Khroai, etc

Quite minute differences, on the order of 5-10 cents, can effect
dramatic, readily perceptible variations in harmony. I have never denied
this. The ear is far more sensitive to the consonance or dissonance of
simultaneously sounded tones than to that of successively sounded ones.
This is elementary musical acoustics, and need not detain us long.

A more vital question is whether tuning systems such as just intonation,
12-tone equal, pythagoreanism, Werckmeister's tuning and 31-tone equal
differ in melodic effect. I concede that there is certainly some
appreciable difference 'at the edges' as it were. I believe someone on
the list referred to the khroai, the Greek 'colorings' or 'shadings'.
Sometimes - as in just intonation for example - these are readily
perceptible, in the sense that we can detect a melodic fifth that has
been badly mistuned, so that it has the familiar 'wolf' effect. _But_
this fifth is still perceived as a melodic fifth (by everyone I know)
not as a unique melodic interval distinct from both diminished and
perfect fifth. _No one_ can deliberately, accurately sing such a badly
mistuned fifth, or remember it clearly. To deliberately tune instruments
to such values is simply to deliberately mistune them, a pleasantly
heterodox but musically useless proposition.

There is a remote possibility that I could be wrong here; some cultures
have apparently used such fifths, although the evidence is contradictory
and unconvincing. Yet use on instruments of fixed intonation has little
to do with use in the living melos of a culture. Use of fantastic
tunings on instruments proves nothing.

But as far as melodic pitch classes, the evidence is monumentally great
that just intonation, 12-tone equal, pythagoreanism, Werckmeister's
system, and _virtually all other known systems_ reduce to 12 pitch
classes in the octave (at most). 31-tone equal also reduces to 12 pitch
classes, with the caveat that its augmented tone (271 cents, 1 2/5
tones) and to a much lesser extent even its doubly augmented prime (155
cents, 4/5 tone of this system) do seem to acquire, in highly favorable
contexts, a very limited melodic independence. These two intervals are
conjunct, and are perhaps possible to sing by pure dead-reckoning, yet
not wide enough to be confused with adjacent pitches. The 31-tone equal
augmented tone is easily confused with the minor third in melody, but
given a harmonic context that supports it - the usual form of the minor
- the thing is possible. And yet as soon as one leaves the realm of
purely conjunct music, these two intervals become very difficult to
preserve melodically, vis-?-vis the tonic.

To add one or at most two pitch classes that require elaborate care to
use at all, and this at the enormous expense and trouble of using 31
tones in the octave, is veritably Horace's mountain laboring to bring
forth a mouse.

The only systems that _do_ add notably to the number of melodic pitch
classes in the octave are the 19-tone equal and those that very closely
approximate to it.

The difference between the number of heptatonic melodies expressible by
choosing from 19 pitch classes versus 12, is theoretically on the order
of 40 to 1 (not 70 to 1, as I stated earlier; I failed to omit the tonic
from the combination). But many factors - the requirement for a modicum
of consonance, the requirement that modes not bunch all their tones
together in one small region of the octave, etc - reduce the difference
to perhaps more like 10 to 1. Still, 10 to 1 is a vast melodic advance.
This is enough to partly explain why Arabs, Turks and Indians - who seem
to use approximations of the 1/3 tone in their popular melody - so often
find Western classical music melodically crude. There are other factors,
notably the appalling predominance of the ionian mode in classical
Western music. But this is doubtless the major factor.

Rock melody - and also chromatic Renaissance polyphony - non-Westerners
do not as a rule find crude or uninteresting. This may be because rock
singers, and also such composers as Gesualdo, sometimes use 1/3 tones
and other 19-tone equal melodic pitch classes.

I realize that some of my ideas and conclusions are somewhat novel. I
expect no one to shout 'Eureka!" and automatically announce I am 100%
correct. These are my own findings; I believe they deserve careful
study. But if someone wants to refute them totally or in large part , he
has a lot of work to do!

When I began work with tunings I had absolutely no predisposition in
favor of the 19-tone equal. The only thing that might have predisposed
me in some way to enneadecaphony (the 19-tone equal temperament) was the
fact, which I observed at the piano as a boy, that I could easily sing
two, but not three notes 'in between the white keys'. But I filed away
this knowledge as a mere curiosity; for obviously such a division would
have been an unequal 17-tone temperament, with all sorts of problems.
When I finally learned of the 19-tone equal it seemed to me rather
cowardly and insignificant to add only 7 tones to the 12-tone equal!
Also, I associated 19-tone equal with Costeley and Salinas, whom I
assumed would have quite superannuated criteria for selecting a
temperament. The less men know, the more arrogantly do they dismiss the
past, and suppose whatever they may find at random to be of
unimpeachable, precious value. I tended to think of 22- and 24-tone
equal as the initial minimalist systems, with a preference for the
latter, for I had been told so many times that 12-tone equal was
wonderful that I believed it.

In all of these assumptions I could not have been more wrong. The great
paradox here is that by proceeding beyond 19 equal tones in the octave,
we tend to return ourselves melodically to our diatonic and 12-tone
equal starting point, with perhaps one or two of the diatonic modes
improved enough so as to be usable, as compared to the very rough
12-tone versions, and perhaps some slight shadings without melodic
significance. But by exercising a little self-control and _stopping_ at
19 tones, we open up Western instrumental music to the chromatic and
enharmonic worlds - and also dramatically improve the diatonic modes,
making them more tonal and coherent both harmonically and melodically,
both as regards consonance and as regards dissonance. And we also
provide our singers with a temperament that can lead them through all
the chromatic and enharmonic possibilities, not just those that their
own unaided imaginations have devised.

All this was at least glimpsed 400 years ago by the great Francisco
Salinas, whom mankind will one day regard with the same awe that is now
reserved for a Newton or an Einstein.

Even if one is determined to remain within diatonicity, 19-tone equal
effects profound improvements over 12-tone equal. Take the much-maligned
locrian mode for example. In 12-tone equal the fourth degree of this
mode, the perfect fourth, is the only degree definitely related to the
tonic by a strongly consonant interval; the minor third and sixth are
markedly deteriorated.

As an example of the other degrees, take the second degree, a diminished
second. This is most directly related to the tonic via the disjunct
consonantal progression B-E(ascending)-C(descending). There are three
other methods of reaching C, using only consonances, but I refrain from
giving them. But the major third is severely weakened in the 12-tone
equal, so that the progression loses part of its force. This is what I
mean when I say - as I often do - that inferior temperaments 'weaken the
tonal fabric'.

I realize that some want their music to sound as random and inarticulate
as possible, and would be ashamed to win any kind of audience - and so
desire any kind of relation between the divers tones of music to be
abolished. I leave them to their folly. To argue with such people is
like trying to convince a sweet old Southern gentleman that he really is
not Napoleon.

In the 19-tone equal, on the other hand, the second degree of the
locrian is obtainable via two consonances neither of which is noticeably
rougher than just, or rather, the additional roughness is so small as to
impart brilliance, but not enough to weaken the mind's conviction that
consonances are present. The result is that even when the most severe
dissonances occur in the 19-tone equal locrian mode - and it has plenty
of them, enough to sate the most dissonance-loving person - these
dissonances 'hang together': they imply consonance, just as true
consonances can lead immediately to strong dissonances. But this is not
so for the 12-tone equal. There, two 'consonances' (such as two major
thirds) can create a theoretical strong dissonance, the augmented fifth,
that is acoustically a weak consonance.

This is one reason why the 12-tone equal, tolerable in the most
consonant diatonic modes, has such an oddly artificial, neutral effect
when used for the chromatic modes - many of the just harmonies are so
radically confounded that the tonal fabric is rent apart, and randomness
- the enemy of all musical order and expression - appears. Musical
randomness does not have an exciting, daring, aggressive, subtle,
pathetic or delicately esoteric effect. The effect is neutral,
frivolous, feeble, insincere, crude, and grovelling. The best examples I
know are from the chromatic Renaissance. Play over one of Gesualdo's
chromatic pieces in 12-tone equal. (Lassus' Prophetiae Sibyllarum will
do almost equally well.) Then play them again in 19-tone equal with an
octave stretch as I have instructed. The former versions are threatened
at every point with randomness; the latter are never so threatened. This
is the best I can do to make my point clear.

Finally, I would like to warn investigators of the 19-tone equal that
unless one stretches the octave as I have prescribed, or thereabouts,
the fifths (and twelfths) somewhat sour the harmony; in certain timbres
this is enough to be decidedly unpleasant. But if a tuning degree of
63.3 cents is used, with an octave of 1202.7 cents, the harmony is
extremely good. Also, the preconfigured 19-tone equal on the Yamaha
DX7-II with the E! sequencer does not work properly; it has a weirdly
mistuned sound the cause of which I have not been able to ascertain. The
user-configured 19-tone equal sounds fine.


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