Re: Historical music and JI: SIJI or SISJI?

Hello, there, and some further comments about just intonation (JI)
systems from various people including JI pianist Dave Hill have
prompted some of the following remarks. While Dave has shown that JI
can be applied to rather modern instruments and compositions such as
the piano repertory, I'd like to address the earlier use of
JI-oriented models in composed European music.

Here it may be helpful to distinguish what I might term SIJI or "small
integer JI" -- where _all_ significant intervals (or at least the
vertical ones) can be defined as "small integer" ratios -- from SISJI,
or "small-integer-stability" JI, where only _stable_ harmonic
intervals necessarily have such small ratios.

Thus both medieval Pythagorean (3-limit) and Renaissance 5-limit JI
schemes come within SISJI, since the most complex _stable_ intervals
and combinations can be expressed with small integers, in other words
an odd-limit not greater than 3 for medieval harmony (3:2, 4:3), or 5
for Renaissance harmony (e.g. 5:4, 8:5).

However, neither system might be taken as strictly SIJI, since
medieval harmony is replete with relative concords like 81:64 and
32:27, while Renaissance harmony often features a rather bold use of
the vertical tritone at 45:32, or the diminished fifth at 64:45.

Why say SISJI rather than just "JI." Here my distinction is that it is
quite possible to devise a "JI" system -- that is, all intervals defined
as integer ratios -- where nevertheless there are no _small_ integer
ratios. For example, a tuning based on a fifth of 16384:10935 (or a fourth
of 10935:8192) very closely approximates 12-tet, as Kirnberger and others
noted in the later 18th and early 19th centuries. Also, a tuning with
equal semitones of 18:17 rather closely approaches 12-tet, as Vincenzo
Galilei (father of the astronomer and physicist Galileo) recognized in the
late 16th century. Such systems might be technically, but not in the usual
musical sense, described as "JI" -- unlike SISJI systems such as
medieval Pythagorean and Renaissance 5-based.

When it is urged that JI is mainly a theory in search of a practice, I
might note that the Continental European polyphony of the era
1200-1400 was very likely mostly conceived in the Pythagorean tuning
described by theorists and found by modern critics such Easley
Blackwood to fit the music. Although some 16th-century theorists
argued that pure JI might not always describe vocal intonation in
practice, nevertheless theorists such as Zarlino took it as the norm
and ideal. In this view, a temperament such as 1/4-comma meantone
(with its pure major thirds at 5:4) might be seen as a slight
compromise of 5-based JI to accommodate the exigencies of a 12-note
keyboard.

However, it bears emphasis that these great eras of JI and quasi-JI
music did _not_ necessarily espouse the ideal of "pure consonance" as
expressed by some 20th-century advocates and critics of JI systems.

Medieval polyphony of the era 1200-1400 is largely based on a subtle
continuum of consonance/dissonance, while Renaissance music depends on
a more subtle and muted but nevertheless vital contrast between
pervasive concords and diverting dissonances such as suspensions. Even
Zarlino, one of the most eloquent exponents of concord as the essence
of good music, notes that dissonances (e.g. at cadences) give the
music a grace it would not otherwise have.

Since the bold use of instability in medieval music has been widely
recognized, and indeed the accentuated contrast between pure 3-limit
concords and sometimes very tense discords (e.g. M7 at 243:128 or
~1110 cents) is a feature of this music further accentuated by the
tuning, I may not need to argue this point at length.

For Renaissance music, it may be worth noting that in a 5-based JI
tuning of the kind described by Zarlino for vocal music, a minor
seventh at 9:5 or a minor second at 16:15 is heard as a definite
discord, not an incidence of "pure consonance."

Thus JI can often mean "an enhanced _polarization_ of concord/discord"
(as in the case of the Renaissance contrast between 9:5 and 5:3,
vis-a-vis the possibly less pronounced contrast of 7:4 and 5:3 in
7-based systems).

An almost-concluding reflection: while both medieval and Renaissance music
are apparently often based on JI systems, the distinctive musical
qualities of these styles may be heard even on a 12-tet synthesizer, for
example; using the historically likely intonations accentuates these
qualities.

Similarly, the same basic Renaissance piece might be sung in 5-based
JI (or, in practice, a close approximation), played on a meantone
keyboard, or indeed on a 12-tet lute (the usual tuning for this
instrument, at least by 1550 or so, if we may judge from the treatises
and some of the lute repertory). While these tunings -- and also
Setharian considerations of timbre, such as the milder effect of a
400-cent M3 on a lute -- may significantly affect the sound of the
music, the basic "Renaissancy" character of the style remains a
constant.

This is not to say that "tuning is irrelevant," only that it is but
_one_ parameter of music, and that "JI" compositions can be as
different in conception and style as Perotin and Palestrina, for
example.

A final note: medieval Pythagorean tuning may have the significant
property among European JI systems that it is easily applied to keyboards
as a consistent principle, whether in the usual 12-note tunings of the era
1200-1450, or in versions of up to 17 notes, with rather free
transpositions available (to speak of "keys" would be an anachronism,
while to speak in terms of "cadential transpositions" might be better, a
topic for another post). In contrast, implementing later JI systems (e.g.
16th-century) with equal flexibility on a fixed-pitch instrument can be a
somewhat daunting task, thus the popularity of "quasi-just" solutions such
as 1/4-comma meantone.

Most respectfully,

Margo Schulter
mschulter@value.net

Excellent Margo! Thanks.

-- Dave Keenan
http://dkeenan.com