source file: mills3.txt Date: Thu, 18 Dec 1997 03:19:01 +0100 Subject: Practical Bases of Just Intonation From: Gregg Gibson One traditional means of deriving just intervals is of course to simply embark upon a series of perfect fifths: 3/2 x 3/2 = 9/4 x 1/2 = 9/8 9/8 x 3/2 = 27/16 27/16 x 3/2 = 81/32 x 1/2 = 81/64 etc Bringing the ratios back within the octave as one proceeds. This is Pythagorean intonation. But does a singer ever sing this many successive fifths (or fourths)? Evidently not. In highly modulated music such a chain might occur, but would be only an almost undiscernable trickle of tonality in the great flood of tones more directly related to the tonic (new or old.) The system is therefore without practical relation to music, and is one of those theoretical delusions which so plague and confuse musicians. But this cycle of fifths does nevertheless have an important r?le to play in the derivation of systems which are fit for practical music. If the cycle of fifths/fourths is used merely to ensure that each one of the tones of a temperament has a consonant fifth above and below it, and that the other consonances are likewise related to the fifths such that consonant chords are present, and the maximum number of consonant melodic intervals likewise present (this is the point I have earlier made in connection with the harmonic and melodic deficiencies of the 22-tone equal) then the cycle of fifths/fourths is very useful - in the context of temperament, not just intonation - to ensure that the cycle of the fifth/fourth shall be harmonically and melodically congruent with the cycles of the major third/minor sixth and minor third/major sixth. That is to say, by congruent, I mean that every pitch of the temperament has both consonant fifths and thirds above and below it. If even one tone lacks this, then we have at a stroke severely restricted both modulation, and also gravely weakened the melodic coherence of music, which depends on a seamless fabric of consonances and dissonances related by consonance. Within limits, music can survive a poor temperament, as the 12-tone equal daily attests, but its variety and expressivity will be found to be greatly constrained. But if the cycle of fifths is used as a basis for just intonation, it excludes the thirds and sixths from consonance, which is absurd. There are two other consonant cycles besides the cycle of the fifths/fourths, to which cycles I referred in the previous paragraph. The thirds and sixths are more common in song than either fifth or fourth, so although they are less consonant, they are but little less likely to be sung accurately, the more so as the thirds are much narrower than the fifth and fourth. It follows that any just intonation which hopes to be of practical use in describing what singers can sing, or what they find easiest to sing, and in describing how closely a given dissonance is related to the tonic, and hence how "tonal" that dissonance is, must include all three consonant cycles in the calculation. These principles follow from acoustics, and not primarily from art or culture. Renaissance polyphony, Rock music, Western 'Classical' music, Arab music, Indian music, etc all share, at least for singers, in these principles. No singer accurately sings the 20th just fifth from the tonic (unless by chance this corresponds to a consonant interval.) As it turns out, even if one does give greater weight to the cycle of the fifth/fourth, calling dissonances "tonal" that are related to the tonic through _two_ successive fifths or fourths (and not one only) we have only two additional tonal dissonances, both of them a mere comma removed from consonances, and so perfectly irrelevant melodically, and worthless for harmony. These are 32:27 (4/3 x 4/3 x 4/3) and 27:16 (3/2 x 3/2 x 3/2), and occur of course among the regular atonal dissonances that I have enumerated. These principles have nothing to do with any one temperament; they hold true for all temperaments and systems; I certainly did not dream them up to promote the 19-tone equal temperament, which however closely conforms to the resulting intervals. No matter how one may try to avoid it, no matter what argument one may devise, it cannot be denied that the 19 degrees of the 19-tone equal correspond remarkably well (though not of course perfectly) to the 7 consonances and 12 tonal dissonances. Of course, as I have just noted, it is possible to discuss _at the margins_ in a manner of speaking. One may discuss whether the septimals should be included as consonances, but even were we so unwise as to take these intervals for consonant, we would do nothing except set up a very limited exception to the general tonality of music. The manner of deriving the tonal dissonances is also debatable, though again, within the realms of reasonable debate, the number of 12 tonal dissonances can be altered only slightly if at all. This is not an arbitrary system, any more than the multiplication table is arbitrary. The tonal dissonances are associated to the consonances through 94 different consonant progressions, which form the fabric of tonality which it is the business of temperament to knit yet more strongly together. By this means dissonance acquires an inevitable, highly memorable character that marvelously contributes to the expressive power of music. I believe I have posted the tonal progressions proper to one of the 19 intervals. I don't think this is the place to post them all, but here is one more: 25/24 , 70.7 cents, the chromatic semitone: 5/4 x 5/3 , 5/4 x 5/6, 5/3 x 5/4, 5/3 x 5/8, 25/18 x 3/2, 25/18 x 3/4, 25/16 x 4/3, 25/16 x 2/3 SMTPOriginator: tuning@eartha.mills.edu From: Aline Surman Subject: 19 tone/20 tone PostedDate: 18-12-97 03:48:00 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: 18-12-97 03:45:54-18-12-97 03:45:54,18-12-97 03:45:31-18-12-97 03:45:32 DeliveredDate: 18-12-97 03:45:32 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 C1256571.000F2EC3; Thu, 18 Dec 1997 03:47:46 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18430; Thu, 18 Dec 1997 03:48:00 +0100 Date: Thu, 18 Dec 1997 03:48:00 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18621 Received: (qmail 9585 invoked from network); 17 Dec 1997 18:46:50 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Dec 1997 18:46:50 -0800 Message-Id: <34989783.57EC@dnvr.uswest.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu