Re: JI, odd-limits, large integers, 16:19:24, etc.

Hello, there, and one immediate spur to this message is a very wise
remark by Bill Alves that the suitability of a given just intonation
system or a tuning for a specific interval should be judged according
to the musical context.

More generally, I'd like to comment on some issues of large integer
JI, odd-limits, and "consonance" limits from a Pythagorean
perspective.

My apologies in advance to anyone who might decide that since I've
expressed many of these views here 22 or so times already, my further
affirmation of them now may be mostly cumulative <grin>.

One notable feature of medieval European Pythagorean JI in the context
of polyphonic composition is that while the prime 3-limit is low, the
"odd limit" is very high when we consider the ratios in practical use,
and especially those of intervals generally deemed by theorists as
_relatively_ concordant although unstable.

After briefly exploring this point, I'll consider the striking
resemblance of one version of the medieval _quinta fissa_ ("split
fifth") sonority, at 54:64:81, to the sonority recently espoused on
this list of 16:19:24.

In 13th-14th century Western European theory, _stable_ intervals are
indeed generally restricted to the obvious "3-odd-limit" forms and
their octave extensions: e.g. unison (1:1), octave (2:1), fifth (3:2),
fourth (4:3), 12th (3:1), 15th (4:1), and for some theorists the 11th
(8:3). Around 1300, "modern" theorists start to espouse the view that
the fourth should be regarded as concordant when supported by the
fifth below, not in itself -- but that's another discussion.

However, a wide variety of intervals with odd-limits far beyond 3 are
regarded as "imperfect" or "medial" concords. The major and minor
thirds (81:64, 32:27) have this status in 13th-14th century practice
and theory, and combinations involving fifths and/or fourths plus
major seconds or ninths, or minor sevenths, also get a somewhat
analogous recognition in certain "dialects" of practice as well as in
two treatises, one of them the monumental _Speculum musicae_ of
Jacobus of Liege (c. 1325). In "modern" 14th-century practice and
theory, the major sixth (27:16) is also regarded as relatively
concordant.

Indeed Jacobus finds that even a bare major second (9:8) has some
degree of "imperfect concord," while 9:4 or 9:2 is a "medial concord"
(comparable to M3 or m3), and 9:1 is actually a _perfect_
concord. Like Jacobus (and evidently some composers of the 13th and
14th centuries), I find 4:6:9 as quite "concordant" and pleasing,
although unstable, at least in this medieval setting; and I find a
1:3:9 as a remarkably stable concord, as Jacobus suggests by calling
9:1 a "perfect" concord. Although a 9:1 (M23) is too wide for
practical Gothic use, as Jacobus himself recognizes in remarking that
a twelfth is about the limit of such practice, we might nevertheless
say that his system has a 3-prime-limit, an 81-odd-limit on _relative_
concord (the "medially concordant" ditone or M3 at 81:64), and a
9-odd-limit on "perfect concord" (maybe roughly synonymous with
stability).

At this juncture, two points may be notable from the perspective of JI
theory.

First, "concord" can be a rather fluid and inexact concept, and
medieval theorists recognize this quite explicitly by speaking of it
in degrees: perfect, medial, imperfect, and the like. The modern
theorist Ludmila Ulehla makes much the same point when she speaks of a
trichotomy between concords, "dual-purpose" sonorities which can be
heard as relatively blending but also to some degree points of
tension, and discords. It seems to me that such a graduated approach
could be valuable for various flavors of JI theory.

Secondly, at least in a Pythagorean setting, "odd-limit" is not
necessarily a good index of concord/discord. Most notably, just about
every 13th-14th century theorist agrees that a bare major or minor
third (81:64, 32:27) is _more_ concordant than a major second (9:8).
If we consider early 15th-century keyboard practice, where major
thirds involving sharps (e.g. e-g#) are frequently realized as
diminished fourths at 8192:6561, this point becomes even clearer. Of
course, one could argue that 8192:6561 is only a schisma (~1.95 cents)
from a simple 5:4, but within a 3-prime-limit system, it is
fascinating how large integers can produce some interesting results.

(In one piece from the Buxheimer Organ Book, if one assumes this
tuning, then we actually have a final sonority of a-db'-e', or
13122:16384:19683. Considering this, I'm tempted to introduce this
keyboard Kyrie setting: "Here's a piece, likely from around the middle
of the 15th century, in 19683-odd-stability-limit JI.")

Now for the medieval 54:64:81, one version of the _quinta fissa_
described by Jacobus of Liege where a fifth is "split" into two
relatively concordant thirds by a middle voice. At 0-294-702 cents,
this 3-limit JI sonority is only about 3 cents from a 16:19:24 at
0-297-702 cents.

Interestingly, while Jacobus himself prefers the major third below and
minor third above, he notes that the converse is also permissible,
citing the opening sonority (a-c'-e') of a motet which has happily
come down to us in the Montpellier and Bamberg codices. Anyone who
looks through music of the period from Perotin to the early Dufay, say
1180-1450, will find that this sonority is treated in practice as well
as theory as unstable but relatively concordant.

What is especially interesting for me is that in early 15th-century
music, where thirds are not only used in a coloristic role (a feature
of 13th-14th century music also), but are becoming more and more
pervasive in the texture, this 54:64:81 (minor third below major
third) can sound quite smooth, I might guess somewhat smoother to my
ears than 64:81:96 (major third below minor).

One contributor has suggested that 16:19:24 or 0-297-702 cents might
be seen as the center of a certain narrow region of the tuning
spectrum, 3 cents or so wide on either side -- which would in effect
place 0-300-700 cents (12-tet) near one boundary and 54:64:81 or
0-294-702 cents near the other.

To sum up, it may be a curious feature of JI systems that
low-prime-limit systems (specifically Pythagorean or "3-limit") may
favor very high odd-limits for some _relatively_ concordant
sonorities, while some relatively high-prime-limit systems may favor
comparatively low "odd-limits," at least in comparison to the prime
limit (or "cap" if one prefers).

Of course, there's also the important lesson emphasized here by Paul
Erlich: a "spiral" system such as Pythagorean tuning, if extended far
enough, can produce a close approximation of just about any ratio or
interval size in cents. Maybe this dialogue also invites discussion of
the "Just Noticeable Difference" (JND), especially for vertical
intervals.

Most appreciatively,

Margo Schulter
mschulter@value.net

>Secondly, at least in a Pythagorean setting, "odd-limit" is not
>necessarily a good index of concord/discord. Most notably, just about
>every 13th-14th century theorist agrees that a bare major or minor
>third (81:64, 32:27) is _more_ concordant than a major second (9:8).
>If we consider early 15th-century keyboard practice, where major
>thirds involving sharps (e.g. e-g#) are frequently realized as
>diminished fourths at 8192:6561, this point becomes even clearer. Of
>course, one could argue that 8192:6561 is only a schisma (~1.95 cents)
>from a simple 5:4, but within a 3-prime-limit system, it is
>fascinating how large integers can produce some interesting results.

Margo,

I have indeed made a case for the odd limit of a ratio as a measure of
consonance, along the lines of Harry Partch, whose "one-footed bride"
shows consonance decreasing steadily as the largest odd factor in the
ratio increases from 1 to 11, and then all other ratios are not
considered consonant at all.

However, speaking of the 81-odd-limit, or even the 27-odd limit, and
expecting this to continue to hold is the best example of what Dave
Keenan calls high-number "nonsense". The accuracy of the ear is such
that these numbers will never come up in the ratios that are important
in describing how we perceive a given interval, unless additonal notes
are involved with the interval in a chord of more than two notes.

The best way to approach the issue of the consonance of intervals is to
imagine a dissonance as a continuous function of the size of the
interval, such as the "one-footed bride" or any of many curves derived
by Hemholtz, Plomp, Kameoka & Kuriyagawa, or Sethares. Some ratios
within the 11-limit might correspond to local minima in the dissonance
curve, but that's all. Odd limits above 19 are certainly irrelevant in
quantifying a bare interval's dissonance or consonance.

>One contributor has suggested that 16:19:24 or 0-297-702 cents might
>be seen as the center of a certain narrow region of the tuning
>spectrum, 3 cents or so wide on either side -- which would in effect
>place 0-300-700 cents (12-tet) near one boundary and 54:64:81 or
>0-294-702 cents near the other.

I would agree that 16:19:24 has a special attraction within the space of
possible minor chords, and attribute that more to difference tones (as
Helmholtz did) and to virtual pitch psychology than to any special
"consonant" character of the intervals 19:16 or 24:19. In fact, 19:16
falls within the local dissonance minimum due to 6:5, and likewise for
24:19 and 5:4, so the chord 16:19:24 is really just a detuned
1/6:1/5:1/4 triad as far as beating and critical band dissonance are
concerned. I know that Daniel Wolf had a problem with this the last time
I said it, but as I see it, if different psychoacoustical phenomena are
responsible for the relevance of small integers, then different sets of
small integers can be relevant in describing the psychoacoustical
properties of a chord.

>To sum up, it may be a curious feature of JI systems that
>low-prime-limit systems (specifically Pythagorean or "3-limit") may
>favor very high odd-limits for some _relatively_ concordant
>sonorities, while some relatively high-prime-limit systems may favor
>comparatively low "odd-limits," at least in comparison to the prime
>limit (or "cap" if one prefers).

I don't know if there's really much to this observation --
high-prime-limit JI systems will be at least as rich as Pythagorean
tuning in intervals that happen to fall near local minima of the
dissonance function despite their high odd-limit. These high odd-limit
intervals are JI by construction only -- they do not have any audible
properties that distinguish them from nearby tempered intervals.