In the
musical system of ancient Greece, genus (Greek: γένος
'genos'' pl. γένη
'genē'' Latin: ''genus'', pl. ''genera'' "type, kind") is a term used to describe certain classes of
intonations of the two movable notes within a
tetrachord
In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) 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 pr ...
. The tetrachordal system was inherited by the Latin medieval theory of scales and by the modal theory of
Byzantine music
Byzantine music ( Greek: Βυζαντινή μουσική) is the music of the Byzantine Empire. Originally it consisted of songs and hymns composed to Greek texts used for courtly ceremonials, during festivals, or as paraliturgical and liturgica ...
; it may have been one source of the later theory of the
jins of
Arabic music
Arabic music or Arab music ( ar, الموسيقى العربية, al-mūsīqā al-ʿArabīyyah) is the music of the Arab world with all its diverse music styles and genres. Arabic countries have many rich and varied styles of music and also ma ...
. In addition,
Aristoxenus (in his fragmentary treatise on rhythm) calls some patterns of rhythm "genera".
Tetrachords
According to the system of
Aristoxenus and his followers—
Cleonides, Bacchius,
Gaudentius,
Alypius, Bryennius, and
Aristides Quintilianus Aristides Quintilianus (Greek: Ἀριστείδης Κοϊντιλιανός) was the Greek author of an ancient musical treatise, ''Perì musikês'' (Περὶ Μουσικῆς, i.e. ''On Music''; Latin: ''De Musica'')
According to Theodore Kar ...
—the paradigmatic tetrachord was bounded by the fixed tones ''hypate'' and ''mese'', which are a
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
apart and do not vary from one genus to another. Between these are two movable notes, called ''parhypate'' and ''lichanos''. The upper tone, lichanos, can vary over the range of a whole tone, whereas the lower note, parhypate, is restricted to the span of a quarter tone. However, their variation in position must always be proportional. This interval between the fixed hypate and movable parhypate cannot ever be larger than the interval between the two movable tones. When the composite of the two smaller intervals is less than the remaining (
incomposite) interval, the three-note group is called ''
pyknon
Pyknon (from el, πυκνόν), sometimes also transliterated as pycnon (from el, πυκνός close, close-packed, crowded, condensed; lat, spissus) in the music theory of Antiquity is a structural property of any tetrachord in which a composit ...
'' (meaning "compressed").
The positioning of these two notes defined three genera: the diatonic, chromatic (also called ''chroma'', "colour"), and enharmonic (also called ἁρμονία
'harmonia''. The first two of these were subject to further variation, called shades—χρόαι (''chroai'')—or species—εἶδη (''eidē''). For Aristoxenus himself, these shades were dynamic: that is, they were not fixed in an ordered scale, and the shades were flexible along a continuum within certain limits. Instead, they described characteristic functional progressions of intervals, which he called "roads" (ὁδοί), possessing different ascending and descending patterns while nevertheless remaining recognisable. For his successors, however, the genera became fixed intervallic successions, and their shades became precisely defined subcategories. Furthermore, in sharp contrast to the Pythagoreans, Aristoxenos deliberately avoids numerical ratios. Instead, he defines a whole tone as the difference between a perfect fifth and a perfect fourth, and then divides that tone into
semitones, third-tones, and
quarter tone
A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each ...
s, to correspond to the diatonic, chromatic, and enharmonic genera, respectively.
Diatonic
Aristoxenus describes the diatonic genus ( grc, διατονικὸν γένος) as the oldest and most natural of the genera. It is the division of the tetrachord from which the modern
diatonic scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) 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 st ...
evolved. The distinguishing characteristic of the diatonic genus is that its largest
interval is about the size of a
major second
In Western music theory, a major second (sometimes also called whole tone or a whole step) is a second spanning two semitones (). A second is a musical interval encompassing two adjacent staff positions (see Interval number for more de ...
. The other two intervals vary according to the tunings of the various shades.
Etymology
The English word ''
diatonic
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 ...
'' is ultimately from the grc, διατονικός, diatonikós, itself from grc, διάτονος, diátonos, label=none, of disputed etymology.
Most plausibly, it refers to the intervals being "stretched out" in that tuning, in contrast to the other two tunings, whose lower two intervals were referred to as grc, πυκνόν,
pyknón, label=none, from grc, πυκνός, pyknós, dense, compressed, label=none. This takes grc, τόνος, tónos, label=none, to mean "interval of a tone"; see Liddell and Scott's
Greek Lexicon' and Barsky (second interpretation), below.
Alternatively, it could mean (as
OED claims) "through the tones", interpreting grc, διά, diá, label=none as "through". See also Barsky: "There are two possible ways of translating the Greek term 'diatonic': (1) 'running through tones', i.e. through the whole tones; or (2) a 'tensed' tetrachord filled up with the widest intervals".
The second interpretation would be justified by consideration of the pitches in the diatonic tetrachord, which are more equally distributed ("stretched out") than in the chromatic and enharmonic tetrachords, and are also the result of tighter stretching of the two variable strings. It is perhaps also sounder on linguistic morphological grounds. Compare ''
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
'' as "across/width distance".
A completely separate explanation of the origins of the term ''diatonic'' appeals to the generation of the diatonic scale from "two tones": "Because the musical scale is based entirely on octaves and fifths, that is, two notes, it is called the 'diatonic scale' ". But this ignores the fact that it is the element ''di-'' that means "two", not the element ''dia-'', which has "through" among its meanings (see Liddell and Scott). There is a Greek term grc, δίτονος, dítonos, label=none, which is applied to an interval equivalent to two tones. It yields the English words ''
ditone'' and ''ditonic'' (see
Pythagorean comma), but it is quite distinct from διάτονος.
The Byzantine theorist
George Pachymeres consider the term derived from grc, διατείνω, diateíno, label=none, meaning "to stretch to the end", because "...the voice is most stretched by it" ( grc-x-medieval, "... σφοδρότερον ἡ φωνὴ κατ’ αὐτὸ διατείνεται").
Yet another derivation assumes the sense "through the tones" for διάτονος, but interprets ''tone'' as meaning ''individual note'' of the scale: "The word diatonic means 'through the tones' (i.e., through the tones of the key)" (Gehrkens, 1914, see ; see also the Prout citation, at the same location). This is not in accord with any accepted Greek meaning, and in Greek theory it would fail to exclude the other tetrachords.
The fact that τόνος itself has at least four distinct meanings in Greek theory of music contributes to the uncertainty of the exact meaning and derivation of διατονικός, even among ancient writers: τόνος may refer to a pitch, an interval, a "key" or register of the voice, or a mode.
[Solon Michaelides, ''The Music of Ancient Greece: An Encyclopaedia'' (London; Faber and Faber, 1978), pp. 335–40: "Tonos".]
Shades or tunings
The diatonic tetrachord can be "tuned" using several shades or tunings. Aristoxenus (and Cleonides, following his example; see also Ptolemy's tunings) describes two shades of the diatonic, which he calls συντονόν (''syntonón'', from συντονός) and μαλακόν (''malakón'', from
μαλακός). ''Syntonón'' and ''malakón'' can be translated as "tense" ("taut") and "relaxed" ("lax, loose"), corresponding to the tension in the strings. These are often translated as "intense" and "soft", as in
Harry Partch
Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...
's influential ''
Genesis of a Music'', or alternatively as "sharp" (higher in pitch) and "soft" ("flat", lower in pitch). The structures of some of the most common tunings are the following:
The traditional
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: ...
of the diatonic, also known as Ptolemy's ditonic diatonic, has two identical 9:8 tones (see
major tone) in succession, making the other interval a Pythagorean limma (256:243):
hypate parhypate lichanos mese
4:3 81:64 9:8 1:1
, 256:243 , 9:8 , 9:8 ,
-498 -408 -204 0
cents
However, the most common tuning in practice from about the 4th century BC to the 2nd century AD appears to have been
Archytas
Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed found ...
's diatonic, or Ptolemy's "tonic diatonic", which has an 8:7 tone (see
septimal whole tone) and the
superparticular 28:27 instead of the complex 256:243 for the lowest interval:
hypate parhypate lichanos mese
4:3 9:7 9:8 1:1
, 28:27 , 8:7 , 9:8 ,
-498 -435 -204 0
cents
Didymus described the following tuning, similar to Ptolemy's later tense diatonic, but reversing the order of the 10:9 and 9:8, namely:
hypate parhypate lichanos mese
4:3 5:4 9:8 1:1
, 16:15 , 10:9 , 9:8 ,
-498 -386 -204 0
cents
Ptolemy, following Aristoxenus, also described "tense" and "relaxed" ("intense" and "soft") tunings. His "tense diatonic", as used in
Ptolemy's intense diatonic scale, is:
hypate parhypate lichanos mese
4:3 5:4 10:9 1:1
, 16:15 , 9:8 , 10:9 ,
-498 -386 -182 0
cents
Ptolemy's "relaxed diatonic" ("soft diatonic") was:
hypate parhypate lichanos mese
4:3 80:63 8:7 1:1
, 21:20 , 10:9 , 8:7 ,
-498 -413 -231 0
cents
Ptolemy described his "equable" or "even diatonic" as sounding foreign or rustic, and its
neutral seconds are reminiscent of scales used in
Arabic music
Arabic music or Arab music ( ar, الموسيقى العربية, al-mūsīqā al-ʿArabīyyah) is the music of the Arab world with all its diverse music styles and genres. Arabic countries have many rich and varied styles of music and also ma ...
. It is based on an equal division of string lengths (thus presumably simple to build and "rustic"), which implies a
harmonic series of pitch frequencies:
hypate parhypate lichanos mese
4:3 11:9 10:9 1:1
, 12:11 , 11:10 , 10:9 ,
-498 -347 -182 0
cents
Byzantine music
In
Byzantine music
Byzantine music ( Greek: Βυζαντινή μουσική) is the music of the Byzantine Empire. Originally it consisted of songs and hymns composed to Greek texts used for courtly ceremonials, during festivals, or as paraliturgical and liturgica ...
most of the modes of the
octoechos are based on the diatonic genus, apart from the ''second mode (both authentic and plagal)'' which is based on the
chromatic genus. Byzantine music theory distinguishes between two tunings of the diatonic genus, the so-called "hard diatonic" on which the ''third mode'' and two of the ''grave modes'' are based, and the "soft diatonic" on which the ''first mode (both authentic and plagal)'' and the ''fourth mode (both authentic and plagal)'' are based. The hard tuning of the diatonic genus in Byzantine music may also be referred to as the ''enharmonic genus''; an unfortunate name that persisted, since it can be confused with the ancient
enharmonic genus.
Chromatic
Aristoxenus describes the chromatic genus ( el, χρωματικὸν γένος or χρωματική) as a more recent development than the diatonic. It is characterized by an upper interval of a
minor third. The ''pyknon'' (πυκνόν), consisting of the two movable members of the tetrachord, is divided into two adjacent semitones.
The scale generated by the chromatic genus is not like the modern twelve-tone
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
. The modern (18th-century)
well-tempered chromatic scale has twelve pitches to the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
, and consists of semitones of various sizes; the
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...
common today, on the other hand, also has twelve pitches to the octave, but the semitones are all of the same size. In contrast, the ancient Greek chromatic scale had seven pitches (i.e. heptatonic) to the octave (assuming alternating conjunct and disjunct tetrachords), and had incomposite minor thirds as well as semitones and whole tones.
The (Dorian) scale generated from the chromatic genus is composed of two chromatic tetrachords:
:E−F−G−A , , B−C−D−E
whereas in modern music theory, a "
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...
" is:
:E−F−G−A−B−C−D−E
Shades
The number and nature of the shades of the chromatic genus vary amongst the Greek theorists. The major division is between the Aristoxenians and the Pythagoreans. Aristoxenus and Cleonides agree there are three, called soft, hemiolic, and tonic.
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
, representing a Pythagorean view, held that there are five.
Tunings
Theon of Smyrna gives an incomplete account of
Thrasyllus of Mendes' formulation of the greater perfect system, from which the diatonic and enharmonic genera can be deduced.
For the chromatic genus, however, all that is given is a 32:27 proportion of ''mese'' to ''lichanos''. This leaves 9:8 for the ''pyknon'', but there is no information at all about the position of the chromatic ''parhypate'' and therefore of the division of the ''pyknon'' into two semitones, though it may have been the ''limma'' of 256:243, as
Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the t ...
does later. Someone has referred to this speculative reconstructions as the traditional
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: ...
of the chromatic genus:
hypate parhypate lichanos mese
4:3 81:64 32:27 1:1
, 256:243 , 2187:2048 , 32:27 ,
-498 -408 -294 0
cents
Archytas
Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed found ...
used the simpler and more consonant 9:7, which he used in all three of his genera. His chromatic division is:
hypate parhypate lichanos mese
4:3 9:7 32:27 1:1
, 28:27 , 243:224 , 32:27 ,
-498 -435 -294 0
cents
According to
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's calculations,
Didymus's chromatic has only 5-
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
intervals, with the smallest possible numerators and denominators. The successive intervals are all
superparticular ratios:
hypate parhypate lichanos mese
4:3 5:4 6:5 1:1
, 16:15 , 25:24 , 6:5 ,
-498 -386 -316 0
cents
Byzantine music
In
Byzantine music
Byzantine music ( Greek: Βυζαντινή μουσική) is the music of the Byzantine Empire. Originally it consisted of songs and hymns composed to Greek texts used for courtly ceremonials, during festivals, or as paraliturgical and liturgica ...
the chromatic genus is the genus on which the ''second mode'' and ''second plagal mode'' are based. The "extra" mode
nenano is also based on this genus.
Enharmonic
Aristoxenus describes the enharmonic genus ( grc,
�ένοςἐναρμόνιον; lat, enarmonium,
enusenarmonicum, harmonia) as the "highest and most difficult for the senses". Historically it has been the most mysterious and controversial of the three genera. Its characteristic interval is a
ditone (or
major third
In classical music, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four semitones.Allen Forte, ...
in modern terminology), leaving the ''pyknon'' to be divided by two intervals smaller than a semitone called
dieses (approximately
quarter tone
A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each ...
s, though they could be calculated in a variety of ways). Because it is not easily represented by
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: ...
or
meantone temperament, there was much fascination with it in the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
.
In the modern tuning system of
twelve-tone equal temperament, ''
enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a writte ...
'' refers to tones that are ''identical'', but spelled differently. In other tuning systems, enharmonic notes, such as C and D, may be close but not identical, differing by a
comma
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
(an interval smaller than a semitone, like a diesis).
Notation
Modern notation for enharmonic notes requires two special symbols for raised and lowered quarter tones or half-semitones or quarter steps. Some symbols used for a quarter-tone flat are a downward-pointing arrow ↓, or a flat combined with an upward-pointing arrow ↑. Similarly, for a quarter-tone sharp, an upward-pointing arrow may be used, or else a sharp with a downward-pointing arrow. Three-quarter flat and sharp symbols are formed similarly. A further modern notation involves reversed flat signs for quarter-flat, so that an enharmonic tetrachord may be represented:
:D E F G ,
or
:A B C D .
The double-flat symbol () is used for modern notation of the third tone in the tetrachord to keep scale notes in letter sequence, and to remind the reader that the third tone in an enharmonic tetrachord (say F, shown above) was not tuned quite the same as the second note in a diatonic or chromatic scale (the E expected instead of F).
Scale
Like the diatonic scale, the ancient Greek
enharmonic scale also had seven notes to the octave (assuming alternating conjunct and disjunct tetrachords), not 24 as one might imagine by analogy to the modern chromatic scale. A scale generated from two disjunct enharmonic tetrachords is:
:D E F G , , A B C D or, in music notation starting on E:
,
with the corresponding conjunct tetrachords forming
:A B C , D, E F G or, transposed to E like the previous example:
.
Tunings
The precise ancient Pythagorean tuning of the enharmonic genus is not known. Aristoxenus believed that the ''pyknon'' evolved from an originally
pentatonic trichord in which a perfect fourth was divided by a single "
infix
An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix.
When marking text for i ...
"—an additional note dividing the fourth into a semitone plus a major third (e.g., E, F, A, where F is the infix dividing the fourth E–A). Such a division of a fourth necessarily produces a scale of the type called pentatonic, because compounding two such segments into an octave produces a scale with just five steps. This became an enharmonic tetrachord by the division of the semitone into two quarter tones (E, E↑, F, A).
Archytas
Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed found ...
, according to
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
, ''Harmonics'', ii.14—for no original writings by him survive—used 9:7, as in all three of his genera; here it is the
mediant In music, the mediant (''Latin'': to be in the middle) is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant note ...
of 4:3 and 5:4, as (4+5):(3+4) = 9:7:
hypate parhypate lichanos mese
4:3 9:7 5:4 1:1
, 28:27 , 36:35, 5:4 ,
-498 -435 -386 0
cents
Also according to Ptolemy,
Didymus uses the same major third (5:4) but divides the pyknon with the
arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
of the string lengths (if one wishes to think in terms of frequencies, rather than string lengths or interval distance down from the tonic, as the example below does, splitting the interval between the frequencies 4:3 and 5:4 by their
harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
31:24 will result in the same sequence of intervals as below):
hypate parhypate lichanos mese
4:3 31:24 5:4 1:1
, 32:31 , 31:30 , 5:4 ,
-498 -443 -386 0
cents
This method splits the 16:15 half-step pyknon into two nearly equal intervals, the difference in size between 31:30 and 32:31 being less than 2 cents.
Rhythmic genera
The principal theorist of rhythmic genera was Aristides Quintilianus, who considered there to be three: equal (
dactylic or
anapestic), sesquialteran (
paeonic), and duple (
iambic and
trochaic), though he also admitted that some authorities added a fourth genus, sesquitertian.
References
Sources
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Further reading
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{{Authority control
Ancient Greek music theory
Byzantine music theory
Greek music
Musical scales
Melody types