Query to Bruce K.

Bruce: Could you post the Gurdjieff scale? I'm not quite sure
how to interpret a "scale of 7 tones/octave, 5 'half tones' and
14 'quarter tones' over a 7 octave range," and I'd like to
see the notes and intervals.


--John


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I have spent a bit of time trying to figure out why there seems to be a
preference for a triadic (4-5-6-8) over a tetradic (4-5-6-7-8) octave
infill in tonal musics. Perhaps this was not exactly the right question,
and it should rather be: why is there a preference for an infill that
leaves a gap at top (i.e. 6-8 is left unfilled by 7)? Could there be an
explaination that does not go into supposed numerological properties of the
number 7?

The best explaination of the infill question I have come up with is that
the infill is heirarchic, following the harmonic series: an octave infill
adds the fifth (2-3-4) and the next infill in of that fifth, not of the
octave (4-5-6-8), and the next infill (theoretically) would be of the third
(8-9-10-12-16)(which can be heard as a stable chord).

It is perhaps trivial to note that these infills are freshman sums.

There seems in this procedure to be a kind of equilibrium building, in that
the infill of the lowest interval counteracts the harmonic series effect of
increasingly smaller intervals at top. Perhaps this spacing equilibrium can
partially account for the prefered use of triads in the resolving position
of tonal cadences.

This procedure inverted appears to be effective for subharmonic sonorities
as well.

Now, if we would prefer to use 12th equivence instead of octave
equivalence, how might our infill procedure change? One immediate
possibility comes to mind: maintain the regular harmonic series and the
infill procedure will be very similar to the octave version 1-3, 1-2-3,
2-3-4-6, 4-5-6-8-12.

And for kicks, here is a version for 7/4 equivalence: 4-7, 8-11-14,
16-19-22-28, 32-35-38-44-56

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As for why there's a historical preference for using triadic rather than
seventh-chord fills within an octave's span, I think that that's pretty
much historical. Seventh chords, especially those with diminished fifths,
have been regarded as dissonances that need to be resolved to triads.

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Gary Morrison wrote:

'' As for why there's a historical preference for using triadic rather
than
seventh-chord fills within an octave's span, I think that that's pretty
much historical. Seventh chords, especially those with diminished fifths,
have been regarded as dissonances that need to be resolved to triads. ''

Yes, but this still begs the question of why this particular preference
appeared and not others. I was trying to argue from first principles about
the infill phenomenon using a spacing model, not an intonational model (for
which Pythagorean and 5-limit just intonations would provide a familiar and
convincing argument for western music through the early baroque). For later
music, where sonorities akin to 4-5-6-7 become common place, it is striking
how these are left in the _dissonant_ category and required to _resolve_.
Even vernacular musics tend not to end on seventh chords. Either this is
evidence of a deep conditioning from common practice music, or there is
some other mechanism at work. Possibilities which spring immediately to
mind include (1) a problem with ratios of seven (or their tolerable
approximations), or (2) a spacing function (which I have called variously
"compacting" or "optimizing"). Any other ideas?

Another possiblity - and one purely from an intonational standpoint - is
that the ET approximation is bad enough to shove the seventh into a
_dissonant_ category and to demand resolution. In this case, however, the
resolutions of seventh chords occuring in real, existing Just intoned
musics - chorales or Barbershop quartets - retain the avoidance of sevenths
in final chords (except for special effects). Is this just conditioning
from the the familiar temperament? Does anyone know of a way to test this?

(Unfortunately, music that I am familiar with using justly intoned seventh
chords (Partch, Johnston, Young, Leedy, my own) is often highly ambiguous
about the whole issue of _resolution_. I do detect some tendency to respect
harmonic series spacing in the voice leading, but this is as yet more of an
impression than an established statistical fact).

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> Yes, but this still begs the question of why this particular preference
> appeared and not others. I was trying to argue from first principles
about
> the infill phenomenon using a spacing model, not an intonational model

Well, I suspect that you're probably better off with a largely
historical model rather than any sort of theoretical one. Once people
became convinced that building chords in thirds was the way to go, they
started with smaller stacks of diatonic thirds and later progressed to
larger ones.

In other words, it may just be a matter of "well, that's just how it
happened" more than there being really clear reasons. But there certainly
is merit to the theoretical possibilities you're suggesting.

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I was rereading old messages.

Daniel Wolf wrote about "Triadic or tetradic infill?":

> Another possiblity - and one purely from an intonational standpoint - is
> that the ET approximation is bad enough to shove the seventh into a
> _dissonant_ category and to demand resolution. In this case, however, the
> resolutions of seventh chords occuring in real, existing Just intoned
> musics - chorales or Barbershop quartets - retain the avoidance of sevenths
> in final chords (except for special effects). Is this just conditioning
> from the the familiar temperament?

[separated for emphasis]

> Does anyone know of a way to test this?

Here's sort of a way:

One of the science fiction stories which most disturbed me (you
might want to skip reading it) was about a guy born
into a society which kept track of the genetic
lineages and predispositions of its members,
and occasionally, when a chid was born with the
potential to become a composer (or other artist) of significance,
the child was removed from the others and most of their cultural
influence, and raised in a separate dwelling, tended to only by remote
control. In the dwelling of this one child was placed an instrument
designed with the potential to produce an almost limitless range of
timbres and tunings, and to be easily programmable.

As this child grew, he discovered
the instrument and began playing around on it with no external
guidance and no preconceived notions of what it was or what
should be done with it. As the child began to produce completely
original unique inspired music, the rest of the society
voyeuristically and vicariously fed off of his captive talents.

Human nature being what it is, such captive artists often
succumbed to deep loneliness with no understanding of what
or why it was. Some would eventually stop producing and
pass into a sort of living death. Occasionally, a rare
one would find the imagination to suppose that there were
something beyond the small world he knew and would effect escape.
The child in the story is one such person. As a young man,
he escapes. After wandering dissoriented and with no idea
of how to meet his own basic needs, he ends up in some local
bar where an old out of tune acoustic piano collects dust in
the corner. The young man sees people, but doesn't know how
to relate to them or perhaps even what they are. They
don't recognize him and after their initial greetings are met
with a confused stare from the young man think him
an idiot, and leave him alone. He
finds the piano, and begins plunking all the notes. He
quickly learns what all the (out of tune) pitches are and
begings playing them in such a way as to make the people in
the bar turn around and be mesmerized by such unusual and
acute beauty. Some of them start to cry without knowing why.

He wanders more. With each hour on the outside, impressions
from radios and other raw noise of the culture begins to
change him.

Soon, his absence from the dwelling is noticed and he is
persued and caught and returned to his old small world.
His brief experience outside has infected his creativity.
People listening in notice strains of familiar melodies
creeping into his improvisations, imitations of vehicle
sounds, and other recognizable noises.

I think he escapes one more time and is caught again.

His priceless
purity of imagination is tainted. Being polluted beyond any
hope of being cleansed, his music has become too ugly to listen
to. The travesty of must stop. He must be punished for
escaping and prevented from producing any more ugliness.
They cut off his hands.

[I know, and I'm sorry. I didn't write it, I just paraphrased it. It
originally appeared in Omni magazine, issue and author forgotten.]

Matt Nathan

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On Thu, 13 Feb 1997, Matt Nathan wrote:
> One of the science fiction stories which most disturbed me (you
> might want to skip reading it) was about [ . . . ]

[loose paraphrase deleted]

The story Matt describes is "Unaccompanied Sonata" by Orson Scott Card,
which is collected in the eponymous anthology, and probably other places
as well. I would _not_ recommend skipping it, though; it is an
excellent story.

> His priceless
> purity of imagination is tainted. Being polluted beyond any
> hope of being cleansed, his music has become too ugly to listen
> to. The travesty of must stop. He must be punished for
> escaping and prevented from producing any more ugliness.
> They cut off his hands.

This isn't quite correct. Actually, at first all that happens is that
one of his Listeners sneaks him a tape of Bach . . . and, afraid of
being discovered, he gives himself away when he stops experimenting with
fugue, and stops inventing new timbres for fear of reproducing a
harpsichord. They just send him away from his home and his Instrument,
his life as a composer, after that. The incident with the piano was his
second offense, and that's when they cut off his fingers. His third
offense is . . . and then after _that_ . . .

Read the story. It _is_ disturbing, but it's also one of the best
damned stories you'll ever read.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Do you like to gamble, Eddie?
-\-\-- o Gamble money on pool games?"

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