source file: mills3.txt Date: Wed, 24 Dec 1997 23:24:26 +0100 Subject: Greek tetrachords in 19 From: John Chalmers 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. SMTPOriginator: tuning@eartha.mills.edu From: John Chalmers Subject: Danielou's theories PostedDate: 24-12-97 23:24:59 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 24-12-97 23:22:51-24-12-97 23:22:51,24-12-97 23:22:21-24-12-97 23:22:22 DeliveredDate: 24-12-97 23:22:22 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256577.007AECC2; Wed, 24 Dec 1997 23:24:36 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA25309; Wed, 24 Dec 1997 23:24:59 +0100 Date: Wed, 24 Dec 1997 23:24:59 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA25295 Received: (qmail 2840 invoked from network); 24 Dec 1997 14:24:42 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 24 Dec 1997 14:24:42 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu