Re: "Newbie" question and responses -- 5:4 or 32:25?

Hello, there, and I'd like to comment a bit on a recent dialogue about
whether and when, in just intonation, substituting a simultaneous or
vertical interval of 32:25 for a 5:4 major third might be either an
obvious "mistake" or an apt solution. Here I suspect that for an
enthusiastic newcomer to Just Intonation (JI), the problem of the
32:25 may be something of an "in-joke," a joke more comprehensible
once explained. Here I'll try to provide a bit of explanation,
warmly inviting further questions, as well as other views.

First, a disclaimer: this discussion seems mainly directed to
Classic/Romantic music in the major/minor key system, or more recent
music, where concepts of "key" and of "modulation" (in the sense of
moving between major/minor keys) are relevant. As someone mainly
involved with medieval and Renaissance/Manneristic tunings -- mostly
Pythagorean (3-limit just) and meantone, I hope that my response may
complement the views of others involved mainly with later styles.

Now, I'd like to begin by explaining the "humor" of the 32:25
problem. Objectively, a vertical or simultaneous interval of 32:25,
e.g. a diminished fourth of E3-Ab3 in 5-limit JI, is simply another
possible interval on a vast spectrum. (Here I use C4 for middle C,
with higher numbers showing higher octaves.) In a typical 5-limit
setting, however, it is often considered what is called a "Wolf," an
interval _not quite_ the same as the one you probably intend. Because
it's "not quite right" if you expect a 5:4 major third, it can sound
figuratively like the "howling of Wolves," and thus the name.

Here's where a special kind humor or irony may come in. In the
intonation problem being discussed, the use of 5-limit JI would
suggest a special priority of having major thirds at a pure 5:4, as
smooth as possible. Using the system of cents, where a cent is 1/1200
of an octave, and 1/12 octave (an equal semitone of 12-tone equal
temperament or 12-tet) is 100 cents, this pure 5:4 interval is around
386.31 cents.

Certainly this 5:4 third can be very smooth, indeed captivatingly so,
and ideal for music where thirds (and sixths) are the most favored
intervals, as is true both in Renaissance music starting around 1450,
and in later music of the kind being discussed in this dialogue.

However, some advocates of JI seem to go further, and will argue not
only that 5:4 is the ideal tuning for major thirds in such music, but
that any tuning substantially different is not really "concordant,"
e.g. the considerably wider major third in 12-tet at 400 cents. Still
wider, and quite active although regarded as _relatively_ concordant,
is the Pythagorean major third formed from four pure 3:2 fifths at
81:64, or about 407.82 cents. In comparison with a 5:4 major third,
one of either 400 cents or 81:64 has prominent beats, at least with
many accustomed timbres of voices and instruments.

Now for the "prank," as it were, of musical geometry: three pure 5:4
major thirds, unfortunately, don't add up to quite a pure 2:1
octave. For example:

5:4 5:4 32:25
c - e - g# - c
386.31 386.31 427.38

We can tune two pure major thirds, c-e and e-g#, but the remaining
interval to form an octave (2:1, 1200 cents), g#-c, must be a
diminished fourth, definitely larger than a 5:4 major third, and in
fact 32:25 or about 427.38 cents. This is about 41.06 cents larger
than pure, a difference called a "diesis" of 128:125, about 1/5-tone.

This "bug" -- or, sometimes, "feature" of Renaissance and later
tunings based on pure major thirds -- can complicate problems of
intonation. If you want a normal 5:4, then hitting a 32:25 can
definitely be a "bug" or "mistake." If you want something very
deliberately "different," however, then it might be a feature.

The 32:25 _can_ be a feature, either in 5-limit JI or in what is
called 1/4-comma meantone for keyboards, a popular Renaissance tuning
where each regular fifth is made slightly narrower than a pure 3:2 in
order to permit 5:4 major thirds. Nicola Vicentino (1511-1576), for
example, uses a sonority with a 32:25 above the bass (B2-F#3-B3-Eb4)
to bring out the Italian word _pianger_, "to weep."

However, if it's _not_ intended as a calculated effect, then singing a
32:25 could be a bit of a humorous predicament, especially as judged
by some bystander not necessarily so enthusiastic about 5-limit JI.

The humor is that the performers are going to such great lengths to
produce _pure_ thirds, and for all their pains, here they are singing
or playing something much more "out of tune" from a conventional
viewpoint than the 12-tet or even the Pythagorean major third! One can
almost hear a proverb: "Extreme (intonational) justice is extreme
injustice."

A bit more than 400 years ago, the lutenist and radical music theorist
Vicenzo Galilei (father of the astronomer Galileo Galilei) indulged in
a similar kind of humor in his _Fronimo_ when he pohed fun of lute
players who would add extra _tastini_ or "little frets" to their
instruments in order to play purer thirds. Lutes were generally tuned
in 12-tet, considered acceptable for this instruments because the
strings somewhat "softened" these thirds, unlike the organ or
harpsichord, where meantone with pure or near-pure thirds was
standard. However, some players -- like modern designers and players
of JI guitars -- weren't satisfied with the "usual" 12-tet, and added
frets for smaller intervals.

In having his bit of fun, Galilei points out that these players
sometimes touch their extra frets so as to produce very audibly impure
fifths -- likely around 40:27 rather than 3:2, another "Wolf" interval
-- thus showing listeners the true "delicacy" of their ears!

Such "humor" seems almost inevitable when people try something new, or
different: it's like seeing an automobile fail to start (let's say a
solar-powered car, for the sake of the environment) and shouting "Get
a horse!", or poking fun at a new and very useful piece of software
because of an obvious bug or two.

To sum up so far: in 5-limit JI or meantone where pure 5:4 major
thirds are the norm, a 32:25 or diminished fourth (spelled, for
example, E3-Ab3 rather than the usual E3-G#3, or A3-Db4 rather than
the usual A3-C#4) is either a rather glaring "bug," or a deliberate
"special effect." It can have great "surprise value," but shouldn't be
an unwelcome surprise to the musicians themselves.

In other kinds of tunings and music, however, an interval at or near
32:25 might be quite routine, whether we choose to call it a "major
third," a "supermajor third," or something else.

In Gothic music of the 14th century, for example, where the usual
Pythagorean major thirds are already quite wide and active, some
performers may have used even wider (super)major thirds at cadences
where they expand to fifths:

G#+3 - A3
E3 - D3

(Here the "+" means that the G# is somewhat higher than the usual
Pythagorean note, making the melodic semitone G#+3-A even narrower
than the usual compact Pythagorean semitone.)

Marchettus of Padua (1318) describes the use of such extra-large
thirds, which _might_ in practice have been tuned somewhere near 32:25
(~427 cents) or 9:7 (~435 cents), making cadences even more dramatic
and intense. In this music, fifths and fourths are richly stable
intervals, contrasting with tense although "tolerable" thirds and
sixths -- either Pythagorean, or in some cases "super-Pythagorean" (if
we follow the likely meaning of Marchettus).

In 20th-century tunings, also, intervals of around 32:25 or 9:7 get
used regularly -- in this case not necessarily as unstable, but also
as pervasive "concords," as in Gary Morrison's music based on his
system of 88-cet (88-cent equal temperament), with its supermajor
third of 440 cents, a bit larger than 9:7. This composer has nicely
demonstrated how a supermajor third can be accepted by the ear as a
concord in the right musical setting.

Also, for example, 17-tone equal temperament (17-tet) has a major
third of 6/17 octave, or ~423.53 cents, quite close to a 32:25.
Depending on one's stylistic inclinations, one could use this interval
either in a "neo-medieval" manner like that of Marchettus, inviting a
resolution to the fifth, or in Gary Morrison's manner, as an
independent concord in its own right.

In addition to explaining the "joke" of hitting a 32:25 in 5-limit JI
(unless it's a deliberate special effect, so to speak), I have
attempted to give some idea of the range of tunings, historical and
recent, available.

Again, I warmly invite questions, knowing that explanations of this
kind can be clearer to the author than to the reader, and also that
while I might not be unique in having musical and intonational biases,
the views of others may at least reflect different biases, thus
possibly lending some perspective to the observer in reaching a
balanced judgment.

Most respectfully,

Margo Schulter
mschulter@value.net