source file: mills3.txt Date: Tue, 23 Dec 1997 21:51:47 +0100 Subject: More on Just Intonation 1 From: Gregg Gibson I apologize for the extemely primitive character of this post, but I desire my meaning to be crystal-clear to all. When a singer sings a note, and wishes to proceed upward or downward by a disjunct interval, he will find the following notes (C used as tonic here) most convenient, because they are consonant (those who choose to wage the war to make 7:4 consonant I advise to do some reading, thinking, and listening; even if they manage the feat, they will find themselves involved in the thankless task of attempting to erect a system of tonal relations outside the senario, and based on very weak consonance at best): Eb E F G Ab A _C_ Eb E F G Ab A 5 6 4 5 3 4 2 3 5 8 3 5 5 6 4 5 3 4 2 3 5 8 3 5 The numbers give the ratios of the intervals corresponding to the ratio of the vibrations. Once a singer has proceeded upward to Eb, say, he can again proceed surely by consonant intervals from Eb to any one of the six intervals above or the six intervals below. Of course, singers do not sing a long succession of disjunct intervals; they outline them with intercalated conjunct dissonances. Occasionally too, they sing disjunct dissonances, but with great difficulty, and less accuracy. Once a singer gets to Eb, here are his new consonant options, as it were: Gb G Ab Bb Cb C _Eb_ Gb G Ab Bb Cb C We can say that a tone such as Gb is related to C, the original tonic here, _through_ Eb. To find the ratio between C & Gb, we multiply 6/5 x 6/5 = 36/25. If we perform a like calculation for every one of the notes at a consonant distance from the initial tonic (here C) we get 12 'tonal' dissonances that are related via one consonant interval to the original tonic. Such dissonances can without exception be reached by at least six (usually eight) different consonant progressions. They, together with the consonant intervals, form a closed, highly coherent tonal system. This 'tonal net' is woven exceedingly tight - though it has a hole, which it is one of the prime objectives of temperament to patch. There can be _no other_ basis of tonal or musical coherence. Let those who question this find another such mathematical basis of music; there is none. >From this system different selections of tones and sub-systems of tones can be made (the modes, and the modal genera consisting of modes related to each other in the same way that the seven diatonic modes are related). As is well-known, each mode possesses a unique ethos, in the sense that the ionian differs in effect from the phrygian, and a melody transposed from ionian to phrygian sounds very different in its general spirit, but in a predictable way. But all these sub-systems are part of the original matrix of just intonation, or rather of that temperament, the 19-tone equal , which most closely corresponds to just intonation. The question arises: could one take Gb in the above example as a new jumping-off point for the voice and understanding? Indeed one can - but the memory of the original tonic fades, and the old tonal relationships give way to new ones ? on a different tonic, but still using the same consonances and tonal dissonances. Furthermore, because the voice and mind can reliably perceive only about 19 melodic pitch classes in the octave, we here involve ourselves in intervals (the 'atonal' disonances such as 6/5 x 6/5 x 6/5 = 216/125 or 3/2 x 3/2 x 3/2 = 27/16) which have no real melodic independence. 27/16 for example is confused with 5/3 melodically ? it sounds like a mistuned 5/3, in harmony a horribly mistuned 5/3. There are many just intonations, but only this version describes with fair accuracy what a singer actually confronts when he tries to sing. Arthur Benade has some interesting passages on the acoustics of why singers can sing consonances with such surprising accuracy, but find disjunct dissonances harder to sing, and this not merely in harmony, but in melody as well. SMTPOriginator: tuning@eartha.mills.edu From: Carl Lumma Subject: new web site PostedDate: 23-12-97 21:58:08 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: 23-12-97 21:55:58-23-12-97 21:55:59,23-12-97 21:55:29-23-12-97 21:55:30 DeliveredDate: 23-12-97 21:55:30 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 C1256576.0072F99F; Tue, 23 Dec 1997 21:57:46 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA24656; Tue, 23 Dec 1997 21:58:08 +0100 Date: Tue, 23 Dec 1997 21:58:08 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA24624 Received: (qmail 1309 invoked from network); 23 Dec 1997 12:58:05 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 23 Dec 1997 12:58:05 -0800 Message-Id: <19971223205639390.AAA224@ascend550.nni.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu