The Sourceset and Its Aggregate Formations

Yale University Department of MusicThe Source Set and Its Aggregate FormationsAuthor(s): Donald MartinoSource: Journal of Music Theory, Vol. 5, No. 2 (Winter, 1961), pp. 224-273Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843226 .Accessed: 16/07/2013 21:35Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected]. .Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize,preserve and extend access to Journal of Music Theory.http://www.jstor.org This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 2 4T h e Sourrce Se ta n d i t sAggre ga t eFobrma ti on sIwi s h t oa ckn owle d ge my i n d e bt e d n e s s t o Mi lt onBa bbi t t , wh os e le ct ure s oncombi n a t ori a li t y d e li ve re d a t Pri n ce t onUn i ve rs i t y i n1952 , a n d wh os es ubs e que n ta rt i cle s on t h e s ubje ctle d me t o i n i t i a t e t h e re s e a rch e s wh i ch a re h e re i npre - s e n t e d . As Mr.Ba bbi t t h a sd e mon s t ra t e d , t we lve -t on eope ra t i on st T h e re a d e r's fa mi li a ri t ywi t h "Se t St ruct ure a s a Compos i t i on - a l De t e rmi n a n t ",by Mi lt on Ba bbi t t(Journ a l of Mus i cT h e ory, Apri l1961) a s we lla s "T we lve -T on e In va ri a n t sa sCompos i t i on a l De t e rmi n a n t s " (T h e Mus i ca l Qua rt e rly,Apri l,1960) a n d"SomeAs pe ct sof T we lve -T on e Compos i t i on "(T h eScore a n d I.M.A. Ma ga - zi n e12 ,1955)by t h e s a mea ut h or, i sn e ce s s a ri lya s s ume d . T e rms d e fi n e d i n t h e a bove a rt i cle s wi lln ot be re d e fi n e d h e re . This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 2 5 B, DO6ALD MAF{T(i .e ., t h os e wh i ch ca nlogi ca lly be d e ri ve d from t h e t we lve - t on es ys t e m) a re we lld e fi n e d a n d t h e i rre s ult s ca n bege n e ra l- i ze d . Spe ci fi cs a re , a s i n a ll mus i c,d e pe n d e n t upont h e i n - t e rpre t a t i onofs uchge n e ra li za t i on s . Akn owle d geofon e 'sma t e ri a ls a n d a na wa re n e s s oft h e i r i m- pli ca t i on swould s e e m t o be a ba s i c con d i t i on for t h e i n t e lli - ge n tcompos i n gof mus i c.T h et we lve -t on es ys t e ma n d t h et on a l s ys t e ma re t h e mos tproce d ura lly s oun d ofa ll pos s i blepi t ch colle ct i on s * 1; but t h e mos tpe rfe ct ma t e ri a lsprod ucepa ra lle lre s ult son ly wh e nt h e y a rei n t e lli ge n t ly h a n d le d .It i sa rgue dt h a t t h epa rt i t i on i n goft h et we lve -t on es e t , a n d con - ve rs e ly, t h econ s t ruct i on oft h e s e tbyope ra t i onon i t s s ubs e t si sa s e s s e n t i a lt o t h e ord e rlycommun i ca t i on ofi d e a s i n d i ge - n ous t o t h et we lve -t on es ys t e m a s t h ee qua lly n e ce s s a ry, a n dThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 2 6 i n s omewa ys a n a logous proce d ureoft on a lmus i c wh e re i nacolle ct i on ofe le me n t sformi n ga ba s i c con s t ruct -t h et ri a dfor i n s t a n ce -ca n be re la t e d wi t h i n or be t we e n i t s cla s s e sby a nope ra t i on - progre s s i on by fi ft h -wh i chprod uce s t h et ot a l s ys t e m- le s s t h a n t we lve- a n d wh i chby e xt e n s i onprod uce st h e t ot a ln umbe r ofe le me n t s a va i la ble - e qua l t ot we lve . Alt h ought h e t on a l s ys t e m, wh os e s ource con s t ruc- t i on s a rei n t e rs e ct i n ga n d wh os eope ra t i on sa rere la t i ve lyfe w, a n d t h e t we lve -t on es ys t e m, wh os e con s t ruct i on s a re n on - i n t e rs e ct i n ga n d wh os eope ra t i on sa rema n y,be lon gt o d i ffe r- e n t cla s s e s ofmus i ca l s ys t e ms ,i mport a n t obje ct i ve s- t h ecre a t i on of n orma t i ve , t h uspre d i ct a ble , a s we lla s n on - n orma t i ve , t h us d ra ma t i cproce d ure swh i ch wi ll h e lp t o d e - t e rmi n e a sma n ypi t ch a s pe ct soft h e t ot a l compos i t i ona s t h ecompos e rd e e ms con s i s t e n t wi t h h i scompos i t i on a l i n t e n t i on s - ma y be re a li ze d i n e a ch ca s ebye xploi t i n gt h os eprope rt i e swh i ch a repe culi a rt o t h es ys t e m. In t h i s a rt i cle Ih a vea t t e mpt e dt opre s e n ti n t a bula r form a ll i n forma t i on e s s e n t i a lt o t h e ca lcula t i on ofmos t ba s i c t we lve - t on eope ra t i on s . Is h a lld e a li n t urn wi t hh e xa ch ord s , t e t ra - ch ord s ,t ri ch ord s , a n don lys umma ri ly wi t h t h eun e qua l t wo- pa rt pa rt i t i on s (1 11), (210), (3 9), (48), (57)oft h e t ot a l s e t , i n e a ch ca s e wi t h a vi e wt owa rdd e li n e a t i n gre la t i on swi t h i n a n da mon g pa rt i t i on s . Comme n t s wi llbe con fi n e d t ot h e cla ri fi ca t i on oft h et a ble s , a n d t o t h ege n e ra ls ubje ctof h a rmon y a s t h e re s ult of a ggre ga t e -formi n gcombi n a t i on s . Exa mple se mployi n gn umbe r n ot a t i on a re a bs t ra ctpi t ch -cla s scompos i t i on sa n d a s s uch a re n ote qui va le n tt o mus i ca lcom- pos i t i on s . Amus i ca lre a li za t i on oft h e s e"compos i t i on s " would n e ce s s i - t a t e t h ea s s i gn a t i onofa t le a s t on e e le me n t of d e fi n i t i on , a n dwould i n volve a s e le ct i on- e i t h e ra rbi t ra ry or mot i va t e dpos s i blyby ot h e r a n d broa d e ra s pe ct soft h e t ot a l compos i t i on-froma mon ga lla va i la blea d jun ct soft h epi t chs t ruct ure .Ad i s cus s i on ofs uchproce d ure scle a rly e xce e d s t h e i n t e n t i on of t h i spa pe r, wh i ch i s t opre s e n tt h epi t chma t e ri a ls a n dpi t chope ra t i on si n fe ra ble from t h efollowi n g pre mi s e s :An y ord e r- i n goft h e t we lve d i s t i n ctpi t ch e soft h ee qua l-t e mpe re dch ro- ma t i c s ca lema y bere ga rd e da s a s e t 'S'oft we lvee le me n t s ; An y colle ct i on of n on -i n t e rs e ct i n gs ubs e t sA,B.......... of 'S' wh i chcon t a i n s a lle le me n t s of 'S' i s apa rt i t i onof 'S'; Anope ra t i onon'S' i s a nope ra t i ononA,B,..........of'S'. (Ope ra t i on sa re d e fi n e de ls e wh e re , a se xpla i n e di n t h e n ot e ont h e fi rs tpa ge .) This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 2 7 T h et h i rt y-fi ves ource h e xa ch ord s a repre s e n t e di n a t a bula rord e ri n g(T a ble I)s e le ct e d s o a s t oe mph a s i zet h e fi ft e e n i n - t e rva l pos s i bi li t i e s un i que ly a s s oci a t e d wi t h e a ch s ource s e ta n d i t scomple me n t *2 . T h e t a ble d i vi d e sve rt i ca lly i n t o fourpa rt son t h e ba s i s oft h e n umbe r ofoccurre n ce s oft h e i n t e rva l s i x: 0,1,2 , or 3t i me s .Wi t h i n e a chpa rt , h e xa ch ord s a reli s t e da ccord i n gt o t h e n umbe r ofoccurre n ce s oft h e i n t e rva l on e *3: 5,4 ,3,2 ,1, or 0t i me s*4 , a n d wi t h i n t h e s es e con d a ry d i vi s i on s , s e t s a re li s t e d i na s ce n d i n gord e r oft h e t e rms wi t hs pe ci a le mph a s i son t h e la s t t e rm: SET 15i n t e rva ls12 34 56 012 4 573332 31 0134 5733332 1 012 4 5832 34 2 1 014 56832 2 4 31 T h us t h ed e gre eofi n t e rva lli cs i mi la ri t y - a s be t we e n s e tn umbe r 1a n d n umbe r 6 - ord i s s i mi la ri t y -a s be t we e n s e tn umbe r 1a n d n umbe r 35 - i si mme d i a t e lya ppa re n t . Gi ve n a s e t wh os e combi n a t ori a l prope rt i e sa reun kn own , fi rs tre d uce i t t o n orma l form, t h e n coun tup t h e fi ft e e n i n t e rva lsa n d con s ult t h e t a ble .On ceh a vi n gi d e n t i fi e d t h e s ources e t , re fe r t o t h e four column s la be le dAggre ga t e T ra n s pos i t i onNumbe rs ; t h e combi n a t ori a l t ypea n dord e r, a s we lla s t h efun ct i on a l t ra n s pos i t i on s , ca ne a s i ly be d i s cove re d .For e x- a mple , s e t n umbe r 4(E)i s t h e All-Combi n a t ori a lT h i rd Ord e rSe t , s i n ce t h re et ra n s pos i t i onn umbe rsa ppe a ri n e a ch column . Se t n umbe r2 3, wh i ch h a s at ra n s pos i t i onn umbe ron ly i n t h e R column , i sobvi ous ly on e oft h ee i gh ts e t sh a vi n gn os pe ci a l combi n a t ori a l prope rt i e s . (Ih a ve la be le d t h e s e R t ype s .) He n ce fort h , t h e a ll-combi n a t ori a ls e t s wi llbe i d e n t i fi e dby t h e cod e le t t e rs A---- F; a llot h e r s e t s wi llbe la be le dI, RI,P, or R plust h e t a ble n umbe r. T h e formula s a s s oci a t e d wi t h Ba bbi t t 's fourca t e gori e sof "s ource h e xa ch ord s "wi llh e re a ft e r be a bbre vi a t e d a s follows : De fi n e t h e t ot a ls e tby i t s t wocomple me n t a ry h e xa ch ord sPa , b. T h e s i x-n ot e con t e n t 'a 'i s d i s t i n ct from t h e s i x-n ot e con t e n t'b' a n d con t e n tord e ri n gi s i mma t e ri a l. i f bta (b=a t ) on ly*5, t h e n Pa. Pa t = A(P t ype )*6 a b0 134 5867910112;a t 6, 67910112 =bThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 2 8 i fb~a(b=a )on ly, t h e n Pa% Ia =A(I t ype ) Sb01 23 465 7 8 910 11;a +,011 109 8 6; a +t 11,110 9 8 75=bi f a Ta +(a =a +t ) [b=b+t follows ]on ly, t h e n PaQ RIb+t =A(RIt ype ) a bS156782 34 910 11; a +, 011765 4 ;a +t l, 108765=ab+, 1098321;b+t 1,111094 32 =bIfa llt h re e re la t i on sh old , t h e s e t i sa ll-combi n a t ori a l;i fn on eoft h e re la t i on sh old , t h e s e t i s on e oft h ee i gh tR t ype s . (Re - me mbe r t h a t P uR=Aa lwa ys .) Ama jori t y offi rs t -ord e r h e xa ch ord s t h e ms e lve s con t a i n n on - i n t e rs e ct i n gs ubs e t s t o wh i ch t h e a bovee qua t i on sa rea ppli - ca ble .(Se e T a bleI, IVor IVa .) (012 )(3 45) b=a t 3, a =a+t 2 , b=a +t 5 (0 14 )(23 6)b=a t 2Not e t h a t i n s e t s of h i gh e r ord e r,(a t =b)t i me s t h e ma xi mumn umbe r ofa ll-combi n a t ori a ls ubs e t se qua ls12 or amult i pleof 12 (i .e ., a ll pri me t ra n s pos i t i onn umbe rs t a ke nt oge t h e rform as ymme t ri ca l con s t ruct wh i chs pli t st h e oct a vee qua lly). (012 )(6 7 8)or(0 27)(6 8 1)a t 6=b,6x2 =12 ,(06) (01) (4 5)(89)or(05)(4 9)(81) e t c.a t 4 =b,4 x3=12 ,(0 48) (0)(2 )(4 )(6)(8)(10) a t 2 =b,2 x6=12 ,(0 2 4 6810) Si n ce t h e h e xa ch ord ca n i t s e lfbepa rt i t i on e d e qua llyby2 ,(32 ), (33) or, con ve rs e ly, d e ri ve dbys ys t e ma t i c ope ra t i on (pri mi n g, i n ve rt i n g,followi n g, e t c.) uponon e(column 1un d e rt ri ch ord s ) or t wo (column 2 ) t ri ch ord s , t h ege n e ra t orswh i ch fulfi llt h e s ere qui re me n t sa re i n clud e d i n T a ble I.T h us our t a ble re ve a lsa llba s i c i n forma t i oncon ce rn i n gt h e s ource s e t 's i n t e rva lli cs t ruct ure , combi n a t ori a l t ypea n dord e r, t h es pe ci fi ct ra n s - pos i t i onn umbe rs fora ggre ga t eforma t i on(s e con d a ry s e tn umbe rs a ree a s i lyd e t e rmi n e d ), a s we lla s t h e t ri ch ord a l mos a i cs con t a i n e d t h e re i n . T h ea ggre ga t e prod uce d by t h es i mult a n e i t y oft wope rmut a - t i on a lly re la t e d s e t forms i s n otn e ce s s a ri ly ord e re d .But t h eprod uct ofs e t un i on would be a t le a s t on e n e ws e t t ype .T h i s"d e ri ve d h a rmon i cs e t ',' wh os ecombi n a t ori a li t yd e pe n d s a tle a s tupon con t e n tpla ce me n t oft h epa rt swi t hre s pe ctt o on eThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and ConditionsSOURCEHEXACHORDSNo.Se t 15In t e rva ls AGGREGAT E T RANSPOSIT ION NUMBERS T RICHORDS (ze ro i s on ~i t t e d t o s a ves pa ce )+12 34 56PI R RI On e Ge n e ra t or T wo Ge n e ra t orsAl012 34 554 32 10611116 12 , 13, 14 , 2 4 12 -15,13-14 ,13-2 5 B2 02 34 5734 32 30611613,15,2 4 ,2 512 -2 7,13-37,14 -2 5 30134 5832 34 3062 15 12 -15,13-14 ,13-37,14 -2 5,14 -37,15-2 7,2 4 -4 8,2 5-37E4 014 589303630 2 ,6,103,7,113,7,112 ,6,10 14 ,15,37,4 8C502 4 57914 32 506336 2 4 ,2 5,2 7,37 13-2 5,15-2 7,2 5-37 6012 34 64 4 32 11111114 12 -13,12 -16,12 -2 6,13-2 4 ,13-2 5,13-2 6,13-36,14 -15,2 4 -2 5 7012 3564 332 2 1112 5) 8012 4 564 32 32 13 3 12 ,14 ,15, 2 5)(12 -15,12 -4 8,13-16,13-2 6,13-37,14 -16,14 -2 6,15-2 4 , 2 4 -2 7) 9012 35734 2 2 31111115)12 -2 4 , 12 -2 7, 13-16,13-2 5,13-2 6,14 -37,16-2 5,2 4 -2 7,2 5-2 610012 4 573332 3111 110134 5733332 12 13) 12 012 4 5832 34 2 11111 12 -14 , 13-15, 13-37, 14 -2 6,14 -36,14 -4 8,15-16,15-2 6,2 4 -37,2 5-37 13014 56832 2 4 313312 -15, 13-14 ,14 -16, 14 -2 6,15-2 4 ,15-2 7,15-4 8,16-37,2 5-37,2 6-37 14 014 5783134 31614 ,36,37) 150134 682 333312 2 5) 160135682 332 4 14 13, 36) 170135782 32 34 162 13, 15,2 7, 37) {12 -2 4 , 14 -2 5,15-2 4 , 15-2 7,16-2 5,16-37,2 5-2 6,2 6-37,2 7-4 8) 18014 5792 2 34 3133 13-14 ,14 -2 4 ,14 -2 5,15-16,15-2 5,15-2 6,2 6-37,2 7-37,36-37,37-4 8 1902 357914 32 4 11137 13-2 4 ,13-2 5,15-37,16-2 7,2 4 -2 5,2 5-2 6,2 5-2 7,2 5-36,2 6-2 7 2 0012 3674 2 2 2 32 11112 6 12 -14 ,12 -16,13-15,13-16,14 -16,15-16,15-37,16-36,2 5-2 7 2 1012 5674 2 12 4 2 4 3 12 ),15,16,2 6, (2 7)(12 -15,13-16,14 -16,15-2 7,16-2 5, 16-37) 2 2 0134 6732 4 2 2 2 5 2 13),14 ,(16, 2 5),2 6, (36)1(12 -36,13-16, 13-37,14 -16,15-36, 2 5-37) 2 3012 368332 2 32 1116) 2 4 012 4 7832 2 332 1116) 2 5012 57832 2 2 4 2 11112 6 12 -13,14 -15,15-16,15-2 5,16-2 5,16-2 7,16-36,16-37,2 7-37 2 6012 4 682 4 14 2 2 111112 -2 4 ,13-2 6,14 -2 6,15-2 4 , 15-4 8,16-2 6,2 4 -2 7,2 5-2 6, 2 6-37 2 702 34 682 4 2 4 12 112 -2 6,13-2 4 ,13-2 6,14 -2 6,14 -4 8,15-2 6,16-2 6, 2 4 -2 5,2 4 -37, 2 6-36 2 802 35682 34 2 2 2 4 4 13,16,2 5,36)12 -36,13-2 5,13-2 6),14 -37,(16-2 6), 2 4 -36,(2 5-2 6,2 7-36) 2 90134 692 2 52 2 2 111116,2 6 13-14 ,13-2 5,13-36,14 -36,14 -37,2 5-36,2 5-37,36-37 300135692 2 4 32 2 4 2 14 ,16, 36,37)13-14 ,13-2 5,14 -2 6,15-36,16-2 6,2 4 -36,2 5-37,2 6-37,36-4 8) 310136892 2 4 2 32 7 2 13,16,2 5),2 6,(36), 37 13-14 ,14 -2 5,15-36,16-2 5,2 7-36) 32 01357914 2 4 2 2 11 13-2 4 ,14 -2 4 ,15-2 6,16-2 6,2 4 -2 5,2 5-2 6,2 6-2 7,2 6-36,2 6-37,37-4 8 D33012 6784 2 02 4 3,95,115,113,9 12 ,15,16,2 716-2 6 + Bra cke t e d t ri ch ord s i n R a n d RI h e xa ch ord s34 0136792 2 4 2 2 35, 115,11 13,14 ,2 5,3716-2 6,16-36,2 6-36 i n d i ca t e t h a t t h e t ri ch ord wi lln otge n e ra t eF302 4 610 06313,91,,7, 135,11,79t h e comple me n t a ry form. F3502 4 68100 60603 1,3,5,7,9,111,3, 5,7,9,111, 3,5,7,9,111, 3, 5,7,9,112 4 ,2 6, 4 8 Con s ult T a ble IVb for t ype R h e xa ch ord s . CD ?K This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 30 a n ot h e r,ma y h a veprofoun d me a n i n g, i f on lyloca lly,t h rought h e forma t i on a n d ma i n t e n a n ceof, a s we lla sprogre s s i ont h rough , h e xa ch ord a lh a rmon i e s *7d i s t i n ct from t h os e wh i chcould beprod uce d by ord e re dfra gme n t a t i onoft h eori gi n a l s e t . Ye t t h e s e t wos e e mi n glyoppos e d proce s s e s- fra gme n t a t i ona n d d e ri va t i on *8- mi gh t ult i ma t e ly re ve a lt h e ms e lve s t obebut d i ffe re n t e xpre s s i on s ofas i n gle con ce pt .(Se eExa mple 3.) Byt re a t i n gt h i s"prod uctofas pe ci fi ccombi n a t i on "a s a n e w s e t ,ca pa bleof a ggre ga t e forma t i on , we ca ngre a t ly e xt e n dt h epos s i bi li t i e soft h eori gi n a l s e t .InExa mple 1, at ype"A" s e tcombi n e s wi t h i t sre t rogra d et o form t woa ggre ga t e sof t ype112 . Exa mple1 P04 11312 5698107 R 17 108 , 965 2 12114 0 112 112(Not e t h a t a d i re ct combi n a t i on wa s s e le ct e d a n d i n t h i s ca s ewa sa bs olut e lyn e ce s s a ry s i n ce t h ere ma i n i n g pa rt i t i on sd o n otge n e ra t es e t s wi t hs pe ci a lprope rt i e s . R14 R146 4 113 1 2 or 04 1131271089657108965 For t h e mos tpa rtt h i s d i s cus s i on ofre s ult a n th a rmon y a s - s ume spoi n t -a ga i n s t -poi n tcombi n a t i on s t h eprod uctof wh i ch , d e pe n d i n g upont h e n umbe r ofvoi ce se mploye d , wi llbe t we lved i ch ord s ,t ri ch ord s ,t e t ra ch ord s ,h e xa ch ord s , or t we lve -n ot ech ord s .Act ua l compos i t i on a lre s ult s , t h ough d e pe n d e n t upont h e s e"ba ckgroun d h a rmon i e s ',' a rea lwa ysa ma t t e r ofi n d i - vi d ua l ch oi ce .) Si n cet ype112 i scon s i s t e n t lyobt a i n e d , we ca n n ow e xploi ti t ss pe ci a l combi n a t ori a l prope rt i e sa n dmult i plyby 2 . Exa mple2P ]0 4113 12g5698107 R710896 512 1 314 0 I192 1100118740 536 RI6354 78110102 91 T o e ffe ct t h i sfour-pa rtcombi n a t i on i t i sn e ce s s a ry t oe mploy t wo s e t forms(I1 a n dRI6) wh i ch a re n ot me mbe rs oft h e s ub- This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 31 group oford e r 8 (s e e t h i sjourn a l:Vol.V,p.75 a n d 79)a s s o- ci a t e d wi t hPo. It wi llbe d e mon s t ra t e d t h a t un d e r ce rt a i ncon d i t i on s a llfour forms a re me mbe rs oft h es ubgroup of ord e r "n "a s s oci a t e d wi t hPo. T h e s e t of Exa mple3i s on es uch i n s t a n ce ("4group"). Be foreca rryi n gout t h e n e xt e xt e n s i on t oe i gh t pa rt s , ord e rch a ra ct e ri s t i cs , n otn e ce s s a ri lyre fle ct i n gacompos i t i on a l s e le ct i on ofve rt i ca lt e xt ure s a n d a t t a ck poi n t s , wi llbe i m- pos e d upont h e combi n a t i on . Exa mple3 P04 11312R7108931 I192 10011 RI6354 7 ,8 134 134Un d e r t h e s e con d i t i on s t h e e xa ctoppos i t e compos i t i on a lope r- a t i onmi gh tbea s s ume d ,n a me ly t h e con s t ruct i on ofli n e a rs e t s ,0 4 11312e t c.,byfra gme n t a t i onofon es e t , 7010 64 135892 11.He re t h e re s ult a n t h a rmon i ch e xa ch ord , h a vi n ga s s ume d ord e rch a ra ct e ri s t i cs ,e me rge se i t h e r a s t h eba s i cs e t , or a s a n e wcon s t ructpos s i blyca pa bleof ch a lle n g- i n gt o s ome e xt e n t t h epri ma cy oft h e ba s i c s e t .Butre ga rd - le s s ofouri n t e rpre t a t i on , t h e mos ts i gn i fi ca n t a s pe ctoft h ea bovee xa mplei s t h ei n t e rd e pe n d e n cy ofve rt i ca la n d h ori - zon t a l pi t che ve n t s . In ord e r t o e xt e n d t h e s t ruct ure t oe i gh t pa rt st h efollowi n gun e qua lpa rt pa rt i t i on sa ren e ce s s a ry: P (2 12 1)R (2 112 ) I (1 2 12 )RI(1 2 21) I(2 121)RI(2112 )P(12 12 )R(12 21) Exa mple4P04 11312R7108965 I192 10011 RI 6354 78 e t c. P2 61534R90101187 I3114 02 1 RI8576910 This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 32T h i sproce s s mi gh t logi ca lly e xt e n d t o a sma n y s i mult a n e ouss e t forms a s t h e re a re me t h od s(i .e .,re gi s t ra l or t i mbra l d e li n e a t i on )n e e d e d t o d e fi n e i t . Acon ce rn forh a rmon y i n t h e s ma ll (i .e ., wi t h i n t h e d e ri ve dh a rmon i c h e xa ch ord s )le a d s t o a n e xa mi n a t i on oft h e ve rt i - ca li t i e s - d i ch ord s ,t ri ch ord s , e t c.-wh i ch re s ult fromd i re ct combi n a t i on s oft wo or more s e t forms *9. As s umi n gan on -d e ge n e ra t eba s i cs e t , e a ch oft h e four cla s s e s oft wo- pa rtcombi n a t i on s *10(Pu P, PuR, PuI, Pu RI) gua ra n t e e sun i queve rt i ca la s s oci a t i on s a n d t h usun i que compos i t i on a l re - s ult s . If PuP=Aobvi ous ly as i n glei n t e rva li s ma i n t a i n e dt h rough outa n d ,d e pe n d i n g uponcombi n a t ori a l ord e r, i n on e oft h e follow- i n gpi t ch -pi t chpa t t e rn s : 0 6;"0 6(3)) 96 )e t c. Wh e nPV R=At h e re s ult a n t i n t e rva ls a repre d i ct a ble on ly wi t hre s pe ctt o agi ve ns e tord e ri n ga n dma y be a ll od d , a ll e ve n , or a mi xt ure ofbot h .In t e rva lsprod uce d by t h e un i onP(0-5)v R(0-5)wi llbecomple me n t e d t h rough re t rogre s s i oni nP(6-11) %R(6-11), but un le s sP(11)=R(0) (Exa mple 5a ) t h epi t ch -pi t chre la t i on sx.+ wi lln ot be ma i n t a i n e d wi t h i n t h e combi n a t i on . (Exa mple5b).In s e t s of h i gh e r ord e r, wh e re more t h a n on ecombi n a t i onbyre t rogre s s i oni spos s i blewi t h i n t h es ub-group oford e r"n ',' t h e i n t e rva ls obt a i n e dby agi ve ncombi n a t i on a rea lt e re d i n t h e la t t e r ca s e s (s e con d ord e rs e t ,t =6; t h i rd ord e rs e t , t =4 or8;e t c.), but t h e ra t i o ofi n t e rva ls od d t o e ve n wi ll bepre s e rve d *11: Exa mple5 Li n e a r Se t s of T ypeD R7 R7 i n t e rva lra t i oP '0 6 7'2 81 453' 91011od d : e ve nR11109354 182 7608:4d =1810113 3 912 4 11 I III' ' IR7R7 P'0 67-2 81'4 53'910118:4R54 39111072 8106 d =72 4 5 93 78105 SIIr iThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 33 T h ecomple t ecombi n a t i on t a ble forExa mple5i sgi ve nbe low. P0OR11P3%jR2 P6QR5P9gR8 P3pR8 P6VR11 P9R2 P0R5 An e xa mi n a t i on oft h e t h re ere ma i n i n g t ra n s pos i t i on soft h e s ecombi n a t i on s wi t h i n t h es ubgroup oford e r 16a s s oci a t e d wi t hourt ype"D"s e t s h ows t h a t : 1)e a ch i n t e rva ls t a t e d wi t h i n agi ve ncombi n a t i on i st ra n s pos e di n t urnby t h es ymme t ri ct e t ra ch ord 0369 a n d t h e re foree i gh t(t wo ba s i c combi n a t i on s t i me sfourt ra n s pos i t i on se a ch )combi n a t i on s a s s oci a t e wi t hP0. e i gh twi t hP1, a n de i gh twi t hP2 ; t a ke nt oge t h e rt h e yyi e lda ll t ra n s pos i t i on sofa lli n t e rva ls obt a i n - a ble wi t h i n t h e combi n a t i on cla s s . 2 )T h et we n t y-four pi t ch -pi t chre la t i on s d i s t ri but e dwi t h i n t h ee i gh tcombi n a t i on s occur t wi ce i n t h e formxa n d t wi ce i n t h e formv.y 3)Ea ch oft h e fourt ra n s pos i t i on sofa combi n a t i on a t agi ve ni n t e rva l pre s e rve st h e ord e r ofoccurre n ce of t h ei n t e rva ls a s we lla s t h e d e ri ve d h a rmon i ca ggre - ga t e s . 4 )T h e i n t e rva lra t i o i s ma i n t a i n e dt h rough out . Wh e n P URI=At h e re s ult s a re s i mi la r t o combi n a t i onby re t rogre s s i on ,e xce ptt h a t t h e i n t e rva ls obt a i n e dby P(O-5)uRI(O-5)=A a rere t rogra d e di n P(6-11) V RI(6-11)=A. Exa mple6 Li n e a r Se t s of T ypeE R7R7 P'04 5'189 '11 23'7 106od d :e ve nRI73610112 4 50891 d =511133111512 :0 1I i1 p4, I Ii12 512 5 P'04 5'1891112 3'7106 RI117102 36894 01512 :0 d =19711 p 11791 IiI- This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 34R7R7 P'04 5189'112 '710612 :0 RI3112 67100184 59 d =9537 151 11759 1 11 d __ Wh e n P uI=A, a lls i xod d i n t e rva ls occurt wi ce , a t ri t on ed i s t a n t , s uch t h a tx.. for a llva lue s ofxwi t h i n as i n glecom- yy bi n a t i on .In fi rs t ord e r a ll-combi n a t ori a l s e t s , -,x wh e n t h eyy combi n a t i on i s move d t o t h ere ma i n i n g t ra n s pos i t i onle ve l wi t h i n t h es ubgroup a n d i n s e t s of h i gh e rord e r t h e re wi llbemorepos s i bi li t i e swi t h i n t h e ci rcui t .T h e combi n a t i on t a blefor t h et ype"E"s e t of Exa mple6i sgi ve nbe low. Exa mple7 s e ri e s xs e ri e s rs e ri e sty s u"d i " #1 PO$ 13P2 2 15P4 717 P6 19P8' IllP10vIi#2P2I1 P4 13P6 V 15 P8 Q 17P10k19P0 Ill #3 P4Ill 6 I1 P8 u13 P10VI5P017P2Q IgAlls i xcombi n a t i on s wi t h i n as i n gleve rt i ca lcolumnpre s e rvepi t ch -pi t ch re la t i on s ,wh i lea lls i xcombi n a t i on s wi t h i n a h ori - zon t a lcolumnpre s e rvet h e ord e r ofi n t e rva lli c occurre n ce("d i ").Not e t h a t e a ch ba s i c i n t e rva lofcombi n a t i ont oge t h e rwi t h i t st ra n s pos i t i on (t =6)i se qui va le n tt o on e oft h e a ll- combi n a t ori a lt e t ra ch ord s 6 =f, (2= e , (1 10)=e . In t h e n e xte xa mpleon e me mbe r ofe a ch oft h e four cla s s e s of combi n a t i on i sa ppli e dt o at ype"A"s e t .Ea ch combi n a t i onyi e ld sd i ffe re n t d e ri ve d combi n a t ori a li t i e s a s we lla s d i ffe re n tre s ult a n t i n t e rva lcolle ct i on s . Exa mple8 R15R15D1348:4 P0 4 1132 '76890:12 PP 132 15 ' 689 R986105712 31114 0P610579811114102 3 385310752 974 9666666666666 This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 35 R16R16RI2 1RI30 4 : 8P132 7510'68912 :0P ' 1132 510689 RI67951081102 4 113I951086712 4 11310 692 8566582 9631115975111379 T h e cri t e ri a for t h e me a s ure me n t ofh a rmon i c d i s t a n ce h a vebe e n d i s cus s e dby Ba bbi t t *12 .Se tt ra n s pos i t i on sa n d s e t formse xi s t i n gwi t h i n t h es ubgroupma i n t a i n ,i rre s pe ct i veof ord e r, t h e h e xa ch ord con t e n t of PO, wh i let ra n s pos i t i onout oft h e colle ct i onbri n gsa bout "mod ula t i on "i n t h a t agre a t e ror le s s e ri n t e rs e ct i on ofcon t e n t obt a i n s .T h e "mod ula t i on "i s e i t h e rd i s t a n t or n e a rd e pe n d i n g upont h ed e gre eofcon t e n t i n t e rs e c- t i on . T ra n s pos i t i on sa n d forms ofd e ri ve d h a rmon i c h e xa ch ord spro- d uce dby t h e combi n a t i on ofre la t e d li n e a r formsma ys i mi la rly be me a s ure d .Wh e n a combi n a t i onprod uce sas i n gled e ri ve ds e t of mi n i ma lly s e mi -combi n a t ori a l t ype , t h e re wi llbe a tle a s t on e ot h e rcombi n a t i on , wh os e cla s s i sd e pe n d e n t uponcombi n a t ori a l t ype , wh i ch ma i n t a i n s con t e n t i d e n t i fi ca t i on of t h e d e ri ve d h e xa ch ord s .T h us i nExa mple 1, (P0 % R7). t ype112 i scon s i s t e n t ly obt a i n e d a n d t h e ot h e r form i s(I1,RI6). But h e ret ra n s pos i t i onoft h e combi n a t i on wi t h i n t h es ubgroup ofi t s ve rt i ca lforms ca us e s at ra n s pos i t i onout s i d e t h e s ubgroup ofi t s li n e a r forms . If,h owe ve r, t h e ba s i c s e t i s d e - ri ve d from as i n gle t ri ch ord ,t ra n s pos i t i ont o as ubgroup me mbe r wi lli n s ure con t e n tpre s e rva t i onwi t h i n li n e a r h e xa - ch ord s a s we lla s wi t h i n d e ri ve dh e xa ch ord s , a n dt ra n s pos i - t i on ofa combi n a t i on out s i d e t h es ubgroup ofi t s li n e a r formswi ll s i mi la rly a ffe ct t h e d e ri ve d h a rmon i ch e xa ch ord s ; t h ust h e cri t e ri a for t h e me a s ure me n t ofh a rmon i c d i s t a n ce ca n bea ppli e d s i mult a n e ous ly t o bot h d i me n s i on s .T h e n e xte xa mplee xploi t st h i sprope rt y i n ord e r t o obt a i ns e con d a ry s e t s a n da ggre ga t e sa t a llle ve ls oft h e s t ruct ure *13.T h emod ula t ory "followi n g" P3 J I2prod uce sa li n e a r i n t e rs e ct i on oft h e t h re en ot e s 678i n on eform, 012 i n t h e ot h e r (012 678i s t h et ypeD s e t ), a n d acorre s pon d i n gi n t e rs e ct i on oft h e t h re en ot e s 2 4 6be t we e na d ja ce n td e ri ve d s e t s (t a ke n wi t h 8100 from t h en on -a d ja ce n ts e t s at ypeF s e t i sprod uce d ). Exa mple9 312 4 510118679R3102 54 11108796P364 578 t ypeB t ypeB t ypeB t ypeB t ypeB 181097610354 2 RI810119670134 2 5I2 1110109 ?I IThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 36 T h us , ad e gre eofh a rmon i cch a n ge cle a rly e xi s t s bot h wi t h i nt h e colle ct i on ofcombi n a t i on s a s s oci a t e d wi t h t h es ubgroup a n dwi t h i n a llcla s s e s ofcombi n a t i on i n clud e d .T h et ra n s pos i t i onofcombi n a t i on s t o le ve ls wi t h i n t h es ubgroupca n , a s i nExa mple 9,pre s e rvecon t e n t ofbot h li n e a r a n d ve rt i ca l s e t s , wh i let ra n s pos i t i ont o le ve ls out s i d e t h es ubgroup e ffe ct ss t ron gs h i ft s i n h a rmon i cs t a bi li t y s i n ce t h e h a rmon i c h e xa - ch ord s wi ll s i mi la rly bet ra n s pos e da n d s i n cepi t ch -pi t chre - la t i on s(e xce pti n t h e ca s e ofPUP)wi llbe a lt e re d .Furt h e r- more , e a ch combi n a t i on cla s sprod uce sd i ffe re n t re s ult a n tcombi n a t ori a li t i e s(s e eExa mple8) e xce ptwh e n t h e ba s i c s e ti st ri ch ord a lly d e ri ve d(Exa mple 9), i n wh i chca s e ,a lt h ought h e combi n a t i on s a re s t i lld i ffe re n t i a t e dby t h ed i ch ord s , PV 1I= PuRIa n d P%R=PPwi t hre s pe ctt ot ype . Oft h et we n t y-n i n es ource t e t ra ch ord s *14 (s e e T a ble II)s e ve n , t h os e wh i ch e xclud e t h e i n t e rva l '4 ', a rei n d e pe n d e n t ly combi n ra t ori a l. T h et we n t y-t wo pos s i blecombi n a t i on s oft h e s es e ve n s e t s a repre s e n t e d(T a bleIIa ) i n t h e form oft e t ra ch ord - a llyd e ri ve d , t e t ra ch ord a lly combi n a t ori a ls ource s e t s of t we lve n ot e s .Ea ch t ot a ls e t i s i n t urnca pa bleof a ggre ga t e(a n ds e con d a ry s e t )forma t i on . Exa mple10 a, 032 1#9P , 032 14 657810119 a +, 74 56 RI, 64 578109112 103 a, 910118 P, 81110902 134 675 ASi n ce t e t ra ch ord a l combi n a t ori a li t yd e pe n d s upont h e e xclus i onoft h e i n t e rva l '4 ', T a ble IId i vi d e s i n t o fourpa rt son t h e ba s i soft h e n umbe r ofoccurre n ce s oft h i si n t e rva l,0,1,2 , or 3 t i me s .T h ea rra n ge me n twi t h i npa rt sfollows t h eproce d ureof T a ble I. T h e re a re t we lvei n d e pe n d e n tcon t e n tord e ri n gsof 's ', t h es e mi -combi n a t ori a ls ource t e t ra ch ord(Pt ype )*15. Oft h e foura ll-combi n a t ori a ls e t s offi rs t ord e r('a ','b','c','d '), t h e fi rs tt h re e h a vee i gh t i n d e pe n d e n tord e ri n gs e a ch , ofwh i ch foura red e ge n e ra t e(P=RI).Se t'd 'h a s s i x i n d e pe n d e n t ord e ri n gsofwh i ch n on e i sd e ge n e ra t e . T h e s e con d ord e r s e t 'e 'wi t ht wot ra n s pos i t i on sfor con t e n tpre s e rva t i on (0,6), h a s but s i x ord e ri n gsofwh i ch a lla red e ge n e ra t e , wh i le t h e t h i rd ord e rs e t'f' wi t h fourt ra n s pos i t i on si se qui va le n ta t a llcon t e n tle ve ls(0,3,6, 9)a n d t h e re a re but t h re e forms- a ll This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 37 t a blesII SOURCET ET RACHORDSNo.Se t 6In t e rva ls12 34 56 &a 012 332 1000 d 2 012 72 1002 1 e 301672 0002 2b4 02 3512 2 010 s 5 0136112 011 c6 02 5702 1030 17 0369004 0028012 4 2 2 1100 90134 2 12 100 10012 52 11110 11012 62 10111 12 01562 0012 1 13013512 1110 *14 014 6111111 1502 36112 101 *160137111111 17014 7102 111 18I 015711012 1 1902 3711112 0 2 002 4 702 112 0 2 1 102 58 012 111 770358 0 1 2 1 20 2 5 '0158 10 1 22 0 2 6 102 4 60 3 0 20 1 2 7!02 6802 02 022 81014 8101310 2 9102 4 802 0301 IIaT HE2 2 T ET RACHORDALLY DERIVEDSOURCE MOSAICSNo. Se tAggre ga t eT r.Nos . PIIRRI 1O'4 8 4 ,8 2a 8b24 ,8 3ab5 c9 4 ,8 4a O d 5e 44 ,8 5 a Oa+5 4 , 86 6 0c2 +11 4 ,8 6 7 d 9s a 84 ,8 6 8 SO 2 +10 4 ,86 9 a 0 a 4 a 8 4 ,80 10 a 0 b5c44 , 83 11 a 0 d 9 a 5 4 ,80 12a 0e 4 a 6 4 ,8330 13 b0 b4 b84 ,80 14 bf4 b6 4 ,8550 15cc4 c8 4 ,80 16 cd 9c1 4 ,80 17 c0e 3c6 4 ,8 77 0 18 d O d 4 d 8 4 , 8 0 19d Obs d 3 4 ,80 2 0e 0 e 4 e 8 4 , 8;2 ,100;62 11e2 e 34 ,8;2 ,107;1710;6 2 2 ff f 4 ,8;2 ,10 00 0 85:,7;1:11T ra n s pos i t i onn umbe rs for I,R, a n d RI a recomput e d bya d d i n gt h e column n umbe r t ot h e P a ggre ga t en umbe rs . If,In s e t s oft wo or t h re e d i ffe re n t t e t ra - ch ord s ,t h e od d me mbe r i s i n a n out e rpos i - t i onon ly PPPa n d PI I combi n a t i on s a rea va i la ble . This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 38 d e ge n e ra t e : t h usgre a t e rcombi n a t ori a l fle xi bi li t y i s a ccompa n i e d by a ni n cre a s i n gre d uct i on a n dd e ge n e ra cy oft h e forms . T h eprod uctoft e t ra ch ord a lcombi n a t i on s i s aqua rt e toft ri - ch ord s wh os e me mbe rs a re n otn e ce s s a ri ly d i s t i n ct .T a ble III d ocume n t s t h efollowi n gd i s cus s i on of qua rt e t s prod uce d by t h ecombi n a t i on ofre la t e d forms ofas i n glet e t ra ch ord : 1.Pri me re la t e d combi n a t i on sge n e ra t efourt ra n s pos i t i on soft h e s a me t ri ch ord :048, 15 9, 2 6 10, 3711.Se t s of h i gh e rord e r h a ve more pos s i bi li t i e s . 2 .Ea chord e ri n gofagi ve nt e t ra ch ord combi n e d wi t h i t s i n - ve rs i on a lly re la t e d formsyi e ld st h e s a me ve rt i ca l pi t chre la - t i on s . Exa mple11 012 30132111098 j 111089 7654 =764 5 3.Un d e r P uRv R, PQ RI Q RI, or P U RU I, a n d i n t h e ca s eof n on -d e ge n e ra t eformst h e re i n , t h e re s ult sd e pe n d uponcla s soft e t ra ch ordord e ri n g(s e e T a bleIII;012 3,02 13,1032 , 1302be lon gt o on ecla s s ,0132 , 0312be lon gt oa n ot h e r cla s s ). 3a .Un d e r P uR~VRa n d PJ RI' RI, a ll ord e ri n gsofagi ve nt e t ra ch ord wh i ch s h a re t h e s a mequa rt e ts h a re t h e s a meve rt i ca lDi t ch re la t i on s . Exa mple12P02 5 3j P52 3 jP0 5 32 j P0 2 5 3jP 0 5 2 3jPO5 324 69 7=P4 9 67=P 49 761 P4 697=P4 967=P4 9 761 LR 111108 R111018 RI111081 RI108111 RI101181R101118 3b.Un d e rPuI uR, a n d i n t h e ca s e of n on -d e ge n e ra t eformst h e re of, ord e ri n gswh i ch s h a re t h e s a mequa rt e ts h a repi t chre la t i on s i ft h os eord e ri n gsa re me mbe rs oft h e s a me cla s s . Exa mple13 0132 031 2 02 31032 1 11108918109# 119810j 118910 6754 6574 ]5764 5674T a ble IIIi n d i ca t e s t h a t e a ch a va i la ble combi n a t i on cla s sThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 39 t a bleIII T ET RACHORDORDERINGS 'T RICHORDALQUART ET UNDER: icoln PolWlPuuRUR PouRIuR'I PulwRcolumn 1column 2 column 3column 4 column 5 at 012 3,02 13,1032 ,1302 (4 8)4 (15,37)2 s e e col.2 s e e col.1s e e col.20132 ,03121 " (2 6)4 (37)4 (16,2 5,2 7,36) 02 31,032 1 " (37)4 (2 6)4(16,2 5,2 7,36) bt 02 35,032 5,2 053, 2 503(4 8)4 (14 ,37)2 s e e col.2 s e e col.1 s e e col.202 53, 052 3" (14 )(2 6)4 (13,16,16,2 5) 0352 , 0532" (2 6)4 (15)4(13,16,16,2 5) ct 02 57, 052 7,2 075,2 705(4 8)4 (14 ,15)2 s e e col.2 s e e col.1e e col.202 75,072 5 " " (14 )4 (2 6)4 (12 ,13,16, 36) 0572 ,0752" (2 6)4 (14 )4 (12 ,13,16,36) d0567,'0657,5076, 5706(4 8)4 (2 6,4 8)2 (14 ,37)2 (14 ,37)2 (12 ,14 ,36, 37)/ (14 ,2 7, 36,37) 0576,5067 I " " (2 4 ,2 6)2 (2 4 ,4 8)2 (2 4 ,2 4 ,2 6,2 6) e t 0167,0617, 0671,0761(4 8)4 /(2 4 )4 (14 )4 /(37)4 /(13,2 5)2 s e e col. 2 s e e col. 1 s e e col.20176,0716 """" "os e e col.2 s e e col.1s e e col.2f t 0369, 0396,0639(4 8)4 /(2 7)4 / (2 4 )4 /(15)4 /(2 4 ,-4 8)2 / s e ecol.2 s e e col.1s e e col.2(2 4 )4 /(15)4 /(12 ,2 7)2 /(15,15,2 7,2 7)/ (12 )4 (12 ,12 ,15,15) a0136,0316,1063,1603(4 8)4(2 4 ,2 6)20163, 0613,1036, 1306" (14 ,15)20361,0631,3016, 3106 (15,37)2T h es ymbolt i n d i ca t e s ad e ge n e ra t e form. T h es ymbol(15)i n d i ca t e s t h e t ri ch ord 015. This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 0 (d e ge n e ra t eforms re d uce t h e n umbe r ofs uch cla s s e s ) gua r- a n t e e sun i quere s ult a n t h a rmon i e s a n dt h us , a s i s t h e ca s ewi t hh e xa ch ord s ,un i que compos i t i on a l re s ult s . T h efollowi n gt a ble furt h e r s umma ri ze s T a ble III. 12 1314 15162 4 2 52 62 736374 8 a xxxxxxxx b xxxxxxxx cxxxxxxxx d xxxxxxx e xxxxxx fxxxxxxx SxXXXXx Acon s i d e ra t i on oft h ere ma i n i n g e qua lpa rt pa rt i t i on(2 2 ) re - la t e s re s ult a n th a rmon y t o s ource h e xa ch ord s .Un d e r agi ve nope ra t i on (combi n a t i on ), a ll ord e ri n gsoft h e s a me t e t ra ch ordwh i ch s h a red ya dcon t e n t wi t h i n t h e i rpa rt ss h a re re s ult a n th e xa ch ord a lcombi n a t ori a li t i e s . Exa mple143110813810318103 1011 81831031081 11964 9114 61194 611694 94 1161164 9 02 572 07502 75052 72 705057216161612 512 512 5 In t h e ca s e ofP vIv R, or i n s e t s of h i gh e r ord e r, t h e re i s ach oi ce of t ra n s pos i t i ona n d t h e re fore a ch oi ce ofre s ult a n t h a r- mon y. Exa mple15 32 9832 989832 9832111054 54 111054 11101110540167016701670167 A A EE Wh e nd i s jun ct d ya d sofagi ve nt e t ra ch ordord e ri n ga re i n t e r- va lli ca llyd e ge n e ra t e . re s ult a n t h e xa ch ord s a re ma i n t a i n e dun d e r PJ IVI,PvRVR, a n dPuvI R, a n d a re a lt e re dun d e r PJPUPa n d PURI RI*16. This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 1 Exa mple16 I111810R111108R111108P810111RI108111 I974 6R7964 I974 6P4 697RI64 79 P02 53P02 53P02 53P02 53P02 53 191919F FT h re e -pa rtt e t ra ch ord a lcombi n a t i on s ,s ubje ctt o t h eprope r- t i e s oft h e i rre s ult a n ts e t s oft h re e or s i x n ot e s ,ma y be e x- t e n d e d t o s i xort we lvepa rt s . T a ble s IIIa n d IVca n be us e d t od e t e rmi n e s uch e xt e n s i on s . Exa mple17 P02 5302 5302 53 P4 6974 6974 697 RI108111108111108111 F F RI5302 5302RI974 6974 6 I111108111108 RI2 1RI2 1R352 0 R7964I111810 12 035 164 79 RI810111 Eve ry t we lve -t on e s e t , con s i d e re d i n t e rms ofi t se qua l t h re epa rt pa rt i t i on (4 3), i sca pa bleofa t le a s t on et h re e -pa rt"obli quecombi n a t i on "*17 (P V URP); gre a t e rcombi n a t ori a l fle xi bi li t yd e pe n d s upon s ymme t ri e swi t h i n a n d be t we e n out e rt e t ra ch ord s .If pa rt sa re d e fi n e d a sx,y, a n dz, t h e n lxyLlz] If X'X+ on ly, t h e nIPiRP/I If Z TZ+on ly, t h e n P R/ RI If X % Z.z tZ on ly, t h e n P R/RI P/Il This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 2T T TIfXx+ , Z Z4 ,XZ, t h e n PI /I/R/RI R/RI P/I/RRITIfX~Z on ly, t h e nIP P/RgP/1RTIf Z~X on ly, t h e nIP IP/R1 IT h et we n t y-t wocombi n a t ori a ls ource s e t s (T a bleIIa ) a llow "d i re ct "a s we lla s"obli que " combi n a t i on s : P0-34 -78-11 110-34 -78-11 1110-34 -78-11 Pri me re la t i on s a rea lwa ys pos s i ble ;re t rogra d ea n d i n ve rs i onre la t i on sa repos s i ble on ly wi t h t h os e s e t s wh i ch re t a i n un i tcon t e n t oni n ve rs i on , re t rogre s s i on , or bot h(s e t s #10,#12 , #14 ,#17, a n d #2 1).T h efollowi n gcombi n a t i on cla s s e s e xi s t : PUPvP=A, PUPUI=A, PJPUvR=A,PvUPRI=A,PvIuR=A, a n d ma y bea ppli e dt o s e t s wh i ch con t a i nt ra n s pos i t i onn umbe rsi n t h eprope r column sofT a ble Ha .(If pri mere la t e d or i n - ve rs i on a lly re la t e d combi n a t i on s a ree mploye d , or i fout e rt e t ra ch ord s a repe rmut a t i on soft h e s a me t e t ra ch ordord e ri n g, T a ble IIIca n be us e d t o ca lcula t e t h e h a rmon i c re s ult s .)Inge n e ra l, t h e t ri ch ord sprod uce d byt h re e -pa rtd i re ct combi n a - t i on s oft h e t ot a ls e t a re e xt e n s i on s of t wo-pa rte ve n t s .Pri mere la t e d combi n a t i on sprod uceas i n glet ri ch ord wh os e ve rt i ca l pi t chrot a t i on s x y za refour t i me st ra n s pos e d . Wh e ny-+ z--x zx y PuRIuP (s e eExa mple10a .)t ri ch ord s of comple me n t a ry ord e r n umbe r a re re la t e dbyt ra n s pos i t i on (i .e ., t ri ch ord s of Ha a rere t rogra d e di nHb).In t h efollowi n g i n ve rs i on a lly re - la t e d combi n a t i on , d i ch ord s a re fi xe d fori n ve rs i on a lly re la t e dforms ; a n d e a ch t ri ch ord i st ra n s pos e d by t h e i n t e rva l 6, i n - ve rt e d a t t h e i n t e rva l 3, a n d t h e nt ra n s pos e d by 6:0369. T h e fourpe rmut a t i on a lly re la t e d t ri ch ord s form a t ot a ls e t . Exa mple18 I I I IP012 3578104 6911 I7654 2 011931108 I11109864 31752 0 1 IL. 1 -1 iI I This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 3 T h e con t e n t wi t h i nt e t ra ch ord s , a s we lla s t h e form oft h e"followi n gs ','ma y beord e re d , a n d t h os eord e ri n gs pe rmut e ds uch t h a t va ri oust ype sofh e xa ch ord a l combi n a t ori a li t y re s ult . Fore xa mple , t h e mos a i c a bcyi e ld sh e xa ch ord a l t ype B,19, a n dR10; bcayi e ld s A,R12 2 ,16, a n dR7; ca byi e ld s C,119,19, RI31, a n d R16.T h e s e t of Exa mple19i sh e xa ch ord a lly, t e t ra - ch ord a lly, a n dt ri ch ord a lly combi n a t ori a l. Exa mple19 a b c02 1114 96753810 BB P02 1114 9 16 7 538 10x0 y 7 I 75683 1011 02 4 119 y 7-x 0 P02 1114 96753810x0w10 I756831010 2 14119 y 7 _z 9 R108357694 1112 0w10x0 RI 9114 2 011038657z9 y 7 P02 1114 96753810 o,o,o+,o+, = s e tI 978105032 4 6111 p,p,p+,p+, = s e tP4 6538110119702q,q,q+,q+, = s e top op+qqoo++q+pq T h e t a ble s ofs e ri e s IV provi d ea lli n forma t i on e s s e n t i a lt ot h e ca lcula t i on oft h ea ggre ga t ea n d d e ri ve d s e t forma t i on swh i che mploy from on e t o four d i s t i n ct t ri ch ord s .T a ble IVapre s e n t s d a t a , s umma ri ze d i n T a bleIV, con ce rn i n gt h e h e xa - ch ord s wh i ch ca n be d e ri ve dbyope ra t i on uponas i n glet ri - ch ord .Fore xa mple , t h e s e t 013followe dby (013)t 6forms at ype134h e xa ch ord ; t h ecomple me n t a ry h e xa ch ordma y be d e - ri ve d t h e re from(mos a i ca a /a +a +)*18 orma ye mploy n e w ge n e ra t ors(mos a i c a a /bb).Mos a i cs oft h et ype a b/a b,a b/a cora b/cda re t h es ubje ctofT a ble IVb.For s e t s oft h et ypea a /bc con s ult bot h t a ble s *19.(T a ble s IVa a n d IVb a re re a d a sfollows : I&v u 2 1 F(2 8) s t o. L ~C~(2 3) T h es ymbol 'a 'a ft e r a s e t n umbe r i n d i ca t e s t h ecomple me n - t a ry form oft h e s e t n umbe r a s li s t e d i n T a ble I (i .e ., 2 9i s t h et a blen umbe r, 2 9a i s t h ecomple me n t ). This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 4t a b 1eIVSOURCET RICHORDST ri -De ri ve d Se t s of12 De ri ve d Se t s of6 ch ord P. t r.n os . 1. t r. n os .He xa ch ord T ype He xa ch ord T ypeAll-comb. I RI PR RI012 0, 3, 6,92 ,5,8,11 A,A,D (8),(2 1) 0130, 65, 11A,B 34 (11)(17),(2 8) (16)(2 2 ),(31) 014 0, 29,11E 6 2 2 (14 a )(8),(30a ) 0,10 9, 7 E 6 2 20, 6 9, 3 A,E340150,63,9B, D, E(8),(17) 0, 2 11,9 E 9 2 10. 107, 9E 92 10,3, 6, 9D 30160, 3 8, 11D 2 9 2 1(2 3)(2 2 a ), (30a ) 0, 9 8, 5 D 2 92 1(2 4 a )(2 8), (31) 02 4 0,1, 6, 7 4 , 5, 10,11 A,C, F0, 5,6,114 ,9,10, 3 A,C,F0, 3, 6, 94 , 7,10,1 B,B, F02 50, 6 3,9B,C 34 (7)(8),(2 8) (15a )(2 2 a ),(31a ) 02 60, 110,.11F 2 0 310, 11 10,9 F 2 0 31 0,510,3F 2 52 20,7 10,5 F 2 52 20,310,1F 2 9 2 1 0,910, 7 F 2 9 2 1 02 70,3,6, 92 ,5,'8,11C, C, D(17),(2 1a ) 036(14 )(2 2 ),(30a ) (16)(2 8),(31a ) 0370, 21, 11E 19 31(14 a )(17),(30a ) 0, 109,11E 19 31 0, 65,11C,E 3404 80, 1,2 ,3 8,9, 10,11 E, E, F4 ,5, 6, 7 0,1,2 , 3 E, E, F8,9, 10, 114 ,5,6, 7 E, E, FAll-combi n a t ori a lt ri ch ord s a re un d e rli n e d . Wh e n s e t n umbe rs a re bra cke t e d ,t h ecomple - me n t a ry h e xa ch ord i s d e ri ve d from d i ffe re n tge n e ra t ors . This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 5 t a bleIVaT RICHORDAL MOSAICS(SINGLEGENERAT OR) 1(B)(2 1)D(2 1)(8)A01o013 B ()m (31)(17)(2 2 ) A014.1 )(1) 3((1) 1 ) @ .62 (2 )E2 2(8) ()r2 2 (2 s oftThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 6 t a bleIVa1(r) .l) i mum93 D 32 1_1'E _"2 a Am-(a_ - --.1 ,I.-Ai r- This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 7 t a bleIVa016 wFI 32 (U)V'(2 2 a ) i i(30a )(31)DA(2 ") (2 4 )2 9a (2 ) t -v S2c BA-wThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 8 t a bleIVa.3(2 2 a ) (31.) c(2 8) (8) S(2 8)(3) (31.)(2 2 ) 3 (134 ) 3" (11.) (7) 2 02 9a2 5K2 342 9. 2 002 6 2 2 2 2 2 1.31F 31 2 1 2 131 F 31. 2 1. 2 2 a . 2 22 2 a2 0 MD.POThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 4 9 t a bleIVa2 12 1F Fvw-- (2 1) c(17) D (17) . C2 a02 7r14 19419 S(s o.)1(31a ) (17) (17)2 111 2111 14 19.419. 14m____ mm aThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 50 t a bleIVa@8rEE34This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 51 t a bleIVbT RICHORDAL MOSAICS(MORE T HANONEGENERAT OR) I I R I AlL6a102 4 r 7 2IB2 7 1,.-i-6,6,7,10 &Ilow-i -1 w2 3. No 2 47t 12 14 .2 4A& 7s12 14 2 4 2 0 1%! 1 1 12 f 12 7. i s_____i 13__Li ln o____-a ____a s __a I _i t I __lI__la A_3_2 013& 6.212 . ,,. A3. 2 1z. A2 12 6.. A0? 2 1a0 - -Pa t a l I 7 S112 0 7 .a11, 2 0a 7&.62 6~~~~62 01014 16 2 010 2 0 2 4 61 2 7 7 Pa112 6a2 79a ...D67 l 2 2 .2 5Fu oa 10 mlIm--- ...-, 0I,151Lx 2 2 3 Ila `2 72 3 ,6&., ft,2 & 2 36,6~/ 1112 42 6,lo-1 .11n .Abei71 96. 2 711 1,. pl-..gThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 52t a bleIVbI 14 P t. .. . I IIa w - e If - 012,,A2 9 i s &14 764 m313730 A11. 2 918.14 7330 31&1&A2 9 11 1 671 17ona 12 7,10 2 011,14101114 2 0. 12 162 4 7162 4i n 2 4 2 0 .72 2 2 2 1a 8h ..-- - ,2 0A7,10 8 -4*2 29.2 2 2 1.8102 4 2 07 ,12 -- i / t 19 6.. 30LC I-&2 9p 2 89C2 0.9& A f7t 2 9.19& 1s6 , .2 6 6.9I 1 91 6,6.1 f 10 L -a2 5i o11. 2 3.1 0.7. &6M 3611f 2 a,2 97a 192 911 16 11U121 .7. 6.3 32 2 & 9 37 II I131 12 7.f9 16v2 2 & ft3 32 2 &., 11 2 &16.16 . 4 61 11l11U11,1462 2 14 14 2 0. 7. 2 94 2 9 117 2 4 . 16 2 1- ,,13-.2 2 A - 2 0 1 76. BO_14 2 1 2 0S 14 2 46. 2 113& 2 151 151L 32 i 2 11 This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 53 t a bleIVbHEXACRORD077 37& pII6. 1t I16a1815103 3117a9 2 612 52 4 . 2 7. == ;1&.30a 2 6 8 U12 ,11 11 82 6, 2 2 4 . o2 30 13s302 "8 12 2 2 7 110151516a 10 16 36 _,122 971 . 12 ,2 712 1lOa 3"I1. 2 94 12 a 7 I 1 13_82 82 2 88.3 .0 111o2 9lo1011 s 2 82 8a 3a s151L2 7124 8 1L 2 7 2 7. 12 . 015 1011a 14 18a 2 5alo 12 14 15 02 13,2 8,17 2 61713a 8 10 l4 M_5.117 14 . 18 18 2 5111-m ot .2 518,2 7,10 7 16a 18a 161014 a 2 51142---2 ,18- 11 32 17a 7c 1 2 733_?a _ _33 7. C13C 17. 9310 1 2 2 26lI,II.,.-- 19I" 2 2 a 3130 16 2 4 a 1612 3 14 191,2 . 37 -._9-_154 .1" 2 3L,6,162 314 . 18.14 . 2 3a 16 2 4 .16. l;s2 8 2 6a 13 This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 54t a bleIVbP0IIRIRI2 3a 4 2 2 3a2 42 3a 2 4 2 31 2 5, 3 , , .., . 17 31. 2 1 9,2 5 12 .17 162 1 |6~fF l7br,ii3i ,|5i -,i,l 9A 1731a 3i 2 3a 162 6 lO22 630D 2 6.2 2 6.2 2 72 4 2 3.342 6m2 6,,2 3.2 6. 32 ,34 2 4 3e 2 3, 2 7A 2 4 32S342 6 o3. D i 1&16.t 2 WO1 165 1.2 6a 16 16 1,16.1 192 53612 6.1 414 . h ot 2 02 6 2 3 SII 0-1 M.-&- 1wj'-l-2 '-5-2 4 a -I V- -- 2 W2 32 534 2 6. 3414 2 52 4 16. 13A 2 1A 17a .14S'7.Illi i ll--lI _ a -,L,2 5 If-346la 19a 522 3 la 1932 a 2 7&-.-."i . 1 ".-19 711-.16.10 T . ' L101616. 15 7"lrs2 61-I-iW1 -7,T & 10-'t ... -- - ' ,,,'",.11.M4 8IN.-M2 6 384a ft30 37 4 6~SThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 55 t a blleIVbP0. 922 6I.17.R...RIAS97A aL"_A2 512 %2 m99,1911. 17a2 6 2 52 8 112 552 19. 9A 2 6 17a 2 8 2 71 1219.l 2 0916 I~ 2 5. I112 0 16,192 "2 5a 2 " 36 10 2 ..IA-a,6 ,?10 2 9 11 i -51 19 2 9A1012 15161"2 9 ,3a 237_ | rIIi r i --i III- 1 I,- Ia 12 l2 2 22 31330C 2 2 12 a 151614 a 2 9A14 8 . 51.5 15 02 14 .152 3l~a 3234 12 3a1 r2 ar- ! 4 " i- 2 3 f .I ,.I L~~-r~~-?~_!~ ._ !- i -= : '=-,, ,,,' F=--+-- ,? ----1- A" 181Z214 14 .2 7,-2 4 34 a 32 2 3a 351 17 3L3 .9 182 4 a 30a ? a sW 15,2 a I'2 :2 "d 1.73;1.3'L31 32 916.2 517. mpe --a =21pw 4 -- Ir Va s4 8u This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 56 t a bleIVbNExACORD T YPEI R RI.036~37 1s 0Iti i. b 141i1618 4t o16I$a4 8 a n3 48i,,1k This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 57 Aggre ga t e -formi n gcombi n a t i on s oft ri ch ord syi e lda t e t ra - ch ord t ri o. Comple t ere s ult s for combi n a t i on s of re la t e dforms ofas i n gleord e re d t ri ch ord a re li s t e d i n T a ble s Va n dVI.Si n ce t h e t a ble s a recros s -re fe re n ce d , on e ca nbe gi ne i t h e r wi t h t h equa rt e toft ri ch ord s (V)or wi t h t h e t ri o oft e t ra - ch ord s (VI).Afe w e xa mple ss h ould i llus t ra t e t h epos s i bi li t i e s : P910 11 xI 11 109 + I 11107 P678P678x 1985 + P34 5I543+ P2 36 P 012 P 0 12 P 0 14fffe fe b c aT h e ba s i c s e t con s i d e re d i n t e rms ofi t se qua l-pa rt pa rt i t i on smi gh tbea ccompa n i e d byi n d e pe n d e n t pe rmut a t i onofi t spa rt s ; t h i s n e e d n otprod ucea d d i t i on a l pe rmut a t i on a lly re la t e d formsoft h e t ot a ls e t .Ad i s cus s i on oft h epa rt i t i on(34 ) s h ould s e r.vea s a mod e lfor a ll pos s i ble e qua l-pa rt pa rt i t i on soft we lve . Ea ch t ri ch ord mus t beca pa bleofat wo-pa rtcombi n a t i on wh i chwi ll prod ucecommon h e xa ch ord a lcombi n a t ori a li t i e s . Exa mple2 0 e mploysa ba s i c s e t ofon ege n e ra t or. Exa mple2 0 RI2 1RI2 1 P 01 51 RI4 89 I 11106R732RI 2 6 71 R31110 P145 9 I 108 I 984 P0 15R 732 RI 61011 R31110I762 RI801P4 59 Foron e ,t wo, or morege n e ra t orst h eproce d urei s a s follows : From T a bleIV, a n d IVa s e le ct t ri ch ord s wh i chyi e ldcommoncombi n a t ori a li t i e sby t h e s a meope ra t i on . Fore xa mple , 013 a n d 014 bot hyi e ld t ypeAbyi n ve rs i on , a n dt ype134by re t rogre s s i on . T a ble IVb li s t s t h epos s i bles ource mos a i cs013, 014 t r.=H. Aft e r a fe wt e s t s t h eprope rmos a i c fort ypeAor134 i s d i s cove re d . This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and ConditionsP PuPPT e t r.P,~uIvl T e t r. POIvIuIT e t r.PvPu RoR T e t r. PuRoRvR T e t r. PuPuRIwRIT e t r. PvRIh RIRIT e t r.PuRIoRI T e t r. 012 03 6 9fff 0 38 11 2 4 f2 4 05 811 17f17s e eCol.II Js e e Col. III s e e Col. I e e Col.I s e e Col.IID6511e fe12 0 I4 710fff I4 710f2 4 2 4 1 4 710 f1717 14 6 9 2 2 f2 2 13692 1f2 1148 112 2 2 4 2 2 158 112 1172 11 37 112 92 4 2 9 13108c2 4 2 5 17410fe e 17392 7f2 7175112 7e 2 7 16 4 11ce c161052 5e 2 5 01306511e fe 06 2 82 7f2 709582 4 f2 40 9112 bfb130 71710fe e 175112 7e 2 7164 11ce c16105 2 5e 2 5301 3928e e f394 10e e e 372 102 5e 2 5 3784 2 3e 2 3 01402 911 bca 0 2 5 7 ccc 0 697 a ce06115e cc6 93 fe e 06511e e e 010911a e a01035ce c14 013 810 ca b 13911 2 6a 2 61 2 811 s a 2I 2 10 92 3a 2 41782 e e f1 793e 2 71683ce c162 92 5e 2 5 4 014 657a bc 4 68102 6b2 6 4 3510d bd4 3 7 8 2 3b2 5 4 10 511e fe 4 108 2 2 7f2 74 752 bfb4 711R 2 4 f2 40I 036#fff03112 M990 8112 15f150 68 2 2 7f2 7 0639fe e 06104 2 7e 2 7011 34 2 3e 2 3 011 9 10a e a02 119bca 02 64 2 8c2 607114 2 5c2 4079 6s cs1S 14 710fff1 4 69 2 2 f2 2 1 36 92 1f2 1 1 7 392 7f2 7 172 8e e f1 739e 2 e 2 7162 9 2 5e 2 5 168 3 ce c1 3108ca b 131192 6a 2 612 1092 3a 2 412 811s a s508112 fff58710a be 54 710f 511 4 10 e fe511104 e fe 5 11 2 8ff5710 8bfb574 2 bfb57 64 a bc 57108bbb 5368bbf 534 10d bd0 ~CD 0 q t i j o0 t T l 0 z ol 1: This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and ConditionsD16038112 4 f2 4 0 32 5bb 098 52 4 f2 40 911 2bi b16014 710f2 4 2 4 1 4 R 112 2 4 2 2 13 711 2 92 4 2 9 13 10 8 c2 4 2 5 I0192 52 4 2 4 f697 10 92 4 964 2 102 92 4 2 9 64 5 7a 2 4 2 30167e e O 011011a 510 11 d e s s e e Col.II s e e Col.III s e e Col.I s e e Col.I s e e Col.II07105ce c0369ff03101bb 071010671T fe2 4 02 3a a 32 389e d 2 37 2 32 3 2 16712 e 17 2 310 112 3a 2 3 2 5101117s 172 1811s a s2 1 9102 3a 2 42 983e cc 2 9 612 55 2 910 52 5e 2 5 2 78 5s e s2 73102 5c2 42 6193e fe 2 839 e fe 2 811 5fff2 63112 4 fs2 8952 4 f2 4S1 8 I ff2 5811fbb 2 5 1 11IOa 2 5 89 2 4 f2 42 36 9 17017 2 510 12 4 b2 4 2 710117s 172 3 R 1 d bd2 3 11102 3b2 5 2 61I 72 5e 2 5 2 651 2 3e 2 3 }S 0639fe e 0610 4 2 7e 2 70 1134 2 3e 2 3 011910a e aIS0 2 817e f2 8 3 9e e e 2 619 2 5e 2 5 2 6732 3e 2 3 2 11104 e fe 11 7 12 7f2 75810 7bn b5P4 12 4 f2 42 6011011s e a 014 52 3e 2 307105 ce c071 4 2 5e 2 5 05103ce c 05 4 9 2 5e 2 2 011109a e a01134 2 3e 2 3 09107bh 09 4 12 4 f2 4 03101 bfh0374 2 4 f2 4a o 2 389e a 2 310112 32 3a 2 3 2 1P11s a s2 19102 3a 2 42 761e cc 2 2 1032 5c2 5 2 5P3 s cs2 51 102 4 c2 5 2 I185fbb 2 11 10 2 4 h 2 4 2 98 7d h d2 95102 5b2 3 s 674 5a s e 67a 93a s 634 9 s a s63 5P,ba c6114 9e cc 6 1181cc 8674 1s cs679 Ra bc634 1bbf6385h bh 611 4 5 d bd6 11R8 ch a4 t0 COCb?o This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions02 7 0369fff 0 3 8112 4 2 4 058111717f0314 Of01104 15f150314 92 4 901014 151715010842 92 4 2 9 010111&2 4 2 3 065 11e e f 06104 2 7f2 708104 2 7e 2 70154 2 3e 2 3 011110s e a2 702 5 811fff2 5692 4 f2 4 2 38 9 17f17 s e e Col.II s e e Col.III s e e Col.I s e e Col.I s e e Col.II2 83 9e fe0371 010119a cb0104 22 6c2 6 05112 s e e0594 2 5c2 406115e e f064 102 7e 2 7011110s e a0154 2 3e 2 3 7031 86cbs 311192 8b2 631089d bd3108112 562 3 396 2 e fe 391152 7f2 73685bfb362 112 4 f2 4703754 2 ba e 7566a a a 714 6 s us712 6s e a71 4 10fe e 7162 s e e 794 2 ce o79106s e a04 8 013 s e e s e e Col. 11s e e Col.III s e eCol. I s e eCol. I s e e Col.II02 35boa0136s cb0 167e d s016 3e a s0316mce0 613a d s0369fe e012 3a s s0 2 1 3s cb012 3a d a01 2 3s e a02 3 5bbb0369fbb02 5 7cCe02 79cd c052 7ce c0 12 7d d d0 12 7 d d b0 1 6 7e s e0617 s *f 0 3 6 9ff * En t ri e s for 04 8a re s umma rIs e d a n d a re li s t e d i n t h e ord e r ofT a ble Ia . 0 OThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 61 t a bleVIT ET RACHORD/ T RICHORDINT ERSECT ION 012 013014 01501602 4 02 502 6o02 71037 04 8 Si I Sa bsbcI a d e 1,2a a a 2 2 2 2 ,3 31 a c a 112 2 .3 1ia da fa1, 2a a a 331 a b c 1,2 ,31,2 ,3. 3 1,2 ,31a d a . Ia e a 11.211.2 ,3 1 1,3 1 b bb 33 1bfb 13311,2 31,2 .3 2 1 c1c c 13 1c d c Ic e c 2 2 1,22 1.2 1,2 ,3 3 1 d d dI d bd 332 2 ,3 2 1 Se e13 11.2 2 31 "fe 1,2 1,2 ,31,2 ,31.2 ,31,2 1,2 ,31,2 1,2 ,3 1 fff1.2 1,2 ,31,2 1,.2i9f9 t1 15f151 17 a 17217 f17 1,22 1.22 1f2 122 2 f2 2 2 22 3a 2 32 22 3 e 2 3312 1,2 111 2 4 b 2 4 2 22 4f2 4 1,2 131,2 ,32 311,222 5 c 2 5 22 5 e 2 52 2 ,32 2 2 2 1 2 6 b2 62 22 6b 2 6322 6c 2 611 2 7e 2 7 2 2 2 1,2 11 2 7f2 72 13 1,23 1212 e 1722 3a 2 4 2 2 3 2 2 11 2 3b 2 532 2 22 4 c 2 52 12 2 2 1 -----------------------------------------------------r 92 4 93 1 151715 2 1172 122 2 2 4 2 2 2 22 9 2 42 92 2 ,3 1 T ri ch ord column n umbe rs re fe r t o rot a t i on ,fi rs t ,s e con d ,or t h i rd(i e ..014 ,14 0,or 4 01) This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 62Exa mple2 1 P3 P013t P4 5 134 134R 9 76R2 1110 P3 T h e n e xts t e p i s t oope ra t e upont h e combi n a t i on : Exa mple2 2P0134 58 R9762 1110 I54 2 109 RI81011367 T h e s e con d h e xa ch ord i s t h e n d e ri ve d a n d ord e r ch a ra ct e r- i s t i csma y bei mpos e d . On e la s te xa mplei llus t ra t e s t ri ch ord a l pe rmut a t i ona t e a chle ve lofa s e tcon t a i n i n gfour d i s t i n ct me mbe rs . Exa mple2 3 013102 578114 69 97 61 1184 52 13010 13454 2 73010961118 81011691034 2 57 T h epos s i blea ll-combi n a t ori a ls ource mos a i cs oft woge n e ra - t ors a re s umma ri ze d be low: T ri ch ord s Source mos a i c Combi n a t ori a lCombi n a t or- t ypea n dope ra t i on i a li t y commonn e ce s s a ry t o bot ht o prod uceat ri ch ord s012 ,015 012 10113Avt AD 012 ,02 7 012 8113BQ t B/B+ D 013, 014 01310112 A't A 134013,02 5 0135810C Vt C 134013, O037 013102 5 Bvt B 134014 , 02 5014 9112 B'% t B134015, 02 70158103 CV t C D 016,02 6 0165711Dvt D12 9 02 5,037 02 53710CVt C134This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 63 T h e combi n a t ori a l prope rt i e sofs ubs e t ss ugge s tt h e a d d i t i on a l pos s i bi li t y of 4 ,6, 8a n d12 -pa rt a ggre ga t e -formi n gcombi n a - t i on s oft h e t ot a ls e t .Ift h e formse mploye da re t o be me m- be rs oft h es ubgroup oford e r"n ", s omere gula t i onoft h e ba s i cs e t i sn e ce s s a ry. T ypeIs e mi -combi n a t ori a l s e t s , wh os ei n ve rs i on a lly re la t e dformsprod uce a ggre ga t e sa t t h e h e xa ch ordle ve l,ma y be e x- t e n d e d t o forma ggre ga t e sa t t h e t ri ch ord le ve l.T ri ch ord sa re d e fi n e d a sa ,b,c, a n d d .Ifca +, d+b+ (a n d i n s omeca s e s b,a ), t h e n PUIh uRu RI.T h e s ource mos a i cs a rea b/a +b+,a a /a +a +. Exa mple2 4P01398574 6102 11 P0/RI8 I 764 10112 031958 17/R 11 RI8591302 11104 67 RI8/P0 R112 1064 7589310 R11/I7 In ve rs i on a lly re la t e d forms ma i n t a i nd ya d s ; t h e fi rs t s i xve r- t i ca lt e t ra ch ord s a rere t rogra d e di n t h e s e con dh a lf; a n d t ot a l "ve rt i ca li za t i on " oft h e'4 'group of POi sa ccompli s h e d . (Con - t i n ua t i on oft h e "ca n on " by e i t h e r oft h e re comme n d e d "follow- i n gs "prod uce ss e con d a ry s e t s .) T ypeRIs e mi -combi n a t ori a ls e t sma ys i mi la rly be e xt e n d e dwh e nb, a , d ~ T c+ (i n s ome ca s e s c,~a ). T h e s ource mos a i csa rea a +/bb+ ora a +/a a +. Exa mple2 5 P02 651193711084RI731104 82 065911 I1195602 84 10137 R4 810173911562 0 In t h i ss pe ci a l ca s ei n volvi n gas i n gle t ri ch ord , ve rt i ca lt e t ra - ch ord s a re fi xe d . T h et ypeP s e mi -combi n a t ori a ls e t combi n e s a s follows : Ps . PuRUR.T h e mos a i cs a rea b/a b ora a /a a . All-combi n a t ori a ls e t s d e ri ve dbyope ra t i on uponon e or t wos ourcet ri ch ord s , re ga rd le s sofmos a i cord e ri n g (i .e .,a b/baor a b/a b)or me t h od ofcon s t ruct i on(i .e .,a a +/bb+ ora b/a b), wi llh a ve a t le a s t t wo re la t e dt wo-pa rt combi n a t i on s wh i ch a reThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 64ca pa bleofe xt e n s i on .T h ePuPor PURcombi n a t i on s ofas e t wh os e s ource mos a i c i sa b/a ba reca pa blei n t h e forme rca s e ofe xt e n s i onbyre t rogre s s i onor i n t h e la t t e rbyt ra n s po- s i t i on (Pu Iwi lle xt e n don ly i n t h es pe ci a l ca s ea =a ta n d /orb=b+t )wh i le e a ch ba s i c combi n a t i on oft h e mos a i ca a +/a +a i sca pa bleofe xt e n s i on i n a n umbe r of wa ys . T h usfour-pa rtcombi n a t i on sd e pe n d -upont ri ch ord ch a ra ct e ri s t i cs a s we lla supont ri ch ord con t e n t a n dpla ce me n t . T h e re a d e rma y d e t e rmi n e for h i ms e lft h e t ot a ln umbe r ofs uchs ource mos a i cs orma y re fe r t o T a ble sI,IV, IVa a n d IVb. (Re me mbe rt h a t t h e combi n a t i on spre s e n t ly un d e r d i s cus s i ond i ffe rs i gn i fi ca n t ly from t h e four a n de i gh t -pa rte xt e n s i on s i l- lus t ra t e d i n s e ct i on on e oft h i spa pe r.) Exa mple2 6 P0312 4 581110967152 4 310 I118109763012 54 P69781011e t c. RI4 52 10367910811R1034 52R7691011854 2 130RI10118769 Not e t h a t un li ke s e mi -combi n a t ori a l combi n a t i on s , n e wformscon t i n ue t h e "ca n on "a n dprod uce s e con d a ry s e t s . Ei gh t pa rt combi n a t i on s , gi ve na s e t wh os ecombi n a t ori a lly re la t e d formsre pre s e n tas ubgroup ofa t le a s t ord e r'8', a reca lcula t e d a s follows : 1)P%IQR URI e xt e n d sby t =6 (PIU IuPvI e xt e n d sby re t rogre s s i on ), wh e ncomple me n t a ry h e xa ch ord s a re ord e re d s ucht h a t'b'='a +t ' fora n yord e ri n gof'a '.0 12 34 5 / 11109 876.T ri ch ordpla ce me n ti sfi xe d , but i n t e rn a lcon t e n tma y be ord e re dfre e ly. (In t h e fi rs t a n d t h i rd ord e rs e t s ,pi t ch e sof comple me n t a ry ord e r n umbe r mus t n ot cre a t e t h e i n t e rva l6: 02 135 4 10 1186 7) d #6 2 ) PuP VR . Re xt e n d s a tt =3, wh e n Hb=Ha t fora n yord e ri n gof'a '.T ri ch ord s a re t h e n ord e re dfre e ly. T h e n e xte xa mplei llus t ra t e s a ne i gh t -pa rte xt e n s i on oft h es e con d ord e r s e mi -combi n a t ori a l s e t , 134 . (Exa mple2 6 ma y be s o e xt e n d e d i fPa n d I pa rt i t i on sa re( 2 112 12 2 1)a n di fRa n dRI pa rt i t i on sa re (12 2 12 112 ). This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions265Exa mple2 7 P0376918114 52 10P691 I1184 52 10307961I52 10 RI169703102 54 811RI703 R1052 4 118196730R4 118 P6910372 5108114 P037e t c. I52 101184 961307I1184RI7031694 811102 5RI169 R4 1181052 730196R1052Ea ch me mbe r oft h es ub-group oford e r 8i sre pre s e n t e d ! T h i rd ord e r s e t s a rea lwa ys ca pa bleofs i xa n dt we lve -pa rtcombi n a t i on s i ft h e i n t e rva ls 4 a n d 8a re e xclud e d from t h ee ve n n umbe re dd ya d s(ord e r n umbe rs 0 1, 23, 45, e t c.)of t h e ba s i c s e t . Exa mple2 8aExa mple2 8bP0185 4 9 1117102 3P0 1854 9 6117102I76112 3101180954 I76112 3101180954R32 10711694 5801P4 50981103112 67 RI4 5908110 3211o6 7I 11103 672 5 04 1 9 8 P89E 4 1052 7361011P894 1052 7361011 I11101367 2 115 04 198I32 710116j94 8510 ERI2 1EEBEEEEEEE Exa mple2 8b combi n e s wi t h i t s e lfa tt ra n s pos i t i on6t oprod uceat we lve -pa rts t a t e me n t wh i ch s a t i s fi e s(62 ),(2 6), a n d(112 ). Ift h e t h re ed ya d soft h e fi rs t h e xa ch ordre pre s e n tt h e i n t e rva ls1(11), 3(9)a n d 5(7)a ta n y e ve n ord e rpos i t i on , a n d t h ecomple - me n t a ry h e xa ch ordt ra n s pos e st h e s ed ya d si n t oa n y e ve n n umbe re d ord e rpos i t i ona t t h et ri t on e , P(xy)----I(yx) a t t h e s a meord e rpos i t i on . *2 0 Exa mple2 9 t =6 P05814 1076112 3 I50894 171011632Ina d d i t i on , i ft h e ba s i c s e t i s i n t h e forma a +/a a +, a n de ve ry e le me n t t a ke n wi t h t h ecorre s pon d i n ge le me n t ofRIi s a n e x- pre s s i onoft h e i n t e rva l 2 ,6, or10, e ve ngre a t e rcombi n a - This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 66 t ori a l fle xi bi li t y (ofwh i chExa mple30i s but a n i n s t a n ce )i sobt a i n e d . Exa mple30 P i113874 5692 101 I 210 569874 1130 R:7 4 830 _1110 12 956 RI6951012 30114 87 P4 370118910162 5 I910612 54 307118 R118074 32 561910 RI2 516910118704 3 P87114 3012 51069 I562 91010118374R304 11876910512RI10192 5674 38011 Exa mple30s a t i s fi e s (62 ) fora d ja ce n tforms ,(34 )for e a chblockofforms re la t e dby a llfourope ra t i on s , a n d(112 ) for a ll formse mploye d .Furt h e rmore , (2 6) i s s a t i s fi e dby PuIl P, I vP, I orby t h ere ma i n i n gs i x forms ; re s ult a n t t e t ra - ch ord s a re ord e re d s uch t h a te i gh t pa rtcombi n a t i on s e xi s t i f e a ch s e t i n blockon e i s i n t h e form (1 2 12 2 121)a n d e a chs e t i n blockt wo i s i n t h e form (2 12 112 12 ).T h e t we lvepa rtcombi n a t i onma y bere pre s e n t e da s a s e ri e s ofove r- la ppi n g a ggre ga t e s : e le'c' c'e cc . e cc c'l ecc'e ec'c cc e ,ecc'. C e Lc--- -1... eI - c. c'e c c c'e 'e :c c'c"e cI~rr-?--l--j,,~,,, a l T h usExa mple30 ma y bere ga rd e da s a con s t ruct i on d e ri ve dbyope ra t i on upon 2 ,4 ,6, or8-pa rtcombi n a t i on s .Not e t h a tt h e e n t i res ubgroup oford e r 2 4 i sre pre s e n t e di fbot h forwa rda n d ba ckwa rdre a d i n gsa re t a ke n .(T h i s i s n otpos s i blewi t ht h e s e t of Exa mple2 9). T h e s e t 's combi n a t ori a l pot e n t i a l i s t h e n a fun ct i on ofre la t i on sThis content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 67 wi t h i n a n d be t we e n i t spa rt s . T o t h e a fore me n t i on e de qua l pa rt pa rt i t i on s (62 ),(4 3),(34 ),(2 6),(112 ), In owa d d t h eun - e qua l t wopa rt pa rt i t i on s : (1 11),(2 10),(3 9),(4 8),(57). T h e me t h odby wh i ch combi n a t ori a l prope rt i e sa re un cove re dre ma i n sun ch a n ge d . De fi n e t h e s e t a s follows :(3)(63)or(36)(3). T h e n PUP=A i fcon t e n t P (9-11)i s at ra n s pos i t i onofcon t e n tP(0-2 ). PU I=Ai fcon t e n t P (9-11)i si n ve rs i on a lly s i mi la r t ocon t e n tP(0-2 ). PUJRI=Ai fcon t e n tP(9-11)i si n ve rs i on a llys ymme t ri c. Allcombi n a t i on s a repos s i blei fout e r t ri ch ord s a re i n ve r- s i on a llys ymme t ri ca n d re la t e d .(02 7.......1116). Exa mple31 P 010111968572 34P867952 4 1310110 P010111968572 34I2 4 315869701110 P010 11 1968572 34RI678352 4 1911010 Un e qua lpa rt pa rt i t i on s ma y bea ppli e dt os ubs e t s ; i n t h e n e xte xa mplet h epa rt i t i on (13)oft h e li n e a r forms re s ult s i n(39) for t h e d e ri ve dh a rmon i e s , a n d t h e combi n a t i on i s e xt e n d e da ccord i n gly: Exa mple320132111089 67542 310 981011 4 576 a n d e xt e n d e d on cea ga i n by (2 1 1): This content downloaded from 177.229.168.144 on Tue, 16 Jul 2013 21:35:18 PMAll use subject to JSTOR Terms and Conditions2 68 132111089 A6754.

The Sourceset and Its Aggregate Formations

Documents

Transcript of The Sourceset and Its Aggregate Formations