Duration :: Where the vertical position of a note on a staff or stave determines its pitch, its relative time value or duration is denoted by the particular sign chosen to represent it. This is the essence of proportional or mensural notation, first developed in the 11th century and about which more information is given in the section below entitled Origin of Music Notation. The coin & paper money you use to pay for goods & services are good examples of relative value. While in England 100 pence = 1 pound and in the United States the cent and dollar are similarly related (100 cents = 1 dollar) in neither case do you know the 'absolute' value of a currency or of its 'denominations'; for example, how many dollars = 1 pound. So it is with musical 'denominations'. The signs do not give duration in units of time, minutes or seconds. That must be given in ways we discuss in lesson 5. The Anatomy of a Note Sign :: In music the denomination of 'coinage' is the note or note sign. One can use either term. Each note sign is a construct of three distinct parts. The note-head, whose position on the stave actually sets its pitch, can be open (white) or closed (black). For all notes except the breve (double whole note) and semibreve (whole note), each note has a stem and, for the notes of shorter time-value, a flag, hook or tail (one flag for a quaver (eighth note), two for a semiquaver (sixteenth note), and so on). The stem can rise from the note-head, in which case it lies on the right-hand side of the note-head, or fall from the note-head, in which case it lies on the left hand side of the note-head (see the two crotchets (quarter notes)). In either case, the flag lies on the right-hand side of the stem (see the two quavers (eighth notes)). When placed on the stave, a note sign will be placed either on a line or on a space between the lines. The position indicates the relative pitch of the note. If the note lies above or below the stave then it will lie on, above or below auxiliary lines called leger or ledger lines. This is illustrated below. Notice how the position of the note on the stave generally determines whether the stem 'rises' or 'falls' from the note-head. Chart of Note and Rest Signs :: If the notes are listed in decreasing time value, longest to shortest, each is half the duration of the one immediately before it. The table of 'denominations' below shows the note with the longest duration at the top and that with the shortest duration at the bottom. Sign number equal to 1 semibreve English American Italian French German Spanish Catalan 1/2 breve or brevis double-whole note breve carrée or brevis or double-ronde (meaning square) Doppeltakt(note) or Brevis cuadrada or breve or doble redonda quadrada (f.) or breu (f.) 1 semibreve whole note semibreve semi-brève or ronde (meaning round) ganze Takt(note) redonda or semibreve rodona (f.) 2 minim half note minima or bianca blanche (meaning white) Halbe(note) or halbe Takt(note) blanca or mínima blanca (f.) 4 crotchet quarter note semiminima or nera noire (meaning black) Viertel(note) negra negra (f.) 8 quaver eighth note croma croche (meaning hook) Achtel(note) corchea or croma corxera (f.) 16 semiquaver sixteenth note semicroma double croche (meaning double hook) Sechzehntel(note) semicorchea semicorxera (f.) 32 demisemiquaver thirty-second note biscroma triple croche (meaning triple hook) Zweiunddreissigstel(note) fusa fusa (f.) 64 hemidemisemiquaver sixty-fourth note semibiscroma quadruple croche (meaning quadruple hook) Vierundsechzigstel(note) semifusa semifusa (f.) 128 semihemidemisemiquaver one hundred and twenty-eighth note centoventottavo (nota) cent-vingt-huitième Hundertundachtundzwanzigstel(note) garrapatea   Rests, periods of silence, are shown in the table below. Rest number equal to 1 semibreve English American Italian French German Spanish Catalan 1/2 breve rest double-whole rest pausa di breve bâton or pause de brève or silence de brève doppel Pause silencio de cuadrada or pausa de cuadrada or silencio de breve or pausa de breve doble pausa (f.) or pausa de quadrada (f.) 1 semibreve rest whole rest pausa di semibreve pause ganze Pause silencio de redonda or pausa de redonda or silencio de semibreve or pausa de semibreve pausa (f.) or pausa de rodona (f.) 2 minim rest half rest pausa di minima demi-pause halbe Pause silencio de blanca or pausa de blanca mitja pausa (f.) or pausa de blanca (f.)  or  4 crotchet rest quarter rest pausa di semiminima soupir Viertelpause silencio de negra or pausa de negra or silencio de semiminima or pausa de semiminima quart de pausa (m.) or pausa de negra (f.) 8 quaver rest eighth rest pausa di croma demi-soupir Achtelpause silencio de corchea or pausa de corchea vuitè de pausa (m.) or pausa de corxera (f.) 16 semiquaver rest sixteenth rest pausa di semicroma quart de soupir Sechzehntelpause silencio de semicorchea or pausa de semicorchea setzè de pausa (m.) or pausa de semicorxera (f.) 32 demisemiquaver rest thirty-second rest pausa di biscroma huitième de soupir Zweiunddreißigstelpause silencio de fusa or pausa de fusa trenta-dosè de pausa (m.) or pausa de fusa (f.) 64 hemidemisemiquaver rest sixty-fourth rest pausa di semibiscroma seizième de soupir Vierundsechzigstelpause silencio de semifusa or pausa de semifusa seixanta-quatrè de pausa (m.) or pausa de semifusa (f.) 128 semihemidemisemiquaver rest one hundred and twenty-eighth rest pausa di centoventottavo cent-vingt-huitième de soupir Hundertundachtundzwanzigstelpause silencio de garrapatea or pausa de garrapatea   Each line in the example below is a single bar (we meet bars in the next lesson ), with the same total time value of notes as every other line. Each line in the example below is a single bar (we meet bars in the next lesson ), with the same total time value of rests as every other line. Dotting and Double-Dotting :: A dot, placed to the immediate right of the note-head, increases its time-value by half; thus a minim (half note) (which is equivalent to two crotchets (quarter notes)) followed by a dot (which in this case is equivalent to a further crotchet (quarter note)) is equivalent to three crotchets (quarter notes), while a dotted crotchet (dotted quarter note) is equivalent to three quavers (eighth notes) and a dotted quaver (dotted eighth note) is equivalent to three semiquavers (sixteenth notes)). If the notehead is located in a space, the dot is placed in that same space. If the notehead is on a line, the dot is placed in the space just above the line. Exceptions sometimes have to be made if several dotted notes share a single stem. A dot placed after a rest or note is called an augmentation dot.   A second dot, placed to the immediate right of the first dot, increases the original undotted time-value by a further quarter. Another way of thinking about the second dot is that it adds the note equivalent to half the note added by the first dot. So, for example, a minim (half note) (equivalent to four quavers (eighth notes)) followed by one dot (equivalent to two quavers (eighth notes)) followed by a second dot (equivalent to one quaver (eighth note)) is equivalent, in total, to seven quavers (eighth notes). Dots after rests increase their time-value in the same way as dots after notes. Beams and Beaming :: When notes with flags lie together in groups they are often linked by one or more lines called beams. The number of beams reflect the number of flags each would have had when an individual note. We illustrate below: a group of four semiquavers (sixteenth notes), first unbeamed and then beamed a group of two quavers (eighth notes), first unbeamed and then beamed a group of one quaver (eighth note) and two semiquavers (sixteenth notes), first unbeamed and then beamed Notice how the beaming reflects the time value of each note Ties :: There are occasions when the duration of a note may not be easily notated using one of the note signs given above. If the duration of a note is longer than a breve (double whole note) or when the addition of dots cannot provide the required duration, groups of notes can be linked by one or more ties. Tied notes are treated as a single unbroken note whose duration is given by the duration of the notes under the tie taken successively. This is illustrated in the example given below where a crotchet (quarter note) tied to a quaver (eighth note) is equivalent to the dotted crotchet (dotted quarter note) that follows. Note that the tie is always written so as to join the note-heads of two notes. The beginning and end of the tie are on the same horizontal level, and the tie is placed between the note-heads (without touching them). Rests are never tied. Origin of Music Notation :: by John Howell, Virginia Tech Department of Music Blacksburg, Virginia, U.S.A. 24061-0240 Vox (540) 231-8411 Fax (540) 231-5034 mail to John Howell John Howell's Web Page When Guido d'Arezzo invented chant notation on a staff in about 1025, the notes were drawn fully black. I imagine that the scribes cut their pens so as to give a broad stroke in one direction and a very narrow stroke in the other, because that's exactly what the manuscripts look like. (Twentieth century music pens - both nibs and fountain pens - were designed exactly the same way.) When late 12th-early 13th century musicians at Notre Dame in Paris needed away to indicate rhythmic values, they came up with the six rhythmic modes. They used existing chant notation (all black notes) with only two note values, short and long (or breve and longa), but in practice also needed a note value equal to a breve plus a longa. The rhythms were indicated not by the appearance of the notes, but by the way they were joined together in multi-note ligatures. When Franco of Cologne, about 50 years later, proposed a system of mensural notation ("measured" notation), his biggest innovation was to assign specific values to specific note shapes. (The notes were still black.) The longa was a square black note with a descending tail on the right. The breve was a square black note with no tail. He added a new note value, the semibreve, which was a square note turned 45 degrees (which is to say a diamond-shaped note). He also added a new longer note, the double longa, which was a rectangular note about twice the width of a longa with a descending tail on the right. Since notation always seems to lag behind what musicians are actually doing and wanting to write down, what happened in the next 150 years was that shorter and shorter note values were introduced to facilitate notating fast passages. As a result the original breve became slower and slower in practice, and the original longa slower still. The notes were still black (except when they were red, but that's another discussion entirely!). By the early 16th century the longa is seldom seen except as the final note of a piece, which is better notated in modern notation as a fermata rather than a note 16 bars long! During the 15th century a major technological innovation came into use. Paper was invented. Or if not invented, it became available at reasonable prices. Instead of the painstaking and labor-intensive business of preparing animal skin (vellum) for manuscript use, paper could be made relatively inexpensively or even bought ready made. Only one problem for the music scribes. Since paper is made of fibers, apparently the ink had a tendency to spread out along those fibers and look like a blob instead of the nice, clean square note shape people were used to. That's when scribes started using outline notes (white notation) instead of solid black notes. Throughout the 16th century, when the first methods for printing music were being worked out, the same white notation using the same square note shapes continued in use in the printed music. As a broad generalization I would guess that the rounded note shapes we are used to did not come into use until the 17th century, when engraving music on copper plates began to replace printing from movable type. Additional Notes on The History of Music Notation by Dr. Brian Blood About the invention of the alphabet, Socrates made the following observation: "This discovery will create forgetfulness in the learner’s souls, because they will not use their memories; they will trust to the external written characters and not remember of themselves. The specific which you have discovered is an aid not to memory but to recollection, and gives only a semblance of truth; they will hear much and learn nothing; they will appear to know much and will generally know nothing; they will be tiresome company, for they will seem wise without being wise." Of course, the same comment applies equally to the notation of music, that traditions become immutable, that the memories of performers weaken, that improvisation is discouraged. However, at the time when our modern notational system was developing, there were pressing practical reasons for wanting to notate certain aspects of a musical lines accurately. These included the systemisation of the performance of religious music in a centrally organised Roman Catholic church, and, later, a desire to fix an intrinsically plastic medium, the better to apply to it the reflections of philosophers and critics. As Richard Restall writes, in his The Notation of Western Music (2nd ed. 1998, Leeds University Press), "the story of musical notation in Western Europe is one of innovations, changes and disappearances." Reference: Isorhythmic Motets and the Art of Memory by Anna Maria Busse Berger It appears that the Egyptians from the 3rd millennium BC practiced some kind of musical notation and that various systems were in use in the Orient in ancient times. A Hittite (or Hurrian) love-song, from about 1800 BC is the earliest example of musical notation we have. In the ancient Greek, Oriental and Jewish traditions, ekphonetic notation, a system of grammatical accents indicating inflections in language or liturgical texts, was in use as early as c. 200 BC. The invention of the Greek system of prosodic signs, from which both ekphonetic and neumatic notation are derived, is generally attributed to the grammarian Aristophanes of Byzantium (257-180 BC). Although many fragments have survived from this period, the Greek Seikilos epitaph is the oldest surviving example of a complete musical composition, including musical notation, from anywhere in the Western world. Although Greek instrumental and vocal music was notated using two different systems of letters, by the time Boethius (c. 470-525) was writing his five books on music theory, the Romans were using a single system based on the first 15 letters of the alphabet. Neumes, themselves derived from Greek ekphonetic notation, were introduced into Gregorian chant probably as early as the 6th century. Although the scholar and music theorist Isidore of Seville, writing in the early 7th century, famously remarked that it was impossible to notate music, fragmentary evidence indicates that staffless neumes were certainly in use by the 8th century where they appear to have been used primarily to jog the memories of singers who would have learned the chants by ear. In their earliest form, neumes were purely adiastematic, that is, they only told the singer whether the pitch went up or down Louis W. G. Barton, in an essay entitled The Neume Notation Project explains that neumes are 'symbolic characters'. Each character is a literal (not pictorial) representation of a note or group of notes, in an analogous way to letters of the alphabet standing for sounds of speech. There is nothing inherent in the shape of the letter 'A' that would lead one to think that it stands for the sound 'ah', and so 'A' is symbolic, not iconic. For this reason the exact form that neumes could take varied from region to region. Reference: Neumatic Notation part of Medieval Liturgy in the Low Countries Black letter styles were a feature of most late Mediaeval hand-scribed manuscripts. Thick nibs scratching away on parchment could not produce light airy forms; rather, and to avoid raising the pen too often from the page, the hand was crowded, filled with numerous ligatures, often close to illegible and very black. This sense of overcrowding persisted even after a further refinement, heightened neumes arranged above and below a line, which appeared sometime towards the end of the 10th century, made the intervals of the melody somewhat clearer. Neumatic Notation through History This notation was not standardised. There would have been considerable local variation Neume Names 9th to 10th Centuries 11th to 13th Centuries Modern Notation Neumes Virga (Latin, literally 'rod') (single note) Tractulus (Latin, literally 'little stroke') Punctum (Latin, literally 'point, dot') (single note) Podatus (Latin, literally 'foot') Pes (groups of two notes) Clivis (Latin, literally 'bend') (group of two notes) Scandicus (Latin, literally 'climber') (group of three notes) Climacus (Latin, literally 'ladder') (group of three notes) Torculus (Latin, literally 'twisted') Pes flexus (group of three notes) Porrectus (Latin, literally 'extended') Flexus resupinus (group of three notes) Compound neumes Scandicus flexus (group of four notes) Porrectus flexus (group of four notes) Torculus respinus (group of four notes) Pes subbipunctis Podatus subbipunctis (group of four notes) Pes, Podatus (from the Latin, pes, foot) - Torculus (from the Latin, torquere, to twist, from its broken form) - Porrectus (from the Latin porrigere, to extend, from the extended form of its lines) - Climacus (from the Latin climax, stair) - Scandicus (from the Latin, scandere, to rise) - Salicus (from the Latin, salire, to jump) - Flexus (Latin, 'bent') - Resupinus (Latin, 'bent back') Ashenafi Kebede, in Music in Black Jewish and Christian Communities, describes the 'dual' system introduced in the 16th century that featured in Ethiopic chant notation. It employed both milikitoch (neumatic signs: curves, dots, dashes, etc.) and siraye (letter notation taken from the Ethiopian Ge'ez alphabet). To keep the two systems apart, the siraye was often placed above, while the milikit was placed below the text of the manuscripts. The siraye consisted of small letters usually written in red ink; the milikitoch were easily recognizable by their distinct shapes, and they were often written in black ink. Although neither the signs nor the letters indicated individual pitches, they clearly reminded the knowledgeable cantor of melodic passages, each with its own relative pitches, each passage being both melodically and textually meaningful. The duration of each pitch, silence, and volume was also measured according to their importance within each passage and the entire chant as a whole. We would recommend the reference below for further information about the development of neumatic notation. References: Gregorian Chant - a survey of neumatic notation Gregorian Chant - examples showing the development of the notation In modern printing of medieval chant usually four line staffs are employed. In medieval manuscripts, however, there might be no staff line, or anything from one to six lines per staff, where each line signified a different voice. Odo, abbot of the great abbey of Cluny from 927 to 942, actively fostered choral music both in his own abbey, where a hundred or more psalms were being sung daily, and in his travels to other monasteries, inspecting their music and instructing their choirs. From documents of the period, we know both of his teaching methods and of the importance attached to music at Cluny. His great innovation was to arrange the notes of the scale into an orderly progression from A to G, in this way developing the earliest effective system of Western musical notation. To aid the pitching the notes in the various Gregorian modes he associated them with string lengths on the monochord. Reproducibility replaced unreliable 'learning by rote'. The failure of the neume system to maintain consistency in the performance of religious chant, inspired Guido d'Arezzo, in about 1025, to extend Odo's methods, and perfect his new staff-based system of music notation. Reference: Graduale [ref: Fol. 236. St. Gall, 1512. Wurttembergische Landesbibliothek, Cod. mus. I.] - an example of early 16th-century notation. Sometime during the 10th century, a red line traced horizontally above the text, gave the singer a fixed note (F=fa), and so helped him to approximate the intervals. A second line in yellow (for C=ut) was added later. The staff arose from the addition of further black lines. It should be emphasised that these development varied from place to place and that a remarkable variety of notations coexisted at any one period. These experiments would result in the assignment of a four-lined staff to sacred music and of a five-lined staff to secular music. The yellow and red colours were later replaced by Guido who indicated the pitch of certain lines on the stave using letters. From one assigned pitch, the pitch of the remaining lines and spaces followed. By the time Petrucci's 1501 Harmonice musices odhecaton appeared, the first book of music printed from moveable type, bass and alto clefs identifying the note F and the note C respectively, symbols virtually identical to those found in earlier manuscript, had become standard. Only in the case where the cantus line was preceded by a C clef marking the bottom line of the stave, would we use today a treble clef marking G on the second line from the bottom of the stave. An example of the Petrucci 'style' may be found on page 160 of A History of Western Music by Grout and Palisca (published by W.W. Norton). As we have noted above, Guido also named the degrees of the scale using the initial syllables of the lines of a Latin hymn (ut, re, mi, fa, sol, la). Originally used for teaching sight singing, these or their derivatives are also used in some languages for naming absolute pitches. However, the term 'absolute pitch' when referring to medieval music should be handled with care. Reference: Pitch, Temperament & Timbre While a staff of five lines was adopted in France for vocal music, one of six lines was used in Italy. Instrumental music employed staves of varying numbers of lines until the 16th century when the five-line staff became the standard. Signs for chromatic alteration of tones appear almost from the beginning but did not assumed their present shapes until the end of the 17th century. After a period during which printers sought to replicate features of handwritten manuscript in their choice of print face, a process of standardisation took place and when new ideas in letter form design were adopted in one country, they became commonplace elsewhere. Those interested in the history of design and form in early printing, much of it unrelated to notation of music, may like to read The Alphabet Abecedarium by Richard A. Firmage published in 2000 by Bloomsbury (ISBN 0 7475 4757 2) - an earlier edition appeared in 1993 published by David R. Godine, Boston, MA, USA. The evolution of rhythmic notation took much longer than that for pitch. It would appear that prior to mensural notation the length of notes would be determined by the ancient rules of proportional rhythm applied to the words accompanying the melody (prosodic feet and proportions). By the 10th century, notation was beginning to develop its own rules. We find, in the anonymous Commemoratio brevis, some striking observations on the relative lengths of different notes. "Breves must not be slower than is fitting for Breves; nor may Longs be distorted in erratic haste and be faster than is appropriate for Longs .... All notes which are long must correspond rhythmically with those which are not long through their proper inherent durations ... for the longer values consist of the shorter, and the shorter subsist in the longer, and in such a fashion that one has always twice the duration of the other, neither more nor less ... for without question all music should be strictly measured in the manner of prosody" Even so, it remains a problem when classifying and analysing the music of the troubadour, associated with Southern France between 1100 and 1300, as to whether the lyrics or the melody form the foundation for interpretation and performance. There is a complete absence of any mensural notation. Mensural or 'measured-time' notation would emerge only slowly and the golden era of the troubadours was a time of evolution. Mensural notation became a necessity as polyphony and, in particular, the motet developed. Time could no longer be elastic. A single invariable time-unit became essential. At first, rhythmic modes were represented by certain patternings of neumes; later, in his Ars cantus mensurabilis (c.1280), Franco of Cologne clearly indicated, for each note, its exact rhythmic length and represented notes of long and short duration by particular specific neumes. In his system, the longer value was in principle equal to three of the shorter values. In the fourteenth century, Philippe de Vitry, author of Ars nova, expanded Franco's 3:1 system to allow duple divisions of the long and short notes, i.e. a 2:1 system. At the various rhythmic levels of a given piece, either relationship was implied, and a system of signs and coloured notes was used to identify which relationship was currently being used or being temporarily set aside. Mensural notation had become, then, a system of rhythmic notation where distinct note-shapes (from about 1260, maxima, longa, brevis and semibrevis - from the fourteenth-century, minima - from the fifteenth-century, semiminima, fusa and semifusa ) indicated the relative lengths of different notes. In mensural notation (or mensuration), the ratio between the semibreve and the long, called the modus, could be 2 (or duple), in which case it was described as being minor, or 3 (or triple), in which case the ratio was said to be major. In a similar way, the ratio of semibreve and breve was called the tempus which when 2, was said to be imperfect and when 3, was said to be perfect. The third ratio, the prolatio, that between the semibreve and the minim, like the modus could be 2, called minor, or 3, called major. Groups of notes could be 'bound' together to form ligatures. This notational system was distinct both from that of rhythmic modes in which the context of the musical line limited any rhythmic flexibity, and unmeasured music which was entirely free from any set rhythmic values. The mensural notation system remained in use until about 1600. Pre-Mensural Notation Symbols Ekphonetic Notation   Accentus Acutus Accentus Gravis   Neumatic Notation 6th to 13th centuries   Virga Virga Jacens (Latin, literally 'thrown rod')     Punctum Mensural Notation Symbols Mensural Notation   Maxima (Duplex Long) Longa (Long) Brevis (Breve) Semibrevis (Semibreve) Minima (Minim) Semiminima Fusa Semifusa 13th century   14th century   15th to 17th centuries Modern Notation from 17th century   Breve (Double Whole-Note) Semibreve (Whole Note) Minim (Half Note) Crotchet (Quarter Note) Quaver (Eighth Note) Semiquaver (Sixteenth Note) By the 15th century, numbers appeared with the appearance of fractions, which in time developed into our time signatures, to mark when one proportionality of rhythmic values was temporarily being substituted for by another. Bar lines, expression signs, and, by the 17th century, the Italian language became the standard language for tempo (e.g. allegro, andante, largo) and, from Giovanni Gabrieli's Sacrae Symphoniae (1597), dynamic (i.e. piano, forte, pp, ff) markings, particularly when used in their abbreviated form. German, French and English composers sometimes indicated, at the top of the page, the mood or speed of a piece in their own language while employing titles drawn from a wide range of languages. With the adoption of equal temperament and the major and minor modes, key signatures indicating a major key or its relative minor became conventional and assumed their present form during the baroque period. Even so there could be the odd example where composers modified standard notation to meet specific needs. This is a portion of the first movement of Partita VII from Biber's collection of scordatura trios, Harmonia artificiosa-ariosa diversimode accordata, a work in 7 movements for two violas d'amore and basso continuo, originally published in 1696; this is a facsimile from the 1712 reprint. The tuning given is for 6 strings. The 9 line stave is Biber's way of extending the fingerings of the ordinary 4 string viola to the viola d'amore. There is one other example of this 9 line stave notation surviving. To those interested in hearing this Partitia by Biber we recommend the superb recording of the complete Harmonia Artificioso Ariosa by The Rare Fruits Council on Auvidis France E 8572. The Auvidis label is now part of Naive Classique. Naive Classique 148 rue du Faubourg Poissonnière 75010 Paris France Tel: +33 (0)1 44 91 64 00 Fax: +33 (0)1 44 91 64 02 Email: info@naive.fr The advent of aleatory music has produced new notational systems, varying from piece to piece, indicating only approximate pitch, duration, and dynamic relations. Notation for electronic music is still not standardized but generally uses traditional reference symbols (staff and clef signs) in conjunction with specially adapted pitch and rhythm notation. Proportionate (proportional) Notation, a graphic method of indicating durations, where instead of traditional notation, the horizontal spacing of symbols represents the intended length of durations has been introduced to handle pulseless music and music in which different and often complicated rhythms progress simultaneously at different speeds. Sagittal is a microtonal notation system by George D. Secor and David C. Keenan. George Secor began development of the Sagittal (pronounced “SAJ-i-tl”) notation system in August 2001. In January 2002 he presented what he had developed to the Yahoo group tuning and offered to consider suggestions for improvements. At that stage the system could notate the equal temperaments with 17, 19, 22, 29, 31, 41, and 72 divisions per octave. Little did he know that he had provided a unifying symbolic principle which would ultimately be developed into a system capable of notating almost any conceivable microtonal tuning. References: A downloadable pdf format file entitled The Written Notation of Medieval Music by Nigel Horne The History of Mechanical Musical Instruments and Musical Notation Sagittal - A Microtonal Notation System