Re: Conversity and inversion

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Conversity and inversion:
Overlapping affinities
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Discussions of tuning theory may often touch on two distinct but
sometimes overlapping forms of relatedness between sonorities:
conversity and inversion.

Simply stated, _conversity_ is a relationship between sonorities which
share the same set of intervals in different arrangements; _inversion_
is a relationship between sonorities sharing the same set of pitch
classes in different arrangements.

While the concept of conversity plays an important role in medieval
and Renaissance theories of multi-voice sonorities, the theory of
inversion gets introduced by early 17th-century theorists such as
Johannes Lippius (1610, 1612) and made a central concept by Rameau
(1722 and later).

Both concepts can be relevant to current tuning theory, and it is my
purpose here to show how two sonorities may sometimes be both
conversities and inversions, sometimes conversities but not
inversions, and sometimes inversions but not conversities.

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1. Basic concepts and notations
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Any multi-voice sonority may be described either as a set of tones or
as a set of vertical intervals. For example, let us consider the
sonority

| D4
| 5
m7 | G3
| m3
| E3

with C4 indicating middle C, and higher notes showing higher octaves.
As the diagram shows, we have three notes (E3, G3, D4) and three
intervals: a minor seventh, minor third, and fifth (m7, m3, 5).

To show the ordering of these intervals, we may use the notation
(m7|m3 + 5), which might be read, "an outer minor seventh divided by
the middle voice into a lower minor third and upper fifth."

If we build another sonority with the same interval set (m7, m3, 5) in
a different arrangement, a conversity results. Let us build such a
conversity while retaining E3-D4 as the outer minor seventh:

| D4 | D4
| 5 | m3
m7 | G3 m7 | B3
| m3 | 5
| E3 | E3

(m7|m3 + 5) (m7|5 + m3)

Our second sonority E3-B3-D4 has the same set of intervals -- but with
the fifth now placed below and the minor third above.

From a perspective of tuning theory, an interesting feature of
three-voice conversities is that the frequency-ratio of a sonority
will be equal to the string-ratio of its conversity, and vice versa.
Thus for our (m7|m3 + 5) and (m7|5 + m3) forms as tuned in 3-limit,
5-limit, and 7-limit ratios:

Limit Ratios for (m7|m3 + 5) Ratios for (m7|5 + m3)
----- ---------------------- ----------------------
3 frequency: 27:32:48 string: 48:32:27
string: 32:27:18 frequency: 18:27:32

5 frequency: 5:6:9 string: 9:6:5
string: 18:15:10 frequency: 10:15:18

7 frequency: 12:14:21 string: 21:14:12
string: 4:6:7 string: 7:6:4

While conversities share the same set of intervals in different
arrangements, inversions share the same set of pitch classes. Thus our
E3-G3-D4 sonority has the pitch class set (E, G, D), and we may, for
example, transpose the lowest note up an octave to form an inversion
of this sonority:

| D4 | E4
| 5 | M2
m7 | G3 M6 | D4
| m3 | 5
| E3 | G3

(m7|m3 + 5) (M6|5 + M2)
(E, G, D) (E, G, D)

Note that while E3-G3-D4 and G3-D4-E4 share the common set of pitch
classes (E, G, D), and thus are inversions, they do not share the same
set of intervals -- (m7, m3, 5) and (M6, 5, M2) respectively -- and
therefore are not conversities.

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2. When is a conversity also an inversion?
------------------------------------------

Some stable and mildly unstable 3-limit sonorities of 13th-14th
century Gothic polyphony in Western Europe have the interesting
characteristic of being analyzable as either conversities or
inversions.

Thus most obvious example might the complete stable trine, with an
outer octave, lower fifth, and upper fourth (e.g. D3-A3-D4) -- or with
the fourth conversely placed below and the fifth above:

| D4 | D4
| 4 | 5
8 | A3 8 | G3
| 5 | 4
| D3 | D3

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

In determining whether (8|5 + 4) and (8|4 + 5) might be inversions as
well as conversities, the test is whether we can build one form from
the other using the same set of pitch classes.

Starting from the (8|5 + 4) form, D3-A3-D4, for example, we can do
this by taking the upper fourth A3-D4 of this sonority, making this
the lower interval of our new sonority, and adding the upper fifth
D4-A4:

| D4 | A4
| 4 | 5
8 | A3 8 | D4
| 5 | 4
| D3 | A3

(8|5 + 4) (8|4 + 5)
(D, A) (D, A)

Both sonorities share not only the same interval set (8, 5, 4) but
the same pitch class set (D, A) -- so that they are inversions as well
as conversities.

This example illustrates an important point about pitch class sets in
general and 3-limit trines in particular. At least for me, the
sonorities D3-A3-D4 and A3-D4-A4 intuitively have different pitch
contents: "two D's an octave apart plus a single A" in contrast to
"two A's an octave apart plus a single D."

From the viewpoint of inversional affinity, however, both D3 and D4 in
the first sonority represent the same pitch class D; and likewise A3
and A4 in the second sonority the same pitch class A. With pitch class
sets, following mathematical set theory, the ordering of elements is
irrelevant and repeated elements are notated only once. Thus the
trines D3-A3-D4 and A3-D4-A4 share the pitch class set (D, A), and are
inversions as well as conversities.

Three-limit trines have three notes (e.g. D3-A3-D4 or A3-D4-A4) and
intervals (8, 5, 4), but only two pitch-classes (e.g. D, A). This last
property distinguishes them from 5-limit triads, which similarly have
three notes (e.g. D3-F#3-A3 or D3-F3-A3) and intervals (5, M3, m3),
but also three pitch-classes (e.g. D, F#, A).

In addition to the stable trines, two relatively concordant but
unstable Gothic sonorities mentioned by Jacobus likewise are at once
conversities and inversions. These sonorities consist of an outer
fifth plus a middle voice dividing this fifth into a fourth below and
a major second above, or the converse:

| D4 | D4
| M2 | 4
5 | C4 8 | A4
| 4 | M2
| G3 | G3

(5|4 + M2) (5|M2 + 4)

Here the fifth and fourth provide euphonious concord, while the
relatively tense major second (9:8) adds color and excitement. The
mirror symmetry of the two arrangements is reflected in the symmetry
of their string-ratios and frequency-ratios:

Ratios of (5|4 + M2) Ratios of (5|M2 + 4)
-------------------- --------------------
Frequency: 6:8:9 String: 9:8:6
String: 12:9:8 Frequency: 8:9:12

Are these conversities sharing the interval set (5, 4, M2) also
inversions? We can demonstrate that they are by starting with our
first arrangement G3-C4-D4 (5|4 + M2), and transposing the lowest note
up an octave to arrive at C4-D4-G4 (5|M2 + 4):

| D4 | G4
| M2 | 4
5 | C4 8 | D4
| 4 | M2
| G3 | C4

(5|4 + M2) (5|M2 + 4)
(C, D, G) (C, D, G)

Since both sonorities share the same pitch class set (C, D, G), they
are inversions.

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3. When is a conversity not an inversion?
-----------------------------------------

In Western European composition, the most familiar example of a
conversity which is not also an inversion may be the two flavors of a
sonority described by Jacobus in a Gothic context as a _quinta fissa_
or "split fifth" divided by a middle voice into two thirds, by Zarlino
in a 16th-century setting as _harmonia perfetta_, and by Johannes
Lippius (1612) as the _trias harmonica_ or harmonic triad.

Whether it is used as a mildly unstable 3-limit sonority, as in the
13th and 14th centuries, or a fully concordant 5-limit one, as in the
Renaissance and later eras, this three-note sonority has the interval
set (5, M3, m3), with the major third placed below and the minor
above, or the converse:

| A3 | A3
| m3 | M3
5 | F#3 5 | F3
| M3 | m3
| D3 | D3

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

Both the 3-limit and 5-limit versions of this combination, and also
the 7-limit versions familiar in some 20th-century music, have the
symmetrical ratios characteristic of three-voice conversities:

Limit Ratios for (5|M3 + m3) Ratios for (5|m3 + M3)
----- ---------------------- ----------------------
3 frequency: 64:81:96 string: 96:81:64
string: 81:64:54 frequency: 54:64:81

5 frequency: 4:5:6 string: 6:5:4
string: 15:12:10 frequency: 10:12:15

7 frequency: 14:18:21 string: 21:18:14
string: 9:7:6 frequency: 6:7:9

While the term "triad" is sometimes used to describe any three-voice
sonority, application of this term to the medieval _quinta fissa_
combination may unfortunately suggest to many people the idea of the
later stable 5-limit _harmonia perfetta_ or triad. Using the term
_quinta fissa_ or "split fifth" in a 3-limit Gothic setting avoids
such an implication.

Whatever the stylistic context and tuning limit, we find that the two
arrangements (5|M3 + m3) and (5|m3 + M3) are conversities but not
inversions. In other words, there is no way of rearranging the pitch
classes forming one arrangement so as to derive a sonority with the
same pitch class set but the converse arrangement.

If we start with D3-F#3-A3, for example, which has the major third
below and minor third above (5|M3 + m3), there is no way we can
rearrange the pitch class set (D, F#, A) so as to produce the converse
flavor with minor third below and major third above (5|m3 + M3):

| A3 | D4 | F#4
| m3 | 4 | M3
5 | F#3 m6 | A3 M6 | D4
| M3 | m3 | 4
| D3 | F#3 | A3

(5|M3 + m3) (m6|m3 + 4) (M6|4 + M3)
(D, F#, A) (D, F#, A) (D, F#, A)

All these arrangements share the identical pitch class set (D, F#, A),
and are therefore inversions; but they have different interval sets,
and thus are not also conversities. We shall shortly return to this
example (Section 4), but the immediate point is that we cannot derive
the two converse flavors of a medieval _quinta fissa_ combination or
later 5-limit triad by a process of inversion.

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4. When is an inversion not a conversity?
-----------------------------------------

Just as some conversities are not also inversions, so some inversions
are not also conversities.

In the 3-limit world of Gothic music, two mildly unstable sonorities
mentioned by Jacobus share the property of being identical to their
own conversities, but of having a variety of nonidentical inversions.
These sonorities have a middle voice splitting an outer major ninth
into pure fifths, or an outer minor seventh into two pure fourths:

| G4 | F4
| 5 | 4
M9 | C4 m7 | C4
| 5 | 4
| F3 | G3

(M9|5 + 5) (m7|4 + 4)

Since the lower and upper fifths of (M9|5 + 5), or the lower and upper
fourths of (m7|4 + 4), are identical, "flipping" the order of these
intervals results in an identical arrangement. Thus either sonority is
its own conversity.

From a mathematical viewpoint, sonorities of this type represent what
is termed a geometric division of the outer interval into two "twin"
intervals with identical ratios -- more informally, we might speak of
a "mean-ratio division." Thus (M9|5 + 5) divides the outer major ninth
at 9:4 into two 3:2 fifths, with a frequency-ratio of 4:6:9 and an
identically proportioned string-ratio of 9:6:4; (m7|4 + 4) divides the
outer minor seventh in two 4:3 fourths, with a frequency-ratio of
9:12:16 and a string-ratio of 16:12:9.

While either of these sonorities has no nonidentical conversities, as
it happens these two sonorities share a relationship of inversional
affinity along with two others we have already met (Section 2) which
are also mutual conversities:

| G4 | F4 | G4 | C4
| 5 | 4 | M2 | 4
M9 | C4 m7 | C4 5 | F4 5 | G3
| 5 | 4 | 4 | M2
| F3 | G3 | C4 | F3

(M9|5 + 5) (m7|4 + 4) (5|4 + M2) (5|M2 + 4)
(F, C, G) (F, C, G) (F, C, G) (F, C, G)

All four sonorities share the same pitch class set (F, C, G), and thus
partake of a "family resemblance" of inversional affinity; the last
two sonorities additionally share the interval set (5, 4, M2), and so
are also conversities as discussed in Section 2.

Of interest in a 13th-century context as mildly unstable 3-limit
sonorities, these four related combinations are also of great interest
in a 20th-century context, where they are often known as "fourth
chords" or "fifth chords," and may be treated as stable concords. They
occur in various world musics with leanings toward vertical fifths and
fourths; for example, the form with a major ninth and two fifths has
been noted in Japanese koto music.

A curious case where two sonorities are conversities but not
inversions, and yet may appear to be "near-conversities," occurs if we
compare the two inversions of a 5-limit triad briefly introduced in
Section 3:

| A3 | D4 | F#4
| m3 | 4 | M3
5 | F#3 m6 | A3 M6 | D4
| M3 | m3 | 4
| D3 | F#3 | A3

(5|M3 + m3) (m6|m3 + 4) (M6|4 + M3)
(D, F#, A) (D, F#, A) (D, F#, A)

Strictly speaking, these two inversions are not conversities, because
they contain different interval sets: (m6, m3, 4) and (M6, M3, 4).
However, if we regard interval categories such as thirds and sixths in
a generic way (treating either M3 or m3 as a generic "third," etc.),
then these sonorities do seem to have an affinity somewhat akin to
conversity, since they share the same generic interval set (6, 3, 4)
in converse arrangements:

| D4 | F#4
| 4 | 3
6 | A3 6 | D4
| 3 | 4
| F#3 | A3

(6|3 + 4) (6|4 + 3)
(D, F#, A) (D, F#, A)

Thus while two saturated 5-odd-limit sonorities which are conversities
cannot also be inversions, and vice versa, they may sometimes be at
once true inversions and "near-conversities."

Most respectfully,

Margo Schulter
mschulter@value.net