Tritriadics, etc.

Re Tritriadics :The idea for tritriadics is from A. J. Ellis's
appendix to Helmholtz's On the Sensations of Tone in which he derives
56 tonal 7-tone "trichordal" modes in 5-limit JI by allowing the tonic,
dominant and subdominant triads to either major or minor and then
circularly permuting each scale (8 x 7 = 56). The concept goes back to
Rameau. What I did was to generalize the type of triad and apply the
idea to MOS and similar scales in ET as well.

One is not limited to only 3 triads or to triads. Some scales are
best chorded with 5 triads, 4 tetrads, etc.

Max Mathews, Linda Roberts and John R. Pierce have also experimented
with this kind of scale, especially those based on 3:5:7 and 5:7:9.
The Pierce-Bohlen scale in based on 3:5:7:9 triads mapped into the
13th root of 3/1 (This ET has also been proposed by Enrique Moreno).
Unfortunately, none of their published papers actually give the
ratios or notes of the 4 scales they discovered. Does anyone have
the M357a, M357b (the P-B scale in JI?), M579 and P3579a scales?

JI: One of the major themes of Brian's recent posts on psychoacoustics
was that subjects tested in the laboratory prefer sharp to JI intervals.
While there is considerable data supporting this idea for single voice
melodic intervals, it is not necessarily true for intervals in chords.
Roberts and Mathews (1984) found that for the middle tone of various
triads (major, minor, 5:7:9 and 3:5:7) some listeners preferred JI and
some preferred intervals either sharp or flat by about 15 cents.
Preference correlated with musical experience, violinists, cellists and tympanists preferred JI, untrained musicians, singers and pianists
preferred the "rich" non-JI intonation. In these experiments the outer
tones were tuned pure and the inner tones varied. In the case of minor
triads, the "rich" intonationists preferred the flat 300 cent interval
to the JI value.

Similar results were obtained for chords in I IV I progressions.
The roots of the triads were 262 hz. In the case of the progressions,
the subdominant was taken from a major or natural minor scale on
262 hz. The 3:5:7 and 5:7:9 progressions appear to be taken from the corresponding tritriadics, but I have not been able to verify this
yet.

The point is that while musical experience has a strong effect
on preference, though how one separates musical experience clearly
from musical talent has never been clear to me, some listeners not
only prefer JI for major and minor chords, they also do so for 7-limit
chords as well. Secondly, while some listeners prefer sharp
intervals, others prefer flat ones or have no preference for the
direction of mistuning. They just like beats. These data imply that
JI has a distinct "mood" which many, but not all, listeners prefer.

The principal reference is Linda A. Roberts and Max V.Mathews. 1984.
Intonation Sensitivity for traditional and nontraditional chords.
J Acoust. Soc. Am. 75(3): 952-959.

--John


Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Wed, 15 Nov 1995 20:29 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id KAA16182; Wed, 15 Nov 1995 10:28:52 -0800
Date: Wed, 15 Nov 1995 10:28:52 -0800
Message-Id: <9511151028.ZM28007@frollo>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

> The 12 tone equal temperament scale is derived from the overtone
> series of a vibrating string.

Of course most of us tuning listers will be prompt to point out that 12TET
doesn't provide a *really great* harmonic series approximation. The 5th
harmonic is way sharp and it misses some really neat harmonics - 7, 11, and 13
for example. It does, however, provide a fairly EFFICIENT mapping to the
harmonics it does match - matching the first 5 harmonics reasonably well on as a
coarse resolution as 100 cents, is in my view an impressive feat.

Buuuut, that's not the central focus of your question. In response to your
question, I suppose it's worth pointing out that Brian is speaking about a
classification of tunings that ... well, I was starting to say "intentionally
avoid" harmonic relationships, but I suspect it would be more appropriate to
say, "make no effort to match" harmonic relationships.

I'm still catching up on Brian's NJNET posts, so I'm not certain of the
context behind his allusion to different vibrational modes. Buuuut, everything
I've read of those posts so far suggest that they are based upon abstract
mathematical formulations (other than those of JI and ET), and not upon "found"
pitch relationships, by which I mean pitch relationships found in natural
vibrational systems.

That is unless of course he's found a NJNET mathematical formulation for the
vibrational modes of spheres or whatever. I kind of doubt if those vibrational
modes are predictable by formulations like the inharmonic series formulations
he's mentioned (such logs of factorials).

So I kinda doubt if vibrational modes of physical structures other than
strings is his main point really. Unfortunately, since I haven't gotten that
far yet, I'm not much good for commenting on what he was speaking of there.


Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 16 Nov 1995 08:30 +0100
Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI)
for id WAA25062; Wed, 15 Nov 1995 22:30:00 -0800
Date: Wed, 15 Nov 1995 22:30:00 -0800
Message-Id: <951116062526_71670.2576_HHB31-4@CompuServe.COM>
Errors-To: madole@ella.mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu