Open letter on "RI" and the open continuum

Hello, there, everyone, and in response to some recent articles, I
would like to make clear my own views on the topic of what has come to
be called "Rational Intonation" (RI), which might be described as the
use of integer ratios both small and large to construct a tuning, or
to serve as one possible reference point for the description of
intervals in various kinds of tempered systems.

Discussions of this kind may be of special interest to newcomers on
this Tuning List, who have an opportunity to observe the diversity of
musical styles and intonational philosophies prevailing here,
sometimes concerning the same tuning.

Before addressing some issues and possible misconceptions regarding
the RI approach, I would like warmly to invite newcomers to ask
questions, give all sides a respectful hearing, and above all to feel
invited to try these tunings and intervals for yourselves and reach
your own musical conclusions. You may arrive at a position different
from any of the "sides" so far established in this debate, and maybe
create some beautiful new music in the process!

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1. Of intervals, ratios, and cents
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In this dialogue, I suspect, the term "RI" can mean two quite
different things, both of which may be found in my posts, and which I
enthusiastically embrace as choices, not as universal imperatives.

First, RI can mean, in its strict or proper interpretation, designing
or using a tuning based on integer ratios only, often with a
characteristic mixture of simple and complex ratios.

Pythagorean intonation, either in its classic medieval European form,
for example, or in some 21st-century extensions and variations, is an
RI system. Discussing the large as well as the small ratios generated
by a chain of pure fifths is a tradition going back at least to the
medieval European theorists, and indeed also to the Chinese theorists
surveyed by James Murray Barbour and others.

Secondly, as illustrated in the recent thread about 46-tone equal
temperament (46-tET) which prompted this dialogue, "RI" seems more
loosely to mean the use of large as well as small integer ratios as
reference points of a kind in describing the vertical or harmonic and
the melodic resources of a temperament based wholly or largely on
irrational ratios.

For example, here I think that there is no disagreement that either
46-tET or the almost identical regular temperament with pure 11:14
major thirds at around 417.51 cents is indeed a _temperament_, not a
tuning based on integer ratios other than the 11:14 third (and 7:11
minor sixth at ~782.49 cents, its inversion or octave complement),
which may or may not be especially relevant depending on the style in
question.

Please note that here, as in other articles, I have specified the
fifth generating this 11:14 tuning as around 704.38 cents, and the
fifth of the 46-tET division as around 704.35 cents, using cents as a
measure rather than attempting any rational approximation. Here the
"sides" are agreed: cents are a very useful measure for rational and
irreational intervals alike.[1]

Now we come to what may be the main disagreement between the "sides":
not the utility of measuring interval sizes in cents, but the
propriety of sometimes using _large_ integer ratios as one possible
kind of emblematic reference or "mapping" of the open continuum.

Specifically, I have observed and observe now that in a neo-Gothic
approach to 21st-century music based on a mixture of 13th-14th century
Gothic styles in Europe with some new elements, the 11:14 tuning or
46-tET nicely provides or approximates a set of rather complex
intervals called the "Four Convivial Ratios."

Two of these ratios, 11:14 and 11:13 (~289.21 cents), are for regular
thirds formed from 4 fifths up or 3 fifths down; the other two, 17:21
(~365.83 cents) and 14:17 (~336.13 cents), are for what might be
termed "submajor/supraminor" thirds formed by 8 fifths down or 9
fifths up.

These ratios are simply possible points of reference on the continuum,
like other ratios often cited here with smaller integers. They do not
limit the range of musical expression, but merely provide some more
"points on the map" for the approximate and imperfect description of a
most subtly gradated sonorous reality.

Thus the debate is not on whether to use cents as a most helpful
measure, nor in general on whether it is appropriate or useful to
describe equal or other irrational temperaments in reference to
integer ratios -- although the latter issue seems one well worth
discussing.

Rather it is a debate as to whether mentioning certain "ratios that
dare not speak their names" -- ratios not fitting within a certain
kind of harmonic outlook -- is appropriate.

My answer is that, at a minimum, it is appropriate to mention them in
conjunction with any style where they are part of the accompanying
theory, as with neo-Gothic music

To do otherwise, in fact, would be a form of self-censorship denying
readers the opportunity to learn about the theory of a music as
formulated by the makers of that music themselves. Such a policy, it
seems to me, would reduce the range of a musical and conceptual
diversity that we should seek to foster and encourage.

Of course, with more complex ratios, it is often an interesting
question whether they are "audibly distinct" intervals or sonorities,
or more like arbitrary reference points or names to describe what are,
in effect, degrees of "tempering by ratio." As I wrote in one of the
posts providing a basis for this controversy:

What I realize here is that how one associates 46-tET intervals
with integer ratios may in part be a matter of style: one
approach is to look mainly for comparatively simple ratios
like 9:11 or 8:13; another is to look for more complex
categories such as 14:17 or 21:34, which might be heard in
many contexts more in effect as shades of temperament than as
distinct relations between harmonic partials.

Having said this, I would also like warmly to support as a helpful
practice the custom of always giving the value in cents for an integer
ratio when it is first mentioned, even if this seems "obvious" to
those of us who happen to be familiar with a given ratio.

Thus I could have and should have noted that 21:34, a supraminor
sixth, has a size of around 834.17 cents; this would help people place
it more easily on the continuum of intervals.

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2. When cents are best
----------------------

A great advantage of using cents as a common measurement, whatever our
views on integer ratios large and small, is that they permit accurate
description while leaving people to analysis or "editorialize" on
possible associations with such ratios as they wish.

For example, in one of the articles prompting this dialogue, I
borrowed a very creative system of Gene Ward Smith and proposed a kind
of matrix to show some of the main "interval families" or "flavors"
available in a given neo-Gothic tuning, with the number of fifths
producing a representative interval in each family shown along with
the size of the interval in cents.

While Gene is in no way responsible for what some might fairly call my
"barbarization" of his system -- any more than the ancient Greeks are
responsible for the interpretations of their modes offered in the
medieval treatises -- this notation may show the advantages of using
cents as the descriptive unit of measure.

For example, let us consider the "Noble Fifth" temperament whose
logarithmic division of the octave is given in Ervin Wilson's famous
Scale Tree, and which was proposed last year by Keenan Pepper on this
Tuning List. This is the regular temperament where the ratio between
the whole-tone (two fifths up less an octave) and the chromatic
semitone (seven fifths up less four octaves) is equal to the Golden
Section, or Phi, approximately 1.618.[2]

Here the value of the fifth is approximately 704.096 cents -- and note
that indeed I give the value in cents, not some rational approximation.
Here an "Interval Family Matrix" for usual neo-Gothic purposes might
look like this:

[1(704.1),4(416.4),9(336.9),15(961.4)]

Here we have first the fifth, then the major third at ~416.4 cents,
then the augmented second at ~336.9 cents, and then the narrow minor
seventh at ~961.4 cents.

This notation leaves readers to make whatever rational associations
they wish, or simply to say, "These are the values in cents, and I'd
prefer to let them stand for themselves in defining this tuning."

Thus one person may take 416.4 cents as a close approximation of
11:14, and another simply as a third some distance from either 4:5
(~386.31 cents) or 7:9 (~435.08 cents), or as a given distance from an
estimated point of maximum "harmonic entropy" around 422-423 cents, or
whatever.

Similarly, someone might take 336.9 cents either as a close
approximation of 14:17, or as something considerably different from
either 5:6 (~315.64 cents) or 9:11 (~347.41 cents), etc.

The advantages of giving intervals in cents, _as well as_ offering
possible interpretations in a given style or theoretical outlook, may
be even more dramatically illustrated if we consider the family of
intervals in this tuning formed by chains of 13, 14, and 15 fifths up
or down.

Here the matrix may not be the most informative one, giving only the
narrow minor seventh at ~961.4 cents, which seems in the general
vicinity of 4:7 (~968.83 cents). Unless one calculates the related
large major third and small minor third based on the size of the
fifth, however, one could overestimate the overall affinity to pure
ratios of 7.

A fuller "family portrait" of this group of intervals might read:

[-13(446.8),14(257.3),15(961.4)]

This makes it clear that the large major third is actually
considerably wider than 7:9, actually somewhat closer to something
like 10:13 (~454.21 cents), the area of "ambiguity" between the
general categories of "large major third" and "small fourth."

Giving a value in cents, 446.8, lets the reader decide how to
interpret this -- and whether, for example, the possible "RI" category
of 17:22 (~446.36 cents) might be helpful in describing this region
somewhere between 7:9 and 10:13.

Similarly, the small minor third at around 257.3 cents might have a
quality rather like that of a 6:7 (~266.87 cents) for some listeners,
and identifying the interval in cents leaves the reader free to make
an assessment.

For some purposes, it might also be interesting to offer larger
rational approximations, but these should supplement rather than
replace the convenient and objective measure of cents.

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3. The continuum, mediants, and diversity
-----------------------------------------

In my view, a vital factor in the musical experience is categorical
perception, the tendency to group intervals into certain familiar or
not-so-familiar categories or regions which may be defined by the
theory and practice of a given style.

Such categories are different ways of viewing a free continuum, and it
is understandable and proper that different people here will attach
significant to different kinds of categories, whether articulated as
integer ratios, as ranges in cents, or as names such as "neutral
seventh" or "supraminor sixth."

One use for "RI" ratios is to give names for certain conceptual
categories of intervals, to emblematize them if one will, without
necessarily implying that they are discrete "harmonic entropy minima"
or "valleys" or the like, as opposed to arbitrary points of reference
on the continuum.

One approach for doing this might be suggested by Erv Wilson's Scale
Tree, taken as a table of interval ratios -- and also to be read as a
table of logarithmic divisions of the octave, with both readings
included within this admirable structure of numerical poetry.

For example, suppose we start with 5:6 and 9:11, the simplest ratio
for a "minor third" and a typical ratio for a "neutral third."

Finding the mediant of these two thirds, we get 14:17, the "supraminor
third" of Manuel Op de Coul's Scala program and also of neo-Gothic
theory. This is not necessarily a discrete "valley" -- the status of a
sonority like 14:17:21 (~0-336-702 cents) as a three-voice combination
maybe being open to further research and discussion -- but may serve
as a nice emblem of the "supraminor third" category.

Suppose we repeat the process again with 5:6 and 14:17, getting 19:23,
or around 330.76 cents. This ratio is yet more complex, and thus
evidently more like a "tempering by ratio" than a discrete aural
relationship of partials, yet it also may have a useful emblematic
role relating to a very beautiful tuning.

In 29-tET, as it happens, the augmented second of ~331.03 cents is
very close to this ratio, and may be taken as either a highly tempered
form of 5:6, or an interval approaching the "supraminor" category
found in the general vicinity of 14:17 (another emblem, not
necessarily a discrete point).

What we have here are rather like multiple compass points, enriching
the lore of the continuum without in any way constraining our
exploration of it. We can articulate "subcategories" in a certain
numerical way, but the continuum itself and its musical beauty
outraces our efforts to categorize it in any event.

One thing RI ratios may also symbolize, in my view, is an interest
among some participants here in portions of the continuum not always
given the most prominent recognition in much "conventional" theory,
even conventional xenharmonic theory.

Sometimes one might guess that the basic disagreements between
xenharmonicists focus on whether a certain set of more or less agreed
upon ratios should be kept pure, or may be tempered by a few cents (or
sometimes more) for certain purposes.

The "subversive" feature of using RI ratios as one optional approach
in describing a temperament may be, not that they exclude specifying
interval sizes in cents or recognizing irrational temperaments as a
fundamental concept and process, but that they tend to "level" the
primacy of certain small integer ratios over large ones.

In short, the "problem" may be not that they prevent one from
recognizing a continuum, but that they imply a worldview where the
small ratios as well as the large ones are possible points on this
continuum rather than necessarily the primary or preferred ones -- a
question which may vary from style to style.

Here I would conclude by advising newcomers, as they may learn from
reading a sample of current and past messages, that my views are
likely minority ones, and that becoming acquainted with the range of
musical outlooks and styles pursued here is the best approach in
evaluating this issues.

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Notes
-----

1. Some people may have mixed feeling about using the specific
logarithmic measure of cents, since for them it tends to imply the
supremacy of 12-tET, and there may propose other units such as "mils"
(thousanths of an octave), but this is a different kind of issue than
the utility of some logarithmic measurement.

2. The "Noble Fifth" tuning is a counterpart of Kornerup's Golden
Meantone, where Phi defines the ratio between whole-tone and diatonic
semitone.

Most appreciatively,

Margo Schulter
mschulter@value.net