TUNING digest 852: _Stability_

I will write a more detailed response when I can get away from some urgent
business.

(1) I am rather apprehensive about the use of _stability_ to characterize a
single chord. From Pauls definition, a /4:/5:/6 minor triad, lacking a
power of two in the lowest voice, is less _stable_ than a Major 4:5:6.
However, when placed in a progression, it may very well have the opposite
reading. For example, if the tonic key is established as minor, than the
Major dominant triad will _resolve_ to the tonic minor triad. Of course the
root motion of the downward fifth (or upward fourth) is a strongly defining
feature, but this ordering of chord qualities is well established in the
_common practice_ repertoire (or else we would find more mixolydian
cadences). In this case, key identity, spacing, and voice leading from one
chord to the next are equally or more determinant of _stability_ than
rootedness. Further, all of these determinants seem largely to be examples
more of conditioning rather than of psychoacoustic properties. (The
existence of alternative practices in other repertoires (eg _modal_
progressions in pop musics) both confirms the locality of the conditioning
described above, and the varieties of progressions encountered in these
repertoires suggest the inapplicabilty of a _stability_ concept outside of
narrowly defined diachronic contexts).

(2) On the other hand, _stability_, in a musical context without
repertoire-defined harmonic progression may be achieved by means of
extended durations or by reiterations. In this case, the conditioning is
taking place in real time (Pavlov meets Stravinskys _assertion_ of
tonality), and the examples of this that I find immediately suggest that
lowest voice, 2^n roots are not a prerequisite to achieving stability.

(3) Back to sine waves. The argument that Paul used against my sine wave
subharmonics can be turned around and applied to regular harmonic-spectra
harmonic pitch complexes, where the reinforcement of a single series will
create masking so that individual pitches will be difficult if not
impossible to discern. (This was my initial rationale for the _sine wave
orchestra_; long experience with synthesized sound and precise tuning had
convinced me that the situation has to be simplified in order to get any
useful results at all).

(4) The intuitionist (like an algorithm maker, or a student of musical
cognition) proceeds in strictly chronological steps. If from the unity 1:1,
the twoity is observed, and from the sum of 1+1, 2 is constructed, yielding
the first harmonic interval, only then may the first subharmonic (the
inversion) be constructed. In constructing the series, the subharmonic will
always lag behind. Of course, another fundamental approach will have all
intervals appear simultaneously, but I am lost in trying to figure out how
that is described cognitively.

(5) I never said that mathematics goes funny at the infinitesimal, I made
the observation that the calculation in this direction is counterintuitive,
and I speculated about why it should be so. Your examples of string lengths
etc. I found curious, as the (pre-wave length) music theory using this
instrumentality was inversionally flexible in the extreme, and ran counter
to Pauls strong harmonic series approach. (Needless to say, classical
harmonics was used to describe a repertoire without chords). There is
indeed a (mathematical and physical) boundary problem for the infinitesimal
(with null or the vacuum) that the infinite does not have.

Daniel Wolf

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> My experience with non-octave scales (and
> I'm still playing with them) is that although you can set up functional
> harmonic relationships and weird but beautiful melodies without octaves,
> once you "modulate" far enough to use an interval which is a near-octave,
> that won't sound like a distant harmonic relationship but like an
> out-of-tune version of the same note.

I absolutely agree with Paul here; off-octaves (in 88CET at least) to my
ears really do sound like out-of-tune octaves rather than pitch relationships in
themselves. Some have disagreed however.

In my 88CET demo tape I have an example of traditional harmonies as rendered
in 88CET. The first ... six I think it was ... chords illustrate this
"wandering tonic" effect. The progression I think was a typical I V ii V V7 I.
The trick is that the second I chord is written with the tonic in a different
octave from the first. The effect I think is really wild and fascinating: If
you're not paying much attention, it just sounds like a boring old chorale
fragment.

But if you listen more carefully (and only a little more carefully), you
notice that the tonic mysteriously migrated by about 1/41 octave. All of our
musical experience since childhood tells us that we just went 360 degrees around
and landed where we started, but mysteriously we somehow landed somewhere else.


Forgive me for pushing one of my proverbial hot-buttons, but some feel that
this wandering tonic effect is a problem to be dealt with, but to me it's a
really interesting, surprising, and useful musical effect.


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