Enharmonic genus
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The enharmonic genus (Ancient Greek: [γένος] ἐναρμόνιον, ἁρμονία; Latin: enarmonium, [genus] enarmonicum, harmonia) has historically been the most mysterious and controversial of the three Greek genera of tetrachords. Its characteristic interval is a ditone (or major third in modern terminology), leaving the remainder of the tetrachord (called the pyknon) to be divided by two intervals smaller than a semitone called dieses (approximately quarter tones, but they could be calculated in a variety of ways). 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 flat with either an upward-pointing arrow or an acute accent above it, or a simple grave accent or downward-pointing arrow ↓. For a quarter-tone sharp, some symbols are h (half sharp), a sharp with either a downward-pointing arrow or a grave accent above it, or a simple acute accent or upward-pointing arrow ↑. Three-quarter flat and sharp symbols are formed similarly.[1] 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.[2][page needed] 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 of the enharmonic
The precise ancient Pythagorean tuning of the enharmonic genus is not known.[3] 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 (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).[4]
Archytas, according to Ptolemy, Harmonics, ii.14 (for no original writings by him survive),[5] as usual gives a tuning with small-number ratios:[3]
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 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 3/4 and 4/5 by their harmonic mean 24/31 will result in the same sequence of intervals as below):[3][failed verification]
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.
References
- ^ Gardner Read, Music Notation: A Manual of Modern Practice (Boston: Allyn and Bacon, Inc., 1964): 143.
- ^ M. L. West, Ancient Greek Music (Oxford: Clarendon Press, 1992): 254–73. ISBN 0-19-814975-1.
- ^ a b c John Chalmers,Divisions of the Tetrachord (Lebanon NH: Frog Peak Music, 1990): 9. ISBN 0-945996-04-7.
- ^ M. L. West, Ancient Greek Music (Oxford: Clarendon Press, 1992): 163. ISBN 0-19-814975-1.
- ^ Thomas J. Mathiesen, "Greece, §I: Ancient, 6: Music Theory, (i) Pythagoreans", The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell (London: Macmillan Publishers, 2001).