation of the diatonic harmony accomp_anying two-part counterpoint can be obtained by means of auxiliary and passing chromatic tones. Of course such altered tones shall not confl ict in any way with the two melodies. For our example we shall take the two-part counterpoint diatonically harmonized from Figure III (2) . �mple of the Chropiatization t. pf Harmonic_ Accompaniment Figure IT. (please see page 9) Diaton ic Harmonization of the Chromatic Counterpoint_ Whose Origin is Diato�ic (Types I and II) The principle of this form of harmonization consists of assigning the diatonic co11sonanc,es as chordal functions. Chromatic consonances as well as all other forms of harmonic intervalsshall be neglected. The quantity of suc cessive consonances corresponding to one H is optional. It is practical to make T or 2T, or 3T correspond to one H. When harmonizing a chromatic counterpoint, whose diatonic original is known, one can assign chordal • 0 9. Figur� IV, ,_ .. . ... ' t I I � JI:= ] !!!!! � ,_ - � � � - ' ... . r-.. � ... - I, u � - .--r .. • - � � � I• ' __. � ·- • ' - �- • • 1lliiiiii iS . • �--::!!! •r [ r � • -t �- - .,_ • . •� ==r '� I• • ' - • - - . • ■ • I •' .. ..-·"" • •► I • • • • • ,. f • .... r tf r r • I• -,. ..... . �r , • • , � •• I'": - J ::t:t::..• ..... 0 � ...... ... � ,.. No. t. l.ooae Lear KIN ■AIID 1r,9:, -e·•�y. M. Y. 0 10. functions directly from the diatonic original . This measure obviously eliminates any possible confusion of the diatonic and the chr.omatic consonances. We shall now harmonize a duet where both parts are chromatic. The theme is taken fro m Figure XXIV of the Two-Part Counterpoint. For clari ty's sake, we shall write out both the or iginal and the chromatized version. We shall choose the following relationship between W and •• which is a modified version of the r3+ 2 , and �which permits to demonstrate the diversified forms of attacks groups of ' Mr and Mrr i n relation to � . Example of Diatopic �rmonization of the Chrom�ti�. Count�rp9int Figure V, (please see page 11) When the diatonic origin of chromatic counter­ point is unknown, the analysi s of diatonic consonances must precede the planning of harmonization. 0 figure ORl�INAL. ., 'l, • � 13 • � r 11. v. • •• �- • • H. 1 � i ,.,_ I• • j t/ ( • 4 7 7 3 £" ; - . .. ' CHROIVlFrflt V�RIRT1otJ • • • �- :.= ,. u � • ..., •• ..• ' �-Ci • 11 !.. •� ·•r � I. £ ■ I . . • ..,. - I ' •• I -s. • �-. � . • �- .t:� �: ..l II ' • ' • - .. . ft '' ... • ... M" ... -· J - -,:1::_ • • • .1• • •• � ' • """ . J. l • *· I.I r I Mi� I • I II. • __l *' =f: . _j - - l :; • • • � .J iS, ..., � • 7S'· • :e-, J � J. . ... • No. 1 . Loose Leaf . KIN 1&8$ e·way. II. Y. 0 12. Lesson CLXXVIII . Symmetric Harmonization of Diatonic . the .. Two-Part Counter�oint (TYPes I1 II, III and IV) . The principle of gmmetric harmonization of the two-part counterpoint consists of assigp.ing all harmonic intervals as chordal functions . The fewer attacks of M r and Mir correspond to one H, the easier it is to perform such harmonization by means of one i:' 13. When a considerable number of attacks (even i n one of the two melodies) corresponds to one H, it becomes necessary to introduce two, and sometimes three L 13 . The forms of the latter should vary only slightly, servir1g the only purpose of rectifying the non-corresponding pitch-unit. For instance, when usirlg !: 13 XIII as � , , correction of tbe eleventh to f tJ gives satisfactory solution for most cases. Thus, -r- 2 in this instance differs from !:_ only with respect to 1 1 . ' The selection of the original 2 13 is a matter of harmonic character . For example, the use of X- 13 XIII attributes to music a definitely Ravelian quality. However, harmonic quality still remains virgin territory awaiting the composer 's exploration . Most of the 36 forms of the L 13 have not been utilized. Whether counterpoint belongs to types I and 0 II, or to types III and IV, it does not give any clue to any particular � 13. And whereas symmetric • 0 13 ., harmonization of the counterpoint of types I and II is a luxury, it is a bare necessity for types III and IV, as the latter correlate two different key-axes. The fact that two different keys with identical or with non-identical scales can be united by one chord is of particular importance. This is so because the quality of a selected Z: 13 is capable of influencing the two melodies ., The ear in our musical civilization is so much conditioned by harmony, that most of our listeners have lost the ability of enjoying melodic line per se. And if the ear of an average music-lover can relate one diatonic melody to some chord progression, the harmonic association of two melodies belonging to two different keys becomes impos sible. Therefore the role of a harmonic master-structure ( -Y: 13 in this case) is one of a synthesizer. 0 The simplest way to assign harmonic functions is by relating the latter to consonan ces first . The master-structure used in the following harmonizations is r 13 XII I . Symmetric Harmonization of the Diatonic Two-Part Counterpoint of Types I and II. F,igµre VI . (please see next page) 0 =[ , --� III p L _. .. , t t I� -7ir -- p"2f, ,. .., • ,_ R �· J --,.,.Aal • ltt; I V � , II& Ir · r • 9"°. �·. - '" . ... ��· . I I! � - � r· • �: :l - I • ' -1 � • !!i I . • .. I • Jr T - • • J • • _,I • • -.• ... • • "" . , I • � -d': ·- • • -. - - , • •• • • . , • ... -� . . • r- • • I I I I• • j • •• • 7P: . "I , -· ,, '• • • I • --. - • c. • • �: .. . J. - C:l • � rS1 �� �- ! rr • �-. . , J• , • � I •• C' J• .. �--- ti I '..,. • -• • • ..u::).:, - ! , - • , • ?P: • . -�. • :!: • ' ,. • I• �- f.l • !. II • ' • �. I . � i, II .,_ ' t � :$1. - " I ] I0 It• , ..Au It - ! . , • , 14. 5 I I ... � • '°' • � ' . I I .,. ! I • :i G: -- , Figure V I . . �r I • No. t. l.oose Lear - • • "'" ° - . . • aAIID I&&$ 8 way, IC.Y. 0 15 . Chromatic variation of r in the above example is obtained through the usual technique: the insertion of passing and auxiliary units. Symmetric Harmonization of the Diatonic T�o-P�rt Counterpoint of Types III and IV. Figure VII. (please see page 16) Symmetric Harmonization of the Chromat�c Two-Par� Counterpo�nt Whose Origin is Diatonic (Types I , II, III and IV) . The principle of symmetric harmonization o f the chromatic two-part counterpoint consists of assigning all the diatonic pitch-units of both melodies as chordal functions of the master-structure ( r 13) and neglecting all the chromatic pitch-units, as not belonging to the scale. It does not matter whether the chromatic units belong to the master-structure or not . When the diatonic original of the two-part counterpoint is unknown, the diatonic units of both melodies should be detected first. Figure VIII. (please see page 17) Counterpoint executed h1 symmetric scales of the Third and the Fourth Group can be harmonized by 0 16. Figure VII . -0 .2. +- t::, - 0 "'c • -I r,- • i.. -I ,,... _ --1•� I _,., -!'- (l) • -p'O . a. • • ] - u- � .I � � I � •·· I I V •► J � ·-• - ,. - w. .,, � I • t• '- +� I ila .- ITI I ,z: ,, iS . ,,.., :,•i,"> ' � � - ·► • "" W- • j t,a. �� • • 1-f?" - " -0 .... �. II � • P.2. � � J • ,I V - 0 .... -- • ,, J l. - ... k- >s CHR0Mf\11e 'IARJAIIOH OF HRR.N\DN'( :i: V • I, ,.,, � " V'-' '""' � �� I • - • I J • - r¥ "'� I� � ' No. t. Loose Leaf .... " • � • It:rs I 0 .. • ft ,: II . • � -�- .. ""• I II H•rI ' I I J , . " . IG J F,t. .... , � ' ...t:::i � :, i�� I ' ·; 1f'; • . '., h� I d , • - I I J -. ... -.l • - . .. 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All forms of contrapuntal continuity as well as complete composit ions in the form of canons and fugues can be harmonized accordingly to this technique. Any of the above described correspondences between counterpoint and harmony can be established by the composer. One sh ould remember that overloading harmonic accompaniments is more a sin than a virtue. For this reason the technique of variable density should receive utmost consideration. 0 • 19. Figure IX, Tl-¼E.N\E: . I I[ " .... ] i- ,�� .. -- -s • ' • - • V" - � .,� I • Iy .l l . •] - , I ,J -e- ,.., � HARN\ON \ Z. f\ii ON C.HROMA11 t. ., . - �- I • I -� \/E\RIA110N POF HARN\OtJ y '•• -. ... .. ....- ,.� J '� •• 4: •► ' •• � .,. I • .L I I �� i� :;= • f: !: .: -. ;, I I I I � -s -s ., - � 1: 1:.. C � '• .... /J :, u ti� - I' .,'" � � '. • . I I � 1J • i"1 ,, I ..,l �1 I '�I Y- � -e , . �. ' f � ' I • ' -·- � :.e I • ., ..f .,. I ., "- � � • • No. 1 . Loose Leaf ... , KIN •AllD 169$ 8 w,y. 11.Y. 0 20. u Lesson CLXXIX. Ostinato Forms of ostinato or ground motion have been known since time immemorial. They appear in different folk and tradit ional music as a fundamental form of improvisation around a given theme. The characteristic of ostinato (literally obstinate) is a continuous repetition of a certain thematic group, which may be either rhythm, or melod y, or harmony. For example, the dance beat of 4/4 in a fox-trot is one of such fundamental forms of ostinato . • As a matter of fact, a rhythmic ostinato is ever present in all the developments in classical symphonies. Take, for example, Beethoven's Fifth Symphony, the first motiI of it consisting of 4 notes, and follow it up through the development (middle section of the first movement). The motif, rhythmically the same, changes its forms of intonation either melodically or in the form of accompanying harmony. Repetitions of groups of chords, as well as repetitions of melodic fragments accompanied by continuously changing chords, are forms of ostinato. Ostinato is one of the traditional forms of thematic growth and, as such, is very well knovm in the form of ciaconna and passacaglia. In many Irish jigs, ostinato appears in forms of pedal point as well as in repetitious melodic fragments. When 0 21. portions of the same melody appear in succession, being harmonized every time anew, (which may be found even in such works as Chopin 's Mazurkas,) we have a case of ostinato. I . Melodic Ostin ato . (BasJ30 Ostdnato) Melodic ostinato, better knovm under the name of "Ground Bass " , is a harmo nization of an ever-repeating melody with continuously changing chords. • Ostinato groups produce one uninterrupted continuity where the recurrence of the bass form produces unity, and the acco�panying harmony - variety. • All forms of harmonization can be applied to the conti nuously repeating melody, and regardless as to whether it appears in the bass or in any of the middle voices, or in the upper voice (above harm ony). As every harmonic setting of chords is subject to vertical permutations, a basso ostinato can be trans­ formed into tenor, or alto, or soprano ostinato, i . e., it may appear in any desirable voice and in any desirable sequence after the harmonization has been completed. In the f9llowing example the ostinato of the theme is a melody in whole notes in the bass (the first itself four bars) , after which it repeats;two more times. The form of harmonization is symmetric in this case, though it could be diatonic or any of the chromatic forms. This device can be used as a form of thematic developme nt, 0 22 .. • and in arranging for the purpose of constructing intro­ ductions or transitions, as any characteristic melodic pattern can be converted into basso ostinato either with the preservation of its original rhythm or in an entirely new setting. * Figure I . Melodic Ostinato Basso Ostinato (Ground Bass) Symmetric Harmonization of the Bass . • u *See: Arensky's "Basso Ostinato" for Piano. • 0 23. Il. Harm onic Ostinato Harmonic Ostinato may be also called, by analogy, "ground harmony". It consists of the repetition of a group of chords in relation to which a continuously. changing melody is evolved. This form of ostinato is the one J. S . Bach employed in his D-minor "Ciaconna 11 for Violin, besides numerous other compositions by Bach and other co�posers. • Among my students, a successful use of this device occurred in an exercise made by George Gershwin, and which later , at my suggestion, was put into the musical comedy, "Let 'Em Eat Cake", of which it became the h�t song ( 11Minen). This form of ostinato can be applied to any type of harmonic progressions. The technical procedure · is exactly the opposite of the first one. we deal with melodization of harmony. In this case As in the previous case, the melody evolved against chords may be transferred to a different position in relation to chord by means o f vertical permutation. Naturally, not every melody will be equally as good under such conditions if it appears in the bass and in the soprano, as th.e chordal functions represented by melody may be more advantageous for an upper part than for the lower , or vice versa. In the following example, the harmonic theme of ostinato emphasizes four different chords (the first two bars ), and is based on a '£ 13 XIII . The melody 0 24 . evolves through the principle of symmetric melodization forming its axi s points in relati on to the chord structure itself. The main resource of variety is the manifold of melodic forms. Figure II. ... IIUI• • Harmon¾£ Ost!neto (Ground Harmony) Symmetric Melodization of Harmony '" ! -� !j ·£ :$,: - �- � •.. t. ,,, ' • �: IIGJ; ... � •!. .. .. k !. I • lii!!!! .i � I• [ r• iii ,,. P.!..,.. ·1 • - - .iiiill ?£i - • t f..... +- �: Pc� L "" .a. • • ' � • � .... j� -t.. ..... � i.. I,. .......l I ,.., � bbf � ,:i � J ·or• , ""' .:. � ,. � III. Con trapuntal Ostinato . • The form of contrapuntal osti nato is well lmown thro ugh the works of old masters, and was usually evolved to a melody known as ncantus firmus n . ,.:f2f: � If a C .F. repeats itself continuously a number of times while the contrapuntal part or parts evolve in relation to it, 0 • 25. produc ing different relations with every appearance of the C. F. , we have a contrapuntal ostinato . In the following example, the theme of ostinato is taken from Figure I , and the accompanying counterpoint is evolved thro ugh Type II, adhering t o a rhythmic ostinato as well (except for a few intentional permutations). Naturally both voices can be exch.anged as well as subjected to any of the variations through geometrical positions � , G) , @ , and ® . Fig!)re III. Contrapuntal Ostinato Basso Ostinato (Gro11nd Bass) • • • - z 5 [ � • j J-o - I II � ,�I - � I 12 I• • ' - • l ,� I L l 0 . � -'- - ' I•• •• I f - • s 1•• ::i , ,� [ 1• ,� t - ] . - C [.-, ] • -e 0 26. Likewise, a counterpoint can be evolved to the soprano voice through the use of the same principle . In Figure IV, the same theme is employed except that it is altered rhythmically, and the counterpoint, in its rhythmic settiqg, produces a constant interference against the C.F ., as it consists of a 3-bar group. The harmonic setting of this example is in Type III : the C .F. is in natural C major, and the counterpoint is in natural A � • major • Firoy.:e IV. C-P 1yPE. nr. • , • �. ,,- • 12 ., • ..... ' .. z �. [ ', •• • � Soprano Ostinato (Ground M�lody) . h � • bI • .I JJ -• - I b • . L• �• r ( • • f , I n. t "f.. t• • § • ,ff!). I • • -� I ..,. • ! . � I ,. i h• • .. • ( � The last two forms of ostinato are extremely adaptable in all cases when it is desirable to repeat one mot if and yet introduce variety into an obligato. These characteristics make the above described device extremel y useful for introductions, transitions and codas, when applied to arranging. • ! _J• 't1J • , 0 J OSEP H S C H I L L I N G ER C O R RE S P O N DE N C E Wit h : Dr . Jerome Gross C O U RS E Subject: Music • Lesson CLXXX. IN STRUMENTAL FORM S OF MELODY AND HARMONY The meaning of Instrumental Form implies a modification of tl1e original which renders the latter fit for execution on a.n instrument . Instrumental can be defined as an applied form of the pure. Depending on the degree of virtuosity which is to be expected from the performers, instrumental forms may be applied to vocal music as well. The main technical characteristic of the instrumental (i. e . , of applied versus pure) form is the development of the quantities (multiplication) and forms of attacks from the origi nal attack. This branch will be concerned only with the first, i . e . , with quantities and their uses in composition, leaving the second, i . e . , the forms of attacks (such as durable, abrupt, bouncing, oscillating, etc.), to the branch of Orchestration. Multiplication of attacks can be applied directly to single pitch-units as well as to pitch­ assemblages. u The quantity of the instrumental forms is dependent upon the quantity of pitch-units in an assemblage. 0 2. When the quantity of pitch-units (parts) in an assemblage is scarce, the number of instrumental forms is low. When the number of pitch-units (parts) in an assemblage is abundant, the number of instrumental forms is high. The latter permits to accomplish greater varie ty in a composition, insofar as its instrumental aspect is ; concerned. The scarcity of instrumental forms derived from one pitch-unit (part) often makes it compelling to resort to couplings. By addition of one co upling to one part we achieve a two-part setting, with all its instrumental < implications. Likewise, the addition of two couplings to one part transforms the latter into a three-part assemblage, etc. This branch consists of an exhaustive study of all forms of arpeggio and their applications in the field of melody, harmony and correlated melodies. Nomenclatur e : ---Score (Group of instrumental strata) s p a ---Stratum (instrumen tal stratum) --- part (function, coupling) --- attack • Pre.liminary Data: =a • p • s = (2) p , s (3) r = s t• � (1) p - 2a ,• • • • p = na = 2S '• • • 2p ,• • • • s = np • z:- = S n 0 3 ., Sources of Instrumental Forms (a) Multiplication of S is achieved by 1 : 2 : 4 : 8 : • • • ratio (i.e., by the octaves) (b) Multiplication of p in S is achieved by coupling or by harmonization. It is applicable to melody (p), correlated melodies (2p, • • • np) and harmony (2p... 4p). The material for p is in the Theory of Pitch Scales and the Tr1eory of Melody. The material for 2p, ... np acting as melodies is in the Theory of Correlated • Melodies (Counterpoint) . The material for 2p, ... np acting as parts of harmony is in the Special Theory of Harmony and in the General Theory of Harmony. (c) Multiplication of a is achieved by repetition and sequence of P ' s (arpeggio). (d) Different S • s and different p • s, as correlated melodies of 2: may have indepe nd e nt instrumental forms ., Definition of the Instrumental Forms: I. (a) •Instrumental Forms of Melody: I (M = p) : repetition of pitch-units represented by the duration­ group and expressed through its common denominator. The number of a equals the number of t. 1 If n'"t = nt, the11 nt = na Rhythmic composition of dur ations assig11ed to each at tack. (b) Instrument�! Forms of M elody: I (M = np): repetition of pitch-units (Pr) and their couplings (Prr , Prrr , ... PN ) and transition (sequence) from one 0 p to another, represented by the duration group and expre ssed through it s common denominator. Instrumental groups of p ' s consisting of repetitions and sequences are subject to permutations. Groups 9f Melody : • (�) Instrumental Forms of the Simultaneous M = PII ,• PI Pr ,• PII P 1rr Pr ,• Prr P 11 Pr ,• PIII Pr Pr • Prrr; , Prr PrrI Prr P rrr Prr Pr Prr • ' Prrr •, Pr Instrumental Forms of the Seguent Groups 9f Melody : I - z • • (�) Instrumental Forms of the Combined Groups of Melody: M = M = P rr Pr P rrr + PI + Prv Prv ; • • • P rv + + + Prrr P r r r P rr -= = + + + P ; • • • Prr r Pr P rv Prrr + + + PrI Prr ... 0 I I . Instrumental Forms of Correlated Melodies: = M (a) I ( rr= P ) : correlation of instrumental forms Mr p melodies (Mr and M11) by means of the two uncoupled of correlating their a ' s . MI (nt = na) ; MI I (nt Mrr ( t = a) MI (t = 2a) �1rr ( t M r (t ■I Mr , = 2na; 3na; ••• mna) = 2a) • , = a) M rr (t Mr (t = a) • Mrr (t = 3a) ' Mr (t = 3a) = a) Mir ( t = 4a) Mrr (t = 3a) M rr ( t = a) • ; a) ; = � t ( • (t ; 4a) 2a) = ( M t MI (t = 3a) ' Mr r ( t = 2a) ( t = 4a) Mrr ( t == 3a)\. Mrr (t = 4a) ( t = 2a) Mrr • -=-:�--- · �=--,---� • �=--,..,---c�• • • • {t = 4a) ' Mr (t = 2a) ' Mr (t = 4a) ' Mr (t = 3a)' • • • Mr I ( t = na) i{1 '{ t"� ma) (b) r Mrr = np Mr • - mp : this form corresponds to combinations of ( ex, ) , ( � ) and ( � ) of I ( b ) . Mrr C °"' ) MI ( o<. )' ; ( ol ) III. Instrumental Forms of Harmony: I b pt I S ( = p, 2p, 3p, 4p) : this corresponds to one part harmony, which is the equivalent of M ; two-part harmony , 0 6. which is the equivalent of two correlated uncoupled melodies ; three-part harmony, which is the equivalent of three correlated uncoupled melodies; four-part harmony, which is the equivalent of four correlated uncoupled melodies. The source of Harmony can be the Theory of Pitch Scales, Special Theory of Harmony and General Theory of Harmony. Parts (p rs) i n their simultaneous and sequent groupings correspond to a, b, c, d. = d. Instrumental Forms of S = p . Material : (a) melody; (b ) any one of the correlated melodies; (c) one-part harmony; (d) harmonic form of one unit scale; (e) one part of any harmony. I = a; 2a; 3a; ma; A var. nt = na Figure I . (please see pages 7 and 8) , 0 0 (a.,) l\1E.N\E. . • ' ., • . , - - -· C! It • • ....) I � ••• M t • • I c I � I - . I -,- - I • F --•. � ., - • • - :it:]t::it:::::l • l .. •- . .- ' - j � •• • - " -19, ,.., I p• I ' � • • • • .., , • '� ...,_ , '-' . ,, J • - . • • I t • J . :f2 llj•.,. ._. �- ·�I •. -s 1- f F. ' • I •• , - .... � � . . � • . I ..-. •I . J m I � . � � � --· - . .- ., •.. f' I • t::Jt::::it:] II •.J • I . f ,F - -E ..... I • F • - EiL �iL!• (:�- .. L._!. ... - I ,_ • - i ,.. • • • � • � ..' • • !. I �� � , � u • • ... I• • � ,# • . -• � .. • I Ci , Figure I. ......... •• • 1 1. - :;g; n - p � � ,. I � No. 1. Loose Leaf KIN 0 0 VAR. I ., If¥ { MM J:.lt ) . 1'· • • J JCll.-1 (c) T1 HE.ME. -•• I I ... • ,- • . ;1 I ., :g. !Jt9- s: ..r- - . • • -� ]-� - Ji I VA-� - I a) : � 0v .:: ;t 1j :jt�; • II f· - ..• •• •i. ... I I -I p� I • � • ,, ' • ,: I ' I � II • � � -- � ,. I • 8. Cr:; jc::J p�... CJ • = . • :i , � ) THE. ME. • I 'I u - �f. I � � I 1� I b:=c I 1 , • ::E I �t:Jt] � - :£Ct -� � � r-� --- J'iiiZ - DC. I I ·� i ..... cA,...,,..C'Aetf-ct, �·� ?l:_ No. 1. l.00110 Leaf KIN ° 1&9� 8 WA)'. N. Y. 0 9. Lesson CLXXXI . peneral Classificatiop of_ I (S = 2p) . (Table of the combinations of attacks for a and b) A = a; 2a ; 3a ; 4a ; 5a; 6a ; 7a ; 8a; 12 a. The following is a complete table of all forms of I (s = 2p). It includes all the combinations and permutations for 2, 3, 4, 5, 6, 7, 8 and 12 attacks. 0 A = 2a; a + b. Total in general permutations: 2 • Total in circular permutations: 2 A - 3a; 2a + b; a + 2b. - p3 31 6 21 ;:; 2 - 3 Eac h of the a bove 2 permutations of the coefficients has 3 general permutations. Total: 3 • 2 = 6 The total number of cases A = 3a • General permutations : 6 Circular permutations: 6 A = 4a Forms of the distribution of coefficients : 4 = 1+3; A = a + 3b ; p'-t 2+2 3a + b . 1 _ 24 _ = 4 3! - 6 - 4 • 0 10. u Each of the above 2 permutations of the first form of distribution of the coefficients of recurrence has 4 general permutations. Total : 4 • 2 == 8 A = 2a + 2b 24 p = 4! = � 2 •2 21 2! 6 The above invariant form of distribution has 6 general permutations. The total number of cases: A = 4a General permutations: 8 + 6 = 14 Circular permutations: 4•3 = 12 A = 5a Forms of the distribution of coefficients: 5 = 1+4; 2+3 . A = a + 4b; 4a + b p � = 5 1 = 120 = 41 24 5 Each of the above 2 permutations of the first form of distribution has 5 general permutations. Total: 5·2 = 10 A = 2a + 3b ; 3a + 2b . _ 5! = 120 = 10 pS" -21 3! 2•6 Each of the above 2 permutations of the second 0 11. form of distribution has 10 general permutations. Total : 10•2 = 20 The total number of cases: A = 5a General permutations: 10 + 20 = 30 Circular permutations: 5•4 = 20 A = 6a Forms of the distribution of coefficients : • 6 = 1+5; 2+4; 3+3 . A = a + 5b; 5a + b . p 6 = fil = 720 = 6 120 51 Each of the abov e 2 permutations of the first form of distribution has 6 general permutations. Total: 6 • 2 = 12 A = 2a + 4b; 4a + 2b . p 6 = 61 21 4 1 720 = 15 2•24 E ach of the above 2 permutations of the second form of distribution has 15 general permutations, Total: 15 • 2 = 30 A = 3a + 3b _ 720 61 6•6 3! 3 ! = 20 The above invariant (third) form of distribution has 20 general permuta tions. The total number of ca ses : A = 6a General permutations: 12 + 30 + 20 = 62 = 30 Circular permutations: 6 • 5 0 12. A = 7a Forms of the distribution of coefficients: 7 = 1+6; 2+5; 3+4 . A = a + 6b; 6a + b . 7 1 = 5040 = 61 720 7 Each of the above 2 permutations of the first form of distribution ha s 7 general permutations . Total: 7 • 2 = 14 A = 2a + 5b; B? = 5a + 2b . 71 2 1 51 = 5040 = 21 2 • 120 Each of the above 2 permutations of the second form of distribution has 21 general permutations. Total: A = 3a + 4b; P.7 - 21 • 2 = 42 4a + 3b - • = 5040 31 41 6 • 24 71 35 Each of the above 2 permutations of the third form of distribution bas 35 general permutations. The total number of cases: A = 7a General permutations: 14 + 42 + 70 = 126 = 42 Circular permutations: 7 •6- V 0 13. A = 8a Forms of the di.stribution of coefficients : 8 = 1+7; 2+6; 3+5; 4+4 A = a + 7b; 7a + b . 8 1 = 40,320 = 8 p = a 5, 040 71 Each of the above 2 permutations of the first form of distribution has 8 general permutations. Total : 8 • 2 = 16 • A = 6a + 2b . 2a + 6b; p 8 = �o,320 = 28 2 •720 21 6 1 a1 = Each of the abov e 2 permutations of the second form of distribution has 28 general permutations. Total : 28 • 2 = 56 A = 3a + 5b ; P8 _ - 5a + 3b . 8! = 40,920 = 56 3I 5 l 6 • 120 Each of the above 2 permutations of the third form of distribution has 56 general permut ations. 56•2 = a12 Total: A = 4a + 4b P8 = 81 _ 40,320 _ 70 4 1 4 1 - 24 • 24 - The above invariant (fourth) form of distribution has 70 general permutati ons. 0 14 .. The total number of cases : A = 8a General permutations: 16 + 66 + 112 + 70 = 254 Circular permutati ons : 8•7 = 56 A = 12a Forms of the distribution of coefficients: 12 = l+ll; 2+10; 3+9; 4+8; 5+7; 6 6 + A = a + llb; lla + b . = 12 1 = 47� 1991� 600 = 12 39 ,916 , 800 11 ! Each of the above 2 permutations of the first form of distribution has 12 general permutat ions. Total: 12 • 2 = 24 A = 2a + lO b; lOa + 2b . _ 479,001.i 600 121 21 101 - �- 3, 628, 800 = 66 Each of the above 2 permutations of the second form of distribution has 66 gen eral permutations. Total : 66•2 = 132 A = 3a + 9b; 9a + 3b . P,l = 311 291 I = 479,091,600 = 220 6 • 362, 880 Each of the abov e 2 permutations of the third form of distribution has 220 general permutations. • • Total: 220•2 = 440 0 15 . A = 4a + Sb; _ - 8a + 4b , 12 1 _ 479,0,01,600 4 1 8 1 - 24 •40,320 =: 495 Each of the abov e 2 permutations of the . fourth form of distribution has 495 general permutati ons . Total : 495 • 2 = 990 A = 5a + 7b; 7a + 5b . 12 1 = 479,001J 600 120 • 5 , 040 51 71 = 792 Each of the ab_ove 2 permutations of the fifth form of distribution has 792 general permutations. Total: 792• 2 = 1584 A = 6a + 6b 00 = 924 479,001,6 _ _ 12 1 ° p,� 720 •720 - 61 61 The above invariant (sixth) form of distribution has 924 general permutations. The total number of 6ases : A = 12a General permut ations: 24 + 132 + 440 + 990 + + 158 4 + 924 = 4094. Circular permutations: 12 • 11 = 132 0 16. Lesson CLXXXII. Figure I I . The interval of octave can be changed to any other interval. For tl1e groups with more than 6 attacks o�ly circular. per�utati9ps are included. (please see pages 17-22) figure I I I . Examples of the polynomial atta cK-groups (coefficients of recurrence) . (please see page 22) • • 0 A = a Figµre I I . A = 3a : 2a+b; a+2b. A = 2a; 2 forms 2 combinations, 17. 3 permutations each. Total 2•3 = 6 A � 4a: 3a+b; 2a+2b; a+3b ' .. ♦ - - ... ♦ ♦ -- -- - --" £ L - - ... ,. I lJII Total : 4+6+4 - 14 A = 5a : 4a+b; 3a+2b; 2a+3b; a+4b ," ,. I I ... .. -- II • .. .,,. - � .,. .... ......... -- - ' • , • .... .. • - - -- +- + ♦ III I•• • 0 18. A = 6 a : 5a+b; 4a+2b; 3a+3b; 2a+4b; a+5b .............. .,. " ft, .... +• ... .... .,..,. • .. .,. .. ♦ + .,. .,. .,. ...,,,...,. - - + .,.. ♦++ .,.. .,..,..,_ . .,. ,.,. +• + .,.,,...,., ++ ........ .. .,. .... .. .,_ .,. - - .,. .,. ....,. ......,. - -I;l .,. <#JI' .... -- .... .... ·�-.. .......... � .,,. . .,. .. II•1 .... • ... .,. ,,,. +-...,..... ... ,�-#- .,. T•• .. .,. • .,. 4- r TI•I arr ,,. .. � .,. Total; 6+15+20+15+6 = 62 No. 1 . Loose Leaf "'" 1691 e·w,y. N.Y. 0 19. A = 7a : 6a+b; 5a+2b ; 4a+3b; 3a+4b;2a+5b; a+6b • . " Ii; • .,.. ... ,,,,. .,,. - .,..,._.,. -- +- .,. ,,,,. . - ,,_.,,,. .,. ... • + • .,,. • .,,..,. .,. .,. • .,. + .,. ,_. .,. I .,. ..,., .,- . . .,. .,,. ....... I -. ... .. . .......,. ,, I .. .... .,,. • � .,. .,. .. ... • .,. .,,. --- 'I m • :, ----- 4- ' , .. \ .... .,..,. ,,,. .,. +•+ -- r t l J t. . I • 1'[ 1I I .I -• ' • TL • .- • r,_ • II • • .,. • 0 �.,,�.. -�. Total � 7+21+35+35+21+7 - 126 No. 1. Loose Leaf KIN 169$ 8'w.a.y. N. Y. 0 /. A. = 8a: 7a+b; 6a+2b; 5a+3b; 4a+4b; 3a+5b; 2a+6b; a+7b I,� - 20. ..,...,...,_.,...,.. � '#- ,. + .,,., ..,.. .,... .,.,.. .,.. .,.. .,..,,. - le ++ -- - II •A .,.. .,.. ... .,...,..,#, .,.,#,.,_.... .,,,, .,...,..,. . .,. .,. .... .,,,, 4 '• • - •+++ .,..,..,#a I- • ••• •. .. ( general) • •• , .....,..,.. .,...,...,. .,...., (general) ' .... .,..� .,...,.,,_.,.. .,. .,.. .... --- ♦ .,. .,. --- .,. .,...,. .,.. -- - .,_ .,. .,. .,,. (general) ,...,..,. • (general) .,. .,., • 0 general) • No. 1 . Loose Lea.t KIN 0 21 • .,. genera Total: 8+28+56 +7o+56+2a+a - 254 A = 12a : lla+b; l0a+2b ; 9a+3b; 8a+4b; 7a+5b; 6a+6b; 5a+7b; 4a+8b; 3a +9b; 2a+l0b; a+llb ,,. , l ' I -- - - - I•I -• ••• I T · /n } '"' 1•� •• I 11"'!1 . I. 1•11 I ,I , • . I' I 1 1 I � , I, ♦ , I I•I I ' 1III II I ' , • . . ..... •--1•----­ • ••• 4 .. �� et, No. t . Loose Lear "'" f �� 169$ l way. N. Y. 0 22 • .,. .,. • Total : 12+66+220+495+792+924+792+495+220+66+12 = 4094 Firoire r;rr . A = Summation Series II A = Summation Series I .,. A = 3(2+1) + (2+1) 2 A = (2+1+1) 2 .. A - (3+1+1) + (1+3+1) + (1+1+3) No. 1. Loose Leaf 0 u 23. Lesson CLXXXIII. Instrumental forms ofS = �E Material: (a) coupled melody: M ( P II ) ' PI (b) harmonic forms of two-unit scales; (c) two-part harmony; (d) two-parts of any harmony. • Pr Prr ' I = a: I = ab, ba: permutations of the higher orders. Coeffic ients of recurren c e : 2a+b; a+2b; . . • • • • ma + n b. Figure IV. (please see pages 24-29) Individual attacks emphasizing one or two parts can be combined into one attack-group of any desirable form. Example : I (S = 2p) : bbbb b bb bbb bb aa ; aaaa ; aaa aa ; aa a aa ; • • • b Figyre V. (please see page 30) 0 24. r1gy;re IV. • ,,,I • ,. 11I........ • �I I �,,. . " I• ,--. I • I I .. • • • • • • l M ; �a 2 . � 4 ,,.. • � I .... , �+ � TI ,- - � ul i', •• j:::::l "'• •.. .. "I � • I "A A + ,c " • 0 • "'• :i j ' .,. . I . ... . "'• + ] ] ] "I j ] ] � • � •.. � j •• • ] j ] l ] - � ] -■ . • I ■ j ] ' 0 25. ' � No. 1. l,oose Leaf ] ] KIN 0 • 26. . • ]t:J � J ] J • Oc::l ]0 � � . ] " j w I I ] j -9.]t::l II � j J ] ' • fl i I - ft ; . ·•• • ( - I• � I "J.t - ... • • ::i::::J • • . . "i • • . I-' ] • • • • , •• • • --,, No. t . Loose Leaf "'" 169$ a·w�y. N. Y. 0 27. -IJ � "• , • IL, I I • I • :i , ' •� I - le-- • -,. . � ti �- ,,,6 -. • ••• ' ' ... ., I .- - I y I • j - • ' ::. • ::J • ·� • I • i ] I •• When the progression of chords (H 1) has an assigned duration group, instrumental form (I) can be carried out through t . ' .,. r\ • 0 . ]�� .., ', � :it . L· • . " : ,...,... ]• - . I • •' " _,, � •• • • \. --- • .. �� '-" ' � . I -tr I• � _,. II �!:J J -• • JL] fc i .. ] 'll •• ] ]t:::J •• I . JG ---A � • i .. • u � ........... - No. 1. Loose Leaf tAIID 169� s·way. !f.Y. 0 28 . 0 ' ---- � . .,, u-s �I � • • • -I • • • ' • j . . ' -- • � 0 jCJ . ' ., I - ' • ]Cj7� I � ]Li f• • • • - ' • I I . --· I - II I ' •• • I ,u" • - � ' -• " I • , I - , •j • • -! I • • . I• . L • ,. -,. � � J • ,J• • -· • J J I , ' -·,., ------ c:i . :iO .. ., i: ,1 ·-• ' ' I -d. . ' ' I' . "i� I ... I , ; . • ] -• -·I .. • • , • � I ' , • - i· �· • I • - ' • • d. �t::i • ' • • r- -I ,..... . - '. • • ,, 0 • .. •j - I 'lj ' ' • • i• I ]0 . No. 1. Loose Leaf ] 0 29. Var. : the two preceding variations combined - .. • . • :it:l • . . • I� ,, J isj - ,� • If¥ • Jt:J .,.. ·� •I ] J C. I • •• • - -· I• • I• • ] JC • • . II I' II . Ii.. J• • "'i "" j •• I. ' -- � - • If· I 1r, � � • j • • I � j ' ' j:::; 11 • . ZJ ' :i:: ,. j J ' j . .� • • • � ,_ ] • • j • - in • J JO • No. t. Loose Leal KIN 169$ s·w-ay. N. Y. 0 30. Figµre V. • , ' .. , ') r ' �-e- ... I [:[ ffi � '-J • = c;; a- 3 .. • - .. ',,. VAR. : ll.J c1J • r ' OJ . • • ]t:J :r:LEt:E I - j CiJL� �t:[i - � J::: : ! I I = - • H = 4 a., ) �... ... .. . � - . • ' = :J j •• ' ' ' -r • • j :l= I ; ' ... I .., • � C. • T-• ::J::JD I I r jtJ No. t. Loose Le•f KIN 100 159� 8·••Y· N.Y. 0 u 31. Lesson CLXXXIV. (• = 3p) General classification of I S (Ta ble of the combinations of attacks for a, b and c) A = a ; 2a ; 3a ; 4a ; 5a ; 6a ; 7a ; 8a ; 12a. The following is a complete table of all forms of IS ( = 3p). It includes all the combinations end per�utations for 2, 3, 4, 5, 6, 7, 8 and 12 sequent attacks� (1) I = ap (one part, one attack). Three invariant form s : a or b or c . A = a p , 2ap, . . • map . This is equivalent to I (S = p). (2) I = a2p (one attack to a part, tv,o sequent parts) Three invariant forms : ab, ac, be • E a ch inva riant form produce s 2 attacks and has 2 permuta ti o ns. This is equivalent to I ( S = 2p). Further combinations of ab, ac, be are not necessary as it corresponds to the form s of (3) . (3) I = a3pt (one attack to a part, three sequent parts). One invariant form : abc. The invariant form produces 3 attacks and has S permutations: a bc, a c b, cab, bac, bca, cba . • 0 All other attack-groups (A = 3 + n) develop from this source by means of the coefficients of recur rence. Figure V"J.. I(S = 3p) : attack-groups for one simultaneous p. (please see page 33) Development of attack-groups by means of the 4 • coefficients of recurrence. A = 4a; 2a+b+c; a+2b+c; a+b+2c. C. 41 P'i = 2 t = �4 = 12 Each of the above 3 permutations of the coefficients has 12 general permutations. Total in general permutations: 12 •3 = 36 Total in circular permutations: 4 • 3 = 12 A = 5a. Forms of the distribution of coefficients : 5 = 2+2+1 and 5 = 1+1+3 A = 2a+2b+c ; 2a+b+2c; a+2b+2c p _ 51 5" - 2 1 21 120 = 30 2•2 . Each of the 3 permutations of the first form of distribution has 30 general permutations. 30•3 = 90. Total: • 0 � = �t 0 ' 33. Figur� VI. -� - �·. ' .I . I• , • - - • JI , A • L ,, ' II - - - n • • - -' •,I r each form with a corresponding number of t>ermutations - - • • ... -.• • :J .., , ft 6 general permutations 'I - 3 circular permutations - - - - - No. 1 . Loose Leaf "'" f lf>9� 1 w»y. N. Y . 0 34 . A = a+b+3c; a+3b+c ; 3a+b+c . P,r 5 ! = 120 = 20 6 31 = Each of the above 3 permutations of the second form of distribution has 20 general permutations. Total: 20•3 = 60. The total number of cases: A = 5a. General permutations : 90 + 60 = 150 Circular permutations : 5 • 6 = 30 A = 6a. Forms of the distribution of coefficient s : 6 = 1+1+4 ; A = a+b+4c; PLV = 1+2+3; 2+2+2. a+4b+c; 4a+b+c. 6 1 = 720 = 30. 41 24 Each of the above 3 permut ations of the first form of distribution has 30 general permutations. Total: 30•3 = 90 A = a+2b+3c ; a+3b+2c ; 2a+3b+c ; 3a+2b+c. 3a+b+2c ; 2a+b+3c ; Each of the a bove 6 permutations of the second form of distribution has 60 general permutations . Total: 60 · 6 = 360. u 0 A = 2a+2b+2c. The third form of distribution (invariant) has 90 general permutat ions . The total number of cases: A = 6a. General permutations : 90 + 360 + 90 = 540. Circular permutations : 18 + 36 + 6 = 60 . A = 7a. Forms of the distribution of coefficients: 7 = 1+1+5; A = a+b+5c; • 1+2+4; a+5b+o; 2+2+3; 3+3+1 5a+b+c . 7 1 = 5040 = 42 120 1 - 51 p _ Each of the above 3 permut ations of the first form of distribution has 42 general permutat ions . Total: 42 • 3 = 126. A = a+2b+4c; a+4b+2c ; 2a+4b+o ; 4a+2b+c. P1 = 4a+b+2c; 2a+b+4c; 7 ! = 5040 = 105 2 ! 41 2 • 24 Each of the above 6 permutations o f the second form of distribution has 105 general permutations. Tot al: 105 • 6 = 630 A = 2a+2b+3c ; p = 7 2a+3b+2c; 7_, ! -- = ..,._.__,;,. 2! 31 21 3a+2b+2c . = 210 0 36. Each of the above 3 permutations of the third form of distribution has 210 general permutations. Total: 210 •3 = 6 30 A = 3a+3b+c; P.., = , 71 31 31 3a+b+3c ; = a+3b+3c . 5040 = 14 0 6·6 Each of the above 3 permutations of the fourth form of distribution has 140 general permutations. • • Total : 140 •3 = 420 The total number of cases: A = 7a General permutations: 126 + 630 + 630 + 420 Circular permutations: 21 = 1806 + 42 + 21 + 21 = 10 5 A = Sa. Forms of the distribution of coefficients: 8 = 1+1+6 ; 1+2+5; 1+3+4 ; �+2+4; 2+3+3 A = a+b+6c; _ PS - a+6b+c; 6a+b�c. 8 1 _ 40,320 _ 56 61 720 - Each of the above 3 permutation.s o f the first form of distribution has 56 general permutations. Total : 56•3 = 168 A = a+2b+5c; a+5b+2c; 5a+b+2c; 2a+b+5c; 2a+5b+c; 5a+2b+c . 8! 21 5 ! =- �0, 320 2 • 120 = 168 Each of the above 6 permutations of the second form of distribution has 168 general permutations. Total : 168 •6 = 1008 0 37. A = a+3b+4c; a+4b+3c; 4a+b+3c ; 3a+b+4c ; 3a+4b+c; 4a+3b+c. = 4 0� 320 = 280 6 • 24 81 P.I _ 31 4 1 F.ach of the above 6 permutations of the third form of distribution has Total : A = 2a+ 2b+4c; • 280 28 0 •6 2a+4b+2c; 81 21 2I 41 Po• = general permutations. = 168 0 4a+2b+2c = = 42 0• ,2 520 420 • 24 Each of the above 3 permutations of the fourth form of distribution has 420 general permutations . Total : 4 20 • 3 A = 2a+3b+3c ; p = g 3a+ 2b+3c ; = 1260 3a+3b+2c 81 = �0,320 2 •6 • 6 21 3 1 31 = 560 Each of the above 3 permutations of the fi fth form of distribution has 56 0 general permutations. Total: 560 • 3 = 1680 The total number of cases : A = Sa General permut ations: 168 + 1008 + 168 0 + 126 0 + + 1680 = 5796 Circular permut ations : 24 + 48 + 48 + 24 + 24 = 168 A = 12a. Forms of the distribution of coefficients : u 8 = 1+1+10 ; 1+2+9; 1+3+8 ; 1+4+7; 1+5+6 ; 2+3+7; 2+4+6 ; 2+5+5; 3+3+6 ; 3+4+5; 2+2+8 ; 4+4+4. 0 38 . a+b+lOc ; A = p , i. _ - • a+lOb+c ; lOa+b+c 121 _ 4791p011 600 3,628,800 101 - = 132 Each of the above 3 permutations of the first form of distribution has 132 general permutations. Total: 132 •3 = 396 • A = a+2b+9c ; a+9b+2c; 9a+b+2c; 2a+b+9c; 2a+9b+c; 9a+2b+c. p _ ,:a. - 121 21 9 1 = 479,ppl,§OO = 660 2 • 362,880 Each of the above 6 permuta tions· of the second form of d istribu tion has 660 general permutations . Total : 660 • 6 = 3960 A = a+3b+8c ; a+8b+3c; 8a+b+3c; 3a+b+8c; 3a+8b+c; 8a+3b+c. 12 1 = 47�,001,6p� 6 •40,320 31 81 = 1980 Each of the abov e 6 permutations of the third £orm of distribution bas 1980 general permutations. Total : 1980 • 6 = 11,880 A = a+4b+7c; a+7b+4c; 7a+b+4c; 4a+b+7c; 4a+7b+c ; 7a+4b+c. p'" = 12 1 41 7 I = 479,001,600 24•5 , 040 = 3960 Each of the above 6 permutations of the fo urth u form of distribution bas 3960 general permutations. Total: 3960•6 = 23,760 0 39 .. A = a+5b+6c; a+6b+5a; 6a+b+5c; 5a+b+6c; 5a+6b+c; 6a+5b+c. = 121 51 61 479,001,600 = 5544 120•720 Each of the above 6 permutations of the fifth form of distribution has 5544 general permutations ., Total: 5544• 6 A = 2a+2b+8c; = 2a+8b+2c; 121 2 1 2 1 81 32,264 8a+2b+2c = 479,001, 600 = 2970 2• 2 • 40,320 Each of the above 3 permutations o� the sixth form of distribution has 2970 general permutations. Total: 2970 • 3 = 8910 A = 2a+3b+7c; 2a+7b+3c ; 7a+2b+3c ; 3a+2b+7c; 3a+7b+2c ; 7a+3b+2c. P,,_, 121 479,001,600 = 7920 = = 2 1 31 7 1 2 • 6 • 5, 040 Each of tr1e above 6 permutations of the seventh form of distribution has 7920 general permutations. Total : 7920•6 = 47,520 A = 2a+4b+6c; 2a+6b+4e; 6a+2b+4c; 4a+2b+6c; 4a+6b+2c; 6a+4b+2c . 12 1 21 41 6 1 = 479,00�,600 = 1386 2 • 24 • 720 Each of the above 6 permutations of the eighth form of distribution has 1386 general permutations . Total: 1386•6 = 8316 0 40. A = 2a+5b+5c; 5a+2b+5c; 121 2 1 51 51 5a+5b+2c . = = 479,001,600 16 ' 632 2 •120 • 120 Each of the above 3 permutations of the ninth form of distribution bas 16,632 general permutations. Total: 16 , 632 •3 A = 3a+3b+6c; = P,4 3a+6b+3c; 12 1 31 3! 6 1 = 49,896 6a+3b+3c _ 479,001,600 6 • 6 • 720 0 - = 18,480 Each of the above 3 permutations Qf the tenth form of distribution has 18,480 general permutation s . Total: 18,480•3 = 55,440 A = 3a+4b+5c; 3a+5b+4c; 5a+3b+4c; 4a+3b+5c; 4a+5b+3c ; 5a+4b+3c. P.·� = 12 1 = 479,oo�,spo 6 • 24 •120 31 41 51 = 27,720 Each of the above 6 permutations of the eleventh form of distr ibution has 27,720 general permutations ., Total: 27, 720 · 6 = 166,320 A = 4a+4b+4c 479 1,6 00 = 34, 650 12 1 ,00 , = = P,:2.. 4 1 4 1 41 24•24•24 The twelfth form of distribution (invariant) has 34,650 general permutations. The total number of case s : A = 12a. General permutations: 396 + 3960 + 11,880 + 23,760 + + 3�64 + 8910 + 47,520 + 8316 + 49,896 + 55,440 + 166,320 + + 34 , 6 50 = 443, 3 12. Circular permut ations : 36 + 72 + 72 + 72 + 72 + 36 + + 72 + 72 + 36 + 36 + 72 + 12 = 660. 0 Lesson CLXXXV . A � 4a; 2a+b+c ; a+2b+c; a+b+2c Figure VII . -. �otal in ge�eral p�mutations: 12+12+1.2 = 36 Total in circular permutations: 4+4+4 = A = 5a; 2a+2b+c; 2a+b+2c ; a+2b+2c - •J - - "I '(I I I I•• •1 - I'I I :1 I J - •• . !"l .,. II.I"...II .l "I I I . !] I 12 .. )l �I I• I •• - ·•I I -i IfI . c:========= - Total in general permutations; 30+30+30 = 90 Total jn cjrc11Jar perrm1tatiaos: 5±5+5 = 15 A = 5a; a+b+3c; a+3b+c; 3a+b +c ,. � ,, 0 - - - - --• "J � .-. � • f'. - - '"' ( 'I I 1'11 �I - • •• •I• • • • . • • ". I• ■III• - . 11"111 I I 1•111111 1 11.111 1 Total in general permutations: 20+20+20 = Total in circular permutations: 5+5+5 = 15 No. t. l.ooae Lear 0 The entire total for 5 attacks: in general permutations: 150 0 42. 1n circular permutations: 30 A = 6a; a+b+4c ; a+4b+c; a+b+4c • • Total in general permutations: 30•3 = 90 Total in circular permutations; 6•3 = 18 A = 6a ; a+2b+3c ; a+3b+2c ; 3a+b+2c; 2a+b+3c ; 2a+3b+c ; 3a+2b+c • • • • , - 'IIIJI I II I Total in general permutations: 60• 6 = 360 I. I � � .......... ­ fl I I t:a • I-. I II"• 1•1II I 11 Total in circular permut ations: 6 • 6 = 36 No. 1. Loose Leaf KIN •&as e·w,y. "· Y. 0 43. A = 6a; 2a+2b+2c • The entire total for 6 attacks: in general permutations: 540 1n circ11Jar perm11tations: 60 A = 7a; a+b+5e; a+5b+c; 5a+b+c • 0 Total in general permutations: 42•3 = 126 Total in circular permutations: 7 • 3 = 21 A = 7a; a+2b+4c ; a+4b+2c ; 4a+b+2c; 2a+b+4c; 2a +4b + c; 4a+2b+c • � -.;;i.� - .......... No. t . l.oose Leaf KIN 159& s·way. N. Y. 0 44. IA,, -- - , ------:c-·....- - ' IIJ.;. • - 1-1 I •••- f 1 -l • 11 _ 1 I _ -_ _ _ -_ _ _ -_ _ _ -_ _ _ -_ _ _ _ _ _ _ _ _ _ _ -_ _ _ _ _ _ _ -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -_ _ -_ _ _ -_ _ _ -_ • Total in general permutations : 105•6 = 630 Total in circular permutations: 1•6 = 42 A = 7a; 2a+2b+3c; 2a+3b+2o; 3a+2b+2o ., - l . - :,,III "' "�-I lJ• • ( ("" '1 1,.IJI -l • IIIJI Total in general perm utations: 210• 3 = 630 Total i•n circ11Jar permutatj otls: 7•3 = 2l A = 7a ; 3a+3b+c; 3a+b+3c; a+3b+3c • Total in general permutation s : 140•3 = 420 Total in circ11Jer pernn1tations: 7•5 = 21 The entire total for 7 attacks : in general permutations: 1806 in circular permutat ions : 105 No. 1. Loose Leaf KIN ° lti9S 'll •�y. If.Y. 0 A = Sa; a+b+6c ; a+6b+c; 6a+b+c 45. • '� - - .. I •• t•I1 I-III ' • •• •, .. Total in general permutat ions : 56•3 Total in circular permutations : 8 • 3 ,, A -- Ba; a+2b+5c; a+5b+2c; 5a+b+2c; 2a+b+5c; 2a+5b+c ; 5a+2b+c ., r = 168 = 24 �. - 11 •• --- - -- . ". ' •;a I II_J_ Ll&.o.l!fl I I Ill■ I ajI III • ll I • •• I 1· 11•■1 I II [ - I I 1 11111! ■ l11fl Ill • �... ....�. Total in general permutations: 168•6 = 1008 Total in circ ular permutations: 8•6 = 48 No. 1 . Loose Leaf "''" auo 169S 11·way. N. Y. 0 • 46. A. ::: 8a; a+3b+4c; a+4b+3c; 4a+b+3c; 3a+b+4c; 3a+4b+c; 4a+3b+c � ' • J '4 • .. 1 .'. ,, . ., -,8 ,, , n1 ...,.,.,,, I� - �1 ---'1l"'\'I"'\- - ,, .,, ij • i--":l"'\1J � •Tl c:,.y-r� I rt.T� - - - I" II·,:.,l"'l 1 • • " ·. - • • Total in general permutations : Total in circular permutations: / I• 1680 48 A = Ba; 2a+2b+4c; 2a+4b+2o; 4a+2b+2c -- ..,- I�('!.JI f"\( -. I"'I ,-LJ - 11I -' 1111 I 111, • � 0 ·,. I - • '"I 1,, I I •" - r, , ,s Total in general permutat ions : 420 •3 = 1260 Total in circular permutations: 8 • 3 = 24 No. t. Loose Leaf KIN ° 169$ 8 way. N. Y. 0 47. A = Ba ; 2a+3b+3c; 3a+2b+3c; 3a+3b+2c - .ti•'-I •1 1::111 •• I .J. III I JI'.I I _J• II •I.a • Total in general permutations : 560 • 3 = 1680 Total in circ11Ja.r perro,1tat1ons: 8•3 = 24 The entire total for 8 attacks: in general permutations: 5796 in circular permutations : 168 A � 12a; a +b+lOc; a+lOb+c; lOa+b+c r- u :.:::: A r, � ' A I '. .I;,,,: LI 1.1II • ... I Total in general permutations : 132•3 = 396 Tota l in circular permutations; 12•3 = 36 -. = I• - I • • 12a ; a+2b+9c ; a+9b+2c; 9a+b+2c; 2a+b+9c; 2a+9b+c; 9a+2b+c • • 0 No. 1 . Loose Leaf 0 48. 0 • • ______--_ ___--_ -_ -_ -_ _ _ -_ ______ -_ _ -_ _ __"'"----- --------- _--_ -_ _____ -_ _ -_ -_ -___ -_ -_ _____ Total in general permutations : 660•6 = 3960 Total in circular permutat ions: 1 2 • 6 = 72 A = 12a ; a+3b+8c; a +8b+3c; 8a+b+3c; 3a+b+8c; 3a+8b+c; 8a+3b+c • • , .... • I .,, I I ( I ....., ,,,. 11 , Total in general permutations : 1980•6 = 11 ,880 Total in circular per.mutations : 72 No. t. Loose Lear KIN 1$9, tfway. N. Y. 0 49. A = 12a; a+4b+7c; a+7b+4c; 7a+b+4c; 4a+b+7c; 4a+7b+c; 7a+4b+c - � --- • .. I . . I ,, ,.. • , . -I • I• - II I':'III • .. • - .... I•. T_I1 • • J'aI'• I - • , �,· u11..- ,. ,..1r1- • Total in general permut ations: 3960 .6 = 23, 760 Total in circular permutations : 12 •6 - 72 A = 12a; a+5b+6c; a+6b+5c; 6a+b+5c; 5a+b+6c; 5a+6b+c; 6a+5b+c • • • ,- - -- - - ' IrI • I •• ' - r• ' .,..,,1....e - .• •• •, 0 No. t. Loose Leaf "'" 169f', uND s·••Y· N. Y. 0 50 • • Total in general permutations : 5544•6 = 32, 264 Total in circular permutations : 12•6 = 72 A = 12a; 2a+2b+8c ; 2a+8b+2c; 8a+2b+2c Total in general permutations : 2970 • 3 = 8910 Total in circ ular permutat ions : 12•3 = 36 A = 12a; 2a+3b+7c; 3a+2b+7c· 2a+7b+3c; 3a+7b+2c · 7a+2b+3c; 7a+3b+2c • • • Total in general permutat ions : 7920•6 = 47, 520 Total in circular permutations : 12 • 6 = 72 No. 1. Loose Lear KIN ■uo 169� e'way. ti. Y. 0 A = 12a; 2a+4b+6c; 4a+2b+6c; 51. 2a+6b+4c; 6a+2b+4c; 4a+6b+2c; 6a:t:4b±2c ' - • - I • - l■I I -· l 11 l�)JI Total in general permut ations : 1386 •6 = 8316 Total in circular permut ations: 12 • 6 = 72 A � 12a; 2a+5b+5c; 5a+2b+5c; 5a+5b+2c -- a II[ ■• r. I ..-•-I ll I 1""1 -11 1 .. Total in general permutations: 1�632 • 3 Total in circular permutations: 12 • 3 • • • = 49,896 = 36 No. t. l.ooae Leaf Kll« 169$ s·•�y. N. Y. 0 0 - 52. A = 12a; 3a+3b+6c; 3a+6b+3c; 6a+3b+3c • -- � t I .. ' • J r,...,., - lz r•-I •JI • ' • ,-nTJCt • Total in general permutations : 18480 • 3 = 55,440 Total in circular permutations : 12 • 3 36 A = 12a ; 3a+4b+5c ; 4a+3b+5c; 3a+5b+4c; 4a+5b+3c ; 5a+3b+4c ; 5a+4b+3c • 0 • • � ,- I • • •.I. • • • Total in general permutations: 27 , 720 • 6 = 166,320 Total in circular permutations: 12·8 = 72 0 __ .,.,,...._ � ,, No. 1. Looae Leaf 0 53. n I ,.. - II: The entire total for 12 attacks� in general permutations : 443, 312 in circular permuta tions: 660 I 0