Triplicates, Cow Tails and Rivers: Periodic Elements of a Thillana

Curtis AndrewsMUS 532: PeriodicityDec.6, 2013

Preamble: 1) India is immensely diverse. It is better seen as a country of countries, each with unique histories, languages, cultures and identities. Thus, when speaking of Indian music, I want to be clear to the reader that I am primarily referring to the classical arts. Further, when referring to historical treatises on music, we must bear in mind that these were written before the division of classical music was even conceived. The ancient but still utilized concepts and terms I will use apply to the classical music of today, especially to Carnatic music (South Indian music).2) This paper deals with a large number of foreign language terms, mostly derived from Sanskrit and a few from Tamil. The transliteration of these terms into our Latin/English script is varied and complicated. I am choosing to simply spell the terms as they normally used by writers but with no use of lexical marks to denote proper pronunciation.3) I consciously avoid use of the term matra to represent smaller units/pulses. The term has various uses today in North and South Indian music that are distorted from the historical use. My own training in Carnatic avoids using it.

Indian music has a rich and expansive history dating from pre-historic times through present, enduring numerous changes and influences yet retaining salient features. In Indian musical systems and thought, we find the concepts of time and rhythm performed, articulated and expanded via elaborate modes of expression. Intensely cyclical in nature and intimately entwined with the concept of tala, the study of Indias play in periodicity offers a glimpse into a world of rhythmic fulfillment and wonder.This paper will examine the aspects of tala and other rhythmic structures evident in a thillana of 20th century Carnatic composer and violinist Lalgudi G. Jayaraman. I will first explain key rhythmic concepts used in Carnatic music. I will then explore how the Western concept of meter can be and applied to the Indian concept of tala, and the a few limitations or problems inherent in using certain Western theoretical ideas onto non-Western musics. I will analyze the manner in which this composition is built upon several uniquely Indian rhythmic concepts, and the various forms of grouping periodicity found therein. I will transcribe the relevant sections, identify and explain its own tala schematic to give the reader a glimpse into organization of periodicity found in this piece, which can also apply to other genres of Carnatic music.

Historical ContextNada Brahma the world (or God) is manifested in and through sound. This neatly encapsulates the Indian concept of music. Appreciation and recognition of the divinity, power, and necessity of sound and music in India is found in a vast body of scriptures, philosophical works, historical epic narratives and poetry ranging from 1500 BCE through present times. Being informed by this continuous stream of attention, time and rhythm in India have flowered into an unparalleled mandala of kaleidoscopic proportions and sophistication. Indian music has its ancient roots in the Vedas (1500-1200 BCE), a massive compendium of verses containing prayers, hymns, and ritual preparations that were meterically set and use of a specific tonal relationship in the process of recitation. One of the four Vedas, Sama Veda, is actually sung rather than recited. The earliest treatise on music would be the Natya Shastra (2nd c BCE -2th c CE) that dealt with among other subjects, music theory, nomenclature, instruments, and performance practice as it related to theatre, which was a nexus for music, dance and narrative and the context for musical practice outside of ritual use (Pesch 2009:459). The concepts and terms used in Natya Shastra informed the later work of Matangas Brhaddesi (9th c CE) and Sarngadevas Sangita Ratnakarna (13th century AD), the latter being highly influential in the formation of current musical theory and practice in the classical realm.

DefinitionsSruti mata, laya pita is an often-quoted musical maxim, essentially translated as sruti (pitch/tone) being the mother and laya (time and/or rhythm) being the father. It stresses the overarching importance of sruti and laya. If one falters, everything falters. Since sruti is not what concerns my discussion, I will not dedicate much space to it except to say that the concept of sruti (lit. that which is heard) practically relates to the smallest perceptible distance (in terms of pitch) between two tones. If one is to develop their sruti sense, they are developing their ability to identify and meld with discrete frequencies. Strength in sruti is essential to all pitch-based aspects of music, especially raga, the characteristic grouping of tones and phrases that are the basis of all melodic aspects of Indian music.Laya is an equally broad concept and in modern parlance is used in different ways by scholars and musicians. Trichy Sankaran, an authority on Carnatic music, defines laya as the practical bedrock of the whole concept of time and is the inward orderly flow of rhythm (Sankaran 1994:24-25). It can refer to tempo or speed, such as 1:2:4 ratio of acceleration and also denotes the measurement of interval between two action (i.e.: duration). More abstractly, musical of concepts groove, time, and feel are related to the idea of laya and having a sense of good sense of laya is essential for any performing musician. To be a great musician however, ones layajnanam (knowledge of laya) must be especially strong.The Indian concept of tala is immense in scale, complexity and expression. The term itself is derived from cosmic sources. The root syllable ta is taken from tandava, the vigorous masculine form of dance by Lord Nataraja/Siva, while la is from lasya, the graceful feminine form of dance by Goddess Parvati (Sen 1994:11). Generally, tala encompasses all that is related to matters of musical time, duration, rhythm, grouping, pulse, repetition, and tempo. Specifically it can refer to an individual metrical cycle with recognized structures and properties and is analogous to the Western concept of meter. One Tamil writer says, If one can see the form of the southern breeze, the form of Siva, the form of scent, the form of Manmatha (Cupid), the form of the flute tone and the form of the Vedas, one can see the subtlety of the tala(Sambamurthy 2001:18). Note the constant reference to form in the above verse, which is what tala does at its most essential level, to give measured form to the flow of time.Since the 16th century, key aspects of tala theory and practice were first documented as Tala Dasa Pranas the ten essential elements of tala. They include: 1)Kala division of time into discrete units from small to large2)Yati arrangement of patterns producing geometric shapes3)Anga component parts or sections of a tala4)Jati classification according to numeric value of unit groups 5)Kriya physical gestures of showing tala6)Kalai subdivisions of angas of a tala7)Laya tempo; ratios of movement between actions8)Marga repertoire of various durational values of angas9)Prastara process of permutation/combination10)Graha relationship between start/end point of music in relation to the tala

An example of a tala would be Adi tala of 8 beats (aksharas). It is structured as 4 + 2 + 2 and consists of 3 angas (4, 2, 2) each of which have kriyas known as a laghu and drutam. Laghu is shown by a clap plus 3 finger counts equaling 4 aksharas in total. Drutam contains a clap (1 akshara) and a wave (1 akshara), a total of 2 aksharas. The wave is shown by an upturned hand. The claps are considered sashabda kriyas (audible action/sounded) while the wave and finger counts are classified as nishabda (silent action/unsounded). Seen this way, tala is an aurally and visually explicit means to show the measurement of time.

Tala as MeterTala and the Western concept of meter are analogous in that they create a periodic and hierarchic temporal foundation. Both establish structurally important points of accents and involve the interaction of two or more levels of beat pulsation. Martin Clayton, in his thorough exposition of rhythm and time in North Indian music, which shares many theoretical and some practical concepts with South Indian music, identifies several commensurate qualities of tala in relation to meter. They include: Tala/meter outlines a periodic and hierarchic temporal framework Tala/meter has interaction of two or more pulsation streams, and beats found on more than one stream are structurally important; marked by hand gestures in tala beats/pulsation at the beat/akshara level are equal in duration rhythm is interpreted and expressed in relation to tala/meter (Clayton 2000:200)There are also several impactful considerations that distinguish tala from meter as well. Lerdahl and Jackendoffs influential work, A Generative Theory of Tonal Music, outlines aspects of meter that effectively apply to tonal systems of music according to a set of well-formedness rules, but one of their rules definitely does not apply to other musics of the world. In their definition, each metrical level must consist of equally spaced beats, especially at the level of strong beats (Lerdahl and Jackendoff 1983: 68-69). This is evidently something that does not happen with Misra Chapu tala as well as several other talas in common use, both in South India and North India. Misra Chapu does have mensural qualities at the level of the tactus/beat (the akshara level) but at the level larger metrical important points, i.e.: strong beats, it does not. It is structured as 3 + 2 + 2, or even further as 3+4. This grouping reflects the angas or important sections that define the tala and serve to orient a musician to their place within the cycling of it. Example 1 shows how Misra Chapu breaks the rule of equal beat duration at each metrical level. Essentially, it is non-isochronous.(see example 1 in Appendix)There are several other important differences between tala and meter. While meter conditions rhythmic behaviour and perception in reference to stressed beats, the structurally important aspects of a tala cycle exist and function as referent points of musical phrasing rather than points of musical stress. Angas tell you where you are, not what to do. One of the dasa pranas that should be mentioned here is graha, which details a key point in a tala where important musical ideas begin or end. This point can be in one of three places: 1) sama: on beat one of a tala, 2) atita: before sama and 3) anagata: after sama. The idea of graha is important in a discussion of stressed/unstressed points of a metric cycle because depending on the compositions graha, beat one (normally considered a very strong beat in tonal music) may never be stressed until perhaps the very end of the piece (when all sound ends on sama). Misra Chapu is interesting still because beat one is actually reckoned with a wave and is considered silent. Another critical difference between tala and meter is that tala is explicit it does not allow for any ambiguity in metric perception. In fact, tala is performed or kept by a musician, by someone in the ensemble designated to keep tala, and/or by members of the audience (or listener at home). If the hands are not used, metal cymbals function in the same manner using ringing and muted sounds to represent sounded and unsounded kriyas. Unlike meter, it is not silent or perceived based on the flow or music, rather it is aurally and visually present and obvious. I will even go further to say that for performers and listeners, a tala like Misra Chapu functions much in the same way that the timelines of African music do, by identifying key points in the flow of time that one can latch onto and use as points of resolution or points of departure. Nigerian musicologist Meki Nzewi uses the apt term phrasing referent to describe a unique theme played as a non-varied of layer ensemble sound that is reiterated for the duration of a piece. He goes on to say that sensibility for the phrasing-referent role becomes an inherent music factor of music thinking (phrasing sense) and composition as musicianship develops (Nzewi 1999:74-75). Sounds like tala to this author.Returning to a comparative stream, applying conventional notions of meter to tala tends to be too restrictive. We should not be surprised. After all, we are trying to take theoretical and tools for analysis from one musical culture (the Western classical) and apply them to musical systems far removed. It is inevitable that a broadening of terms is essential if any progress is to be made in this venture. Justin London's formulation avoids one of the problems of Lerdahl and Jackendoffs when he says that meter minimally consists of two levels: B [beat] and M [measure] (where M = some modular ordering of Bs)(1995:68). This more expansive view of meter applies much better to non-isochronous meters. London also provides us with an alternate mode of representing meter as a stable, recurring pattern of attentional energy, one that correlates suitably with the cyclical nature of tala, that of the circle (London 2004:80). I have developed a circle representation of Misra Chapu (see example 2 in Appendix) that represents the seven aksharas of the tala (what London would refer to as the N cycle), the angas ( his time points) and surprisingly shows some amount of symmetry (for an odd cycle) when dashed lines are drawn between the angas, forming a trapezoid.

Rhythmic FormsWhile the tala dasa pranas are exhaustive in scope, there are further developments in Carnatic rhythm that have developed in recent centuries and are extensively used at present. The most prevalent are the two cadential forms known as mora and korvai. Both have developed considerably in the hands of percussionists and dance exponents and use phrase patterns that are juxtaposed against the tala to create cross-rhythmic tension and release. These devices are used by musicians to signal the end of the development of an idea, the end of section in a composition and/or the composition itself. Moras are distinguished by one major characteristic: a three-fold repetition of a phrase, with or without pauses between each phrase. The basic structure can be represented as XYXYX, where X=phrase and Y=pause (known as karvai) This gap as it is sometimes called, can be a combination of sound + karvai, sound (that fills the gap) or just karvai. Through the process of prastara, enormous variety is possible within this form. The X and Y phrases can be homogenous, or they could logically increase or decrease in value such as X Y 2X Y 3X, 3X Y 2X Y X or X Y X 2Y X (See example 3 in Appendix).Korvai (lit. strung together) is the other major cadential form and is generally more complex in structure than mora. Whereas mora is unitary in nature and defined by a three-fold repetition of similar material, korvai entails at least two major sections, of which the latter can included a mora in its design. Further, a korvai can undergo prastara to generate variation to fit any desired length of time. Finally, it is quite normal for any korvai to be repeated 3 times, either in exact repetition or with variance of thematic material and/or durations of constituent parts (See example 4 in Appendix).Evident in the design of these devices is a discernible sense of shape. This is known as yati. There are 6 different yatis denoting patterns that 1) expand (gopuccha), 2) reduce (srotovaha), 3) stay the same (sama), 4) contract then expand (damaru), 5) expand and then contract (mridanga), 6) expresses randomness (visama). Examining the designs of mora and korvai, we can see examples of the first 5 of these yati. They are useful both as compositional devices as well as mnemonic devices (see example 5 in Appendix)All of the above, tala, dasa pranas, korvai, and mora figure prominently in my analysis of Jayaramans thillana and generate ideas about meter and grouping that illuminate the discussion of how these concepts fit into the Carnatic world.

Dhim Dhim TananananaLalgudi G. Jayaraman (1930-2013) was one of the most influential Carnatic violinists and composers of the 20th century. Born into a musical family, he quickly rose in fame and accompanied all major vocalists of his era and created his own bani (school/tradition) of violin playing. But perhaps his longest lasting contribution will be the dozens of compositions he introduced into the Carnatic field, which continue to be celebrated and performed by current artists. These compositions span several genres of Carnatic music, but his thillanas deserve special mention for the sheer number that he composed (over 30) for both concert and dance contexts and his innovations in using uncommon ragas and talas (www.lalgudis.com).The thillana is a major component of any recital of Bharatanatyam, the classical dance of South India, and is characterized by quick tempo, lively rhythmic phrases and predominance of solkattu (spoken rhythms) rather than sahityam (lyrics). It has a variant on the musical concert platform and is included in every concert performance. According to musicologist S.R. Janakiraman, thillanas follow the same formal structure of the kriti, another major genre of Carnatic music, by using 3 sections: pallavi, anupallavi, and charanam. They generally use certain classes of raga and talas but exceptions to this are certainly evident (Janakiraman 2008:223-4). Historically, the thillana has been inspired by the North Indian genre of tarana but shares characteristics with other genres with roots going back as far as the 9th century (Pesch 2009:283). The thillana I am using in this analysis is known by the first syllables used in the pallavi or opening section, dhim dhim tanananana. It uses the raga Revati as its melodic material, is set to the tala of Misra Chapu of 7 beats and is sung by Sudha Raghunathan. I chose this thillana for several reasons: 1) it uses a tala of an asymmetric nature that creates tension with the musical phrase and broadens our Western concept of meter, 2) the musical phrases themselves are built upon certain rhythmic formulas found Carnatic music and generate interesting grouping structures, and 3) it is simply a beautiful piece of music.

Grouping Structures in Dhim Dhim Tanananana(akshara/beat will be taken as q, unit as e;sub-unit as x; one cycle is 7 qs/14 es,28 xs; sections and structures will be referred to by location in min:sec)

A thillana has inherent grouping structure at the level the section, of which there are three. Each section has certain characteristics, varying slightly by genre. In a thillana, the pallavi is the opening section that contains the main theme of the piece, uses solkattu as text and stays in the lower and middle sthayi (octave) of the raga. The pallavi also serves as a refrain that is stated numerous times throughout the course of a composition. The anupallavi contrasts with the pallavi both in terms of rhythmic and melodic content, and also register, as it is usually where the upper sthayi is introduced for the first time. The charanam is the final section where lyrics are used along with sollkattu as text and usually contains one or more rhythmic designs with a cadential function. All musical phrases in this thillana are grouped into two or four cycles of the tala. This grouping is a result of the cadential forms that the composer has decided to use the foundation for majority of this piece, as we shall see. The only section where we find smaller and continual repetitions of a pattern is in the anupallavi, whose main phrase covers one cycle in terms of rhythmic shape, but two cycles when analyzed melodically.

PallaviThe main part of the pallavi (0:01-0:06) is built upon a mora of the XYXYX form. X=6 units (3+3). Y=5 units. Added together it is 6(3+3)+5+6(3+3)+5+6(3+3)=28 units/2 cycles. Also notated as X6Y5X6Y5X6 (see example 6a in Appendix). The second major part of the pallavi (0:21-0:30) extends of over 4 cycles and resembles a korvai, a concatenation of 2 events that dovetail into one another, each of which is actually a mora. The purvanga (0:21-0:26) is grouped as 6(2+2+2)+6+6(2+2+2)+6+6(2+2+2) = 30 units (X6(2+2+2)Y6 X6(2+2+2)Y6 X6(2+2+2)).The uttaranga (0:26-0:31) is grouped as 6(3+3)+4+6(3+3)+4+6(3+3) = 26 units (X6(3+3)Y4 X6(3+3)Y4 X6(3+3)). Together they equal 56 units/4 cycles (see example 6b in Appendix). From a simple analysis of numbers, listening or even performance of the constituent parts of the mora and korvai found in the pallavi, one can see how each phrase does anything but align with a 3+2+2 structure. In fact, pronounced tension and release with the tala is desried.

AnupallaviThe anupallavi (1:50) presents grouping that contrasts with that of the anupallavi and shows clear alignment and lack of tension with the underlying grouping of the tala.Each phrase of the anupallavi is grouped 3 (6{2+4}) + 4 (8{3+5}) with stressed syllables of the text falling on angas (see example 7 in Appendix). Listening to this while keeping tala is very comfortable and reaffirms the structure of the tala with little tension. Interest is generated and sustained by melodic ornamentation and variation of the phrase with each repetition while maintaining the 3+4 structure.

CharanamThe charanam (3:13) presents a combination 3+4 phrasing as found in the anupallavi and phrases that generate tension with the tala as found in the pallavi. The charanam also contains two remarkable examples yati-driven design. The sahityam sections of the charanam outline the 3+2+2 structure of the tala, using two sentences of text, each spread over four cycles of tala and embellished with subsequent repetitions. From 4:34-5:02 the aforementioned yati-driven structure appears. It is composed of one cycle phrase (A) followed by another cycle (B), the first of which is actually a cycle of silence. Every re-iteration of A is identical but each B becomes systematically filled in from the end of the cycle. In terms of yati, each A is sama yati (isomorphic) while each B is a continually unfolding srotovaha yati (expanding) (see example 8 in Appendix).Though the previous yati-based structure contains a beautiful logic, Jayaraman has saved the best for last. The final and most dense grouping structure of the whole piece is found at the end of the charanam (5:17). While most rhythms thus far have been at the unit (e-note) level with brief embellishments at the sub-unit (x-note) level, this last group of phrases explicitly stay in the sub-unit subdivision for four whole cycles. The complete structure, which is quite ingenious, is one large XYXYX mora, X expanding in value and Y being constant. Each X is actually designed as a nested sub-mora (mora within a mora). The first sub-mora (5:17), without karvai, is xxx, where x=4 sub-units. The second sub-mora (5:19) is contains a karvai and is grouped xyxyx, where x=8 sub-units and y=4 sub-units. The last sub-mora (5:22) is also grouped xyxyx where x=12 sub-units and y=8 sub-units. The full structure is represented:X12(x4x4x4) Y8 X32(x8y4x8y4x8) Y8 X48(x12y8x12y8x12) where the superscript numbers represent the sub-unit values of letter and in total equaling 112 sub-units (four cycles of tala). This may look complex but is really just another example of prastara frequently practiced by drummers. 7+7+7=21, but so does 5+7+9, simply take two from the first 7 and add to the last. If we equalize the previous mora, that is, make each sub-mora isomorphic, we find this structure: X32(x8y4x8y4x8) Y8 X32(x8y4x8y4x8) Y8 X32(x8y4x8y4x8), totaling 112 sub-units. What Jayaraman has done is logically re-arrange his numbers by reducing each letter of his first sub-mora by a value of four and increase each letter of his last sub-mora by the same value. The solkattu used for these moras also reflect this overall srotovaha yati design. (see example 9a/9b in Appendix).

Final MoraIn the foregoing analysis of Dhim Dhim Tanananana, we have seen how a short dance-inspired composition reflects many fundamental features expounded in tala theory and practice. Normally the domain of layavidwans (masters of rhythm), these features are also utilized to great effect by composers of modern Carnatic music such as Lalgudi G. Jayaraman. Further, the grouping structure found in this composition is a result of the inherent grouping found in the structural design of the moras, korvais and yati-based structures used in Carnatic music. My peregrination has touched briefly on the metric aspects of tala, illuminating some key aspects of conventional definitions of meter that do not conform to Carnatic ideas of meter. While tala contains facets of meter, meter does not contain all that tala is. But our comparative tendencies do at least tell us one thing: if we want to generate a broad theoretical language, which can be applicable to musics of the whole world, we have to start by broadening our scope and softening our rules. Since we have created these rules, can we not bend, break or eradicate them as we see fit?

BIBLIOGRAPHYClayton, Martin. 2000. Time in Indian Music: Rhythm, Metre, and Form in North Indian Rag Performance. Oxford: Oxford University Press.

Janakiraman, S. R. 2008. Essentials of Musicology in South Indian Music. Chennai: The Indian Music Publishing House.

Lerdahl, Fred, and Ray Jackendoff. 1996. A Generative Theory of Tonal Music. Cambridge: The MIT Press.

London, Justin. 1995. Some Examples of Complex Meters and Their Implications for Models of Metric Perception. Music Perception: An Interdisciplinary Journal 13 (1): 5977.

. 2012. Hearing in Time: Psychological Aspects of Musical Meter. 2nd ed. New York: Oxford University Press.

Nzewi, Meki. 1999. Strategies for Music Education in Africa: Towards a Meaningful Progression from Tradition to Modern. International Journal of Music Education 33: 7287.

Pesch, Ludwig. 2009. The Oxford Illustrated Companion to South Indian Music. 2nd ed. Oxford and New York: Oxford University Press.

Sambamurthy, P. 2001. South India Music Book II. 14th ed. Vol. 2. 6 vols. Chennai: The Indian Music Publishing House.

Sankaran, Trichy. 1994. The Rhythmic Principles and Practice of South Indian Drumming. Toronto: Lalith Publishers.

Sen, Arun Kumar. 1994. Indian Concept of Rhythm. Delhi: Kanishka Publishers, Distributors.

www.lalgudis.com. Accessed Dec.5, 2013

APPENDIX

Ex.1 Misra Chapu with dot analysis of Lerdahl and JackendoffO O X X O1 2 3 4 5 6 7 1 O = wave. . . . . . . . .akshara/ beat level X = clap. . . . . .anga/structural level. . .anga 2nd level. .avartana/cyclical level

Ex.2

Circle representation of Misra Chapu

Ex.3Examples of a mora and elaboration using prastara X= ta din gin a tom (5 units)Y= tan . gu (3 units)

a) ta din gin a tom tan . gu ta din gin a tom tan . gu ta din gin a tom 21 units (5 + 3 + 5 + 3 + 5) or X5 Y3 X5

b) ta din gin na tom tan . gu ta din gin na tom ta din gin na tom tan . gu ta din gin na tom ta din gin na tom ta din gin na tom 36 units - 5 + 3 + (5 + 5) + 3 + (5 + 5 +5) or X5 Y3 X10(5+5) Y3 X15(5+5+5)

c) ta din gin na tom ta din gin a tom ta din gin a tom tan . gu ta din gin a tom ta din gin a tom tan . gu ta din gin a tom 36 units (5 + 5 +5) + 3 + (5 + 5) + 3 + 5 or X15(5+5+5) Y3 X10(5+5) Y3 X5

d) ta din gin a tom tan . gu ta din gin a tom tan . gu tan . gu ta din gin a tom 24 units - 5 + 3 + 5 + (3 + 3) + 5 or X5 Y3 X5 Y6(3+3) X5

a) Homogenous form: X y X y X b) increasing X form: X y 2X y 3X c) decreasing X form: 3X y 2X y X d) increasing y form: X y X 2y X

Ex.4 Simple korvai(purvanga)Ta . Di . Takajonu Tom . . .(12)Di . Takajonu Tom . . .(10)Takajonu Tom . . .(8)Jonu Tom . . .(6)Tom . . .(4)

(uttaranga using mora)Tadin . ginatom(6)Taka Tadin . ginatom(8)Takadiku Tadin . ginatom(12)64 units

Ex.5Six yatisGopuccha SrotovahaSama DamaruMridangaVisama12345112345 12345 1 12312341212345 1234 12 1234512312312345 123 123 1121234 12 1234 12341123451 12345 12 12 1234 123 123 1234 12 12345 1

Ex.6a Mora and grouping in 1st half of pallaviX6(3+3) Y5 X6(3+3) Y5 X6)3+3) = 28 units

X Y X Y XEx.6b Korvai and grouping in 2nd half of pallavi X6(2+2+2) Y6 X6(2+2+2) Y6 X6(2+2+2) = 56 units

Ex.7 Phrases that outline tala structure in anupallavi

O O X X O O X X

Ex.8 Yati-based phrasing in charanam

Ex.9Final mora/sub-mora in charanamX12(x4x4x4) Y8 X32(x8y4x8y4x8) Y8 X48(x12y8x12y8x12) = 112 units