Greek tetrachords in 19

I found the table of tetrachords in 19-tet quite interesting.
Most of them, however, are rather un-Greek in their contours and
arrangements, though the Islamic writer Safiyu-D-Din presented all
the permutations of the intervals of his tetrachords too. I've played
most of the over the years and find many useful for building
novel scales.

I have some qualms about calling the genus 1 1 6 (C C# Db F,
0 1 2 8) enharmonic as it appears to me to be a blend of the
hemiolic chromatic and true enharmonic because of the deviations
of 19-tet's intervals from JI and the 0 mod 12 temperaments in which
Aristoxenos's tetrachords are conveniently expressed. I do not
think that just the mere presence of intervals of 1 degree of
19 justifies calling such tetrachords enharmonic. Nineteen-tet
has a fair approximation to the Chromatic and Diatonic genera of
Archytas (and the corresponding genera of Ptolemy), but both have
an interval of 1 degree.

I have presented much of the argument for this before, but I
repeat it at the end of this post in new guise for those who might
have missed it or want to read more about the topic.

As for Pythagorean tuning, I doubt singers were often required to
jump to tones 4 or 5 fifths removed from their current notes except
in the old enharmonic with its ditonal leaps or the archaic
harmoniai of Aristides Quintilianus. However, the voice was usually
supported by either the aulos or a plucked string instrument such
as the lyra or kithara. Melodies tended to be conjunct and quite narrow
in range, so I don't imagine the vocal demands were excessive. In
the absence of harmony, some intonational lattitude may have been
acccepted, though we know little of performance practice in Greek
music or the much earlier Sumero-Babylonian music which was similarly
tuned. The same is probably true for early medieval music in Europe.

--John

Enharmonic in 19-tet:

Whether the enharmonic is expressed in 19-tet depends somewhat
on one's definition of the enharmonic genus. Historically, it
began as a pentatonic scale in Pythagorean tuning and the important
intervals were not microtones, but the incomposite ditones
(408 cents, 81/64). Toward the end of the 5th century BCE, the
limmas at the bottom of each trichord were split to form two rather
small "quarter tones" and in the process enlarge the scale to seven
tones. In the 4th century, a process of "sweetening" took place
in which the ditone was shrunk (or softened) first to a 5/4 (Archytas,
390 BCE), then to various more or less neutral thirds, to end up
as a minor third, probably a 32/27 according to Aristoxenos (~330).
Aristoxenos was the son of a famous musician and probably knew what
he was talking about in terms of the general effect of the tunings then
in use.

The size of the pyknon, or dense region of two dieses at the bottom
of the tetrachord served to charactarize the genus or type of tetrachord.
In his system, thus, the enharmonic consisted of two 1/4 tones and
a ditone, the soft chromatic of two 1/3 tones and flat major third of
367 cents, the hemiolic chromatic of two 3/8 tones and neutral third
(350 cents), and finally the intense chromatic of two semitones and
a minor third (or trihemitone). Aristoxenos grudgingly admits that
there is an enharmonic genus with an "irrational" large lowest interval
and a chromatic with the pyknon divided in the proportion of 1:2
(Winnington-Ingram. It is clear that these are not the preferred
tunings in his day.

Now, the "enharmonic" in 19-tet is a hybrid between the true
enharmonic and the soft chromatic. Because of the sharp fourth
and the flattish major third of 19, the tetrachord 1 1 6 has the
1/3 tones of the hemiolic chromatic, but a major third that is
intermediate between the incomposite 367 cent interval of the hemiolic
chromatic and the ditone of the enharmonic, though it is closer
to the former. This tuning somewhat resembles Archytas's enharmonic,
except that the middle interval is too large and the upper one too
small. Furthermore, the 1/3 tone of Archytas's genera was chosen to
make a 7/6 with the subtonic (Erickson, W-I, etc.), but the
corresponding interval of 4 degrees in 19 is not as consonant.
Thus the 19-tet genus 1 1 6 is neither enharmonic or chromatic in sensu
stricto.

Melodically, however, it sounds quite enharmonic because of the major
third, though it lacks the clarity of Aristoxenos's Pythagorean
or pseudo-tempered tuning and the consonance of Archytas's ratios.

The best Greek chromatic tuning in 19-tet is the 1:2 genus,
which closely resembles Archytas's tuning, though he used the
32/27 instead of the 6/5 as the uppermost interval. The tetrachord
0 1 3 8 (1 2 5) also approximates Ptolemy's Intense chromatic. The more
Aristoxenian tuning with an evenly divided pyknon, 2 2 4, does not
closely resemble any ancient Greek tuning, though Byzantine and Islamic
writers give similar tetrachords in JI.

There is also the diatonic genus with the hemiolic chromatic diesis
of Aristoxenos. In 19-tet it would be 1 4 3 degrees or 0 1 5 8. This
tetrachord approximates Archytas's diatonic.

Aristoxenos's soft diatonic genus, 0 100 250 500 cents is not
expressible in 19-tet, as 1 3 4 is not a good approximation.

Anyway, I do not think 19-tet is a very good temperament for playing
ancient Greek scales or music written in them unless the corpus is
restricted to just a few tetachords or specific 19-tet tetrachordal
music is composed. I would think the same would be true for any
music in which the fourth and fifth and their compounds are the
most prominent intervals and the same is true for other negative systems.
For such musics, 17 or some multiple of 12 would be my choice if
straight JI were not considered.


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From: John Chalmers
Subject: Danielou's theories
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