source file: mills3.txt Date: Sun, 14 Dec 1997 01:47:51 +0100 Subject: Just Intonation From: Gregg Gibson The great principle of music theory in its pitch aspect is to define only those theoretical systems which we are profoundly convinced that singers and practical musicians can actually use, or which are necessary for the derivation of such systems. Disjunct intervals which are highly dissonant with the tonic, and which cannot be reached by consonant intervals, or by conjunct intervals themselves defined by consonant intervals, have _no_ musical existence. They are as irrelevant to music as the third planet of alpha Centauri. Consonances are not playthings or inventions of man, which we can hopefully extend to the 9th or 11th harmonic; they have an objective, physical existence, and music without them is inconceivable and impossible. They are as necessary to music, either melodically or harmonically conceived, as the numerals are to mathematics, and their existence is just as certain. I apologize if all this seems too elementary to be worth stating. But these principles have been repeatedly denied by a certain school of modern theory, at least by implication. Having now set the stage, I should like to inquire what just intonation is, and what is the form of just intonation that a singer (any singer, not just a Western singer of some particular tradition) can use, and which it is the business of temperament to imitate, both for instruments and for the training of singers. Within the octave the consonances are six in number, and _only_ six: 3:2 4:3 5:3 5:4 6:5 & 8:5. The septimal intervals 7:4 7:5 & 7:6 are dissonances, both because their partials beat with each much more severely than is the case for the consonances, and because they are melodically confused with closely adjacent dissonant intervals directly derived from the consonances (e.g. 16:9 with 7:4.) The septimals have no harmonic connection with the six consonances, and have undoubtedly dissonant inversions 8:7 10:7 & 12:7. (Septimal 'harmony' therefore gives rise to dissonances.) But the six consonances do have a mathematical interrelationship with each other, as will be seen. All music begins with a tonic, the initial note. The singer in creating for himself a body of tones that are interrelated acoustically, and which he can remember and internalize, can proceed by the narrower, conjunct intervals (closer than 6:5, say,) and here he is limited only by the skill of his vocal chords and mind, which make it very difficult to sing or remember pitches closer than about 55-60 cents. As soon as music becomes disjunct, however (a compass of more than about 6:5) the singer (or listener or instrumentalist) runs the very real risk of becoming totally confused; pitches cease to have anything in common that one can remember, and music readily disintegrates into a welter of random noise, that all sounds much alike, and so can have little power to illustrate language or affect our emotions. There is a great deal of evidence that non-Western cultures are at least as intolerant of what they conceive to be aimless or random as the most hidebound Western conservative, although certainly they define this somewhat differently. None of these ideas are new. I seek only to make myself the purveyor of what the common sense of mankind knows almost instinctively. There is no objection if some wish to challenge these principles. If they can find an enthusiastic audience for music that violates these principles, I certainly do not object. But they should be intimately aware of these principles. So far composers who have sought to enshrine randomness or pure dissonance have only discredited the musical art. Fortunately, disjunct music (or what ultimately amounts to the same thing, music of compass wider than about 6:5) is possible. It is the phenomenon of consonance that makes it so. In our century consonance has acquired a reputation for vaguely indicating hypocrisy or insincerity or immaturiy in those who love it; one thinks of sugary-sweet childrens' choirs singing one of the more uninspired polyphonic masters - or something much worse. This is a very dangerous mistake. Consonances are not the enemy of dissonance. They only are what make dissonance itself emotionally meaningful. Not only that, but consonances alone make disjunct melody itself possible. And this is as true for the Indian or Arab or Turk as for the cultural heirs of the Greeks and Romans. Providentially - or perhaps this is the result of human evolution itself - just as melody becomes too disjunct to keep in order by the 'dead-reckoning' of conjunct intervals alone, we find an undoubted consonance, 6:5. We then find a series of consonant, beat-free intervals that provide us with sure resting points for the voice (and for the understanding) and which melody can invest with the most profound emotional significance. Again, this is as true for the rock singer as for the opera singer or an Indian singer. Once we reach any one of these consonances, we can take it for a new tonic, though in subordination to the original tonic. (I have no space here to discuss how the tonic can be changed, but I will consider that elsewhere in a more appropriate forum.) By this means we can, for example, if we wish, sound a minor third above the original minor third, so that we reach 6/5 x 6/5 = 36/25 which is dissonant with the original tonic, and disjunct, but - and this is a kind of miracle - nevertheless singable with fair accuracy and comprehensible to us. Conjunct intervals intercalated between these disjunct consonances further assist the voice and the understanding. Such dissonances as 36:25 (the diminished fifth) number twelve within the octave. They hold a quite unique place among the dissonances, because they are, in a sense, 'consonant with the tonic at one remove', as it were, because they can be sung entirely by singing consonances, though since a subtonic (6:5 in this example), that is to say, an intermediate tonic, separates them from the original tonic, they are not so closely related to the tonic as are the six consonances. I call these intervals 'tonal dissonances', because they more than anything else bind disjunct music together and make it tonal, that is to say coherent and consistent with itself. I am not of course referring to the so-called 'tonal' era of Western music, which might be better called the period of its decadence. The human voice does not sound these intervals with the same confidence and accuracy that it sounds the consonances, or even those conjunct dissonances made familiar to it by cultural training. Nevertheless it can readily distinguish the tonal dissonance 25:18 (the augmented fourth) from the tonal disonance 36:25 (the diminished fifth) because these are 63 cents apart, because they indicate quite different relations to the tonic, and also because there is enough 'tonal space' between 4:3 and 3:2 so that two (but not three) pitches can be kept melodically distinct, both from each other and from the adjacent consonances: 4:3 25:18 36:25 3:2 The 25:18 occurs in the lydian mode, and has an entirely different psychological effect eery and weird, but relaxed) from 36:25, which has a menacing, tense effect, and occurs in the locrian mode. The septimal dissonance 7:5 (the septimal tritone) also lies between 4:3 and 3:2, but becasue it lacks any relation with the consonances, and has a dissonant inversion 10:7, it is melodically and harmonically confounded with 25:18 (only 14 cents narrower than 7:5.) The tonal dissonance 25:18 is in fact much more easy for a singer to sing than 7:5, because the former is part of the fabric woven from the undoubted consonances, whereas 7:5 occupies the shadowy, weird realm between consonance and dissonance. Here are the six consonances and twelve tonal dissonances, together with the prime and octave: 1:1 25:24 16:15 10:9 9:8 6:5 5:4 32:25 4:3 25:18 36:25 3:2 25:16 8:5 5:3 16:9 9:5 15:8 48:25 2:1 Or in cents: 0 71 112 182 204 316 386 427 498 569 631 702 773 814 884 996 1018 1088 1129 1200 or by name: augmented prime, diminished second, minor tone, major tone, minor third, major third, diminished fourth, perfect fourth, augmented fourth or tritone, diminished fifth, perfect fifth, augmented fifth, minor sixth, major sixth, subminor seventh, superminor seventh, major seventh, diminished octave, octave Every music theorist has these values by heart, as well he should have - they are part of the basic vocabulary of music. Many of these intervals are inconveniently close for the voice. For example, 25:24 and 16:15 are separated by only the doubly diminished second, alias the diesis, 128:125, 41 cents. The melodic limen of 55-60 cents suggests that the voice (and mind) cannot reliably use more than about 20 pitches in the octave. Perhaps our species will one day come into contact with creatures who have a different limen. Now it happens that the consonances and tonal dissonances number 19 pitches within the octave. The obvious (and only) solution is to divide the octave into about 20 equal degrees. It happens that only one temperament within 12 and 31 equal tones in the octave gives reasonably accurate values for the consonances, and also conciliates the three consonant cycles in such a manner that all consonances are available not merely on one tonic, but on all possible tonics. This is the 19-tone equal temperamnt. The 12-tone equal temperament gives reasonable (or halfway reasonable) values only for the consonances. The 19-tone equal gives much better values for the consonances, although its fifths are rougher, and also gives good values for the tonal disonances, which is no less important, for music is not only consonance and conjunct dissonance, but disjunct dissonance also. It may be instructive to pursue the method further, and ask ourselves, could singers reliably relate dissonances to the tonic through a sub-subtonic? For example, could one proceed from 6:5 to 36:25 and again by a minor third to 216:125, the greater diminished seventh? Probably not. But even if one could, the resulting 18 additional dissonances - I call them 'atonal dissonances' - are so close that no singer could ever keep them melodically distinct. The 31-tone equal is the temperament that corresponds to the introduction of the 'atonal dissonances'. But the 19-tone equal accepts four of these into its fabric, namely the diminished third 144:125 and the augmented minor tone 125:108, which it merges into a single tone that is also confounded with 7:6; and the greater diminished seventh 216:125 and augmented sixth 125:72 which it also merges, and likewise confounds with the septimal interval 7:4. Here are the 18 atonal dissonances: 128:125 27:25 144:125 125:108 75:64 32:27 125:96 27:20 45:32 64:45 40:27 192:125 27:16 128:75 216:125 125:72 50:27 125:64 and in cents: 41 133 245 253 275 294 457 520 590 610 680 743 906 925 947 955 1067 1159 or by name: doubly diminished minor tone, diminished major tone, diminished third, augmented minor tone, augmented major tone, subminor third, augmented third, superperfect fourth, superaugmented fourth, subdiminished fifth, subperfect fifth, diminished sixth, supermajor sixth, lesser diminished seventh, greater diminished seventh, augmented sixth, submajor seventh, augmented seventh Note that four different, incompatible terminologies are used: the 10:9/9:8 and 16:9/9:5, as the commatically separated intervals, by tradition have each their own terminology, as do the thirds and sixths on the one hand (major and minor) and the fourths, fifths, primes and octaves on the other (perfect). No authorities use quite the same terminology, and none that I have found include quite all the intervals. There is a direct and exact relationship between just intonation and the three best temperaments of all, namely 12- 19- & 31-tone equal. The 12-tone equal loosely approximates the consonances, but radically distorts the tonal dissonances, so much so indeed that certain of these (e.g. the diminished fourth 32:25) are confounded with consonances. This particularly disturbed Helmholtz, and he was right to be disturbed. Certain nineteenth century composers used this in their modulations, with results that quickly became a clich?. The 19-tone equal gives close approximations to the consonances and the tonal dissonances, while confounding the atonal dissonances with the tonals. Finally, the 31-tone equal gives close approximations to the consonances, tonal dissonances and atonal dissonances, the last to no good or useful effect. The method can of course be pursued yet further, to 24 'ultra-atonal dissonances', and leads to 50-tone equal temperament (virtually Zarlino's 2/7 comma mesotonic.) The 43- & 55-tone equal are also fully cyclic, and constitute alternate, though less logical methods of conciliating the divers categories of intervals. Our notation is closely in accord with 19- 31- & 50-tone equal. The first of these may be called the 'system of the flat and sharp', the second the 'system of the double flat and double sharp', and the last the 'system of the triple flat and triple sharp'. The 12-tone of course confounds both flat and sharp, and also, at two points, accidental with natural. The 43- & 55-tone equal are also mixed systems. There are many systems of just intonation. But I daresay this system corresponds most nearly to what is actually possible for the voice, and present to the mind. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Reply to Graham Breed PostedDate: 14-12-97 02:27:27 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: 14-12-97 02:25:34-14-12-97 02:25:35,14-12-97 02:25:16-14-12-97 02:25:17 DeliveredDate: 14-12-97 02:25:17 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 C125656D.0007D103; Sun, 14 Dec 1997 02:27:18 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA14772; Sun, 14 Dec 1997 02:27:27 +0100 Date: Sun, 14 Dec 1997 02:27:27 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA14769 Received: (qmail 29027 invoked from network); 13 Dec 1997 17:27:24 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 13 Dec 1997 17:27:24 -0800 Message-Id: <3493987D.855@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu