# Aristoxenus

**Aristoxenus** of Taranto ( ancient Greek Ἀριστόξενος *Aristóxenos* ; * around 360 BC ; † around 300 BC ), son of Socrates' student Spintharos , was a Greek philosopher and music theorist . He was first a pupil of the Pythagorean Xenophilus of Chalkidike and the music teacher Lampros , later in Athens a pupil of Aristotle and a member of the Peripatetic school. He is the oldest ancient writer on music , of whom extensive writings have survived.

Aristoxenos defined the following terms on a purely musical basis: interval , tone system , tone , semitone , third tone , quarter tone , ..., diatonic , chromatic and enharmonic tone system , duration , rhythm . He shaped essential parts of the later music terminology in late antiquity and in the Middle Ages. These terms have survived to this day, but in some cases with a modified meaning.

## Music theory

### Empirical-mathematical method

Aristoxenus was a strict empiricist and built music theory strictly on perception with the ear and is therefore considered the leading harmonist . He formulated a sharp antithesis against all of his predecessors. In particular, he rejected the acoustic music theory of the Pythagoras school , which defined intervals by means of numerical proportions, as a stray into a foreign field, criticized their inaccurate flute and string experiments as well as the uncheckable hypotheses of Archytas . Even so, as a former Pythagorean he remained a strict mathematician ; he even increased the deductive principles in the field of music theory: he ridiculed the missing or vague definitions of his predecessors as an "oracle" and demanded "meticulous" definitions, axioms and proofs in the field of music. He implemented this complete music-mathematical concept in its *harmonic elements* and its *rhythmic elements* without a model . Both writings are not completely preserved; The definition of terms is largely complete, as are the axioms in terms of harmony, while the proof with lengthy verbal meticulous deductions breaks off at some point. Mathematically, he used the theory of magnitudes of Eudoxus by Knidos , which are handed down in the similarly titled *elements of * Euclid . His music theory is therefore an early example of applied mathematics from classical antiquity shortly before Euclid.

### Harmony

The oldest precise definition of **intervals** comes from Aristoxenus: He defined an **interval** (διάστημα) in the sense of a closed interval of a set of tones linearly ordered by "higher" and "lower" (φθόγγος). He defined a tone system (σύστημα) as a composite interval according to the usual composition [A, B] [B, C] = [A, C] in ancient geometry for routes. Each interval has a size ; If one writes the size of the interval [A, B] as for segments with AB, then for Aristoxenus the size rule AB + BC = AC of the Pythagorean Philolaus applies . Aristoxenos also calculated incommensurable quantities with irrational proportions. He used the **tone** (τόνος, whole tone ) as the unit of interval sizes . He divided it into any number of equal parts and formed the semitone, third tone, quarter tone, ... Through a traditional, comprehensible listening experiment, he ensured that the intervals generated by the consonances (octave, fifth, fourth) are multiples of the semitone. Since he did not recognize the traditional acoustic interval proportions, the following equations apply to him (as on the modern piano) regardless of the acoustic mood:

Octave = fifth + fourth | (Definition) |

Tone = fifth-fourth | (Definition) |

4th = 2½ tone | (Experiment) |

5th = 3½ tone | (derivable) |

Octave = 6tone | (derivable) |

To classify the intervals, Aristoxenus applied Plato's method of Dihairesis mathematically and defined five distinctions (in modern terms, the negated equivalence relations ), including the distinction between the **form** (σχημα) of the intervals based on the sequence (τάξις) of the size of their unassembled intervals; for two-tone intervals [A _{1} , B _{1} ],…, [A _{n} , B _{n} ] this corresponds to the following equation:

- Form of [A
_{1}, B_{1}]… [A_{n}, B_{n}] = A_{1}B_{1}… A_{n}B_{n}

He defined four-tone **tetrachords of** the form A B C with A + B + C = fourths as special intervals and classified them according to special forms into the tone sexes “diatonic”, “chromatic” and “enharmonic”. He started from an infinite number of forms in each tone gender and explicitly named the following six forms:

Enharmonion: | Quarter tone Quarter tone 2 tone |

Chroma malacon: | Third tone Third tone 11/6 tone |

Chroma hemiolon: | 3/8 tone 3/8 tone 7/4 tone |

Chroma toniaion: | Semitone semitone 3/2 tone |

Diatonon malacon: | Halftone 3/4 tone 5/4 tone |

Diatonon syntonon: | Halftone tone tone |

On this conceptual basis, Aristoxenus was the only music theorist in history to devise an axiomatic sound system theory in which he characterized melodic systems using axioms and then derived their form with evidence. First he derived the forms of the consonances and showed that there are cyclic permutations of a form for tetrachords of the form A B C:

Forms of the fourth | Forms of the fifth | Shapes of the octave |
---|---|---|

ABC | ABC tone | ABCABC tone |

BCA | BC clay A | BCABC clay A |

CAB | C tone AB | CABC tone AB |

Tone ABC | ABC tone ABC | |

BC clay ABCA | ||

C tone ABCAB | ||

Tone ABCABC |

As a **complete system** (σύστημα τέλειον) he then defined the smallest melodic system that contains all these consonance forms. Its derivation is missing in the original source; based on later sources, he calculated systems of the following form:

- Tone ABCABC Tone ABCABC for tetrachords with interval sequence ABC

A connection to today's music has only its complete system, which is based on the diatonon syntonon and has the form of today's minor scale over two octaves:

- Tone ½ tone tone tone ½ tone tone tone tone ½ tone tone tone ½ tone tone tone

### rhythm

Aristoxenus built his rhythms largely analogous to harmonics. Here he used **duration** (χρόνος) as a type of magnitude , also incommensurable durations with irrational relationships. As an analogy to the prime numbers (πρωτος αριθμος) he defined the prime duration (πρωτος χρόνος) as a perceptible duration that cannot be broken down into several perceptible durations. He determined perceptibility through feasibility when speaking, singing or moving the body (the smallest prime duration can therefore be determined individually by experiment). As he emphasized, this results in an infinite number of prime durations, which he used as duration units; In addition to the smallest prime duration p, this includes all durations between p and 2p.

He defined a **rhythm** as a continuous sequence (χρόνων τάξις) parallel to the interval form of the harmony. He considered the composition of the rhythms on the three rhythmic levels “speaking”, “singing” and “body movement” with syllables, tones and figures as rhythmic elements. He set these rhythmic elements in durations (quasi as ordered pairs) and discussed a complex three-layer rhythm formation. This is the first attempt to define polyrhythmic structures. The tetrachords correspond to two- to four-limbed feet in rhythm, which have nothing to do with verse feet of the metric , but are explained on the level of body movement (think of dance steps). He classified the feet with seven differentiations, including a distinction between the rhythmic genders “ dactylic ”, “ iambic ” and “ paionic ”. The fragmentary evidence of his rhythm with deductions of the feet possible in rhythm formation can no longer be reconstructed without hypotheses, because certain axioms are lost here.

## Influence on later music theorists

All of the later ancient music theorists in the area of harmony adopted the musical terminology from Aristoxenus. This applies not only to his followers, the so-called Aristoxeneers (see below), but also to his opponents among the younger Pythagoreans. During the lifetime of Aristoxenus, one of them was the mathematician Euclid , who in his musical work *Teilung des Canon offered* a Pythagorean-modified version of the diatonic Aristoxenian tone system, but at the same time proved a series of sentences against the harmony of Aristoxenus, including the negation of the divisibility of the tone, of the experimental axiom fourth = 2½ tone and the equation octave = 6 tone. These sentences only show the mathematical incompatibility of the Pythagorean-acoustic position with the empirical-musical position in relation to the level of development of ancient mathematics. Mathematically, however, because of Euclid's mathematical authority, the Pythagorean theory largely prevailed; Aristoxenus only remained authoritative in terms of terminology. Representatives of this Aristoxean-Pythagorean compromise line were also Eratosthenes and especially Ptolemy . Because of Aristoxen's criticism of the inaccurate string experiments (string = gut!), Ptolemy suggested improvements to the canon or monochord and criticized Aristoxen's experiment as imprecise, which is acoustically and experimentally sound. He continued to work through Boethius , who passed on the dispute between the Pythagoras and Aristoxenos schools in the Latin-speaking area and had a major influence on the medieval and contemporary theory of sound systems. In medieval music, the aristoxenic forms of the octave, which are also known as octave genres, got a practical meaning for the church modes .

In the field of rhythm, Aristoxenus had only a slight formative effect on the later theory. Here the linguistic metric remained dominant, in whose terminology Dionysius Thrax only included the term “foot” and its two- to four-part structure and reinterpreted it in terms of the rhythm of the language.

## Aristoxene

**Aristoxeneer** is the name of those music theorists who orientated themselves on the teaching of Aristoxenus and stayed away from the Pythagorean direction. They include Kleoneides (= Pseudo-Euclid), Aristeides Quintilianus , Bakcheios Geron , Psellos and some anonymous authors of musical treatises , some of which are wrongly ascribed to Aristoxenus (Pap. Oxy. 9). The Aristoxeneer were nothing but epigones who did not reach the level of their model by far and who diluted his teaching very much. They removed all mathematics from his teaching, that is, all axioms and proofs and many definitions, as well as the whole experimental, perceptual foundation. Often the teaching of the Aristoxenians is not clearly separated from the teaching of Aristoxenus. In general, the reception of Aristoxenos was already heavily influenced by misunderstandings in antiquity. His antipythagoreanism was often misinterpreted as an inability to calculate and his empirical attitude as anti-mathematical. These misunderstandings persist to this day. An example of this is Johann Mattheson , who wrote pamphlets against any kind of musical mathematics under the pseudonym "Aristoxenus the Younger". Since the Aristoxenian renaissance by Vincenzo Galilei , the twelve-level tempered tone system has often been misinterpreted as being Aristoxean, because the size equations of Aristoxenus apply in it; however, a temperature always presupposes the Pythagorean system with the perfect fifth 3: 2 as the archetype, which Aristoxenus did not have.

## Text output (partly with translations)

- Rosetta Da Rios (Ed.):
*Aristoxeni elementa harmonica.*Publicae officinae polygraphicae, Rome 1954. - Giovanni Battista Pighi (Ed.):
*Aristoxeni rhythmica.*Patron, Bologna 1959 (Greek text with Italian translation). - Lionel Pearson (Ed.):
*Aristoxenus, Elementa rhythmica. The Fragment of Book II and the Additional Evidence for Aristoxenean Rhythmic Theory.*Texts edited with Introduction, Translation, and Commentary. Clarendon Press, Oxford 1990, ISBN 0-19-814051-7 (Greek text and English translation). - Stefan Ikarus Kaiser (ed.):
*The fragments of Aristoxenus from Tarent*(=*Spudasmata.*Vol. 128). Newly published and supplemented, explained and translated. Olms, Hildesheim u. a. 2010, ISBN 978-3-487-14298-2 (Greek, Latin and German)

## Translations

- Heinrich Feußner (Ed.):
*Aristoxenos. Basics of rhythm. A fragment in the corrected original with a German translation and explanations, as well as with the preface and Morelli's remarks, newly published*Edler, Hanau 1840. -
Paul Marquard (Ed.):
*Αριστοξενου Aρμονικων τα Σωζομενα. The Harmonic Fragments of Aristoxenus.*Greek and German with critical and exegetical commentaries and an appendix, containing the rhythmic fragments of Aristoxenes. Weidmann, Berlin 1868, ebookdb.org . - Paolo Segato:
*Gli elementi ritmici di Aristosseno.*Panfilo Castaldi, Feltre 1897 (Italian translation). -
Rudolf Westphal :
*Aristoxenus of Taranto. Melik and rhythm of classical Hellenism.*2 volumes. Abel, Leipzig 1883–1893 (reprographic reprint. Olms, Hildesheim 1965). - Stefan Ikarus Kaiser:
*The "harmonious elements" of Aristoxenus of Taranto. Interpretation and translation.*Salzburg 2000 (Salzburg, university, diploma thesis, 2000; German translation). - Stefan Ikarus Kaiser:
*The fragment from the second book of the "Rhythmic Elements" by Aristoxenos from Taranto. New German translation.*In:*Querstand. Contributions to art and culture.*Vol. 4, 2009, ZDB -ID 2222733-7 , pp. 133-135.

## literature

**Overview representations**

- Bruno Centrone:
*Aristoxène de Tarente.*In: Richard Goulet (ed.):*Dictionnaire des philosophes antiques.*Volume 1:*Abam (m) on à Axiothéa.*Center National de la Recherche Scientifique, Paris 1989, ISBN 2-222-04042-6 , pp. 590-593 -
Fritz Wehrli , Georg Wöhrle , Leonid Zhmud :
*The Peripatos until the beginning of the Roman Empire.*In: Hellmut Flashar (ed.):*Outline of the history of philosophy . The philosophy of antiquity.*Volume 3:*Older Academy - Aristotle - Peripatos.*2nd edition, Schwabe, Basel 2004, ISBN 3-7965-1998-9 , pp. 493–666, here: 576–582

**Investigations**

- Oliver Busch:
*Logos syntheseos. The Euclidean Sectio canonis, Aristoxenos, and the role of mathematics in ancient music theory*(=*publications of the State Institute for Music Research.*Vol. 10 =*studies on the history of music theory.*3). Olms, Hildesheim u. a. 2004, ISBN 3-487-11545-X - Wilfried Neumaier:
*What is a sound system? A historical-systematic theory of the occidental tonal systems, based on the ancient theorists, Aristoxenus, Eucleides and Ptolemaios, presented with the means of modern algebra*(=*sources and studies on the history of music from antiquity to the present.*Vol. 9). Lang, Frankfurt am Main a. a. 1986, ISBN 3-8204-9492-8 (Tübingen, University, dissertation, 1985). - Wilfried Neumaier:
*Ancient rhythm theories. Historical form and current substance*(=*heuremata. Studies on the literature, languages and culture of antiquity.*Vol. 11). Grüner, Amsterdam 1989, ISBN 90-6032-064-6 ( books.google.de ).

## Web links

**Wikisource: Aristoxenos**- Sources and full texts

- Literature by and about Aristoxenus in the catalog of the German National Library

personal data | |
---|---|

SURNAME | Aristoxenus |

ALTERNATIVE NAMES | Aristoxenus |

BRIEF DESCRIPTION | Greek philosopher and music theorist |

DATE OF BIRTH | around 360 BC Chr. |

PLACE OF BIRTH | Tarent |

DATE OF DEATH | around 300 BC Chr. |