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The Structure of Standard Music Notation Roberto Casati

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The Structure of Standard Music Notation Roberto Casati The Structure of Standard Music Notation Roberto Casati To cite this version: Roberto Casati. The Structure of Standard Music Notation. Zaibert, Leo. The Theory and Practice of Ontology, Springer, pp.187-201, 2016, 978-1-137-55277-8. 10.1057/978-1-137-55278-5_10. hal- 01508479 HAL Id: hal-01508479 https://hal.archives-ouvertes.fr/hal-01508479 Submitted on 24 Dec 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 10 1 The Structure of Standard Music 2 Notation 3 Roberto Casati 4 Western musicians and musically educated people acquire most of their 5 repertoire through reading musical scores. Learning to read music is a 6 long and time-consuming process. Some crucial conventions must be 7 mastered, and implemented according to sensorimotor patterns that are 8 specific to the instruments one plays. This chapter explores some aspects 9 of these conventions related to time representation. It presents a syntactic 10 characterization of a fragment of Standard Music Notation, and discusses 11 some cognitive consequences of principles that govern the syntax. A pre- 12 liminary hypothesis about obstacles to reading is put forward. A conse- 13 quence of the hypothesis is that certain musical styles appear to be very 14 much in synch with Standard Music Notation, whereas others do not 15 find an easy representation within it. 16 The chapter has two parts. In the first part I provide a characteriza- 17 tion of the temporal fragment of Standard Music Notation (SMN). 18 R. Casati (*) Centre National de la Recherche Scientifique, Paris, France AU1 Ministère de la Recherche, Paris, France © The Author(s) 2017 187 L. Zaibert (ed.), The Theory and Practice of Ontology, DOI 10.1057/978-1-137-55278-5_10 188 R. Casati 19 I treat SMN like a formal language and describe its syntax and semantics. 20 The main theoretical notion is that of an invisible “raster” whose abstract 21 properties make the notation possible. Some principles governing the 22 notation are spelled out. In the second part I draw some considerations 23 of cognitive import about the peculiarities of the notation. 24 1 The Temporal Fragment 25 Standard Music Notation (SMN) is a notation that primarily represents 26 the evolution of pitch in time. In its present forms, pitches are repre- 27 sented as locations on a five-line staff. Here I present a characterization of 28 the time representation dimension only. This is meant to be a fragment 29 of the complete set of SMN notation symbols. As such, the fragment 30 abstracts not only from the pitch dimension, but also from many nota- 31 tional peculiarities and, in particular, redundancies. We shall start from 32 a simplified version of the fragment, and then add a few principles that 33 capture many of the idiosyncrasies of SMN. 34 The simplified fragment includes a set of primitive symbols, notes and 35 ties. 36 Notes, such as 37 38 and a functional expression, the tie 39 40 Notes are aligned on what we shall call a raster. A time signature (3/4, 41 6/8, etc.) is provided, so that notes aligned in the appropriate way on a 42 time-signed raster constitute a music score. 43 The notion of a raster is central for the possibility of using the sys- 44 tem of Standard Music Notation (as well as many other systems, as we 45 shall see) to represent time. A raster is an abstract organization of loca- 46 tions on the score. It is characterized in terms of its spatially relevant 10 The Structure of Standard Music Notation 189 features. A raster R is an ordered set of linearly arranged discrete spatial 47 locations 48 s1, s2, ..., sn 49 such that s1 is the leftmost element on the raster, and each sm+1 location is 50 to the immediate right of its predecessor sm. 51 A connected segment of the raster is a set of positions of the raster such 52 that if sm is the leftmost element of the set and sn is its rightmost element, 53 all si such that m≤i≤n are part of the set. 54 An initial connected segment is any connected segment which includes 55 the first location of the raster, s1. 56 Given a raster R, a score on R is the result of filling every location in a 57 given connected initial segment of R by a note. 58 Let us pause and see what happens here. First, some observations on 59 the “spatial” structure of the raster and the shape of symbols. Right-to-left 60 orientation (customary in SMN) is actually immaterial to the organiza- 61 tion of the raster; the notation could be otherwise oriented. But nothing 62 of importance hinges on it being oriented the way current practice has 63 it, so let it be. Spacewise, the raster could unfold in a spiral; again, noth- 64 ing depends on this—so let it be visually arranged on a straight line. 65 Notes could have triangular or pentagonal shape, or different colors to 66 differentiate them; nothing depends on this, hence let us use symbols 67 that are sufficiently similar to those used in current practice. Second, 68 we need a characterization of the raster structure because we want it to 69 represent time. What can a raster represent? The representation provided 70 by the raster is topological only: the simple semantics of the raster is the 71 following: 72 if a location sm is to the left of a location sn, then the symbol placed on sm 73 denotes an event that precedes the event denoted by the symbol placed on sn. 74 Figuratively, a raster is a kind of sequence of locations: 75 x x x x x x x x x x ... 76 190 R. Casati 77 Given that the only relevant properties are order properties, the raster is 78 not to be read as a implying a pace. In particular, the referents of the loca- 79 tions on the raster are not requested to be equally spaced in time. 80 Third, the raster structure is discrete. Locations in a raster are atomic; 81 there is no such thing as a half location, or as the leftmost part of a 82 location. 83 Raster structures are little noted but important structures. They under- 84 lie other types of notation. Alphabetic writing systems, for instance, are 85 in part a representation of the unfolding of phonetic events in time. In 86 some, particularly transparent writing systems, if a letter symbol is placed 87 to the left (or the right) of another symbol, then the referent of the for- 88 mer must be pronounced before (or after, respectively) the referent of the 89 latter symbol. The raster for writing in most Western alphabets: 90 x x x x x x x x x x … 91 can be filled in by replacing empty position by letters: 92 d x x x x x x x x x … 93 d o x x x x x x x x … 94 d o g x x x x x x x … 95 d o g s _ a n d _ m i c e …1 96 Here, too, the order is purely topological. A character count (includ- 97 ing spaces) of a written text does not tell how long it will take to pro- 98 nounce the text. As it happens, the pronounced length depends on the 99 particular assignments of lengths to the referents of the letter as well as 100 on many other factors. Some vowels in given contexts take more time 101 to pronounce than others. The spacing of locations on the raster does 102 not make one pronounce the letters at any pace. Here as in the case of 103 musical notation, the raster is necessary to represent the unfolding in 104 time; it is discrete; and the only relevant spatial structure is topological 105 structure. 10 The Structure of Standard Music Notation 191 2 Molecular Expressions 106 An important feature of SMN is that it allows for molecular expressions. 107 Using the primitive symbols opportunely placed on a raster, one builds 108 molecular expressions by the use of ties connecting symbols at adjacent 109 locations. 110 If note a occupies position sm and note b occupies position sm+1, say that a 111 is adjacent to b. 112 A dyadic molecular expression is constituted by two adjacent notes con- 113 nected by a tie. 114 Now, stipulate that ties can take as their argument both atomic and 115 molecular expressions. Thus a molecular expression can be constituted by 116 a note a at sm linked by a tie to a molecular expression in turn composed 117 by note b at sm+1 and note c at sm+2. 118 However, we do not have any reason to distinguish the molecular 119 expression constituted by a note a at sm linked by a tie to a molecular 120 expression composed by note b at sm+1 and note c at sm+2, from the molecu- 121 lar expression constituted by a molecular expression composed by a note 122 a at sm linked by a note b at sm+1, and note c at sm+2.
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