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.