tuning in the head [was: Hypermeantones]

In responding to Paul Erlich I shall touch on a subject that was also
discussed a few days ago: a hypothetical scale (or scales) that
would exist in the unconscious process of music perception. This
subject might appear off-topic, but on second thought I suppose it
is relevant to the Tuning List. For one, if such scales
exist, don't we all want to find out their tunings? Besides
that, this conjecture arises out of questions related to tuning,
in more than one way.

I am afraid there is too much "theory" in this message. My apologies
in advance. I had to mention some theory to show how it is related
to tuning -- then I had to reply to Paul.

First, let me clear up a misunderstanding

>> (The stack of three fourths is another story.)
> Why?
> I think in that case the consonance of using a 7:3 for the outer
> interval is a strong motivation to use this temperament.

Good Grief, Paul (=: -- according to our own postings we are in
agreement over the tuning _and_, to a remarkable degree, over the
function of a stack of three fourths.
I have written:
: I have not found either a theoretical or an
: experimental grace in (or about) 27:36:48:64, yet. I have played four
: tones, varying continuously two of them; result: nothing special about
: the above proportion. Actually the sonority got more "interesting",
: albeit more Gesualdo-like, as I contracted the total interval.
: (Might it be C-D-F = {1/8 : 1/7 : 1/6} (common overtone C) with a
: drone G?)
I said "drone" because I have not come to terms with "added" yet.
I note that this separate tone is consonant with the root of the rest.
You have written
> Essentially, the previous chord is 1/1:7/6:7/4. Then, holding these
> notes, the root moves down a "fifth." The temperament is essential
> since it allows the 7/4 to function as a smooth suspended "fourth"
> over the new root [...]

My trouble here is that the notion "function as a suspended x" seems
at odds with dualism. I have found examples in Common Practice that can
be rationalized as suspensions, in the sense that an irrational chord
can be split into two parts, the one belonging to the previous chord
and the other to the next chord. Then, in scanning Gesualdo and some
similar music I got into trouble (I was looking for it, it seems)
because I found irrational chords that could not be rationalized in
the above way. Perhaps I am ripe for a paradigm shift.

I have not managed to listen to your TIBIA yet (hey, Greeks: he means
"AULOS") but it sounds interesting altready. Next, I focus on a
remark you have made about what defines the root of a chord.
I am taking it not in its context but as implying a general conjecture,
that direct acoustical considerations are predominant in the
perception of music, instead of allowing a cognitive representation to
have the last word, say, on what is the root of a chord:

> By emphasizing the low bass note, I'm clearly implying a normal,
> otonal root, rather than a high-pitched common overtone which would
> be the utonal "root".

On the other hand, I do resort to inversional invariance when only
with inversions we can arrive at harmonic functions that fit dualism.
Not that I do not find direct acoustical clues very effective
sometimes! For example, the effect of a minor third on the piano
seems very different in the higher range than in the lower range.
In the first case it sounds like an incomplete major triad, but in
the second case it sounds like the lower part of a minor chord (and
it seems to me like it should be turned into its inverse, the major
sixth). Yet, since I focus on dualism, such clues can at most clear
any inherent ambiguity in my harmonic analysis, not determinine
harmonic functions by themselves.

In other words, I assume the existence of some cognitive "plane"
at which the tones are reperesented as pitches on a cyclical
8ve, and on which the "root" is decided. (This assumption is
unavoidable if one assumes inversion invariance; however,
it might not apply to all kinds of music.) Now it is time to ask
whether that interpretation would be continuous or discrete
(categorical). At this early time I cannot exclude either alternative.
It is like comparing a digital with an analog computer.

The case of a discrete cognitive scale is tricky. So far I have
employed only a 12-ET scale (I am ashamed to admit). For example,
if C is in a chord, the representation of its significant
overtones would be (assuming "small numbers" 3,5,7,9,15, and 17):
{C, G, E, Bb/A#, D, B, Db/C#}, and the process of finding the
"lowest" common overtone of the chord would be reduced to a simple
comparison. Similarly, the process of checking for fundamental bass
would be reduced to comparing lists of undertones, like
{C, F, Ab/G#, D/Ebb, A#/Bb, Db/C#, B}.

If I repeat the same procedure in meantone (say, 19-ET) I sometimes
find awkward intervals between successive "roots". Assuming some
temperament at this point is not a simple matter, because these
roots are already discretized; one would need to assume a
separate cognitive plane to represent roots in a coarser scale.
Therefore I have retreated to exclusive 12-ET for the time, but I
want to revisit the issue before long.

The next quote was near the previous one.

>> Your theory may not allow this interpretation but believe me, as a
>> musician I know it is real.

Of course I take your word seriously and I am deeply obliged. On the
other hand, we all know the sun rises every morning but the verdict of
millions did not hold Copernicus back (hey, Greeks: I meant Pythagoras,
but let us not confuse them now.) Assuming "my" theory does not
contradict your _practice_ (this is still an open issue in the
case of three stacked fourths) then all is well. Perhaps most people
think of music theory as mainly history, but to me theory means
coming up with new suggestions to be tested.

> [...] the seventh approaching a 7:4 [...]
> despite the apparent contradiction
> that the fourths, still heard as 4:3s, would mathematically imply a
> 16:9 seventh.

I understand you like this:
If we could assume that this temperament was adopted in the cognitive
plane, a fourth would be defined as an interval of so many steps rather
than the ratio 4:3, therefore there would be no contradiction.

I just wanted to emphasize the authority of such a process after it
has started running.

Good day!
- George Kahrimanis anakreon@hol.gr