Melodic Limen and Definition of Pitch Classes

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'.


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