Sectio canonis

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Sectio canonis (Latin; in the original Greek Κατατομή κανόνος , German The division of the canon ) is a work of music theory in ancient Greece , which the mathematician Euclid of Alexandria around 300 BC Has written.

Classification and delimitation within the Greek music theory

The concept of consonances ( ancient Greek σύμφωνος = sýmphōnos , sounding together), which is so important in harmony theory , was developed in the long history of music in ancient Greece . The consonances are special intervals that are characterized by the fact that the number of vibrations of the primary tones are in simple numerical ratios ( octave 1: 2, fifth 2: 3, fourth 3: 4 ...) and that they sound particularly pleasant to the human ear. The number of vibrations could not be measured by an ancient musician, but when realizing the consonant intervals on the monochord (also known as canon ), the string lengths were (inversely) proportional to the pitches and were therefore in the same simple numerical relationships. That already happened in the 6th century BC. BC the Pythagoreans were encouraged to develop a mathematically sound theory of harmony. However, they came into conflict with music theorists who followed a hearing psychological approach and rejected acoustic / mathematical thinking as alien to the discipline and speculative. Aristoxenos presents this basic attitude in his Elementa harmonica , which he developed at the end of the 4th century BC. Chr. Composed. About a generation later, Euclid created - probably in contradiction to Asristoxenos - with the Sectio Canonis a concise, precisely formulated and mathematical evidence-based representation of the most important theses of the Pythagoreans.

Structure and content

In the following, the text passages are localized according to the edition by Heinrich Menge . The translation into German largely follows Wilfried Neumaier ( Was ist ein Tonsystem ). In some cases, the translation into English by André Barbera is used.

The text is divided into 4 parts:

  • Brief introduction to general and Pythagorean music theory
  • 9 mathematical auxiliary theorems
  • 9 music theory sentences
  • 2 chapters with the sound system representation

Brief introduction to general and Pythagorean music theory

Euclid attributes sounds to movement, high sounds to fast, deep sounds to slow ones. By raising or lowering the sounds, they merge, so they are made up of parts. But what is made up of parts can be described in terms of numbers. So sounds are described by numbers or ratios. But the ratios are many times that over-piece (lat. Superparticularis ), the multi-part (lat. Superpartiens ). Consonants are specified by the fact that they are a pleasant-sounding mixture of 2 sounds and that their numbers are multiples or parts.

9 mathematical auxiliary theorems

In the following 9 chapters, sentences about intervals (ancient Greek διάςτῃμα = diastema, space, distance) are established and proven. But, even if every chapter begins with this term ( if there is a multiple interval ... , have multiple intervals ... ), it is not defined and was not mentioned in the introduction. It is an undefined term, the meaning of which must be determined from the context. Then an interval can be assigned to every 2 tones (A, B), which are determined by natural numbers , and this is characterized by the ratio of the higher to the lower tone (AB). Behind this quantification of the tones is the idea of ​​the string lengths that result when they are implemented on the monochord. However, this is not addressed at any point. The intervals can be related to Euclid's elements (Book V, Definitions). While sizes and the ratio of sizes are generally dealt with there, the Sectio canonis is limited to multiples (ratio n: 1) and parts (ratio (n + 1): n). In the elements (Book V, Def. 6) the proportion is also defined, which plays an important role in the sectio canonis without being defined: in proportion are sizes that have the same relationship . 2 intervals with a common midrange are added by multiplying their ratios (e.g. fourth 4: 3 + fifth 3: 2 results in 4/3 * 3/2 = 2: 1 octave).

Chapters III and IX are particularly important. Anicius Manlius Severinus Boethius took this over into his work De institutione musica - Chapter III:

In the case of an over-holy interval, neither one nor more middle numbers can be placed proportionally between them (translation by Oscar Paul )

Explanation: For each of the smallest natural numbers (A, B) in an over-dividing ratio (n + 1): n (like 3: 2, 4: 3 :, ...) there is no natural number C between A and B could be set. A: C = C: B = (n + 1): n is therefore not possible. For a larger interval in the same ratio (such as 6: 4, 12: 8, ...) there are natural numbers D between A and B, but these are not proportional - so A: D = D: B - because the smaller interval has no such proportional number either. This proof, however, is left to the elements (VIII, 8). But since the equation A: D = D: B can easily be transformed into , or , it can be seen that there are definitely such numbers, but they do not belong to the natural numbers. Boethius ascribes these ideas, which lead out of the space of natural numbers to irrational numbers, to the music theorist Archytas of Taranto . The connection between his surviving writings and this work is disputed.

Chapter IX is reproduced by Boethius as follows:

Six sesquioctave proportions are greater than double the interval

Explanation: 6 intervals (AB, BC, CD, DE, EF, FG), each in the divisional ratio 9: 8, are larger than the multiple interval AF '(F' between F and G) in the ratio 2: 1 because the total size of the intervals is calculated by multiplication , and that is greater than 2. Euclid calculates this step by step, starting from A = 262144 (= ) and B = 294912 (= A * 9/8).

9 music theory sentences

This is followed by 9 movements with specific musical statements. The note names are used (mése, néte, proslambanomenos ...), as well as the musical interval names ( diapason (octave), diapente (fifth), diatessaron (fourth), ..). For the evidence, reference is made to the preceding auxiliary sentences.

In Chapter XIII the interval τόνων (tone, whole tone ) appears for the first time in writing , whereas previously only tone and interval were used. In Chapter VIII the sesquioctav (9: 8) was introduced as the difference between the intervals fifth (3: 2) and fourth (4: 3). This is now transferred to music intervals in Chapter XIII and the tone is defined as the difference between Diapente and Diatessaron. In Chapter XVI it is stated that, according to Chapter III, the tone with its divisional ratio (9: 8) can not be placed in between a middle number proportionally, i.e. the tone cannot be divided into two equal parts. This clearly reproduces the Pythagorean statement: a musical representation is not possible if it cannot be grasped with the available mathematical definitions. So Euclid stands in contradiction to the acting musician, who generates any tones on his monochord and judges them with his trained ear, and in particular to the music theory of Aristoxenus, which halves the tone, divides it into thirds, etc. With the same argumentation, the division of the pyknon is made in Chapter XVIII as made by Aristoxenus, discarded.

In Chapter XIV the subordinate clause IX is transferred to musical intervals and postulates that the octave is smaller than 6 tones (in a ratio of 9: 8). Building on this, and building on the preceding sentences on fourths and fifths, it is shown in Chapter XV that the fourth is less than 2½ and the fifth less than 3½ tones; also in contrast to Aristoxenus. Euclid's statement is completely correct from the standpoint of a natural, acoustically- based tuning of the sounds, while Aristoxenus bases it on a balanced tempered tuning .

Sound system representation

Chapters XIX and XX contain the representation of a tone system, a division of the canon . In a nutshell, the string of the canon is halved, divided three or four times, thus creating the consonant sound of the octave, fifth or fourth. We do not learn anything about the underlying structure, tetrachords or clays , they are assumed to be known. If you follow the somewhat clumsy explanations (Chapter XIX: ... if you divide CB in point Z into two equal parts, then CB is twice as large as ZB, and CB and ZB are in the ratio of the octave ... ) and order the mentioned Tones and intervals according to size, this results in a tone system with the structure: fourth, fourth, diazeuxis (separating intermediate tone), fourth, fourth, bombyx (lowest tone). The fourth has the structure: tone, tone, remaining interval ( Limma ). In Chapter XV it was shown that the remaining interval is less than a semitone, but Euclid does not go into its calculation.

Tradition and survival

In Porphyry's commentary on Ptolemy's theory of harmony, there are numerous parallels to the work of Euclid. Anicius Manlius Severinus Boethius took in his work De institutione musica (Book IV, 1–2) a translation of the introductory text and the first IX chapters into Latin . However, he does not name his source.

In 1895 Carl von Jan edited the text. He used several manuscripts for this, but also the commentary by Porphyry. On the basis of this edition, Heinrich Quantity translated into Latin in 1916 and Andrew Barker translated into English in 1989.

Text editions and translations

  • Andrew Barker: The Euclidean Sectio Canonis in Greek Musical Writings Volume II, Cambridge 1989.
  • Carl von Jan: Musici scriptores graeci , Leipzig 1895.
  • Heinrich Quantity: Sectio canonis in EUCLIDIS Phaenomena et Scripta Musica , Leipzig 1916.

literature

  • André Barbera: The Euclidean Division of the Canon. Greek and Latin Sources , Lincoln / London 1991.
  • Annemarie Jeanette Neubecker : Ancient Greek Music , Darmstadt 1977.
  • Daniel Heller-Roazen: The fifth hammer , Frankfurt am Main 2014.
  • Wilfried Neumaier: What is a sound system? , Frankfurt am Main 1986.

Individual evidence

  1. ^ Heinrich Husmann : Vom Wesen der Konsonanz , Heidelberg 1953
  2. ^ Heinrich Husmann: Fundamentals of ancient and oriental music culture , Berlin 1961, p. 9
  3. ↑ Annemarie Jeanette Neubecker: Ancient Greek Music , p. 16f
  4. ↑ to the end of the chapter: Wilfried Neumaier: What is a sound system? , P. 171f
  5. ↑ Wilfried Neumaier: What is a sound system? , P. 119
  6. ↑ Wilfried Neumaier: What is a sound system? , P. 120
  7. ↑ Annemarie Jeanette Neubecker: Ancient Greek Music , p. 115
  8. ↑ Wilfried Neumaier: What is a sound system? , P. 121
  9. ↑ Wilfried Neumaier: What is a sound system? , P. 121
  10. ^ Andrew Barker: The Euclidean Sectio Canonis , p. 195, notes 12, 13
  11. ^ WR Knorr: The evolution of the Euclidean Elements , VII, I The Theorem of Archytas and the epimoric ratios
  12. ↑ Daniel Heller-Roazen: The fifth hammer , third chapter
  13. ↑ Andrew Barker: The Euclidean Sectio Canonis , p. 202, note 51
  14. ↑ Rudolf Westphal : The music of ancient Greece. Euclid , pp. 244f
  15. ↑ to the end of the chapter: Wilfried Neumaier: What is a sound system? , Pp. 133-137
  16. ^ André Barbera: The Euclidean Division of the Canon. Greek and Latin Sources , pp. 46ff, pp. 80-94
  17. ^ André Barbera: The Euclidean Division of the Canon. Greek and Latin Sources , p. 38
  18. ^ Heinrich Quantity: Sectio canonis , Praefatio VII-IX