19-tone equal divagations

I am pleased that my postings have elicited so many responses, some of
them quite thoughtful.

It is pointless to reply to those who have convinced themselves that
extremely minute intervals (narrower than a quarter-tone) can acquire
true melodic independance. Many years of hopeful experiment in this
direction have led me to conclude decidedly otherwise.

Even 24-tone equal, despite decades of determined exposition by divers
revolutionaries, remains utterly irrelevant to the main course of
musical evolution. While the starry-eyed among us attempt to distinguish
what is not distinguishable, the 12-tone equal continues to effect its
ravages on our ears. Indeed this miserable temperament has never been so
universal as it is today. And so the pretended best becomes the enemy of
what is actually usable (19-tone equal), and the ally of the worst.

Turning now to some of the comments that have been made in response to
some of my ideas. To suggest that the use of the 12-tone equal has
induced subjects to fix on the 50-cent quarter tone as the limen of
melodic perception is superficially plausible, but fails when it is
recalled that the musically naive tend to be less, not more sensitive to
the narrower intervals. See for example Szasz. This also assumes that it
is possible to internalize the 12-tone equal to such an extent that it
can replace the just intervals as the criterion of intervallic hearing,
than which nothing is less certain.

Bill Alves (quoted by Paul Ehrlich) made a very interesting comment to
the effect that the four chromatic modal genera (scales) comprising 28
modes, that I have discovered and posted:

C D E F G A B C
C D E F G Ab B C
C D E F G Ab Bb C
C Db E F Gb A Bb C

can be played in the 12-tone equal. But before I comment on this I would
like to observe that the three most usual forms of the modern
(post-Baroque) minor scale are present among these 28 chromatic modes.
John Chalmers, in a private communication, has commented on this to me
(I hope I do not misrepresent him), and it is certainly a very
intriguing circumstance that I have long since noticed.

But to return to the matter of playing these in 12-tone equal. Yes, this
is of course possible, but the resulting modes conflate a number of
aurally distinct modes into single, neutral modes, and lack much of the
distinct melodic character of the 19-tone versions.

For an example of what I mean, take the most common form of the minor
scale (on C for convenience):

C D Eb F G Ab B C

This can be played in 12-tone equal, but loses much of its piquancy
thereby. This is partly because the highly characteristic augmented tone
Ab-B is confounded with the minor third. Historically, the 'minor scale'
derived much of its attractiveness from the use of the mesotonic, which
preserves the augmented tone close to the just values (there are
several). I have elsewhere referred to the fact that this is one of the
few respects in which 31-tone equal is melodically quite distinct from
12-tone equal. But 19-tone equal is still more distinct.

Again, to give a better idea of exactly why one cannot adequately play
chromatic modes in 12-tone equal, let me offer up Yasser's old (but very
good) analogy between 12-tone equal and 7-tone equal. One can reproduce
the seven diatonic modes in 7-tone equal, but they are merged into a
single, neutral mode, for 7-tone equal has no semitones. This is very
much what occurs when a chromatic mode such as C D Eb F G Ab B C is
played in 12-tone equal - it is largely sterilized of its unique modal
flavor, because the two species of semitones are confounded.

Indeed, even the _diatonic_ modes are severely enfeebled in 12-tone
equal, both because this temperament has such poor consonances, and
because its dissonances are not intense enough to provide the requisite
tension between consonance and dissonance, which is (within a suitably
tempered, coherent, closed system) what propels music forward both
melodically and harmonically.

Paul Ehrlich asserts that 22-tone equal is the only path to escape
diatonicism. Not so. His categorical attitude is refreshing, but 22-tone
equal is not a temperament at all, but a mere tuning artefact that
reproduces the worst defects of just intonation. Not that I wish to
question the validity of the just ratios as standards for musical
thought.

The two principles that constitute genuine alternatives to diatonicism
are:

1) The chromatic genus, where harmony is required. I have defined the
four modal genera comprising 28 modes that have enough consonant chords
on enough degrees (4 of 7) to be harmonically usable in chromaticism.
Here is as good a place as any to remark that there are two additional
chromatic genera that possess consonant chords on 4 of the 7 degrees
(none have consonant chords on 5 or 7 of the degrees, and only the
diatonic has such chords on 6 of the degrees):

C _Db_ E F G A B C
C Db Eb F G Ab _B_ C

These however each contain a degree (underlined) that is a member of no
consonant chord within the modal genus, and therefore the modes of these
modal genera (scales) could never be established in harmony.

2)the enharmonic genus, where harmony is not an important element. This
involves the use of 1/3 tones, which are never written in our music, but
which fill the living rock melos of our people. No theorist should ever
presume to discount the importance of what the musically untutored
produce from their own melodic inspiration. They stand in need of our
guidance, but we also of theirs.

The 19-tone equal temperament _alone_ can give access to either of these
two genera, just as it alone gives adequate access to the diatonic
genus. This follows from the principle of the melodic limen, and from
the incredibly close harmonic congruence between just intonation and the
19-tone equal. I have not leisure here to treat this in the depth that
it deserves, but would like to observe something that I do not believe
has ever been clearly noticed before. If the consonances of the senario
are each taken as new tonics, we arrive at 19 just intervals within the
octave:

1:1 25:24 16:15 10:9 9:8 16:15 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

These are the intervals that singers can actually sound accurately
singing pure consonances, although there is some debate concerning the
ability to reliably distinguish between the commatically separated
species of tones. Of these, the 19-tone equal merges the minor and major
tones, and also the two species of minor seventh, and intercalates the
augmented tone/diminished third and the augmented sixth/diminished
seventh. Thus this temperament directly corresponds to the most
fundamental intervals of just intonation. No other temperament enjoys
such a direct correspondence, although the 31- and 50-tone (_not_ the
17- 22- 43- or 53-tone) equal are next in the sequence.

Finally, on a lighter note, Carl Lumma was somewhat incoherent in his
remarks. Perhaps he would care to extend on them. Does he have a tuning
system that he thinks is not ridiculous and not
'un-good'?


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