Nabokov, Nicolas [Nikolay]

II. Notational systems

1. Materials: general.

2. Letters of the alphabet.

3. Syllables.

4. Syllables and vowel acoustics.

5. Words.

6. Numbers.

7. Graphic signs.

8. Hybrid systems.

BIBLIOGRAPHY

Notation, §II: Notational systems

1. Materials: general.

A musical notation requires, in essence, two things: an assemblage of ‘signs’ and a convention as to how those signs relate to one another. A written musical notation requires further a spatial arrangement of the signs on the writing surface that makes a ‘system’ of the assemblage; it is this system that forms an analogue with the system of musical sound, thus enabling the signs to ‘signify’ individual elements of it.

Only rarely has music fashioned its own sign systems. It has generally been content to take over systems in use for other purposes (such as the representation of arithmetical values, of speech inflection or of the sounds of natural language). In so doing it has often discarded part of the system and modified the shapes of the signs to suit its purpose. Such signs, the ‘materials’ of notation, can be broadly classified into two categories: the phonic and the graphic. Phonic signs include letters, syllable-signs and word-signs (signs that convey both the meaning of the word and its sound in speech – known as ‘logo-syllabic signs’). Certain systems of numerals also come into this category: systems that assign names to at least the lower range of numbers. Graphic signs include geometric shapes, lines, dots, curves, grids and the like.

Phonic signs are by their nature already representational of sounds outside music. They can be ‘spoken’ as well as written, which increases their communicative power. But they have an all-important additional quality: either they have meaning (like word signs and numbers) or they belong to some system of ordering (like letters and in some cases syllables). These are the properties that were implied above in speaking of the adoption for other purposes of systems already in use.

Notation, §II: Notational systems

2. Letters of the alphabet.

For the requirements of an alphabetical notation, it is not in fact the phonic – or perhaps ‘phonemic’, since each letter at least in principle signifies a single sound of language – quality of a letter that is important but rather its position within a conventional order: an alphabet. The ordering of letters in an alphabet offers a ready-made base for notation, as it can be directly related to the intrinsic acoustical order of musical sound. It thus becomes an analogue of musical order: an item in the musical order is specified by reference to its place on the analogous system.

As stated above (§I, 2), the earliest-known alphabetic writing dates to the middle of the 2nd millennium bce. The first known to have an established order of letters is the Hebrew alphabet, traceable back at least to the 6th century bce. This order corresponds to the acrostics in the Bible (Lamentations, Proverbs, Psalms). Until the 17th century alphabetic writing existed in only a small area of the world: the Middle East, the Mediterranean countries, Eastern and Western Europe, South Asia and Korea. The earliest alphabets – Hebrew, Phoenician, Aramaic and North Semitic – all developed between 1000 and 500 bce. From these developed the Greek, Latin, Cyrillic and early Indian alphabets. As to order of letters, the Greek alphabet is close to the Hebrew, the Latin close to the Greek.

One of the advantages of an alphabet for music notation is that it consists of single rather than compound signs – signs that are distinctive and at the same time compact. Another is that it contains a convenient number of signs (alphabets range from about 20 to 50 letters, most having between 20 and 30) to represent a chromatic double octave or a diatonic triple octave; fewer can be selected to represent a single octave in a repeating scheme or the frets on a stopped-string instrument. Moreover, the letters of alphabets are generally assigned names (the fourth letter of the Hebrew alphabet being called daleth, the Greek delta, the Latin and modern Western European de and so forth), so that the notation can be spoken as well as written.

The alphabet was used for pitch notation in ancient Greece, and then around the 10th century in western Europe before being formalized in shape and absorbed into staff notation as clefs (C, F, G) and accidentals (‘b’, ‘h’). The alphabetic system is implicit still in staff notation, since in most European countries the placing of notes on the staff is translated into spoken letter-names (except in France, where they are translated into fixed solmization syllables; see Pitch nomenclature). The Western system is a repeating one, since the letters refer only to pitch classes, not to specific pitches; therefore the 19th-century German philosopher and scientist Hermann von Helmholtz developed a scheme of dashes to indicate pitch register (the dashes deriving from Greek notation but the letters coming from the Latin alphabet): A, B, C–B, C–B, c–b, c' (middle C)–b', c''–b'', c'''–b''' etc. The alphabet has also been used to denote keys, finger positions or frets in many Western tablature systems.

There are many examples of verbal abbreviation in Western notations: the letter p, for example, is used as an instruction to play softly (piano) and, in a rather elaborate formalized fashion (as an alternative to ‘Ped.’), below the staves, to indicate application of the sustaining pedal of the piano. ‘Significative letters’ were used in conjunction with some early Western neumatic notations to indicate duration (c to stand for cito or celeriter, ‘quickly’, i.e. ‘short’ value) and direction of movement (l to stand for levare, s for sursum, both meaning ‘upward’).

In all these non-alphabetic uses of letters, the notation can be described as ‘secondary’: that is, the letters signify words that in turn signify musical elements, rather than signifying musical elements directly. However, such is the force of tradition that formalized letters often cease to be recognizable: by this means a pedal mark has become a graphic sign that refers directly to the pianist’s foot movement. The same is true even for alphabetic uses of letters: the treble and bass clefs are now scarcely recognizable as formalized letters ‘G’ and ‘F’, and have become instead graphic signs for the two fixed pitches g' and f with a range of special technical connotations associated.

Notation, §II: Notational systems

3. Syllables.

As with letters, syllable notations fall into two categories: those that operate by reference to an established order of syllables, and thus relate directly to a musical order (‘primary’ notations), and those that use syllabic abbreviations of words, and operate by reference to meaning or name (‘secondary’ notations). Cutting across this categorization is the orthographic one: that some of these syllable systems are expressible as single symbols (ideograms or ‘characters’) while others have to be spelt out in letters.

A classic case of the first (‘primary’) category is the set of Japanese syllables i, ro, ha, ni, ho, he, to. These are the initial seven syllables of an established order of some 48 Japanese characters closely analogous to the order of an alphabet – that is, it is a conventional order rather than an intrinsic one. In Western music terminology in Japan, these first seven function exactly like the Western letters A, B, C, D, E, F, G, with repetition for each octave in the same way. Thus a C major scale is represented as ha–ni–ho–he–to–i–ro–ha, each having a single character to represent it in written form. (A more extensive set from this series was used in 17th-century shamisen tablature to represent successive finger positions from the open bass string to the highest position on the treble string.) A simpler example is the set of syllables for the Balinese five-note slendro scale, a set that rotates through five vowel sounds: ding–dong–dèng–dung–dang. It is almost an alphabetical system using only vowels, save for the fact that Balinese literary script uses characters rather than letters and therefore has no alphabet. The characters for these five notes are shown in fig.3 (see also Indonesia, §II, 1(ii)(b), Table 1).

Similar to this is the set of Chinese syllables for the pentatonic scale: gong–shang–jue–zhi–yu (see fig.4a, with the parallel set of Korean syllables using the same Chinese characters, fig.4b).

The Chinese gongche notation is a more complex system of the same type. It consists of ten characters, or ideograms, each representing a syllable that stands for a note on a largely diatonic scale extending over a 9th. Fig.5 shows these syllables and their characters, with he arbitrarily set to the pitch c. Octave positions are sometimes shown by the addition of an affix or small mark. A chromatic scale could be produced from this by the use of the prefixes gao- (‘high’) to raise a note, or xia- (‘low’) to lower it, by a semitone; but after the 11th century gao- ceased to be used. Korean musicians in the 15th century adopted the ten basic characters, applying their own pronunciation: hap, sa, il, sang, ku, ch’ŏk, kong, pŏm, yuk and o. The Korean notation is called kongch’ŏkpo and it does not use affixes or marks, allowing sa to denote d or d, and similarly with il, kong and pŏm. It is noteworthy that four of the characters in gongche notation are numerals (si is four, yi is one, liu is six and wu is five); thus the notation is partly numerical.

The South Asian system of syllabic solmization is usually written down in Devanagiri script in North India, or Tamil or Telugu script in the South (fig.6). Although notation is generally considered to be of little importance in what are predominantly oral traditions, it is widely used as an aid to memory or as a learning tool. This is particularly true of Karnatak music, which relies to a much greater extent on a body of compositions than does Hindustani music. The syllables themselves may describe the duration of a pitch through the use of a short or long vowel: usually a short vowel stands for a pitch of one mātrā (‘beat’) or less and a long vowel for two beats or more. Symbols modifying the pitches vary from system to system but common devices include a short vertical line above the syllable denoting a sharpened pitch, or a short horizontal line below the syllable showing a flattened pitch. The syllables are arranged on a framework which shows the rhythmic cycle (tāla), one line of notation being equal to one cycle of the tāla (see fig.7).

Rather different, but not unlike the Indian solmization syllables, are the Western medieval ut, re, mi, fa, sol, la. They are indeed syllables in written form, being the initial syllables of the first six lines of a seven-line hymn to St John, the text of which is attested from about 800 and would have been well known in the 11th century when Guido of Arezzo created a solmization system from them. The syllables were by chance distinctive, and operated by reference to a textual order. But their referential character was much strengthened by the fact that the first six lines of the hymn’s melody began successively on the degrees of the scale c–a, and they thus operated by reference also to an established external musical order – though whether the melody existed before the solmization system, or whether it was designed as a supporting aid, is not known. The derivation is shown in ex.1. Out of this succession of notes was created the ‘natural hexachord’, which was flanked by a ‘soft hexachord’ of the same succession transposed a 5th lower and a ‘hard hexachord’ transposed a 5th higher, the three forming together the underlying musical system known as musica recta. This total system was transposable to other relative pitch levels, and isolated hexachords of ‘alien’ pitch levels could be introduced, each hexachord having the identical set of syllables (see Solmization, §I, 1; Hexachord; Musica ficta; and Guido of Arezzo).

Javanese titilaras kepatihan (‘cipher notation’) whose seven syllables, ji, ro, lu, pat, ma, nem and pi, are abbreviations for the numbers 1 to 7: siji, loro, telu, papat, lima, nĕm and pitu.

In addition to their referential power and their capacity (as abbreviations) to refer to the meanings of words, syllables have a further quality: onomatopoeia. The degree of openness or closedness of the vowel sound, the presence or absence of initial and terminal consonants, and the character of any such consonants (dental, labial, nasal etc.) is frequently used to reflect tone-colour, attack or rhythmic value. A simple case is ‘scat singing’ in jazz, where doo is used for a stressed and sustained note, bee for a short unstressed note and bop for a staccato note, stressed but often off the beat. Thus the pattern bop bop bee-doo-bee-doo-bee-doo-bee can be sung to the rhythmic pattern shown in ex.2 by a scat singer almost as if it were a rhythmic solmization; it can also be used as a verbal communication of the rhythmic pattern and is thus halfway to being a notation of a rudimentary and imprecise kind.

Onomatopoeic syllables are used by Ewe drummers in Ghana. Two strokes of the butts of the hands in succession at the centre of the drumhead are represented by the syllables ga-da, the softer sounds of the hands brushing across the centre of the drum by ka-tsa, and the use of splayed fingers to produce a combination of round drum tone and sharpness of attack by ga-tsya. But the relationship between drum sounds and syllables goes beyond representation: it is an identity – the drums are themselves thought of as producing the syllables, and when syllables are spoken to the drums they are spoken at the same pitches as the drums. Oral drum notations are widespread in South Asia and are described at length elsewhere in the dictionary (see India, §III, 6(iii)(a)–(b); Mrdangam §1; and Tabla, §3).

Notation, §II: Notational systems

4. Syllables and vowel acoustics.

Whereas the syllabic systems discussed above (§II, 3) represent specific pitch classes, scale degrees or performance techniques, other syllable systems, less formalized but highly regular, tend to use vowels and consonants in accordance with their acoustic phonetic features to reflect iconically relative pitch, duration, resonance, loudness and so on. The relations between such syllables and musical features are thus far from arbitrary.

Vowels, in particular, are often used in accordance with what phoneticians call their intrinsic pitch, intensity and duration (see Hughes, 1989). For example, the vowels i, e, a, o, u in their approximate Spanish or Japanese pronunciations are often perceived as constituting a descending pitch sequence (reflecting their ‘second formant’ pitches). Many cultures exploit this intrinsic pitch ordering of vowels in teaching instrumental music. Thus the fixed melodic repertory of the Japanese nōkan flute is taught by singing mnemonics such as ohyarai houhouhi, in which successive vowel pairs reveal melodic direction with over 90% accuracy: the sequences ohya, rai, uho and uhi all represent melodic ascents, with uhi signifying the largest leap because its two vowels are at opposite ends of the pitch spectrum; iho and hou represent melodic descents (see Japan, §VI). In several such systems in Japan (where scholars call them shōga) and Korea (yukpo or kum), exceptions to this relationship between vowels and melodic direction often result from the competing acoustics of intrinsic duration and intensity, whereby a is favoured for comparatively long, loud or metrically important notes, while i and u are used for weak or short notes, with e and o in between.

Consonants also play a role. In the sequence teren for Japanese shamisen lute, t indicates a normal, resonant down-pluck; r signifies a gentler sound (never the initial note of a phrase), either an up-pluck or a left-hand pizzicato; and n shows that the second sound is longer than the first. In many drum mnemonics throughout the world, a final k – a stopped sound – represents a damped stroke, while a final nasal or vowel shows that the sound is left to resonate and decay naturally.

Such systems could be called ‘acoustic-iconic systems’. Their oral origins are reflected in the lack of any indigenous explanations for their patterning; their iconic symbolic power (teachers emphasize their importance) lies precisely in their acoustic naturalness. Today, however, such systems are often written down. In many Japanese and Korean written notation systems (which tend to be different for each instrument), each line of tablature or pitch notation is accompanied by a line of acoustic-iconic syllables (see below fig.16b). The fact that this may happen even when this line adds no information to the tablature, as in shamisen bunka-fu notation, confirms the traditional importance of such syllables in transmission.

Notation, §II: Notational systems

5. Words.

Words have assumed a place in Western staff notation only during the last 350 years or so. They have done so with the rise of the score and of the desire of composers to specify the instrumental forces for their music; and this has happened simultaneously with the desire also to specify tempo, mood, character and detailed matters of tone production and attack (see Tempo and expression marks). Thus, for tempo, words such as largo and allegro were introduced, and a set of modifiers was applied to them to express shades of meaning: molto, assai, non troppo, -etto and so on. Such words, together with others expressing mood and character – such as andante, scherzo and scherzando, dolente – generally appear at the beginnings of sections or whole movements (even serving as titles). It is no coincidence that their introduction occurred in that part of the Baroque period during which the doctrine of the Affections (Affektenlehre) was the predominant aesthetic, and that a great expansion of the range of terms, and of the languages from which they were drawn, took place during the Romantic era. Other words, such as rallentando, ritenuto and stringendo for tempo, and pizzicato, leggiero and flautando for attack and tone production, control temporary changes and localized features, and thus appear in the course of the musical notation.

The most striking aspect of the Western use of words is its consistently auxiliary nature. Words are almost never on the staff, but above or below it, or in the margin. They were not integral to the system when Western staff notation was being formulated during the late Middle Ages, when even the part-names tenor and contratenor were not always supplied and when a name was almost never given to the top voice. They have since become indispensable to staff notation, but have retained their auxiliary position, so that a music copyist will enter the note symbols representing pitch and rhythm before finding the most convenient places in which to add the verbal elements of the notation so that they can easily be read. This situation is not merely the result of historical circumstance. There is the more pragmatic ground that Western words are written alphabetically and thus have two disadvantages for notational use: they occupy a lot of space, and (more important) they take time to read and understand.

These disadvantages are not present in most East Asian writing systems, where characters represent syllables or words. The classical Chinese language is in essence made up of monosyllabic words that do not change or acquire prefixes or suffixes under different grammatical conditions as they do in most Western languages; the most that they do is become incorporated into compounds of monosyllables (e.g. nü-ren means ‘female’ + ‘person’, thus ‘woman’). So when, in the Chinese fixed-pitch system of the 12 lü, the names of individual pitches are written down, each pitch is represented by a pair of ideograms. Moreover, when the note names huangzhong (‘yellow bell’, pitch c), linzhong (‘forest bell’, g), yingzhong (‘answering bell’, b) and jiazhong (‘pressed bell’, d) are written down, the second ideogram is always the same. In fact, when the names of all the chromatic pitches are written down their first ideograms are distinctive (i.e. they do not require the second ideogram to distinguish them from others): ‘yellow bell’ (c), ‘greatest tube’ (c), ‘great frame’ (d), ‘pressed bell’ (d), ‘old purified’ (e), ‘mean tube’ (e or f), ‘luxuriant vegetation’ (f), ‘forest bell’ (g), ‘equalizing rule’ (g), ‘southern tube’ (a), ‘not determined’ (a) and ‘answering bell’ (b). Thus in notation the names are abbreviated to their first words, as shown in fig.8; see also China, §II, 4, Table 2

Words are often used as ‘labels’ or memory aids for standard melodic formulae. The so-called neumatic notation of Japanese karifu relies on words beneath the graphic symbols to indicate a large amount of the melodic inflection. The same is true of Tibetan Buddhist notation, whose neumes have written above them verbal instructions as to vocal production, directional movement and ornamentation. In oral traditions, groups of words and whole phrases are used as mnemonics for standard patterns. Ex.3 shows an African instance: the sentence ‘b’o tan ma tun ro’ko Baba ma j’iyan tan’ (‘else I must go back for more, Father, don’t finish the yam’) is broken up into syllables in the piece of music for a pair of hourglass drums and a small kettledrum, from the Yoruba in Nigeria.

Notation, §II: Notational systems

6. Numbers.

Numbers would perhaps seem to be the most readily adaptable of all materials for notational purposes. They provide a reference system that can control any or all parameters of musical sound, as the pioneers of integral serialism demonstrated. In particular, pitch can be controlled by assigning numbers to the notes of a scale, to the keys of a keyboard, to the finger positions or frets of a string instrument, or to the holes or valves of a wind instrument (or the fingers of its players), and pitches can be represented in this way individually or relative to each other by the measurement of interval in a melody or chord. Duration lends itself most naturally to numerical representation because the hierarchy of beats in musical metre involves subdivision of a large time unit or multiplication and addition of small units and is thus intrinsically arithmetical. Any other parameter, such as loudness, attack or tone-colour, can in theory be measured as a scale of values and then be represented by those values as numbers (e.g. 1 for extremely soft, 5 for moderate and 10 for extremely loud, with the intervening numbers for gradations between these), but such systems have tended to be restricted to the coding of music for computers.

In practice, the measurement of pitch by numbers (other than for scientific purposes) has been very rare, and is a predominantly modern phenomenon. Perhaps the most important was the Galin-Paris-Chevé method from the mid-19th century. The numbers 1–7 represented pitches, with a dot below for lower octave and above for upper. The numbers were purely visual: they were spoken as ut, re and so on. This system was adopted in modified form in China, Japan and other countries. The abbreviated number system of the Javanese kepatihan notation has already been discussed as a syllabic notation (§3 above). Notational systems for the Japanese koto use the numbers 1–13 in Japanese characters (though the characters for 11–13 are not true numbers). But these are secondary systems in the sense that the numbers refer to the 13 strings on the instrument rather than directly to the pitches that they produce: the pitches will depend upon the scale to which the instrument has been tuned. Fig.9 shows the 13 characters and their Arabic numeral equivalents, together with the notes that they represent in the most common tuning (hirajōshi); because of the pentatonic scale in use the number of any note is five away from that of its octave. A similar system exists for the 25-string Chinese se, using the Chinese numbers 1–25. An even more extended number notation for pitch (not fixed-pitch) is the pitch representation of the Ford-Columbia computer input language for music. There, the numbers 1–49 designate leger lines and staff lines and their intervening spaces: thus 1 is the tenth leger line below the staff, 2 the space above that, and so on. The entire set of numbers is dependent on the clef governing the staff. One type of modern Japanese shamisen notation uses three kinds of numeral: Arabic numerals form a direct pitch notation using 1–7 for an ascending scale in the central octave and the same numbers with a dot to the left and the right respectively to represent the notes of the lower and higher octaves: Roman numerals I–III to the right of these numbers show the three strings of the instrument; and Japanese characters for the numbers 1–3 indicate which finger is to be used.

Probably the earliest, and at the same time the most complex, number notation is the jianzi pu for the Chinese Qin. Like the notation for the Japanese Koto, its numbers refer directly to the means of production and only indirectly to the sound produced. The strings of the qin can be stopped at studs which serve as frets, or at points between them. Numbers are used to indicate all three of these: 1–7 for strings, 1–13 for the studs (hui, in ascending order), and 1–10 as a guide to the distance between two studs (fen). The three (often only two, because there is not always a fen number) are gathered together into a complex note symbol, with the string number in the lower half and the other two in the upper half, together with other symbols to indicate the stopping finger, the plucking finger and certain technical details. Fig.10a shows the Chinese numerals, and Fig.10b shows a single note symbol made up of five elements, of which three are numbers and the remaining two special symbols.

Western notations use Arabic numerals in keyboard tablatures and Italian lute tablature of the Renaissance. They are also used in staff notation to indicate metre and to show unusual rhythmic groupings. Thus time signatures have a denominator that represents a level of note value (on a scale from semibreve = 1 to minim = 2, crotchet = 4, quaver = 8, semiquaver = 16 etc.: these numbers are used in American and German parlance to describe the levels of value, with semiquaver being ‘16th-note’ and ‘Sechzehntel’) and a numerator that indicates the number of units of that level in a bar. A triplet in a duple metrical context is indicated by a number 3 within a slur mark, and in Chopin’s music, for example, this is extended to groupings of 11, 21 and so on.

Notation, §II: Notational systems

7. Graphic signs.

The act of writing a succession of notational syllables is graphic because it traces a path across the writing surface. That path is the analogue of the passage of music through time. The direction of the path tends to follow the prevailing direction of writing for the language of the country concerned. The Chinese, Korean and (to some extent) Japanese languages have been written from top to bottom, in columns beginning at the right-hand side of the page: consequently most Chinese and Korean notations have been written in columns in the same way, and so have Japanese instrumental notations. On the other hand, Japanese neumes (karifu, meyasu) are written horizontally from right to left. Tibetan, Javanese, Balinese, Greek and Latin are all written horizontally from left to right. Consequently Tibetan neumes and Javanese and Balinese ideographic notations all read in that direction, as do Western neumes, alphabetical and staff notations, and tablatures.

This path across the writing surface may be more precisely defined by the spacing out of notational symbols so that each space represents a beat of the prevailing metre. Thus in Chinese gongche notation the ideograms representing pitches are equidistant down their columns; and when there is a gap in the column of ideograms the previous pitch is assumed to continue to sound for a second beat. Alternatively, beats may be marked by a graphic symbol. One such is a dot – as in Japanese gagaku notation, which uses small dots for the basic beat and large dots for every fourth or eighth beat – defining two levels of metre (such dots often indicate the sound of percussion). Another such symbol is a line drawn at right angles to the path – as in Korean ‘mensural’ chŏnggan notation (which encloses its symbols in a grid with thin and thick horizontal lines to show their places within two levels of metre), in modern Japanese Ikuta-school koto notation (which uses short and long horizontal lines to show the same), or in the bar-lines of Western staff notation. Such graphic marks have the economic advantage that the spaces allocated for beats need not be equal in size: metrical units containing several symbols can be given more space than units with few or none.

So far, the path discussed has been one-dimensional. But it is also possible to define a broad path across the writing surface and to treat the width of the path as a second dimension. This dimension can be made the analogue of some other parameter of music: in particular, of a technical aspect of an instrument – the string or course of a zither or lute, for example, or the keys of a metallophone – or of pitch (as in diastematic neumatic notations) or volume (as in some electronic scores).

A system of notation recently discovered in Mongolia and used in Nomyu Khan monasteries in the 18th and 19th centuries is thought to describe melodic pitches arranged according to the tuning of the half-tube zither (yatga). This notation takes the form of lines tracing the broad tonal contours of the melody rather than a series of discrete notes and should probably be regarded as signifying the ten strings of the yatga running horizontally across the surface of the page (fig.11). Much more research is needed into this system, however, before definite conclusions can be drawn about what precisely it represents.

A simple way of using the second dimension for pitch in vocal music without need for new signs is to ‘height’ the syllables of text themselves, as in dasian notation; however, this does not work for music with any degree of melisma.

Western staff notation is another form of the same procedure. The dots, however, are made void or full and supplied with stems and flags or beams to represent grouped durations in such a way that the horizontal dimension between two bar-lines can be treated flexibly. In other words, the exact proportional use of space to time is obviated by the application of duration symbols to the dots. Such duration symbols are themselves graphic signs; moreover, their beaming into groups conveys other information such as accentuation, phrasing, differences of dynamic level and the application of syllables.

Such graphic signs as these last belong to a reference system – in this case a system representing duration and comprising only five elements: a stem, a flag, a dot and two kinds of note head; if the void head can be regarded as an ‘absent’ head then they constitute four signs, each of which operates in a binary way (see fig.12) as present (+) or absent (−) in appropriate positions. Similar graphic reference systems are the signs of Japanese goin-hakase notation and its later modifications, karifu and meyasu, and also the ‘teardrop’ notation, gomafu, and its later development bokufu. In the first three of these, a notched-stick shape is rotated through eight positions that correspond to eight pitches of a pentatonic scale, thus spanning a 10th (fig.13). They are linked together to form a graphic trace extending leftwards from the text syllable. The trace is not however an exact representation of pitch since the notation relies on the names of standard melodic formulae written beneath. In gomafu and bokufu marks are put to the left or right of syllables to indicate such standard formulae.

A comparable system is that of so-called dasian notation from the 9th century. The materials for this constitute a spatial matrix with pitch as the vertical axis and time as the horizontal, and the Greek prosōdia daseia in two transformations: first modified into four distinct forms to designate the four pitches of the tetrachord; and then with each form reversed, inverted, and reversed and inverted to represent the higher pair of tetrachords, with the first two also shown facing downwards giving 18 signs in all (fig.14; see also Organum, §2).

A rather special case of a notation that is graphic and operates on binary principles is Braille notation for the blind. The basic material is a display of six dots arranged in a matrix two (across) by three (down). These dots are raised from the surface of the paper by embossing, so that they can be felt. Each dot is either present (embossed) or absent. The pattern of the upper four dots designates pitch and the pattern of the lowest two designates duration. There are special patterns for octave register, accidentals and other notational devices.

Other graphic signs do not belong to such a system. They represent movement and shape in music, and thus display elements in relation to each other. They cannot specify individual musical elements, as can referential notations. Notations that rely on graphic relationship have only relative pitch significance, even when they have taken over an existing sign system, such as the accentual signs of the 5th-century Syriac writers (nine principal signs denoting main and subsidiary pauses, interrogative accents and so on, and made up of dots in different placings and groupings), or those of the 9th-century Tibetan scribes, or the classical Greek prosodic accent signs from which Byzantine ekphonetic notation evolved, or the signs of the Roman grammarians from which Western neumes are sometimes alleged to have developed (see Ekphonetic notation). That is because, without the imposition of a grid system, distance is difficult for the eye to judge, for both reader and writer. The line of text to which a melody was to be sung could be used as a pitch demarcation, with dots above and beneath syllables signifying higher and lower pitches, as in some Vedic chant books.

Neumes are stylized contour shapes. Their rises and falls and level lines represent rises and falls and level passages in a melodic line. Neumes thus differ from ekphonetic notations (though the dividing-line is sometimes difficult to draw) in that they are not concerned with inflection of voice between high, medium and low, but with groups of sung pitches rising and falling over a quite narrow range: a neume may represent a pattern of intervals whether it lies high or low in the voice’s compass. Each neume is thus self-contained; the pitch relationships between a neume and its neighbours are not necessarily graphically shown, though in the ‘heighted’ neumes that appear in Western European sources from about the 10th century some attempt is made to show this.

The neumes of Tibetan Buddhist notation are made up of curves and undulations of varying amplitudes that represent directional movement of the voice, together with crosses or circles representing the sound of drums or cymbals (see Tibetan music, §II, 4).

Notation, §II: Notational systems

8. Hybrid systems.

Many notations are hybrid in that they use more than one type of material. Japanese karifu, for example, has already been discussed above (§§5 and 7): the notation is generally called ‘neumatic’, but is equally a verbal notation in that Japanese characters under the graphic neume shapes give essential information about melodic turns of phrase (see fig.15). Tibetan Buddhist chant notation has also been discussed in these two contexts, since verbal instructions as to vocal production and other aspects of performance appear above the line of neumes. The jianzipu notation for the Chinese qin has also been shown to contain special symbols as well as numbers. In the following discussion, three notations will serve to illustrate the interaction of materials.

Occasionally two materials interact in a tautologous way – that is, they call for the same musical result but by different visual means. But most interactions are in some way complementary.

A notation that combines tautologous and complementary uses of different materials is the notation for the kŏmun’go or Korean zither. The notation is known as hapchabo and dates from the 15th century. It is an adaptation of Chinese jianzipu, but whereas the Chinese notation uses numbers for the designation of both string and stopping-point, the Korean notation assigns names to its six strings (see fig.16a) and uses the string name in conjunction with a number for the stopping-point. Added to the left of this name and number is a graphic symbol indicating the left-hand stopping finger, and where necessary symbols for direction of stroke, ornaments and so on. The central part of the notation is thus a complementary hybrid of word, number and graphic signs. This compound symbol is placed in the middle of three columns. In the right-hand column appears the central scale degree kung from the Korean oŭmyakpo ‘five-note abbreviated notation’, and in the left-hand column appear a group of Korean letters that signify one of the Korean solmization syllables from series such as tŏng, tung, tang, tong and ting, or rŏ, ru, ra, ro and ri (see fig.16b). All these notational elements, with double tautology as to pitch, point to B stopped with the left thumb and plucked with an outward stroke (fig.16c).

The most fully hybrid of all notations is the staff notation of the West. It uses all the types of material discussed above. Fig.17, the beginning of the Prelude from Liszt’s first book of Etudes d’exécution transcendante, contains examples of letter notation in (1) the clefs, which are formalized letters G and F; (2) the accidentals, which are formalizations of ‘b’ () and ‘h’ (, ); and (3) the dynamic marking f, which is an abbreviated verbal notation. It also contains syllabic notations, both of them abbreviations for words: (1) the pedal application Ped., so formalized as almost to be a pure graphic symbol; and (2) the technical instruction rinforz., for rinforzando. It also contains two examples of full verbal notation: (1) the general designation ‘Presto’ for the tempo and character of the Prelude as a whole; and (2) the localized technical instruction energico. It has several examples of numerical notation: (1) the tempo specification, which supplements the tempo aspect of the verbal instruction ‘Presto’; (2) the indication of octave transposition; (3) the fingering in bar 2, which is a technical notation; and (4) the indication ‘19’ for rhythmic grouping. But its main constituents are graphic notations: (1) the staves, bar-lines and brace; (2) the note symbols and rests; (3) the time signature ‘’, which derives from the medieval half-circle designating duple division of breve and semibreve (and thus is not in origin a verbal abbreviation of ‘common time’, though it has acquired this status in more recent times); (4) the phrase mark, which is partly a graphic duplication of pitch and partly an indication of phrase articulation that duplicates the beaming of note symbols; (5) the pause sign; (6) the pedal release sign; (7) the staccatissimo signs; and finally two suggestively graphic signs, (8) the spread-chord indication in bar 1, and (9) the decrescendo and crescendo signs.

From this it can be seen that staff notation is a complex multiple hybrid system with very low redundancy, partly technical and tablature-like, partly representational.

Notation, §II: Notational systems

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Notation