Genus (music)
Genus is a term used in the Ancient Greek and Roman theory of music 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".
Tetrachords
According to the system of Aristoxenus and his followers—Cleonides, Bacchius, Gaudentius, Alypius, Bryennius, and Aristides Quintilianus —the paradigmatic tetrachord was bounded by the fixed tones hypate and mese, which are a perfect fourth 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 interval, the three-note group is called pyknon. The positioning of these two notes defined three genera: the diatonic, chromatic, and enharmonic. The first two of these were subject to further variation, called shades—χρόαι —or species—εἶδη. 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", 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 tones, to correspond to the diatonic, chromatic, and enharmonic genera, respectively.Diatonic
Aristoxenus describes the diatonic genus as the oldest and most natural of the genera. It is the division of the tetrachord from which the modern diatonic scale evolved. The distinguishing characteristic of the diatonic genus is that its largest interval is about the size of a major second. The other two intervals vary according to the tunings of the various shades.The word is derived either from the Greek, dia and tonos, meaning "proceeding by whole tones", or "through, at the interval of" a "tone", or from Greek dia and tonikos, meaning "at intervals of a tone".
Shades or tunings
The diatonic tetrachord can be "tuned" using several shades or tunings. Aristoxenus describes two shades of the diatonic, which he calls "soft" and "syntonic". The structures of some of the most common tunings are the following:The traditional Pythagorean tuning of the diatonic, also known as Ptolemy's ditonic diatonic, has two identical 9/8 tones in succession, making the other interval Pythagorean limma 256/243:
hypate parhypate lichanos mese
| 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's diatonic, or Ptolemy's "tonic diatonic", which has a 8/7 tone and the superparticular 28/27 instead of the complex 256/243 for the lowest interval:
hypate parhypate lichanos mese
| 28/27 | 8/7 | 9/8 |
-498 -435 -204 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. It is based on an equal division of string lengths, which implies a harmonic series of pitch frequencies:
hypate parhypate lichanos mese
| 12/11 | 11/10 | 10/9 |
-498 -347 -182 0 cents
Byzantine music
In Byzantine music most of the modes of the octoechos are based on the diatonic genus, apart from the second mode 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 and the fourth mode 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 as a more recent development than the diatonic. It is characterized by an upper interval of a minor third. The pyknon, 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 modern well-tempered chromatic scale has twelve pitches to the octave, and consists of semitones of various sizes; the equal temperament 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 to the octave, and had incomposite minor thirds as well as semitones and whole tones.
The scale generated from the chromatic genus is composed of two chromatic tetrachords:
whereas in modern music theory, a chromatic scale is:
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, representing a Pythagorean view, held that there are five.Tunings
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 does later. Someone has referred to this speculative reconstructions as the traditional Pythagorean tuning 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 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's calculations, Didymus's chromatic has only 5-limit 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 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 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, leaving the pyknon to be divided by two intervals smaller than a semitone called dieses. Because it is not easily represented by Pythagorean tuning or meantone temperament, there was much fascination with it in the Renaissance. It has nothing to do with modern uses of the term enharmonic.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:or
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 was not tuned quite the same as the second note in a diatonic or chromatic scale.
Scale
Like the diatonic scale, the ancient Greek enharmonic scale also had seven notes to the octave, not 24 as one might imagine by analogy to the modern chromatic scale. A scale generated from two disjunct enharmonic tetrachords is:with the corresponding conjunct tetrachords forming
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 additional note dividing the fourth into a semitone plus a major third. 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 .Archytas, according to Ptolemy, Harmonics, ii.14—for no original writings by him survive —as usual gives a tuning with small-number ratios :
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 but divides the pyknon with the arithmetic mean of the string lengths :
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.