Yahoo Tuning Groups Ultimate Backup tuning Re: JI and integer ratios -- large and small

Hello, there, and I'd like to comment briefly on the question of large
and small integers in just intonation (JI). Incidentally, my warm
greetings to Jacky Ligon, who appears, like me, to have something of a
taste for _large_ integer ratios.

Anyway, for music such as medieval-Romantic Western European music
which makes a distinction between stable and unstable sonorities
(stable sonorities being fully concordant, and unstable sonorities
sometimes ranging on a subtle scale from relatively concordant to
acutely tense or discordant), I would say that "JI" implies that
_stable_ sonorities have _small_ integer ratios. Unstable sonorities
may often have large integer ratios, and this is a part of JI also.

The stability or "odd" limit determines how far the expectation of
_small_ integer ratios may go. For Gothic music, this is the 3-limit
(2:1, 3:2, 4:3). For Renaissance-Romantic 5-limit music, it also
includes 5-based ratios such as 5:4 and 6:5. Beyond these simple
intervals, forming the stable or fully concordant sonorities of a
given era, large and complex ratios seem to me a normal feature of JI.

First, let's consider the kind of case that the "small integer ratio"
rule may be seeking to exclude, where a tuning may be "rational"
without being "JI" in any usual sense.

For example, suppose we define a fifth at 16384:10935 -- certainly an
integer ratio. As it turns out, this "just tuning" is virtually
identical to 12-tone equal temperament (12-tet), as Kirnberger and
other late 18th-century theorists pointed out (sometimes using instead
a fourth at 10935:8192). Why isn't this tuning "JI"? The answer is
that just as with 12-tet, we won't get any _small_ integer ratios.

In contrast, let's look at 3 JI systems which do have small integer
ratios for stable concords as defined in a given system. Here I'll
list the most complex stable sonority for each system -- in each case,
featuring two special varieties of these simple ratios.

A _multiplex_ ratio has one string-length or frequency an exact
multiple of the other, e.g. 2:1, 3:1, etc. -- generally n:1.

A _superparticular_ ratio has one string-length or frequency equal to
the other plus an integral portion of it, e.g. 3:2, 4:3, etc. --
generally n + 1:n.

Now let's compare some of these ratios and sonorities -- and also some
unstable intervals with larger integer ratios -- in three JI systems,
the 3-limit, 5-limit, and 7-limit:

---------------------------------------------------------------------
system type stable sonority example of complex ratio
---------------------------------------------------------------------
3-limit trine (2:3:4) 81:64 (major third)
5-limit triad (4:5:6) 32:25 (diminished fourth)
7-limit tetrad (4:5:6:7) 49:40 (neutral third?)
---------------------------------------------------------------------

Note that in the 3-limit or Pythagorean JI of the Gothic era (around
1200-1420), the stable three-voice sonority of the complete trine has
pure intervals: the 2:1 octave, 3:2 fifth, and 4:3 fourth. However, an
unstable interval like the major third (81:64) has a rather complex
ratio -- nicely fitting its musical role of often resolving to a pure
unison or fifth, which resolves the tension.

In a 5-limit system, we have the pure 3-limit intervals plus also the
pure major third (5:4) and minor third (6:5), which with the 3:2 fifth
make up a triad. However, again, we find that an unstable interval
like the diminished fourth (32:25, two 5:4 thirds) has a complex
ratio, and is used in Renaissance music as a definite point of
tension, resolving to some concordant sonority with small integer
ratios.

In a 7-limit system, we additionally have such intervals as the pure
7:4 minor seventh and 7:6 minor third. In some musics using these
intervals, they may participate in a stable "tetrad" of 4:5:6:7. Once
again, we also encounter more complex ratios such as 49:40, a kind of
"neutral third" about midway between major and minor (although I'm not
sure how 7-limit JI people would describe this interval).

Note that in moving from 3-limit to 5-limit to 7-limit and beyond, we
find some complex intervals at a lower stability limit which may have
simpler equivalents at a higher limit. Thus the complex 81:64 of
medieval 3-limit style gets replaced by the simpler 5:4 of Renaissance
5-limit style -- each tuning fitting its own style. Similarly, if we
go as high as the 11-limit, we might compare the 49:40 of 7-limit with
the smaller and simpler ratio 11:9 (a common neutral third).

In other words, the motto for JI systems of this historical kind might
be: "Integer ratios only, small _and_ large." However, if there are
_only_ large ratios like Kirnberger's 10935:8192 without any small
ones like 3:2 and 4:3, then we are dealing with a "rational tuning" as
opposed to a "JI system" in the usual sense.

Note that some people choose to consider only 5-limit or higher
systems as "JI," possibly because such systems involve a distinctive
complexity while 3-limit JI is a regular tuning with a structure much
like that of other regular tunings such as Renaissance meantones.

In such regular tunings, we have a "plane" of two dimensions: 2-limit
octaves (2:1) and chains of fifths all the same size (a pure 3:2 in
3-limit or Pythagorean JI, smaller than pure in meantones, larger than
pure in some neo-Gothic regular tunings). All intervals can be
generated either from a chain of identical fifths, or from octaves.

In contrast, with 5-limit or higher JI system, we have a complex
"polyhedral" geometry where a pure 5:4 major third (unlike a 3-limit
81:64), for example, _cannot_ be generated from four pure 3:2 fifths,
but represents a separate "dimension" of organization.

Personally, speaking in good part as an advocate of 3-limit JI, I
would say that the planar/polyhedral issue is distinct from the basic
concept of classic JI: a system with all intervals defined as integer
ratios, and _some_ of these intervals defined as _small_ integer
ratios. More typically, in 3-limit, 5-limit, and 7-limit systems
alike, small integer ratios are used for stable concords, but often
impressively large integer ratios for various unstable intervals and
sonorities.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:
> http://www.egroups.com/message/tuning/12459
>
> [in conclusion]... Personally, speaking in good part as an
> advocate of 3-limit JI, I would say that the planar/polyhedral
> issue is distinct from the basic concept of classic JI: a system
> with all intervals defined as integer ratios, and _some_ of
> these intervals defined as _small_ integer ratios. More
> typically, in 3-limit, 5-limit, and 7-limit systems alike,
> small integer ratios are used for stable concords, but often
> impressively large integer ratios for various unstable intervals
> and sonorities.

Hello Margo. A wonderful summing-up! And thank you, for
unintentionally coaxing me to explain myself a little further.

I suppose partly the reason I'm arguing for the 'low-integer'
definition of 'just-intonation' is my linguistic interests.

The very term 'just' refers specifically to the 'purity' of
the *harmonious connection* between low-integer ratios and
a high degree of consonance.

Certainly, even within a relatively simple 5-limit system,
there is the possibility of combining pitches in such a way
as to acheive magnificently large-integer ratios in the
intervals.

But 'just' is specifically meant to describe the 'stable
sonorities', with the implication that the complexities we
may revel in would be merely by-products.

I suppose that's the main reason I prefer to reserve 'just'
for small lower-integer systems, and to use 'rational' when
the system gets larger by extension of the scope of *either*
the exponents or prime-factors.

And, as I pointed out before, 'rational' is better as a generic
term to describe any tuning composed of 'ratios'. (get it?... ;-)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Re: JI and integer ratios -- large and small

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Pierre Lamothe <plamothe@aei.ca>