source file: mills3.txt Date: Thu, 18 Dec 1997 05:43:58 +0100 Subject: Melodic Limen and Definition of Pitch Classes From: Gregg Gibson I have now spoken enough of the melodic limen as a general phenomenon. There remains the question of how the melodic limen interacts with the particular intervals of music to produce the actual pitch classes of melody. The seven consonances and twelve tonal dissonances constitute points within the octave, each at the center of its own melodic region. Taking these regions at +/- 30 cents (the 60-cent melodic limen) we have the following regions: -30 ? 30 prime 1:1 41 ? 101 chromatic semitone 25:24 82 ? 142 diatonic semitone 16:15 152 ? 212 minor tone 10:9 174 ? 234 major tone 9:8 286 ? 346 minor third 6:5 356 ? 416 major third 5:4 397 ? 457 diminished fourth 32:25 468 ? 528 fourth 4:3 539 ? 599 augmented fourth 25:18 601 ? 661 diminished fifth 36:25 672 ? 732 fifth 3:2 743 ? 803 augmented fifth 25:16 784 ? 844 minor sixth 8:5 854 ? 914 major sixth 5:3 966 ? 1026 lesser minor seventh 16:9 988 ? 1048 greater minor seventh 9:5 1058 ? 1118 major seventh 15:8 1099 ? 1159 diminished octave 48:25 1170 ? 1230 octave 2:1 Remember that I am here considering melody, not harmony; obviously an octave of 1230 cents would be something terrible in harmony. The minor and major tones greatly overlap, and so do the lesser and greater minor sevenths; as far as melody is concerned these pairs merge. At the other extreme, the prime is absolutely distinct from the chromatic semitone, the diatonic semitone is clearly distinct from the minor tone, the minor third is very easily distinguished from both the major tone and the major third, the perfect fourth is quite distinct melodically from both the diminished and augmented fourths, the augmented fourth is distinct from the diminished fifth, the perfect fifth is highly distinct melodically from the diminished and augmented fifths, the minor sixth is quite distinct from the major sixth, the major sixth is quite distinct from the lesser minor seventh, the greater minor seventh is distinct from the major seventh, and the diminished octave is distinct from the perfect octave. To my own ears at least, all the above predictions of theory are correct. The augmented fourth and diminished fifth, for example, are as different in character as the locrian and lydian modes (when sung just), the former melancholy and 'minor', the latter weirdly major and hilarious. But in the 12-tone equal, where these modes must use the same 600-cent interval, this difference in character is far less pronounced. A third class of justly-intoned interval-pairs would be predicted to be but little distinct when played on instruments; singers must either exaggerate the difference between these intervals in order to sing them, as in 19-tone equal, or else merge them altogether, as in 12-tone equal. These are: chromatic semitone & diatonic semitone (faintly distinct to my ears, although with great effort; perhaps the very conjunct nature of these intervals permits their separation by sheer 'dead-reckoning'.) major third and diminished fourth (not distinct to my ears) augmented fifth & minor sixth (distinct to my ears with great effort, the former slightly expansive and 'major', the latter slightly contractive and 'minor') major seventh & diminished octave (not distinct to my ears) Finally, there are two "holes" in the octave big enough to definitely escape from the melodic orbits of the consonances and tonal dissonances. These are 234 ?286 cents (center: 260 cents) and 914 ? 966 cents (center: 940 cents). Like faint stars that come out at night while the sun no longer shines (forgive the poetaster), the atonal dissonances that occupy these two "black holes" enter into prominence. The augmented second/diminished third "hole" has three atonal dissonances, of which 125:108 at 253 cents is nearest the center. 7:6 is at 267 cents, so may exert some influence. The augmented sixth/diminished seventh "hole" has likewise three atonal dissonances (necessarily so, since these two "holes" are complements of each other), of which 216:125, the complement of 125:108, is nearest the center. 7:4, at 969 cents, is not part of this "hole", and moves definitely within the tonal orbit of 16:9. It is to be noted also, that 7:4 is very disjunct, and this further increases the difficulty in singing it. 7:4 is in fact the very type of the interval that when sung "just" sounds badly mistuned; indeed, the effect of 7:4 when played just, very well corresponds to the typical Western notion of a singer unable to carry a tune. The third septimal interval, 7:5, is swallowed up by 25:18, which it closely resembles in melodic effect; the latter is however even more intensely weird, cool, laid-back, etc. If we now insert 125:108 and 216:125, merge the two pairs 10:9/9:8 & 16:9/9:5, divide the narrow 10-cent gaps remaining between some of the pitch classes, and divide the narrow overlaps equally, we have the following division of the octave: -30 ? 35 perfect prime 36 ? 91 augmented prime/diminished second (chromatic semitone) 92 ? 147 minor second (diatonic semitone) 148 ? 228 major second 229 ? 285 augmented second/diminished third 286 ? 351 minor third 352 ? 406 major third 407 ? 462 diminished fourth 463 ? 533 perfect fourth 534 ? 600 augmented fourth 601 ? 666 diminished fifth 667 ? 738 perfect fifth 739 ? 793 augmented fifth 794 ? 849 minor sixth 850 ? 915 major sixth 916 ? 971 augmented sixth/diminished seventh 972 ? 1052 minor seventh 1053 ? 1109 major seventh 1110 ? 1164 diminished octave 1165 ? 1230 octave The major second and minor seventh pitch classes are predicted to be much wider than the average width, whereas ten other pitch classes are slightly narrower than average, and seven pitch classes about average in width. I do not want to make too much of this, for there is evidence to the contrary, but it is a common observation that the major second and minor seventh are the most variable in widths of the intervals, and not only in Western music... The pitch class comprising the consonances may however well be slightly narrower than those of the disjunct tonal dissonances, and especially the atonal disjunct dissonances, for it is well-proven that singers can sing consonances much more accurately than disjunct dissonances, and audiences may be more able to hear errors. These tables are nothing more than the roughest of estimates; nevertheless they do permit us to predict quite well the melodic character of the several degrees of the usual temperaments. The nearer the degree of a temperament to the centers of the 19 pitch classes, the more surely will that degree sound melodically distinct, and possessed of a unique character, for the more perfectly will its degrees reproduce the tonal fabric of just intonation, without however the intonational flaws of just intonation (namely the inconsistency of the three consonant cycles), which a good temperament resolves. There is however, another most important point to be made. As has been seen, ten of the pitch classes of just intonation may be but roughly 55 cents wide, and my own ears at least find that at least a few of the just intervals require some considerable attention to keep distinct from adjacent tones in melody. One would expect that two tones as 25:24 and 16:15, separated by only 41 cents, would be far less melodically distinct than 25:18 and 36:25, separated by 63 cents. In fact, whatever may be the corrections which greater study may make to the above tables, it is quite certain that the 19-tone equal, which equalizes the intervals at 63.3 cents (63.16 cents without a stretched octave) is melodically richer than just intonation itself, because in this temperament almost all the last instances of melodic confusion of adjacent intervals are removed, a fact that becomes apparent on the most cursory examination of this temperament. In the 19-tone equal four interval pairs that are susceptible of being melodically confused in just intonation (chromatic vs. diatonic semitone, etc) become quite distinct. It is instructive to use the above determination of the melodic pitch classes of just intonation to analyze the intervals of 12-tone equal, which are widely known. The 12-tone equal has, melodically considered, excellent fourths and fifths, and its major seconds and minor sevenths also are not bad, becasue they are relatively near the center of the just ranges for these intervals. Much less satisfactory are the minor seconds, minor thirds, major thirds, minor ixths, major sixths and major seventh. Although highly imaginative persons very often _try_ to hear additional intervals in the 12-tone equal, this temperament has no chromatic semitone, augmented tone, diminished fourth, tritone, diminished fifth, augmented fifth, augmented sixth or diminished octave. Consequently, it is these intervals of the 19-tone equal, or of just intonation for that matter, that are likely to sound strangest to the typical Western musician. But they sound strangely beautiful, not merely strange. Perhaps the most atonal of the 12-tone intervals is its 600-cent interval, which has neither the even-more-than-septimal weirdness of the augmented fourth nor the intensely menacing quality of the true diminished fifth, but rather an utterly neutral, colorless ethos. 19-tone equal has two atonal intervals, its augmented second/diminished third and its augmented sixth/diminished seventh, of which the former is far less objectionable than the 600-cent atonal interval of 12-tone equal, but the latter even more so. Finally, I give the centers of the 19 melodic pitch classes, together with the degrees of the 19-tone equal: Just 19-tone Equal (with octave stretch) 64 63.3 119 126.6 188 189.9 257 253.2 319 316.5 379 379.8 435 443.1 502 506.4 567 569.7 633 633.0 702 696.3 766 759.6 822 822.9 884 886.2 943 949.5 1012 1012.8 1081 1076.1 1137 1139.4 1200 1202.7 Here is as good as anywhere to announce that I am in the habit of calling enneadecaphony (19-tone equal temperament) with an octave stretch of 2.7 cents 'eumely', from the Greek for 'sweetly-sounding'. SMTPOriginator: tuning@eartha.mills.edu From: Gregg Gibson Subject: Comments to Andrew Kaye PostedDate: 18-12-97 06:18:55 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 06:16:54-18-12-97 06:16:55,18-12-97 06:16:32-18-12-97 06:16:32 DeliveredDate: 18-12-97 06:16:32 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 C1256571.001CFF7F; Thu, 18 Dec 1997 06:18:40 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA18674; Thu, 18 Dec 1997 06:18:55 +0100 Date: Thu, 18 Dec 1997 06:18:55 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA18673 Received: (qmail 20422 invoked from network); 17 Dec 1997 21:18:52 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 17 Dec 1997 21:18:52 -0800 Message-Id: <34991428.1F0C@ww-interlink.net> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu