μονόχορδον, όπως φαίνεται από το όνομά του, ήταν ένα όργανο με μια χορδή. Μερικοί μελετητές το τοποθετούν στην οικογένεια του λαούτου, δηλ. με βραχίονα, χέρι (Th. Reinach La mus. gr. 127). To μονόχορδο χρησιμοποιήθηκε για τον καθορισμό των μαθηματικών σχέσεων των μουσικών ήχων, δηλ. ήταν ένας κανών , και συνήθως ονομαζόταν "ο Πυθαγόρειος κανών", γιατί απέδιδαν την εφεύρεσή του στον Πυθαγόρα . Ο Πολυδεύκης (IV, 60) λέει ότι το μονόχορδο ήταν αραβική εφεύρεση· εφευρέτης του οργάνου, στη μυθολογία, φέρεται ο Απόλλων, ο οποίος μάλιστα το πρόσφερε υπό μορφή αψίδας στην Αρτέμιδα (Censorinus). Ο Νικόμαχος , πάλι, γράφει πως το μονόχορδο ονομάζεται από πολλούς ("φάνδουρος ", ενώ οι Πυθαγορικοί το λένε "κανόνα" (πρβ. λ. πανδούρα). Μεγάλοι μαθηματικοί, όπως ο Αρχύτας , ο Ερατοσθένης και ο Δίδυμος , εργάστηκαν τους λόγους των διαστημάτων στο μονόχορδο. :
Βλ. τα λ. ελικών και κανών .
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Monochord.
An ancient single-string instrument first mentioned in Greece in the 5th century bce, and said to have been an invention of Pythagoras. The monochord remained a viable musical device, used mainly for teaching, tuning and experimentation, until the advent of more accurate instruments in the late 19th century.
In its earliest form the monochord's single string was stretched across two fixed bridges which were erected on a plank or table. A movable bridge was then placed underneath the string, dividing it into two sections. The marks indicating the position of the fixed bridge were inscribed on the table beneath the string. The resonating box, seen in drawings after the 12th century, was a late medieval addition which increased the portability in addition to enhancing the tone of the monochord. After 1500 one of the end bridges was replaced with a nut, the attendant lowering of the string enabling the user to press it directly on the belly of the instrument. Although simple to use, this modified monochord was considerably less accurate. The name monochord was usually retained for multi-string instruments when the strings were tuned in unison or when the instrument was used for the same purposes as a monochord. The medieval instrument varied from about 90 to 122 cm in length. During the Middle Ages the selection of a monochord's basic pitch was influenced by its size and by the voice range of the user rather than by any existing standards.
1. Acoustical systems.
The divisions of the monochord are usually presented in terms of proportions, string lengths or cents. A fourth method, that of expressing string lengths by means of logarithms, was often used in the 18th century, but this system, like the cents system derived from it, is not proportional and cannot be used on the instrument without further calculation. The first two can be directly applied and are the only kinds of division to have attained any practical significance before the 20th century; this kind of division is designated a manual division.
Aristotle and Euclid followed Pythagoras's lead when discussing intervals; Aristoxenus, however, used a six-interval scale and a non-proportional arithmetic, or fractional, division of the string. Pythagorean techniques of dividing the monochord were introduced to the Middle Ages by the late antique writers Theon of Smyrna, Aristides Quintilianus, Nicomachus of Gerasa, Cleonides, Gaudentius, Ptolemy, Porphyry, Bacchius and the author of the treatise known as Bellermann's Anonymous.
Diagrams showing divisions of the monochordThe Pythagorean concept of division by proportions is based on the relationship of the harmonic and arithmetic means as they are represented by the numbers 6, 8, 9 and 12. The ratio 12:6 produces the octave; 9:6 and 12:8, the 5th; 8:6 and 12:9, the 4th; and 9:8, the major 2nd. Reduced to their lowest terms these ratios are dupla (2:1), sesquialtera (3:2), sesquitertia (4:3) and sesquioctava (9:8). They can be applied to a string in two ways. For example, in fig.1a , one whole tone (D down to C) can be produced by dividing half the string length (AY) into eight parts (DY) and then adding an equal ninth portion (sesquioctava) to form the second pitch (CY). Conversely (fig. 1b ) a subsesquioctava proportion (8:9) can be used if the string length AY is divided into nine parts and the second is sounded with only eight of them (BY).
In fig.1a the monochord is divided in a descending manner, from the higher pitches to the lower. The second division (1b), moving from the lower to the higher pitches, is an ascending division. It is of course possible to use both techniques alternately in one division. The more complex ratio, like that of the Pythagorean semitone (256:243), can be determined by calculation with simple intervals, for example, the sum of two whole tones (9/8 x 9/8 = 81/64) is subtracted from the fourth (4/3 – 81/64 = 256/243) – an extremely simple manoeuvre when done on the instrument.
The completion of either of the above divisions in the manner of the Middle Ages would give a two-octave scale in the Pythagorean tuning whose lowest note would be given by the entire length of the string. In general it may be said that the Greek writers up to 500 CE used the descending division. Medieval scholars began with the descending division and subsequently adopted the ascending division. The technique of the latter, originally attempted by Boethius, was first successfully described by Odo of St Maur (Cluny) in about 1000. Writers of the Renaissance and the post-Renaissance eras preferred the ascending division.
The selection of the technique to be used in working out a specific division was often dependent on its intended usage. Although all medieval divisions achieve the same end and utilize the same four proportions, the method of division selected depended on whether it was for a speculative (descending division) or a practical (ascending) treatise. The popularity of the ascending division parallels the rise of the practical treatise in the late Middle Ages.
2. System of string lengths.
The cumbersome nature of the proportional system together with the difficulty of using a compass to divide the string caused some investigators to adopt the system of string lengths, an accurate and simple method of proportional pitch representation. The only problem with the string lengths lies in the number of units encountered. For example Johann Neidhardt in 1706 specified a string length of 1781·82 units for the second step of his scale which he based on a division into 2000 units. Other advocates, like Marpurg, suggested the use of only three digits to represent the total length of the string; however, this was a compromise rarely admitted by the users of the technique.
3. Division of the chromatic scale.
Semitones can be determined on the monochord by three methods: by extending the superparticular divisions, arithmetically dividing the tone, or by mean-proportional division. In superparticular divisions two complete and different (different even for notes which are enharmonically equivalent) sets of chromatic notes are available. These may be obtained by the successive application of the sesquialtera proportion (beginning with the note F) or of the subsesquialtera proportion (beginning with B). The former will produce a series of perfect 5ths in descending order (called ‘flat semitones’), and the latter a set of ascending perfect 5ths (‘sharp semitones’). Arithmetical semitones are determined by an equal division of the difference between the string lengths of two pitches a step apart. This method was frequently used in post-medieval times even though the semitones are of unequal size. The mean proportional string lengths necessary for single equal semitones are usually determined by means of the Euclidean construction (a perpendicular erected at the juncture of two string lengths which are used as the diameter of a semicircle will equal the proportional length). To determine two or more mean-proportionals, a mechanical device like the mesolabium (a series of overlapping square frames) can be used to substitute for the mathematical function of the cube root; multiple mean-proportionals can also be formed by means of the sort of geometrical figures used by Lemme Rossi in the 17th century.
4. Uses.
In addition to its value as an experimental device, the monochord served throughout the Middle Ages as a teaching instrument. Monochord-based diagrams and sets of directions for determining the consonances abound in both speculative and practical treatises of this era. Until the adoption of sight-singing methods based upon the hexachord system, the monochord was used to produce pitches for rote singing; from then until the 13th century it was used mainly to check correct reproduction of intervals. The decline of its pedagogical use after this time is probably due to the introduction of keyboard instruments. The use of the monochord by teachers in the Renaissance was restricted to those few who rigidly maintained the Pythagorean scale as the basis of their musical instruction.
Because so much of the early use of the monochord was didactic, its users attempted to make the division as efficient and accurate as possible. The efficiency of a monochord division depends on the relation between the number of separate measurements and the number of notes produced. The results of these efforts are particularly noticeable after 1450 because after this date each new division often produced a new variation of a given tuning. Often the musician wished to change the tuning but not infrequently he was only seeking a simpler method of division. It would seem that the appearance of an altered tuning bothered the Renaissance musician little, for because of the monochord's inaccuracy, a variation of a few cents (in some cases as much as 22 cents) was a small sacrifice to make for a more efficient division. A case in point is the division of Ramos de Pareia whose monochord tuning varied widely from the accepted Pythagorean standard. Ramos, however, was apparently not bothered by the pitch deviation as long as he was able to simplify the division. To this end he stated: ‘So therefore we have made all of our division very easy, because the fractions are common and not difficult’. In many cases this desire is not stated expressly, as it was by Ramos, but it may be suspected that it served as an underlying cause of many tuning variations in the Renaissance and later eras.
The other areas in which the influence of the monochord is evident are in its instrumental applications and its use as a symbolic device. In the former instance the use of the monochord in ensembles is cited in both Greek and medieval writings. In later times, however, the descendants of the monochord, the clavichord (sometimes called monochordia by 15th- and 16th-century writers), hurdy-gurdy and trumpet marine, were more frequently used. Throughout the late Middle Ages and the Renaissance the monochord is often mentioned as a basic tool in the design or measurement of bells and organ pipes. Finally, until about 1700 the monochord was commonly used to show the unity existing between man and the universe. It is represented as a divided string whose pitches may represent the solar system (musica mundana), the muses, the zodiac, or even bodily functions; often this is being tuned by the hand of God.
For Jacques de Liège's division of the monochord, see Theory, theorists, [not available online]
Bibliography
S. Wantzloeben : Das Monochord als Instrument und als System entwicklungsgeschichtlich dargestellt (Halle, 1911)
J.M. Barbour : Tuning and Temperament: a Historical Survey (East Lansing, MI, 1951/R, 2/1953)
K.W. Gümpel : ‘Das Tastenmonochord Conrads von Zabern’, AMw, xii (1955), 143–66
C.D. Adkins : The Theory and Practice of the Monochord (diss., U. of Iowa, 1963)
J. Chailley : ‘La monocorde et la théorie musicale’, Organicae voces: Festschrift Joseph Smits van Waesberghe angeboten anlässlich seines 60. Geburtstag, ed. P. Fischer (Amsterdam, 1963), 11–20
C. Adkins : ‘The Technique of the Monochord’, AcM, xxxix (1967), 34–43
M.-E. Duchez : ‘Des neumes à la porteé’, Musicologie médiévale: Paris 1982, 57–60
J. Smith : ‘The Medieval Monochord’, JMR, v (1984), 1–34
C. Meyer, ed.: Mensura monochordi: la division du monocorde (IXe–XVe siècle) (Paris, 1996)
Cecil Adkins
Grove
Επίσης, S. Wantaloeben, Das Monochord, 1911.
http://www.musipedia.gr/