Yahoo Tuning Groups Ultimate Backup tuning Paul Erlich's "melodic guidance" and 14th-century cadences

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Paul Erlich's paradigm and late Gothic cadences:
"Signposts" in 14th-century trinic music
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[Please note that this paper is written on the basis of Paul Erlich's
germinal article in _Xenharmonikon_ 17 (1998), and that I have not yet
read his recent paper on _The Forms of Tonality_. My interpretations,
however accurate or otherwise, thus may not necessarily reflect his
current views as expressed in the new statement.]

As one comprehensive introduction to physics points out: "Physicists
like to find areas that seem to have nothing to do with each other and
to show that they are related if you look at them closely enough."[1]

Remarkably, Paul Erlich's famous article on "Tuning, Tonality, and
Twenty-Two-Tone Temperament"[2] suggests such a connection between the
favorite cadences of two very different eras of Western European
music: the 14th and 18th centuries.

Many listeners would likely agree that the cadences of these eras have
different sounds. First, here's a favorite final cadence from the 14th
century, the era of Guillaume de Machaut and Francesco Landini, in a
standard Pythagorean intonation:

http://value.net/~mschulter/py3ei001.mid

Compare this with a typical 18th-19th century final cadence, here
tuned in Werckmeister's Temperament 3, one likely choice in the era of
Johann Sebastian Bach:

http://value.net/~mschulter/wm3i001.mid

Despite the differences in style, Erlich's approach suggests that both
cadences use "signposts" or "melodic guidance" to reinforce the
definition of a "strong" vertical center.[3]

Following and somewhat enlarging upon Erlich's approach, these
signposts may tend to take on different shapes depending on at least
two variables: the basic type of scale structure, and the type
of saturated vertical sonority, favored in a given era or style.

Erlich's paper, published in 1998, mentions medieval European music,
but looks more closely at some other styles. However, when his
concepts are applied to the complex polyphony of the 13th and 14th
centuries, they seem actually to predict the "strongest" and most
popular cadences especially in vogue during the latter portion of this
era.

------------------------------------------
1. Closest approach and "position finding"
------------------------------------------

Looking at a three-voice example of a 14th-century cadence from a
style-specific point of view, it's easy enough to explain this
progression, here shown in a notation with C4 as middle C:

C#4 D4
G#3 A3
E3 D3

http://value.net/~mschulter/py3ei003.mid

We might first note that the destination sonority or goal of the
cadence, D3-A3-D4, is the characteristic "perfect" or saturated
sonority of the era, with outer 2:1 octave (here D3-D4), lower 3:2
fifth (here D3-A3), and upper 4:3 fourth (here A3-D4). Johannes de
Grocheio (c. 1300) refers to this complete sonority as manifesting
_trina harmoniae perfectio_, "the threefold perfection of harmony,"
suggesting the modern English name "trine."[4]

An anonymous treatise of the same epoch, recently ascribed to the
great theorist Jacobus of Liege, states that the outer octave of this
sonority is best arranged to follow the order of numbers which Nature
has "serially" (_seriatim_) disposed, "2-3-4." In this natural series,
the 2:3 fifth is placed below the 3:4 fourth.[5]

Focusing on the unstable sonority E3-G#3-C#4 leading to this ideally
concordant resolution, we see that it has two unstable Gothic
intervals: the major third E3-G#4, and the major sixxth E3-C#4. In the
resolution, the major third expands to a fifth (M3-5) and the major
sixth to an octave (M6-8), with these resolutions mutually reinforcing
each other.

More generally, the 14th century favors "closest approach"
progressions in which an unstable interval resolves to a stable one
with one voice moving by a whole-tone and the other by a diatonic
semitone. This preference favors two varieties of our (M6-8 + M3-5)
cadence, featuring either ascending or descending semitonal motions:

C#4 D4
G#3 A3
E3 D3

http://value.net/~mschulter/py3ei003.mid

D4 E4
A3 B3
F3 E3

http://value.net/~mschulter/py3er004.mid

Typically the form with ascending semitonal motion is preferred as a
final cadence,, while the form with descending semitones is preferred
for internal cadences.

While these cadences admirably epitomize the specific patterns of a
Gothic style, they also fit a more general guideline proposed by
Erlich, the principle of "melodic guidance"[6]:

"The rarest step sizes are only found adjacent
to notes of the tonic chord, acting as 'signposts'
if not necessarily leading tones _per se_."

In Gothic music, the relevant "step sizes" are those of a diatonic
scale, often supplemented in the 14th century, as in later eras, by
additional inflected notes. Here, as typically in diatonic music, the
"rarest" sizes are the diatonic semitones, at 256:243 (~90.22 cents)
in Pythagorean tuning.

Thus Erlich's guideline calls for these semitones of a seven-note
diatonic set to be located adjacent to the notes of our "tonic chord"
or cadential center, here the complete Gothic trine D3-A3-D4 or
E3-B3-E4. Both cadences indeed satisfy this condition, and represent
the _only_ available patterns using regular diatonic intervals and
melodic steps for satisfying it:

C#4 D4 D4 E4
G#3 A3 A3 B3
E3 D3 F3 E3

(D-G#: G#-A, C#-D) (F-B: F-E, C-B)

The notes used in each progression fit within a seven-note diatonic
set: D-G# in the first example, and F-B in the second. We can confirm
that in the first example, the two diatonic semitones G#3-A3 and
C#4-D4, are both adjacent to the resolving trine D3-A3-D4. In the
second example, the semitones F3-E3 and C4-B3 -- only the first in
this instance actually used as a cadential step -- are likewise
adjacent to the trine E3-B3-E4.

In 14th-century terms, these progressions follow the rule of "closest
approach" that thirds expanding to fifths or sixths to octave should
be major. In Erlich's 20th-21st century terms, they also follow the
more general rule that for the strongest definition of a vertical
center, the "rarest steps" of a scale should be adjacent to the notes
of this stable center.

From either perspective, the standard Pythagorean tuning of the era
enhances relevant musical contrasts. Fifths and fourths are pure,
while unstable major thirds and sixths have rather complex ratios of
81:64 (~407.82 cents) and 27:16 (~905.87 cents), thus adding to the
sensen of vertical tension and excitement, and resolving it in a most
satisfying way.

At the same time, the contrast between generous whole-tones at 9:8
(~203.91 cents) and the narrow 90-cent semitones may make the role of
the latter as Erlich's melodic "signposts" yet more compelling.

As our cadence on D3-A3-D4 illustrates, accidental inflections (here
G# and C#) are sometimes required to follow the rule of "closest
approach," or of Erlich's "melodic guidance." In Section 3, we
consider this facet of 14th-century accidentalism in more detail.

----------------------------------------------------
2. Erlich's "melodic signposts": from trine to triad
----------------------------------------------------

Curiously, our 18th-century cadence given at the beginning of this
article also follows Erlich's guidance, the main difference being a
different standard of saturated stability, here the complete triad at
4:5:6 (outer 2:3 fifth, lower 4:5 major third, upper 5:6 minor third):

F5 E5
B4 C5
G4 G4
D4 C4
G3 C3

http://value.net/~mschulter/wm3i001.mid

This cadence, in approaching the triadic center C3-C4-G4-C5-E5, also
features the resolution of an unstable interval by stepwise contrary
motion: the diminished fifth B4-F5 contracts to the prime concord of
the major third C5-E5. Additionally, the unstable minor seventh G3-F5
further accentuates the level of tension.

As in our 14th-century trinic cadences, so in this 18th-century
triadic cadence, the progression fits within a seven-note diatonic set
in which both diatonic semitones are adjacent to the vertical center.
Here the set is F-B, with the semitones B-C and F-E both adjacent to
notes of the triad C3-E3-G3.

Werckmeister's well-temperament used in the MIDI file for this
example, while intended in good part to permit a closed tuning in only
12 notes on a fixed-pitch instrument, does so while fitting the
intonational parameters of this kind of style: major and minor thirds
in the most common transpositions are rather close to the simplest
ratios of 5:4 (~386.31 cents) and 6:5 (~315.64 cents).

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3. Closest approach and accidentalism: originizing and ultimatizing
-------------------------------------------------------------------

While both 14th-century and 18th-century cadences follow the paradigm
of Erlich's "melodic guidance," the former do so in a way involving an
expressive element giving the music a special suppleness and flavor:
fluid accidentalism.

Using only the seven notes of a diatonic scale without accidentals, we
can build closest approach cadences of the (M6-8 + M3-5) variety on
only two degrees: F or E:

E4 F4 D4 E4
B3 C4 A3 B3
G3 F3 F3 E3

F - C - G - D - A - E - B F - C - G - D - A - E - B

Cadence on F: <http://value.net/~mschulter/py3ei004.mid>
Cadence on E: <http://value.net/~mschulter/py3er004.mid>

From a closest approach perspective, the trines F3-C4-F4 and E3-B3-E4
are the only ones in which this diatonic set provides a _major_ third
which can expand by stepwise motion to the fifth of the trine (G3-B3
to F3-C4; F3-A3 to E3-B3).[7]

From Erlich's perspective of "melodic guidance," similarly, only the
trines F3-C4-F4 or E3-B3-E4 satisfy in this diatonic set the property
of having both semitone steps (B-C, E-F) adjacent to their notes.

Dr. Marshall Tuttle, a music theorist and Wagner scholar, has observed
that in the first type of cadence (here on F3-C4-F4), the fifth of the
resolving trine is the _first_ or _lowest_ fifth of the chain making
up the diatonic set, here F-C-G-D-A-E-B.[8]

Taking a cue from this observation of Dr. Tuttle, we may note that the
second type of cadence (here on E3-B3-E4) involves a trine whose fifth
is the _last_ or _highest_ of the chain (here again F-C-G-D-A-E-B).

From this 21st-century viewpoint, we might say that an (M6-8 + M3-5)
cadence with ascending semitonal motion "originizes" a trine such as
F3-C4-F4, placing its fifth at the origin of a diatonic chain.

A similar cadence with descending semitonal motion "ultimatizes" a
trine such as E3-B3-E4, placing its fifth at the "ultimate" position
in such a chain.

The desire to make closest approach cadences on various steps of an
octave species or mode in the course of a piece -- in our current
terms, to originize or ultimatize various trines -- provides a
powerful motivation for 14th-century accidentalism.

Erlich's 1998 paper seems to focus on a kind of model where cadences
are largely derived from the usual tones of a diatonic or other
melodic set (e.g. his decatonic scale based on stable vertical tetrads
of 4:5:6:7).

In contrast, the 14th-century approach is to use a variety of octave
species with accidental inflections available as needed or desired to
achieve closest approach cadences, as in the final cadence of
E3-G#3-C#4 to D3-A3-D4 in an octave species or mode of D-D.

Not only the "native" steps of an octave species or mode, but the
routine inflections called for at cadential progressions, may
contribute to its characteristic qualities. Consider, for example,
this approach to the center D3-A3-D4 as realized in 22-tone equal
temperament (22-tET), an appropriate choice given Erlich's able
advocacy of this tuning through practice as well as theory:

F4 E4 D4 C#4 D4
C4 B3 A3 G#3 A3
A3 G3 F3 E3 D3

http://value.net/~mschulter/22tei001.mid

The melodic diminished fourths F4-C#4 and C4-G#3 outlined by this type
of descending cadential figure take on a special quality in 22-tET,
where they have a size of about 327.27 cents, actually a kind of large
minor third!

--------------------------------------------
4. Closest approach and the diagonal tritone
--------------------------------------------

One obvious difference between our favorite 14th-century cadences and
the most popular 18th-century form is that only the latter involves
the vertical interval of the tritone (augmented fourth) or diminished
fifth (e.g. F3-B3 or B3-F4), which Erlich terms a "characteristic
dissonance" involving the same number of scale steps as a stable
concord, here the usual fourth or fifth (e.g. G3-C3 or C4-G4).

In a seven-note diatonic set, such intervals uniquely involve the two
notes at the extremes of the chain of fifths, e.g. F-B or B-F in the
set F-C-G-D-A-E-B.

In vertical terms, we might say that the defining unstable interval of
our trinic (M6-8 + M3-5) cadences is the major third, available within
the same diatonic set at two locations where it can expand by stepwise
contrary motion to a stable fifth included in the same set, e.g. F-A
or G-B in the set F-C-G-D-A-E-B.

At some level, possibly, the availability of _two_ such intervals, one
expanding to the first fifth of the chain (G3-B3 to F3-C4) and the
other to the last fifth (F3-A3 to E3-B3), may tie in with the
characteristic polarities between these two types of resolutions in
14th-century cadences and musical forms.

Does the unique interval of the tritone play any role in these same
cadences? One reason it may figure more prominently in 18th-century
triadic progressions is that there, with thirds and sixths as stable
intervals, it can resolve to such intervals by stepwise contrary
motion (e.g. d5-M3 in our example using a Werckmeister tuning).

However, we find that the tritone does also occur in our 14th-century
cadences as a kind of "diagonal" interval:

C#4 D4 D4 E4

G#3 A3 A3 B3
\ /
\ /
E3 D3 F3 E3

D-A-E-B-F#-C#-G# F-C-G-D-A-E-B

In the first progression, we have a diagonal tritone between G#3 in
the unstable sonority and D3 in the resolution; similarly, in the
second, we have the tritone relation F3-B3.

One might say that this tritone marks the "metes and bounds" of a
diatonic set including all the notes involved in the progression:
here D-A-E-B-F#-C#-G# or F-C-G-D-A-E-B.

It should be noted that this "local cadential set" is not necessarily
that of the prevailing octave species or mode, with certain degrees
often taking on a fluid quality.

In the octave species of D-D, the medieval Dorian mode, for example,
we can expect frequently to encounter both forms of G/G# and C/C#; the
fluidity of the degree B/Bb (or German H/B) in the traditional
medieval gamut may provide a precedent for this flexibility, along
with the 13th-century use of various accidentals (Eb, F#, C#).

Thus while Erlich's "melodic guidance" points to a common feature of
14th-century and 18th-century cadences, this principle may promote
either the restriction of the range of modes largely to those
permitting such cadences with "native" notes; or the use of a variety
of modes with fluid accidental inflections as required or desired to
achieve these cadences.

-------------------------------
5. Some historical implications
-------------------------------

While many 20th-century histories of Western European composition have
tended to assume a gradual process of evolution, or even "progress,"
toward the tonal system of the 18th-19th centuries, medievalists such
as Richard Crocker have asserted that 14th-century music has its own
"tonal plans" and schemes of long-range organization based on a
compelling cadential logic.

Following the general kind of analysis which Erlich's paper suggests,
we can indeed conclude that both 14th-century and 18th-century
cadences represent especially efficient and compelling progressions
for defining a stable vertical center, with the 2:3:4 trine or 4:5:6
triad respectively defining such a saturated center.

Both types of cadences involve Erlich's "melodic guidance," with both
semitones of a diatonic set adjacent to notes of the resolving
sonority.

Additionally, both types involve the element of unstable intervals
resolving to stable ones by stepwise contrary motion -- in Erlich's
terms, one might speak of such intervals also as "adjacent" to those
of the resolving sonority -- thus M3-5 and M6-8 in a trinic context,
or d5-M3 in a triadic context.

Such comparisons, of course, do not necessarily "explain" the
stylistic choices of a specific era or genre, but can point to some
organizational themes which may manifest themselves in music of
various eras and styles.

Also, while Erlich's paradigm predicts that certain types of
progressions such as those favored in the 14th and 18th centuries may
be especially "strong" in defining a vertical center, musicians are
free to use or even at times to prefer more "gentle" or "pluralistic"
formulae. One charm of 13th-century music, for example, is its
frequent use of cadences with all voices moving by whole-tones, as in
this example:

G3 A3
E3 D3
C3 D3

http://value.net/~mschulter/py3o001.mid

Since Erlich's 1998 paper does not itself address a specific analysis
of complex 13th-14th century Western European polyphony, I emphasize
that while my indebtedness to him is obvious, the responsibility for
any (mis)interpretations is of course mine.

Specifically, I have approached Gothic polyphony as a trinic/diatonic
type of style where the fluid cadential inflections of the 14th
century may define a "local" seven-note diatonic set not necessarily
identical to that of an overall "native" octave species or mode.

Whether or how Erlich's original formulations might fit such a
scenario may remain an open question, with his response much invited.

At the least, however, the concept of "melodic guidance" does neatly
connect the very different cadential styles of the 14th and 18th
centuries, suggesting that one can make interesting comparisons while
recognizing the unique integrity and beauty of each era.

-----
Notes
-----

1. David Halliday and Robert Resnick, _Fundamentals of Physics: Third
Edition Expanded_ (John Wiley and Sons, New York, 1988), p. 331.

2. Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17:12-40 (Spring 1998).

3. Ibid., pp. 19-20 and n. 24, citing a statement by Richmond Browne:
"Rare intervals aid position finding."

4. Johannes de Grocheio's (or Grocheo's) treatise has variously been
known as _Theoria_, _De musica_, or _Ars musicae_. For Latin text, see
E. Rohloff, _Der Musiktraktat des Johannes de Grocheo_ (Leipzig 1943),
with passage on _trina harmoniae perfectio_ at p. 44; for English
translation, see Albert Seay, _Johannes de Grocheo Concerning Music_
(Colorado College Music Press Texts/Translations 1), Colorado Springs:
Colorado College Music Press, 1967, at p. 6.

5. For the passage deriving this sonority from the series of ratios
2-3-4, see _Jacobi Leodiensis Tractatus de consonantiis musicalibus,
Tractatus de intonatione tonorum, Compendium de musica_, ed. Joseph
Smits van Waesberghe, Eddie Vetter, and Erik Visser (Divitiae musicae
artis, A/IXa), Buren: Knuf, 1988, 88-122 at 122; the Latin text is
available on the World Wide Web, Thesaurus Musicarum Latinarum,
Indiana University,
<http://theme.music.indiana.edu/tml/14th/JACCDM_TEXT.html>.
Although I am not aware of any explicit mention of frequency ratios in
Western European theory until around the 16th century, or of the
harmonic series until around 1600, the series "2-3-4" interestingly
could in later terms express the trine as the frequency ratio or set
of harmonic partials 2:3:4. Similarly, without positing the harmonic
series, Zarlino takes the _senario_ 1-2-3-4-5-6 as stating the ideal
vertical arrangement of consonant intervals: first the 1:2 octave,
next the 2:3 fifth and 3:4 fourth, and then the 4:5 major third and
5:6 minor third (e.g. C2-C3-G3-C4-E4-G4).

6. See Erlich, n. 2 above, pp. 19-20 and n. 24. Following an approach
based in part on the writings of Joseph Yasser and Harry Partch,
Erlich applies this and other elements of his paradigm to pentatonic
music in the 3-odd-limit (most complex stable intervals 3:2 and 4:3);
diatonic music in the 5-odd-limit (with 5:4 and 6:5 in this role); and
decatonic music in the 7-odd-limit (7:4, 7:5, and 7:6). Complex Gothic
polyphony may represent a different category: it is in these terms a
"3-limit diatonic" system based on the complete 2:3:4 trine, and using
mostly diatonic steps combined with various types of accidental
inflections, regularly required in the 14th century for "closest
approach." Gothic schemes of concord/discord often focus on the usual
13 or 14 diatonic intervals from unison to octave inclusive (depending
on whether one counts the tritone and diminished fifth separately, as
Jacobus of Liege does), but certain 14th-century theorists also
consider certain chromatic intervals such as the augmented sixth
(Jacobus, c. 1325) or diminished fourth (Johannes Boen, 1357).
Marchettus of Padua (1318) and his school use the chromatic semitone
or apotome as a direct melodic step.

7. This diatonic set also includes the major third C-E, which however
has no stable fifth within the set to which it can resolve: the
adjacent steps B-F form a diminished fifth. In the eight-note regular
medieval gamut of Bb-B, a resolution to the fifth Bb-F is possible
(and thus cadences such as C4-E4-A4 to Bb3-F4-Bb4). In such a cadence,
one might posit a different "local diatonic set" Bb-F-C-G-D-A-E. This
kind of approach involving seven-note diatonic sets, while distinct
from the medieval system of hexachords and mutations based on six-note
sets, may be somewhat congenial to it. On hexachords and accidental
inflections, see <http://www.medieval.org/emfaq/harmony/hex.html>.

8. Dr. Tuttle's e-mail address is <inotmark@aol.com>.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hello, there, everyone, and thanks to people such as Graham, Joe
Pehrson, the Monz, and of course Paul himself for their recent
responses to my article on Paul's paradigm as it relates to
14th-century Western European cadences.

This discussion provides me an opportunity not only to comment on some
of the stylistic issues raised, but to express a bit of humility about
the style-specific orientation of much of my own musical practice and
theory. Maybe one moral is that we all need to compare notes (and
intonations), as is happening in this thread.

---------------------------------------------------
1. Paul's article and my commentary: some questions
---------------------------------------------------

First, Paul, in responding to your germinal article of 1998 in
_Xenharmonikon_ 17, I might say that in its scope it reminds me
something of the writings of Joseph Yasser, whose _Medieval Quartal
Harmony_ was a major influence on me during my college years. Of
course, you cite various other sources also, at once gleaning valuable
concepts and showing that these concepts of diatonicity or the like
are not specific to 12-tone equal temperament (12-tET), as has
sometimes been supposed in 20th-century presentations.

In my cadential analysis inspired by this paper, I wanted to emphasize
possible common ground, and might now note some possible distinctions
between related but not necessarily equivalent concepts.

For example, your "characteristic dissonance" and my "directed
instability" are obviously related, and in 18th-century tonality they
happen to be synonymous. However, while major-third-to-fifth is
certainly a form of "directed instability" in the 14th century, can it
be called a "characteristic" one, since in any seven-note diatonic
scale -- or six-note hexachord, for that matter -- there are at least
_two_ major thirds, and the major third is not produced from the same
number of scale steps as any stable consonance?

A neo-Yasserian might cleverly argue that the dynamic role of the
major third in 14th-century verticality suggests a "pentatonic"
element in this music, since as you show the major third is a
"characteristic dissonance" in a pentatonic scale.

Maybe one point here might be the advantages of recognizing two
variables or dimensions for this kind of analysis: the type of scale
structure, and the point of stable saturation in a given style.

For example, Yasser's scheme setting some of the background for your
paper seems to suggest in Partchian terms "3-limit pentatonic; 5-limit
diatonic; 7-limit decatonic." This appears to me to be the main
theoretical ground you cover, and it's an immense territory in itself,
especially for one paper of less than 30 printed pages which not only
addresses such comparative and historical issues but maps out the
basis for a new musical practice.

However, an artistic road I happen to be on, maybe with some curious
company, is to keep the saturation point fairly simple but use an
intricate scale system with lots of shades of instability and
chromatic or enharmonic "surprises."

For example, while a Yasserian paradigm might predict that the point
of saturation will be different in 19-tET or 31-tET than in 12-tET,
all three systems can be used, for example, for Renaissance or
"5-limit modal" styles with thirds and sixths as the most complex
stable concords. This was Vicentino's approach, and is also mine in a
24-note version of 1/4-comma meantone when I'm following a
"Xeno-Renaissance" style.

An even more dramatic example, of course, is later Gothic polyphonic
based on the stable trine of 2:3:4 ("3-limit" saturation) and using
all 12 notes of a Pythagorean chromatic tuning -- or a larger one, as
with the 15 notes called for by Solage around 1400 (Gb-G#), or the
17-note Pythagorean gamut proposed by Prosdocimus and Ugolino in the
early 15th century (Gb-A#).

Neo-Gothic music carries this kind of "complex 3-limit" development
yet further with tuning systems where a diatonic semitone is divided
into a cadential "diesis" serving as a kind of semitone in its own
right, for example, and a smaller commalike interval. Here, too,
there's a kind of precedent in Marchettus of Padua (1318), and his
"fivefold division of the tone" -- however problematically interpreted
-- provides inspiration for lots of neo-Gothic experimentation.

Another basic issue: should the intervals and melodic motions used at
a decisive cadence generally be present within a given diatonic scale
or mode itself, for example, or may they result from fluid accidental
inflections? In a very large portion of the music I play, improvise,
and compose, the second kind of situation applies.

While people trained mainly in a major/minor kind of tonal system
might be able to speak to this point better than I can, I do prefer to
use terms other than "key" or "modulation" because it seems that they
may imply the first kind of situation where the second actually
applies. Instead, I may speak of "transpositions" or "mutations" or
the like.

Also, for 16th-century music, my concept of "directed instability" may
be rather less clear than for 14th-century music -- or 18th-century
music, for that matter. It's easy to show how 14th-century or
18th-century cadences use unstable intervals to make a sense of
inherent vertical tension very clear; with 16th-century cadences,
things can be more ambiguous, especially if we disregard the vital
role of the suspension, for example.

Of course, that very "ambiguity" may be part of the beauty of
16th-century music: in Partchian terms, one might speak of traditional
"3-limit" resolutions guiding motion between "5-limit" sonorities. I'm
almost tempted to speak of thirds and sixths in a kind of orbital
free-fall, and Zarlino's doctrine of the imperfect and perfect
concords seems to me not unconsonant with such an appreciation.

----------------------------------------
2. Stylistic viewpoints: comparing notes
----------------------------------------

Graham, you have raised an important question on which others have
commented also: the relationship between stylistic particularities and
general patterns or musical commonalities.

First, as someone mainly focused specifically on medieval and
Renaissance European music and modern offshoots, I might suggest that
a musician and scholar such as Daniel Wolf, with his awesome
cross-cultural and historical scope, might have a unique kind of
perspective to contribute here.

Also, a musician such as Haresh Bakshi trained in the practice and
theory of the 22 srutis of India, and the musical system of which they
are a part, has an invaluable perspective to share, one informed not
only by intonational concepts but a revered oral tradition of teaching
and learning.

Further, an artist such as Judith Conrad, who can not only expertly
perform a composition of Bach on the harpsichord, but tune the
instrument by ear in an appropriate well-temperament, may have special
insights to share in a dialogue intimately embracing within its scope
the music of this era.

What I would want to emphasize at the beginning of this kind of
discussion is that each style has its own logic. As someone who has
sometimes felt less than satisfied with attempts to analyze my beloved
14th-century music in 18th-century terms, I would emphasize that for
me to attempt to reverse this process would likely prove equally
unsatisfactory.

Sometimes styles can differ rather dramatically in their patterns of
concord/discord and intonation, for example, while sharing important
common assumptions.

For example, over the last three decades and a bit more, I've found
myself in a kind of delicate balance between Gothic and Renaissance
leanings, each style exerting its own pull and fascination.

If asked to say what elements these styles might share in common,
maybe I'd reply that it's a kind of "Polyphony" -- a term I use in a
possibly new sense by analogy with Partch's "Monophony" -- a system
where intervals and cadences can be organized around a subtle balance
of centers.

This might be in contrast to either "Monophony" -- tonality with all
notes ideally focused on a single center -- or what I'm tempted to
call "Isophony," a kind of isotropic musical space where all points
seem of identical and symmetrical importance, with "pantonalism" as
one possible example (whether based on 12 tones or some other number).

To be prudent, I'll stress that "monophony" (music for a single
melodic line) and "polyphony" (music with more than one note or
melodic line sounding at the same time) have more conventional
meanings; but given the Partch centennial, I couldn't resist a bit of
play with his use of "Monophony."

In the process of comparing notes, we may find that some viewpoints
are more common than others, but each style-specific viewpoint is
equally precious for describing its own most familiar music -- and
potentially equally invalid when taken as a universal grammar of
music.

By the way, Monz, I want enthusiastically to agree with your point
about the benefits of technique, a technique sometimes informed by
theory as well as vice versa.

For example, a couple of years ago or so, I was fortunate enough to
learn a bit of the four-voice keyboard approach of Tomas de Santa
Maria (1565). He is describing especially the art of _fantasia_,
likely meaning mainly "improvisation," in four parts.

One basic idea is to focus on the intervals formed by the outer voices
at an octave, tenth, twelfth, fifteenth, etc., with the inner voices
making "differences" or additional intervals which add consonances and
fill in the space.

In addition to enriching my analysis and improvisation, I find that
this approach can also be helpful in learning a new piece,
complementing rather than substituting for musical empathy with
another composer based on sheer experience in writing or playing in
similar styles.

Graham, I suspect that in some ways Classic and jazz styles may share
common assumptions different from those of medieval styles, although
if we took a comparative perspective, we might hear interesting
similarities and differences with other world musics.

This leads me to conclude on a cautious note. While period-specific
assumptions may have impeded the development of a paradigm for common
patterns shared by different eras of Western European composition, I
wonder if in some ways this might be a fortunate thing.

My concern would be that a "unified theory of [Western European]
music" might tend to marginalize other world musics even more than the
less theoretically tidy canon of analysis during the past century.

One effect of the marginal status of the European medieval music I
love was to encourage me to explore other world musics also, in part
to show that "strange" features of Gothic composition (as described in
some of the literature to which I was exposed) are found in other
cultures, and in oral as well as notated traditions. Both in asserting
that medieval verticality is a system in its own right, and by seeking
out cross-cultural comparisons (for example with Chinese music),
Joseph Yasser lent me some encouragement in this kind of exploration.

May I caution to say that such a sampling does not necessarily avoid a
certain ethnocentricism: "This Burmese song is so like Machaut." Once
again, the "common patterns" one recognizes may say as much about the
listener as about the musical language being heard.

In response to your comments, Joe Pehrson, I would therefore say that
analyses at different levels are fine -- as long as we recognize the
hazards of generalizations which may sometimes distract from as well
as clarify stylistic particulars.

Above all, as a medievalist in the European tradition, I would caution
against "universal grammars" of music which are in fact partial
explanations of European composition. A truly "world class" theory
must have a worldwide scope, with worldwide participation,
emphatically including that of Indigenous musical traditions.

A commendable aspect of your article, Paul, is that it does look into
some different cultural traditions, including the 22 srutis of India
(with mention of Ramon Satyendra as an advisor, possibly in this
connection?) and Thai and Cambodian music in some approximation of
7-tET (with a footnote quoting Daniel Wolf).

Especially for intonationally aware people, such an open perspective
is both appropriate and rewarding.

Comparing notes and intonations is invaluable -- as long we respect
the integrity of the various styles and cultures represented, and seek
to minimize the hazards by expanding the circle of dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net

Paul Erlich's "melodic guidance" and 14th-century cadences

🔗mschulter <MSCHULTER@VALUE.NET>

🔗graham@microtonal.co.uk

🔗jpehrson@rcn.com

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>