Musical system of ancient Greece

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The musical system of ancient Greece evolved over a period of more than 500 years from simple scales of tetrachords, or divisions of the perfect fourth, into several complex systems encompassing tetrachords and octaves, as well as octave scales divided into seven to thirteen intervals. [1]

Contents

Any discussion of the music of ancient Greece, theoretical, philosophical or aesthetic, is fraught with two problems: there are few examples of written music, and there are many, sometimes fragmentary, theoretical and philosophical accounts. The empirical research of scholars like Richard Crocker, [2] [3] [4] C. André Barbera, [5] [6] and John Chalmers [7] has made it possible to look at the ancient Greek systems as a whole without regard to the tastes of any one ancient theorist. The primary genera they examine are those of Pythagoras and the Pythagorean school, Archytas, Aristoxenos, and Ptolemy (including his versions of the genera of Didymos and Eratosthenes). [8]

Overview of the first complete tone system

As an initial introduction to the principal names and divisions of the Ancient Greek tone system we will give a depiction of the "perfect system" or systema teleion, which was elaborated in its entirety by about the turn of the 5th to 4th century BCE.

The following diagram reproduces information from Chalmers. It shows the common ancient harmoniai, the tonoi in all genera, and the system as a whole in one complete map.

Depiction of the ancient Greek Tone system Systema-teleion-english.png
Depiction of the ancient Greek Tone system

The central three columns of the diagram show, first the modern note-names, then the two systems of symbols used in ancient Greece: the vocalic (favoured by singers) and instrumental (favoured by instrumentalists). The modern note-names are given in the Helmholtz pitch notation, and the Greek note symbols are as given in the work of Egert Pöhlmann  [ de ]. [9] The pitches of the notes in modern notation are conventional, going back to the time of a publication by Johann Friedrich Bellermann  [ de ] in 1840; in practice the pitches would have been somewhat lower. [10]

The section spanned by a blue brace is the range of the central octave. The range is approximately what we today depict as follows:

The central octave of the ancient Greek system Ancient-greek-middle-octave.png
The central octave of the ancient Greek system

Greek theorists conceived of scales as descending from higher pitch to lower (the opposite of modern practice).

The earliest Greek scales were tetrachords, which were series of four descending tones, with the top and bottom tones being a fourth apart in modern terms. The sub-intervals of the tetrachord were unequal, with the largest intervals always at the top, and the smallest at the bottom. The 'characteristic interval' of a tetrachord is its largest one.

The Greater Perfect System (systema teleion meizon) was composed of four stacked tetrachords called (from lowest to highest) the Hypaton, Meson, Diezeugmenon and Hyperbolaion tetrachords. These are shown on the right hand side of the diagram. Octaves were composed from two stacked tetrachords connected by one common tone, the synaphe.

At the position of the paramese, the continuity of the system encounters a boundary (at b-flat, b). To retain the logic of the internal divisions of the tetrachords and avoid the Meson being forced into three whole tone steps (b–a–g–f), an interstitial note, the diazeuxis ('dividing'), was introduced between the paramese and mese. This procedure gives its name to the tetrachord diezeugmenon, which means the 'divided'.

To bridge the inconsistency of the diazeuxis, the system allowed moving the nete one step up, permitting the construction of the Synemmenon ('conjunct') tetrachord – shown at the far left of the diagram.

The use of the Synemmenon tetrachord effected a modulation of the system, hence the name systema metabolon, the modulating system, also called the Lesser Perfect System. This was considered apart, built of three stacked tetrachords — the Hypaton, Meson and Synemmenon. The first two of these are the same as the first two tetrachords of the Greater Perfect System, with a third tetrachord placed above the Meson. When all these are considered together, with the Synemmenon tetrachord placed between the Meson and Diezeugmenon tetrachords, they make up the Immutable (or Unmodulating) System (systema ametabolon). The lowest tone does not belong to the system of tetrachords, as is reflected in its name, the Proslambanomenos, the adjoined.

In sum, it is clear that the Ancient Greeks conceived of a unified system with the tetrachord as the basic structure, but the octave as the principle of unification.

Below we elaborate the mathematics that led to the logic of the system of tetrachords just described.

The Pythagoreans

After the discovery of the fundamental intervals (octave, fourth and fifth), the first systematic divisions of the octave we know of were those of Pythagoras to whom was often attributed the discovery that the frequency of a vibrating string is inversely proportional to its length. Pythagoras construed the intervals arithmetically, allowing for 1:1 = Unison, 2:1 = Octave, 3:2 = Fifth, 4:3 = Fourth. Pythagoras's scale consists of a stack of perfect fifths, the ratio 3:2 (see also Pythagorean Interval and Pythagorean Tuning).

The earliest such description of a scale is found in Philolaus fr. B6. Philolaus recognizes that, if we go up the interval of a fourth from any given note, and then up the interval of a fifth, the final note is an octave above the first note. Thus, the octave is made up of a fourth and a fifth. ... Philolaus's scale thus consisted of the following intervals: 9:8, 9:8, 256:243 [these three intervals take us up a fourth], 9:8, 9:8, 9:8, 256:243 [these four intervals make up a fifth and complete the octave from our starting note]. This scale is known as the Pythagorean diatonic and is the scale that Plato adopted in the construction of the world soul in the Timaeus (36a-b). [11]

The next notable Pythagorean theorist we know of is Archytas, contemporary and friend of Plato, who explained the use of arithmetic, geometric and harmonic means in tuning musical instruments. Euclid further developed Archytas's theory in his The Division of the Canon (Katatomē kanonos, the Latin Sectio Canonis). He elaborated the acoustics with reference to the frequency of vibrations (or movements). [12]

Archytas provided a rigorous proof that the basic musical intervals cannot be divided in half, or in other words, that there is no mean proportional between numbers in super-particular ratio (octave 2:1, fourth 4:3, fifth 3:2, 9:8). [11] [13]

Archytas was also the first ancient Greek theorist to provide ratios for all 3 genera. [1] The three genera of tetrachords recognized by Archytas have the following ratios:

These three tunings appear to have corresponded to the actual musical practice of his day. [13]

The genera arose after the framing interval of the tetrachord was fixed, because the two internal notes (called lichanoi and parhypate) still had variable tunings. Tetrachords were classified into genera depending on the position of the lichanos (thus the name lichanos, which means "the indicator"). For instance a lichanos that is a minor third from the bottom and a major second from the top, defines the genus diatonic. The other two genera, chromatic and enharmonic, were defined in similar fashion. [14]

More generally, three genera of seven octave species can be recognized, depending on the positioning of the interposed tones in the component tetrachords:

Within these basic forms, the intervals of the chromatic and diatonic genera were varied further by three and two "shades" (chroai), respectively . [16] [17]

The elaboration of tetrachords was also accompanied by penta- and hexachords. The joining of a tetrachord and a pentachord yields an octachord, i.e. the complete seven-tone scale plus a higher octave of the base note. However, this was also produced by joining two tetrachords, which were linked by means of an intermediary or shared note. The final evolution of the system did not end with the octave as such but with the Systema teleion, a set of five tetrachords linked by conjunction and disjunction into arrays of tones spanning two octaves, as explained above. [1]

The system of Aristoxenus

Having elaborated the Systema teleion, we will now examine the most significant individual system, that of Aristoxenos, which influenced much classification well into the Middle Ages.

Aristoxenus was a disciple of Aristotle who flourished in the 4th century BC. He introduced a radically different model for creating scales, and the nature of his scales deviated sharply from his predecessors. His system was based on seven "octave species" named after Greek regions and ethnicities – Dorian, Lydian, etc. This association of the ethnic names with the octave species appears to have preceded Aristoxenus, [18] and the same system of names was revived in the Renaissance as names of musical modes according to the harmonic theory of that time, which was however quite different from that of the ancient Greeks. Thus the names Dorian, Lydian etc. should not be taken to imply a historical continuity between the systems.

In contrast to Archytas who distinguished his "genera" only by moving the lichanoi, Aristoxenus varied both lichanoi and parhypate in considerable ranges. [19] Instead of using discrete ratios to place the intervals in his scales, Aristoxenus used continuously variable quantities: as a result he obtained scales of thirteen notes to an octave, and considerably different qualities of consonance. [20]

The octave species in the Aristoxenian tradition were: [21] [18]

These names are derived from:

Aristoxenian tonoi

The term tonos (pl. tonoi) was used in four senses, for it could designate a note, an interval, a region of the voice, and a pitch. [22]

The ancient writer Cleonides attributes thirteen tonoi to Aristoxenus, which represent a transposition of the tones of the Pythagorean system into a more uniform progressive scale over the range of an octave. [17]

According to Cleonides, [22] these transpositional tonoi were named analogously to the octave species, supplemented with new terms to raise the number of degrees from seven to thirteen. In fact, Aristoxenus criticized the application of these names by the earlier theorists, whom he called the "Harmonicists". [18]

According to the interpretation of at least two modern authorities, in the Aristoxenian tonoi the Hypodorian is the lowest, and the Mixolydian is next-to-highest: the reverse of the case of the octave species. [17] [23] The nominal base pitches are as follows (in descending order, after Mathiesen; Solomon uses the octave between A and a instead):

fHypermixolydianalso called Hyperphrygian
eHigh Mixolydianalso called Hyperiastian
eLow Mixolydianalso called Hyperdorian
dHigh Lydian
cLow Lydianalso called Aeolian
cHigh Phrygian
BLow Phrygianalso called Iastian
BDorian
AHigh Hypolydian
G (and A?)Low Hypolydianalso called Hypoaeolian
GHigh Hypophrygian
FLow Hypophrygianalso called Hypoiastian
FHypodorian

The octave species in all genera

Based on the above, it can be seen that the Aristochene system of tones and octave species can be combined with the Pythagorean system of "genera" to produce a more complete system in which each octave species of thirteen tones (Dorian, Lydian, etc.) can be declined into a system of seven tones by selecting particular tones and semitones to form genera (Diatonic, Chromatic, and Enharmonic).

The order of the octave species names in the following table are the original Greek ones, followed by later alternatives (Greek and other). The species and notation are built around the template of the Dorian.

Diatonic

TonicNameMese
(A)(Hypermixolydian, Hyperphrygian, Locrian)(d)
BMixolydian, Hyperdoriane
cLydianf
dPhrygiang
eDoriana
fHypolydianb
gHypophrygian, Ionianc′
aHypodorian, Aeoliand′

Chromatic

TonicNameMese
(A)(Hypermixolydian, Hyperphrygian, Locrian)(d)
BMixolydian, Hyperdoriane
cLydianf
dPhrygiang
eDoriana
fHypolydianb
gHypophrygian, Ionianc′
aHypodorian, Aeoliand′

Enharmonic

TonicNameMese
(A)(Hypermixolydian, Hyperphrygian, Locrian)(d Doubleflat.svg )
BMixolydian, Hyperdoriane
c-Lydianf-
d Doubleflat.svg Phrygiang Doubleflat.svg
eDoriana
f-Hypolydianb
g Doubleflat.svg Hypophrygian, Ionianc′-
aHypodorian, Aeoliand′ Doubleflat.svg

The oldest harmoniai in three genera

In the notation above and below, the standard double-flat symbol Doubleflat.svg is used to accommodate as far as possible the modern musical convention that demands every note in a scale to have a distinct, sequential letter; so interpret Doubleflat.svg only as meaning the immediate prior letter in the alphabet. [24] This is a complication unnecessary in Greek notation, which had distinct symbols for each half-flat, flat, or natural note.

The superscript symbol - after a letter indicates an approximately half-flattened version of the named note; the exact degree of flattening intended depending on which of several tunings was used. Hence a three-tone falling-pitch sequence d, d-, d, with the second note, d-, about 12-flat (a quarter-tone flat) from the first note, d, and the same d- about 12-sharp (a quarter-tone sharp) from the following d.

The (d) listed first for the Dorian is the Proslambanómenos, which was appended as it was, and falls outside of the linked-tetrachord scheme.

These tables are a depiction of Aristides Quintilianus's enharmonic harmoniai, the diatonic of Henderson [25] and Chalmers [7] chromatic versions. Chalmers, from whom they originate, states:

In the enharmonic and chromatic forms of some of the harmoniai, it has been necessary to use both a d and either a d or d Doubleflat.svg because of the non-heptatonic nature of these scales. C and F are synonyms for d Doubleflat.svg and g Doubleflat.svg [respectively]. The appropriate tunings for these scales are those of Archytas [26] and Pythagoras. [24]

The superficial resemblance of these octave species with the church modes is misleading: The conventional representation as a section (such as C D E F followed by D E F G) is incorrect. The species were re-tunings of the central octave such that the sequences of intervals (the cyclical modes divided by ratios defined by genus) corresponded to the notes of the Perfect Immutable System as depicted above. [27]

Dorian

GenusTones
Enharmonic(d) e f- g Doubleflat.svg a b c′- d′ Doubleflat.svg e′
Chromatic(d) e f g a b c′ d′ e′
Diatonic(d) e f g a b c′ d′ e′

Phrygian

GenusTones
Enharmonicd e f- g Doubleflat.svg a b c′- d′ Doubleflat.svg d′
Chromaticd e f g a b c′ d′ d′
Diatonicd e f g a b c′ d′

Lydian

GenusTones
Enharmonicf- g Doubleflat.svg a b c′- d′ Doubleflat.svg e′ f′-
Chromaticf g a b c′ d′ e′ f′
Diatonicf g a b c′ d′ e′ f′

Mixolydian

GenusTones
EnharmonicB c- d Doubleflat.svg d e f- g Doubleflat.svg b
ChromaticB c d d e f- g b
DiatonicB c d e f(g) (a) b

Syntonolydian

GenusTones
EnharmonicB c- d Doubleflat.svg e g
ChromaticB c d e g
1st Diatonicc d e f g
2nd DiatonicB c d e g

Ionian (Iastian)

GenusTones
EnharmonicB c- d Doubleflat.svg e g a
ChromaticB c d e g a
1st Diatonicc e f g
2nd DiatonicB c d e g a

Ptolemy and the Alexandrians

In marked contrast to his predecessors, Ptolemy's scales employed a division of the pyknon in the ratio of 1:2, melodic, in place of equal divisions. [28] Ptolemy, in his Harmonics, ii.3–11, construed the tonoi differently, presenting all seven octave species within a fixed octave, through chromatic inflection of the scale degrees (comparable to the modern conception of building all seven modal scales on a single tonic). In Ptolemy's system, therefore there are only seven tonoi. [17] [29] Ptolemy preserved Archytas's tunings in his Harmonics as well as transmitting the tunings of Eratosthenes and Didymos and providing his own ratios and scales. [1]

Harmoniai

In music theory the Greek word harmonia can signify the enharmonic genus of tetrachord, the seven octave species, or a style of music associated with one of the ethnic types or the tonoi named by them. [30]

Particularly in the earliest surviving writings, harmonia is regarded not as a scale, but as the epitome of the stylised singing of a particular district or people or occupation. [31] When the late 6th-century poet Lasus of Hermione referred to the Aeolian harmonia, for example, he was more likely thinking of a melodic style characteristic of Greeks speaking the Aeolic dialect than of a scale pattern. [32]

In the Republic , Plato uses the term inclusively to encompass a particular type of scale, range and register, characteristic rhythmic pattern, textual subject, etc. [17]

The philosophical writings of Plato and Aristotle (c. 350 BCE) include sections that describe the effect of different harmoniai on mood and character formation (see below on ethos). For example, in the Republic (iii.10–11) Plato describes the music a person is exposed to as molding the person's character, which he discusses as particularly relevant for the proper education of the guardians of his ideal State. Aristotle in the Politics (viii:1340a:40–1340b:5):

But melodies themselves do contain imitations of character. This is perfectly clear, for the harmoniai have quite distinct natures from one another, so that those who hear them are differently affected and do not respond in the same way to each. To some, such as the one called Mixolydian, they respond with more grief and anxiety, to others, such as the relaxed harmoniai, with more mellowness of mind, and to one another with a special degree of moderation and firmness, Dorian being apparently the only one of the harmoniai to have this effect, while Phrygian creates ecstatic excitement. These points have been well expressed by those who have thought deeply about this kind of education; for they cull the evidence for what they say from the facts themselves. [33]

Aristotle remarks further:

From what has been said it is evident what an influence music has over the disposition of the mind, and how variously it can fascinate it—and if it can do this, most certainly it is what youth ought to be instructed in. [34] )

Ethos

The ancient Greeks have used the word ethos (ἔθος or ἦθος), in this context best rendered by "character" (in the sense of patterns of being and behaviour, but not necessarily with "moral" implications), to describe the ways music can convey, foster, and even generate emotional or mental states. Beyond this general description, there is no unified "Greek ethos theory" but "many different views, sometimes sharply opposed." [32] Ethos is attributed to the tonoi or harmoniai or modes (for instance, Plato, in the Republic (iii: 398d–399a), attributes "virility" to the "Dorian," and "relaxedness" to the "Lydian" mode), instruments (especially the aulos and the cithara, but also others), rhythms, and sometimes even the genus and individual tones. The most comprehensive treatment of musical ethos is provided by Aristides Quintilianus in his book On Music, with the original conception of assigning ethos to the various musical parameters according to the general categories of male and female. Aristoxenus was the first Greek theorist to point out that ethos does not only reside in the individual parameters but also in the musical piece as a whole (cited in Pseudo-Plutarch, De Musica 32: 1142d ff; see also Aristides Quintilianus 1.12). The Greeks were interested in musical ethos particularly in the context of education (so Plato in his Republic and Aristotle in his eighth book of his Politics), with implications for the well-being of the State. Many other ancient authors refer to what we nowadays would call psychological effect of music and draw judgments for the appropriateness (or value) of particular musical features or styles, while others, in particular Philodemus (in his fragmentary work De musica) and Sextus Empiricus (in his sixth book of his work Adversus mathematicos), deny that music possesses any influence on the human person apart from generating pleasure. These different views anticipate in some way the modern debate in music philosophy whether music on its own or absolute music, independent of text, is able to elicit emotions on the listener or musician. [35]

Melos

Cleonides describes "melic" composition, "the employment of the materials subject to harmonic practice with due regard to the requirements of each of the subjects under consideration" [36] —which, together with the scales, tonoi, and harmoniai resemble elements found in medieval modal theory. [37] According to Aristides Quintilianus (On Music, i.12), melic composition is subdivided into three classes: dithyrambic, nomic, and tragic. These parallel his three classes of rhythmic composition: systaltic, diastaltic and hesychastic. Each of these broad classes of melic composition may contain various subclasses, such as erotic, comic and panegyric, and any composition might be elevating (diastaltic), depressing (systaltic), or soothing (hesychastic). [38]

The classification of the requirements we have from Proclus Useful Knowledge as preserved by Photios [ citation needed ]:

According to Mathiesen:

Such pieces of music were called melos, which in its perfect form (teleion melos) comprised not only the melody and the text (including its elements of rhythm and diction) but also stylized dance movement. Melic and rhythmic composition (respectively, melopoiïa and rhuthmopoiïa) were the processes of selecting and applying the various components of melos and rhythm to create a complete work. [39]

Unicode

Music symbols of ancient Greece were added to the Unicode Standard in March 2005 with the release of version 4.1.

See also

Related Research Articles

In music theory, a diatonic scale is any heptatonic scale that includes five whole steps and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other.

In music theory, the term mode or modus is used in a number of distinct senses, depending on context.

<span class="mw-page-title-main">Pythagorean tuning</span> Method of tuning a musical instrument

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In music theory, a tetrachord is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion —but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system.

<span class="mw-page-title-main">Enharmonic equivalence</span> Distinct pitch classes sounding the same

In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. This relation extends to pitch classes, chords, and key signatures. The term derives from Latin enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').

<span class="mw-page-title-main">Aristoxenus</span> 4th century BC Greek Peripatetic philosopher

Aristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and meter. The Elements is the chief source of our knowledge of ancient Greek music.

Articles related to music include:

Dorian mode or Doric mode can refer to three very different but interrelated subjects: one of the Ancient Greek harmoniai ; one of the medieval musical modes; or—most commonly—one of the modern modal diatonic scales, corresponding to the piano keyboard's white notes from D to D, or any transposition of itself.

Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek harmoniai or tonoi, based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic scale, related to the medieval mode.

The Phrygian mode can refer to three different musical modes: the ancient Greek tonos or harmonia, sometimes called Phrygian, formed on a particular set of octave species or scales; the medieval Phrygian mode, and the modern conception of the Phrygian mode as a diatonic scale, based on the latter.

<span class="mw-page-title-main">Enharmonic scale</span>

In music theory, an enharmonic scale is "an [imaginary] gradual progression by quarter tones" or any "[musical] scale proceeding by quarter tones". The enharmonic scale uses dieses (divisions) nonexistent on most keyboards, since modern standard keyboards have only half-tone dieses.

In the musical system of ancient Greece, genus is a term used to describe certain classes of intonations of the two movable notes within a tetrachord. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of Byzantine music; it may have been one source of the later theory of the jins of Arabic music. In addition, Aristoxenus calls some patterns of rhythm "genera".

The Hypophrygian mode, literally meaning "below Phrygian ", is a musical mode or diatonic scale in medieval chant theory, the fourth mode of church music. This mode is the plagal counterpart of the authentic third mode, which was called Phrygian. In the Middle Ages and Renaissance this mode was described in two ways: the diatonic scale from B to B an octave above, divided at the mode final E ; and as a mode with final E and ambitus from the A below to the C above. The note A above the final had an important melodic function. The melodic range of the ecclesiastical Hypophrygian mode therefore goes from the perfect fourth or fifth below the tonic to the perfect fifth or minor sixth above.

In the musical system of ancient Greece, an octave species is a specific sequence of intervals within an octave. In Elementa harmonica, Aristoxenus classifies the species as three different genera, distinguished from each other by the largest intervals in each sequence: the diatonic, chromatic, and enharmonic genera, whose largest intervals are, respectively, a whole tone, a minor third, and a ditone; quarter tones and semitones complete the tetrachords.

<span class="mw-page-title-main">Diatonic and chromatic</span> Terms in music theory to characterize scales

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

<span class="mw-page-title-main">Incomposite interval</span>

An incomposite interval is a concept in the Ancient Greek theory of music concerning melodic musical intervals between neighbouring notes in a tetrachord or scale which, for that reason, do not encompass smaller intervals. Aristoxenus defines melodically incomposite intervals in the following context:

Let us assume that given a systēma, whether pyknon or non-pyknon, no interval less than the remainder of the first concord can be placed next above it, and no interval less than a tone next below it. Let us also assume that each of the notes which are melodically successive in each genus will either form with the fourth note in order from it the concord of a fourth, or will form with the fifth note from it in order the concord of a fifth, or both, and that any note of which none of these things is true is unmelodic relative to those with which it forms no concord. Let us further assume that given that there are four intervals in the fifth, of which two are usually equal and two unequal, the unequal ones are placed next to the equal ones in the opposite order above and below. Let us assume that notes standing at the same concordant interval from successive notes are in succession with one another. Let us assume that in each genus an interval is melodically incomposite if the voice, in singing a melody, cannot divide it into intervals.

<span class="mw-page-title-main">7-limit tuning</span> Musical instrument tuning with a limit of seven

7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 is not.

Pyknon, sometimes also transliterated as pycnon in the music theory of Antiquity is a structural property of any tetrachord in which a composite of two smaller intervals is less than the remaining (incomposite) interval. The makeup of the pyknon serves to identify the melodic genus and the octave species made by compounding two such tetrachords, and the rules governing the ways in which such compounds may be made centre on the relationships of the two pykna involved.

Elementa harmonica is a treatise on the subject of musical scales by Aristoxenus, of which considerable amounts are extant. The work dates to the second half of the 4th century BC. It is the oldest substantially surviving work written on the subject of music theory.

References

  1. 1 2 3 4 Chalmers 1993, ch. 6, p. 99.
  2. Crocker 1963.
  3. Crocker 1964.
  4. Crocker 1966.
  5. Barbera 1977.
  6. Barbera 1984.
  7. 1 2 Chalmers 1993.
  8. Chalmers 1993, ch. 5, pp. 48–51.
  9. Pöhlmann 1970.
  10. Pöhlmann and West 2001, 7.
  11. 1 2 3 Huffman 2011.
  12. Levin 1990.
  13. 1 2 Barker 1984–89, 2:46–52.
  14. Chalmers 1993, ch. 5, p. 47.
  15. Cleonides 1965, 35–36.
  16. Cleonides 1965, 39–40.
  17. 1 2 3 4 5 Mathiesen 2001a, 6(iii)(e).
  18. 1 2 3 Mathiesen 2001a, 6(iii)(d).
  19. Chalmers 1993, ch. 5, p. 48.
  20. Chalmers 1993, ch. 3, pp. 17–22.
  21. Barbera 1984, 240.
  22. 1 2 Cleonides 1965, 44.
  23. Solomon 1984, 250.
  24. 1 2 Chalmers 1993, ch. 6, p. 109.
  25. Henderson 1942.
  26. Mountford 1923.
  27. Chalmers 1993, ch. 6, p. 106.
  28. Chalmers 1993, ch. 2, p. 10.
  29. Mathiesen 2001c.
  30. Mathiesen 2001b.
  31. Winnington-Ingram 1936, 3.
  32. 1 2 Anderson and Mathiesen 2001.
  33. Barker 1984–89, 1:175–176.
  34. Aristotle 1912, book 8, ch. 5.
  35. Kramarz 2016.
  36. Cleonides 1965, 35.
  37. Mathiesen 2001a, 6(iii).
  38. Mathiesen 2001a, 4.
  39. Mathiesen 1999, 25.

Sources

Further reading