Re: Otonal/utonal pairs: 3-limit trines, 5-limit triads

Hello, there, and I'd like briefly to reply to a question here raised
as to whether 3-limit otonal and utonal sonorities are the same. My
conclusion is that, as in the 5-limit case, these sonorities are
distinct, in theory and in musical practice.

While my grounding is more in medieval and Renaissance approaches to
the arrangements of intervals in sonorities than to the more recent
otonal/utonal approach, I do not necessarily see these approaches as
irreconciliable, although they are distinct, and may sometimes focus
on different nuances or shades of musical discernment. Since the
question raised is framed in terms of otonal/utonal concepts, I will
try to reply mainly from this point of view.

In medieval European composition (c. 1200-1420) based on a 3-limit
system, the three-voice unit of complete sonority is the _trina
harmoniae perfectio_ or "threefold perfection of harmony" (Johannes de
Grocheio, c. 1300), including an outer octave plus an adjacent fifth
and fourth. Note that this sonority includes all three of the
stable nonunisonal concords not exceeding the range of an octave: the
octave (2:1), fifth (3:2), and fourth (4:3). For example, using a MIDI
notation with C4 as middle C and higher numbers showing higher
octaves, here is a complete trine on D:

D4
A3
D3

As two other theorists of the same era, Coussemaker's Anonymous I
(c. 1300) and Jacobus of Liege (c. 1325), take note, in this saturated
sonority either the fifth may be placed below and the fourth below,
the more smooth and pleasing arrangement, or they may be arranged _e
converso_ ("conversely") with the fourth below:

3-limit trines (8, 5, 4)
------------------------

otonal utonal

| D4 | D4
| 4 (4:3) | 5 (3:2)
8 (2:1) | A3 8 (2:1) | G3
| 5 (3:2) | 4 (4:3)
| D3 | D3

(8|5 + 4) (8|4 + 5)

Note that in this diagram I use the set notation (8, 5, 4) to indicate
either kind of trine: this notation shows a sonority with the three
intervals of octave, fifth, and fourth, but leaves open the
arrangement of these elements. To indicate the different arrangements,
I use the form (outer|lower + upper): thus (8|5 + 4) is the more
blending and conclusive form with the fifth below and fourth above,
while (8|4 + 5) is the less conclusive form with the fourth below.

From the Latin _e converso_ of Anonymous I and Jacobus, we can derive
the term "conversity" to describe two sonorities which present the
same set of intervals in different arrangements or orderings. Here the
otonal (8|5 + 4) trine and the utonal (8|4 + 5) trine are
"conversities" of the common set (8, 5, 4).

In practice, a composer such as Perotin (c. 1200) may use both forms
of trines prominently, but with the more blending otonal form
consistently preferred as a concluding sonority. Thus otonal and
utonal 3-limit forms are indeed musically distinct in practice and
theory alike.

Likewise, the 5-limit practice of 16th-century Europe is based on a
complete three-note sonority which Zarlino (1558) describes as
_harmonia perfetta_ or "perfect harmony," and Lippius (1612) as the
_trias harmonica_ or "harmonic triad," a name which has stuck. This
combination consists of an outer fifth divided into a major third and
a minor third, e.g.

D3
B3
G3

As with the 3-limit trine, so with the 5-limit _harmonia perfetta_ or
triad, the otonal and utonal forms are conversities sharing the same
set of intervals (5, M3, m3). In the otonal form, deemed more
"natural" and "joyful" by Zarlino and Lippius alike, the major third
is placed below and the minor third above; in the utonal form, deemed
more "sad" or "languid," the intervals are arranged conversely:

5-limit triads (5, M3, m3)
--------------------------

otonal utonal

| D4 | D4
| m3 (6:5) | M3 (5:4)
5 (3:2) | B3 5 (2:1) | Bb3
| M3 (5:4) | m3 (6:5)
| G3 | G3

(5|M3 + m3) (5|m3 + M3)

Zarlino emphasizes that the "variety of harmony" caused by the
contrast between these two arrangements in the course of a composition
is a vital musical factor; various theorists express a preference for
the more "natural" or sonorous arrangement with the major third below
at points of cadential arrival.

If we consider the harmonic series, it is interesting that the otonal
3-limit trine consists of the three adjacent partials 2,3,4; and the
otonal triad likewise of the three adjacent partials 4,5,6. As I will
discuss in another article, Zarlino's series of "sonorous numbers"
yields essentially the same result. Both of these forms have as their
lowest note an even octave of the fundamental (2 for the trine, 4 for
the triad). In contrast, the utonal trine (3,4,6) or triad (10,12,15)
has for its lowest note a partial which is not an even octave of the
fundamental.

Most respectfully,

Margo Schulter
mschulter@value.net