Re: Trines and triads (Paul Erlich and Carl Lumma)

Hello, Paul, and Carl, and everyone.

Please let me begin by thanking you, Paul, for making a very important
point. If Partch defines "otonal/utonal" as a relationship between
sonorities which cannot be obtained from each other by Rameau's
process of inversion (octave transposition of voices), then indeed
this concept does _not_ properly apply to 3-limit trines.

Accordingly, respecting Partch's premises and definitions, I would
amend my original position to say that the contrast between the two
"flavors" of trines I describe may be in some ways musically
_analogous_ to that reflected by the "otonal/utonal" concept with
which I am familiar mainly from discussions on this list.

At the same time, Carl, I would like to thank you for sharing your own
musical perceptions and helping me to validate my own feeling that the
contrast between the two basic flavors of 3-limit trines and 5-limit
triads are indeed analogous.

Rather than using the "otonal/utonal" concept where Paul has taught me
it does not properly apply, I might better present an analysis using
the terms and concepts of medieval and Renaissance theory, the basis
of much of my own outlook. This process may require a number of posts,
somewhat technical but I hope also readable.

Here I would like mainly to answer a few of Paul's questions about
matters of musical feeling. The question of artistic perceptions and
motivations may be at least as important as the mathematical logic of
a given approach to theory, and may thus be an ideal place to start.

First, I would say that in a Gothic setting, a 3-limit trine with
octave, fifth, and fourth sounds indeed more "complete" to me than a
simple fifth or fourth -- rather like a 5-limit triad is more
"complete" than a simple major or minor third. A medieval theorist
such as Johannes de Grocheio (1300) likewise says that _three_ voices
are required in order to "perfect a consonance" -- that is, to achieve
full saturation by sounding the musical "Trinity" of the octave,
fifth, and fourth simultaneously.

Further, I would say that a simple fifth has a flavor musically akin
to that of a complete trine with the fifth placed below and the fourth
above; and a simple fourth to the converse arrangement with the fourth
below and fifth above. This is analogous to the kinship of the simple
major third to a triad with the major third below and the minor third
above; and conversely for the simple minor third.

Using my notation for three-note/interval sonorities where the
intervals are shown as (outer|lower + upper), these affinities might
be shown as follows, with examples of trines and triads given in MIDI
notation where C4 indicates middle C:

Limit Simple interval Kindred trinic/triadic flavor
----- --------------- -----------------------------

3 5 (8|5 + 4), e.g. D3-A3-D4
3 4 (8|4 + 5), e.g. D3-G3-D4

5 M3 (5|M3 + m3), e.g. G3-B3-D4
5 m3 (5|m3 + M3), e.g. G3-Bb3-D4

Note that a complete 3-limit trine includes two richly stable 3-limit
intervals of a similar but distinct quality, the fifth and fourth,
plus an interval from the next lower limit, the 2-limit octave.
Similarly, the complete 5-limit triad includes two similar but
distinct 5-limit intervals, the major and minor third, plus an
interval from the next lower limit, the 3-limit fifth.

In either 3-limit or 5-limit music, I would make a distinction between
a "simple" two-voice interval and a "bare" or "open" one. A fifth or
fourth in 3-limit (trinic) music, and likewise a major or minor third
in 5-limit music, represents a richly stable interval. Thus I would
call such an interval "simple" (in comparison to a complete trine or
triad), but not "open" or "bare."

In contrast, from a musical as well as logical point of view, I might
indeed speak of a "bare" or "open" octave in trinic music, and
likewise of a "bare" or "open" fifth in triadic music -- respectively
a 2-limit interval in a 3-limit setting, or a 3-limit interval in a
5-limit setting. As the term "open" may suggest, in either case such
an interval seems to invite the addition of a third middle voice
dividing it into the richly euphonious fifth and fourth of a complete
trine, or major and minor third of a complete triad.

From a psychological point of view, I suspect that the 2-limit octave
may serve as an integral interval of a 3-limit trine because it is
only one step below the limit of saturation; thus it has a greater
sonorous impact than when added to a 5-limit triad, where it is two
steps removed from the limit of saturation.

Terminology and math aside, the importance of the octave in a complete
trine really hit me when I heard a performance of one of Perotin's
three-voice organa on an album called _Vox humana_. To open the piece,
the ensemble played first the lowest voice alone, then added the
second voice at the fifth -- and only after allowing some time to
appreciate this process did they finally add the third voice at the
octave, triumphantly completing the trine! The richness and resonance
was transcendent.

Thus in trinic music I regard the octave as an integral interval not
only because it fits medieval theory and has mathematical attractions,
but because it concords with my musical experience. In 5-limit or
higher music, treating the octave as more of a mere "doubling" or
"replication" may reflect its lesser perceived sonorousness vis-a-vis
a unison as the texture becomes more dense, the norm of stable
saturation more complex.

Indeed, such differences should caution us that analogies are just
that, as opposed to equations: 3-limit and 5-limit musics are
different systems, and one can focus on either the differences or the
similarities.

Most respectfully,

Margo Schulter
mschulter@value.net

>Ultimately, I have to agree with Margo, and the psychoacoustical explanation
>cannot be found in the concepts of roughness or tonalness. There appears to
>be an additional phenomenon, "rootedness", which gives the lowest sounding
>note an extra probabilistic boost in its likelihood of being interpreted as
>octave-equivalent to the fundamental. This would explain why, for example,
>4:5:6 is more "consonant" than 3:4:5.

Are we sure that 4:5:6 will not have higher tonalness than 3:4:5, when
triadic harmonic entropy appears?

Further, since tonalness is defined as a measure of the "clarity" of the fundamental, should rootedness be a new concept, or should our tonalness
metric include a fix for this phenomenon (the fundamental is more clearly
resolved when the lowest note sounding is octave-equivalent to it)?

Lastly, would removing all factors of 2 from a chord, except those
which preserve the ordering of its identities (leaving the factors
whose removal would invert the chord) before performing tonalness
measurements be a workable fix?

-Carl

>>Lastly, would removing all factors of 2 from a chord, except those
>>which preserve the ordering of its identities (leaving the factors
>>whose removal would invert the chord) before performing tonalness
>>measurements be a workable fix?
>
>Ugly.

Agree, but if you want it separate from tonalness, I fear it's going
to be ugly.

-Carl