Yahoo Tuning Groups Ultimate Backup tuning Circles and orbits (for Joe and Paul)

Hello, there, Joe Pehrson and Paul Erlich, and please let me clarify
the meaning I intended for "orbit," which has aspects and implications
which both of you picked up from different viewpoints.

Thank you for giving me the opportunity better to clarify my use of
this term, while taking note of something linguists sometimes call a
"Gricean implicature" after Paul Grice, a philosopher who developed
the concept of a "conversational implicature."

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1. Circles and orbits
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The first guideline I might offer is that all circulating tunings and
temperaments, or "circles," are also "orbits" -- but the latter
category also includes some schemes with some "circle-like" qualities
which nevertheless are a bit too "eccentric" (an astronomical
metaphor) to meet the usual requirements for a "well-temperament" or
the like.

Let's consider some types of systems qualifying as "circles," or
"circulating" schemes as they are often called:

1. Equal temperaments are precisely symmetrical
closed circles in the mathematical as well as
musical sense;

2. Unequal well-temperaments use generators of
different sizes (e.g. the fifths of a 12-note
or 17-note well-temperament), but within
certain constraints so as to produce "playable
concords" or the equivalent at all positions.

3. Loops use a single regular generator (aside
from an octave or other interval of repetition),
and produce a scheme with a slight asymmetry,
but with the requirements met for a "circle"
(playable concords in all positions).

Examples of the "loop" type of circle are a 53-note Pythagorean system
(with the odd 53rd fifth about 3.62 cents narrow); or 1/4-comma
meantone (with the odd 31st fifth about 0.69 cents wide).

In the latter example, by the way, we have a kind of "anti-Wolf fifth"
notably _closer_ to 3:2 than the others (narrow by around 5.38 cents).

All of these "circles" are types of the larger category of "orbit,"
which includes all of these circulating schemes plus others which have
some "near-circular" properties but might, for example, have "Wolf"
fifths or fourths in one or more positions.

Thus either 17-EDO or George Secor's 17-note well-temperament is a
kind of "17-note circle," a category which in turn falls within the
broader set of "17-note orbits."

However, the latter category of orbits also includes, for example,
schemes where all 17 steps are "minor semitones" or "thirdtones"
within 15 cents or so of 17-tET (~70.588 cents), and intervals such as
thirds and sixths keep within a range from around Pythagorean to
septimal (e.g. major thirds from 81:64 or ~407.82 cents to around 9:7
or ~435.08 cents or a bit larger), but some fifths could be wide by as
much as a full 64:63 or Archytan comma (~27.26 cents), as George Secor
well calls it.

In such a scheme one can use every step as a diatonic semitone, and
rely on 17-note circle-like equivalences such as A#=Cb and Gb=E#, but
in usual harmonic timbres would find certain fifths quite unlikely
landing places for cadential progressions to stability.

In contrast, in a 17-note circle such as 17-EDO or George Secor's
unequal well-temperament, one _can_ comfortably land on a fifth at any
position in the circle.

Here two points may be helpful: the line between a "circle" and some
more eccentric type of "orbit" can be a matter of fine judgment and
sometimes differing tastes; and while the "orbit" category includes
both "circles" and some more eccentric schemes, it has also has
certain expectations (however loose or vaguely defined). In other
words, not every tuning scheme with n notes is an n-note "orbit."

One fine point of defining a "circle" is that the requirement of
"acceptable concords in all positions" can depend both on the types of
sonorities regarded as "stable concords" (and especially saturated
ones, e.g. medieval European trines at 2:3:4, or Renaissance-Romantic
triads at 4:5:6), and one how far one permits such "concords" to vary
from pure and yet be deemed "playable."

For example, some people might regard Kirnberger II's fifths at 1/2
syntonic comma narrower than pure (~10.75 cents) as closer to outright
Wolves than to "comfortably playable," although they come within the
sometimes quoted limit of 1/2 Pythagorean comma (~11.73 cents), or
Dave Keenan's ingenious definition of a "Wolf fifth" as one which,
when placed in a chain of 11 or less, can generate an interval closer
to 3:2.

In a French _temperament ordinaire_ of the 17th-18th centuries, all
fifths are comfortably close to 3:2, but by Baroque-Classic standards,
the widest major thirds at somewhat larger than Pythagorean might go
beyond the limits of "playability" as stable concords. Thus the status
of this type of scheme as a "well-temperament" -- by 18th-century
standards -- is sometimes debated.

The broader "orbit" category includes not only typical or more
marginal circles, but also tunings with some "Wolf fifths" or the
like; nevertheless, even this category excludes lots of tunings.

For example, we would not describe a 17-note Pythagorean tuning as a
"17-note orbit," since some adjacent steps are limmas or diatonic
semitones at 256:243 (~90.22 cents) while others are Pythagorean
commas at 531441:524288 (~23.46 cents) -- roughly 4/9-tone and
1/9-tone. Such an arrangement is clearly distinguishable from a
"quasi-circle" of 17 "thirdtones" or the like, however liberally
defined (say all steps at least 50-55 cents).

In short, orbits include all the more or less usual "circles" or
"circulating" schemes, plus some more eccentric schemes with
"quasi-circular properties."

How an "orbit" of the more eccentric type is likely to differ from a
"circle" may depend on the stylistic constraints.

Thus in a neo-Gothic system, for example a 17-note orbit, Wolf fifths
are the likeliest factor. This is true both because fifths and fourths
are the primary concords, and because relatively concordant intervals
such as thirds and sixths have a much wider range of "interchangeably
playable" variations in this type of style. Thus I would happily
accept anything in the range of 408-440 cents as a "usual neo-Gothic
major third," but would expect "interchangeable" fifths to remain
within about 8-9 cents of pure (and preferably closer).

Note that "noninterchangeable" intervals such as fifths at 32:21
(~729.22 cents, a 64:63 wider than 3:2) can be a special attraction of
these "eccentric orbits," and are precisely what is excluded by the
usual criteria of "circularity" or "well-temperament."

With an 18th-century type of 12-note system, either Wolf fifths or
major thirds deemed to be "noninterchangeable" (i.e. substantially
larger than Pythagorean) might make a system an "orbit" of the type
other than a "circle." These intervals, also, are available for
special effects.

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2. A Gricean implicature
------------------------

Strictly speaking, any "circular" system (equal temperament, unequal
well-temperament/tuning, or loop) is also a type of orbit.

However, if one describes a new or unfamiliar system as "a 22-note
orbit," for example, this may invite the listener or reader to draw an
inference that the system is an orbit of the more "eccentric" type
rather than an equal or unequal "circle."

The reason for this is what Paul Grice has described as the
"Cooperative Principle" in conversation: to give the most informative
description as economically as possible.

Thus it would be usual to describe either 22-EDO or Paul Erlich's
unequal 22-note well-temperament as a "22-note circle," which conveys
both that it is necessarily also an "orbit," and also that it
circulates by usual standards.

If one says that a given tuning system is an "orbit," but _not_ that
it is a circle (a more specific and thus informative description),
then one might draw a "conversational implicature" that the system is
an orbit other than a circle.

Similarly, the following types of "Gricean implicatures" are typical
of everyday conversation:

* "Last night I saw you at the party with someone
(presumably not a mutually known family member,
friend, or acquaintance).

* "We were somewhere in New York State" (likely
somewhere other than Times Square or another
especially familiar place).

Thus if a system is deemed to "circulate" by usual standards for a
given style or genre, we tend to communicate this information by
speaking of a "circle" or the like; to describe a system more
generally as an "orbit" can implicate that it is not within the
narrower and more specific category.

Of course, it can be useful to consider circles as a subset of orbits:
for example, "17-note orbits, conventionally circulating or
otherwise."

Again, the "orbit" may be a rather new concept in tuning systems; like
an astronomical orbit, it can vary greatly in the degree of
"eccentricity."

While all circles are also orbits, it is especially the
"quasi-circular" systems outside the usual standards for a circulating
system that promoted this broader category.

Most appreciatively,

Margo Schulter
mschulter@value.net

Circles and orbits (for Joe and Paul)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗jpehrson2 <jpehrson@rcn.com>