Re: [tuning] Re: Trines and triads (Paul Erlich and Carl Lumma)

Margo!
As fate would have it . Jim French has been making Dulcitars, a drone like instrument with
10 strings in which one side plays a harmonically based scale and the other side a
subharmonic. On one side, the open strings is tuned c-g-c while the other side is tuned g-c-g.
Even though these are the same, having them considered as part of reciprocal series makes the
fretted scales possible. This made me realize that such a chord thought of in a subharmonic
context opens the possibilities toward new modulations as well as harmonic progressions. This
is somewhat similar to the case of the full dim. chord in 12 et. the spelling and conception
(as well as the context) determines how it is used.

"M. Schulter" wrote:

> 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
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

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. Unlike roughness and tonalness, this
component of consonance appears inextricably linked to octave-equivalence,
so (wildly speculating) may reflect some non-linear processing in the brain,
in which period-doubling behavior may be linked to the perception of
especially the lowest sounding note -- and perhaps to melodies as well --
something that could have evolved as an aid to pattern recognition???????