Re: Harmonic/arithmetic contrasts: strings and frequencies

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Harmonic/arithmetic contrasts:
Strings, frequencies, and conversities
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In a recent dialogue on the Tuning List, some of us have been
discussing the topic of sonorities which share the same set of
intervals in different arrangements suggesting a process of "mirror
reflection."

Typical examples in Western European composition from around 1200 to
1900 are the Gothic 3-limit trine and Renaissance-Romantic 5-limit
triad, each with two mirrorlike arrangements (here I use a MIDI
notation where C4 is middle C and higher note numbers show higher
octaves):

3-limit trines (8, 5, 4) 5-limit triads (5, M3, m3)
------------------------ --------------------------
| D4 | D4 | D4 | D4
| 4 | 5 | m3 | M3
8 | A3 8 | G3 5 | B3 5 | Bb3
| 5 | 4 | M3 | m3
| D3 | D3 | G3 | G3

Flavor 1 Flavor 2 Flavor 1 Flavor 2

It will be seen that a complete 3-limit trine (octave, fifth, fourth)
may have either the fifth below and the fourth above or, _e converso_
as two medieval treatises say, the fourth below and the fifth
above. Similarly, a complete 5-limit triad (fifth, major third, minor
third) may have the major third below and the minor third below, or
the converse.

From the medieval Latin _e converso_ of Coussemaker's Anonymous I
(c. 1290 or 1300) and Jacobus of Liege (c. 1325), I derive the English
term "conversity" to describe these mirror sonorities.

Section 1 of this article places these trinic and triadic conversities
on a concrete basis: the ratios between the frequencies of their
tones, or between the lengths of strings or organ pipes producing
these tones. We encounter two basic divisions, the "mean-frequency"
and "mean-string" divisions, which when applied to the outer intervals
of the octave and fifth produce the two mirrorlike flavors of 3-limit
trines and 5-limit triads respectively. The symmetry of these
divisions can be expressed graphically by a "mirrored" ratio notation.

Section 2, taking an "historically correct" medieval and Renaissance
approach, derives both flavors of trines and triads from string-ratios
alone, introducing the "harmonic/arithmetic" contrast.

Section 3 draws some connections between these approaches, suggesting
some possible notational conventions.

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1. Trines and triads: mean-frequency and mean-string divisions
--------------------------------------------------------------

When we speak of a pure fourth as having a ratio of 4:3, two concrete
interpretations are possible. We might be comparing either the
frequency-ratio or vibrational rates of the upper and lower notes of
the fourth, e.g. 400 Hertz (cycles per second) and 300 Hertz; or the
string-ratio between the lengths of strings or organ pipes producing
the lower and upper notes, e.g. 40 centimeters and 30 centimeters.

We may note that in a frequency-ratio reading, the first number of 4:3
refers to the _higher_ note (having the greater frequency); in a
string-ratio reading, it refers to the _lower_ note (produced by the
string with the greater length). Some readers might compare this
reciprocal relationship to that between frequency and wavelength in
physics.

In either interpretation, however, the ratio 4:3 describes the same
musical interval, a pure fourth. As the noted writer Gertrude Stein
might say, "A 4:3 is a 4:3 is a 4:3."

Moving beyond simple two-voice intervals to combinations having three
or more notes and intervals, however, we find that these two
interpretations can produce kindred but contrasting sonorities with a
musical as well as mathematical difference.

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1.1. Medievalists bearing synthesizers: 3-limit trines
------------------------------------------------------

Let us suppose that we have a set of tone-generators, each of which
may be set digitally to any desired frequency. We tune two of these
generators to 200 Hertz and 400 Hertz, forming a pure 2:1 octave.

Let us now add a third note whose frequency is equal to the _average_
or mean of these two frequencies -- that is, to 300 Hertz. We then get
the following sonority:

| 400
| 4:3 (4)
2:1 (8) | 300
| 3:2 (5)
| 200

In addition to the two outer notes at an octave of 400:200 or 2:1, we
now have an interval between the lower two voices of 300:200 or 3:2, a
pure fifth. Between the two upper voices we have a ratio of 400:300 or
4:3, a pure fourth.

This three-note division of the octave into a fifth below and a fourth
above -- Jacobus of Liege speaks of the third middle voice as
"partitioning" or "splitting" the outer octave -- represents the ideal
stable sonority of 13th-14th century European music. Johannes de
Grocheio (c. 1300) describes it as manifesting the _trina harmoniae
perfectio_ or "threefold perfection of harmony" -- thus the modern
English "trine" -- while Jacobus remarks that it is the best way of
ending a polyphonic composition.

Although frequency-ratios were not then in use, we may in modern terms
describe this division of the octave as a "mean-frequency" division:
that is, the added third note has a frequency equal to the _mean_ or
average of the two outer notes of the 2:1 octave.

While our specific realization of this mean-frequency division of the
octave happens to involve a frequency-ratio of 200:300:400, the more
general pattern may be expressed by a ratio of 2:3:4, using the
simplest possible integers.

Note that in writing frequency-ratios of three or more notes, an
_ascending_ order of numbers such as 200:300:400 or 2:3:4 fits the
common convention of describing a sonority from the lowest note
(lowest frequency, slowest vibration rate) to the highest. Thus our
2:3:4 sonority has a fifth (2:3) below and fourth (3:4) above.

Interestingly, a medieval theorist such as Jacobus also seems to
prefer an ascending order for reckoning the intervals in a sonority:
in this arrangement, he speaks of the lower fifth being placed
"before" (_ante_) the upper fourth, or of the fourth being placed
"after" (_post_) the fifth.

Now shifting to a more traditional technology, let us suppose that we
have two strings with lengths of 400 centimeters and 200 centimeters
which produce a pure octave. We now add a third string with a length
equal to the average or mean of these two, 300 centimeters:

| 200
| 3:2 (5)
2:1 (8) | 300
| 4:3 (4)
| 400

As with our three digital frequency-generators, we find that our
division of the octave has produced a complete trine: outer octave,
fifth, and fourth. However, our new sonority has a different
arrangement of these intervals.

Here the lower pair of strings, with a string-ratio of 400:300,
produce a 4:3 fourth; while the upper two strings, with a ratio of
300:200, produce a 3:2 fifth. The fourth is now below, and the fifth
above -- a "mirroring" of our earlier division.

We may refer to our new division as a "mean-string" division: our
third string has a length equal to the average or mean length of the
two strings forming the outer octave.

Proceeding from the lowest note to the highest, we might express this
mean-string division as a 400:300:200 division, or more generally a
4:3:2 division. Since the _longest_ string produces the lowest note,
it seems natural to write such string-ratios in a _descending_ order.
Here we have a 4:3 fourth (400:300) _below_ and a 3:2 fifth (300:200)
above.

Jacobus discusses this alternative arrangement, with the fourth placed
"before" (_ante_) the fifth, at some length, concluding that while
concordant, it is less pleasing than its counterpart with the fifth
below and fourth above, where the more concordant fifth "supports" and
"fortifies" the less concordant fourth.

In sum, our mean-frequency division of 2:3:4 and our mean-string
division of 4:3:2 yield mirrorlike trines with converse arrangements
of the fifth and fourth.

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1.2. Renaissance musical fashions: 5-limit _harmonia perfetta_
--------------------------------------------------------------

Having derived the two flavors of Gothic trines from our
"mean-frequency" and "mean-string" divisions of the 2:1 octave, let us
now apply the same procedure to the pure 3:2 fifth.

Returning to our tone generators, we tune two of them to such a fifth,
say at 200 Hertz and 300 Hertz, and add a third generator at the mean
frequency of 250 Hertz:

| 300
| 6:5 (m3)
3:2 (5) | 250
| 5:4 (M3)
| 200

This is the ideal harmony of the 16th century, described by Gioseffo
Zarlino in 1558 as _harmonia perfetta_ or "perfect harmony," and in
1612 dubbed by Johannes Lippius the _trias harmonica_ or "harmonic
triad." The two lower tones with a ratio of 200:250 or 5:4 form a pure
major third, while the upper two with a ratio of 250:300 form a pure
6:5 minor third.

We can describe this "mean-frequency" division of the fifth as a
200:250:300 division, or more generally a 4:5:6 division.

Moving again from digital generators to strings, we start with two
strings with lengths of 300 and 200 centimeters, forming another pure
3:2 fifth. Adding a third string with a length equal to the average or
mean of these, we arrive at another popular Renaissance sonority:

| 200
| 5:4 (M3)
3:2 (5) | 250
| 6:5 (m3)
| 300

In a mirrorlike reflection of our previous sonority, we have an outer
fifth again divided into two thirds -- but this time with the lower
two strings forming a 300:250 or 6:5 ratio, a pure minor third; and
the upper strings forming a 250:200 or 5:4 ratio, a pure major third.

We can express this "mean-string" division of the fifth as
300:250:200, or more generally 6:5:4.

Zarlino, Lippius, and other theorists of the 16th and early 17th
centuries tell us that the arrangement with the major third below is
more "natural" and "joyful," while that with the minor third below is
more "sad" or "gentle." In the latter arrangement, Zarlino concludes,
the consonances are "out of their natural order." This effect on the
ear in arranging "a mass of consonances" he elsewhere compares to that
on the eye of placing the ceiling of a building where the foundation
is expected.

Thus our two divisions of the fifth, the mean-frequency division of
4:5:6 and the mean-string division of 6:5:4, produce a pair of
mirrorlike sonorities or conversities analogous to our trinic 2:3:4
and 4:3:2 divisions of the octave.

The parallel is musical as well as mathematical: just as Jacobus
expresses a preference for a trine with the 3:2 fifth below and the
4:3 fourth above, so Zarlino prefers a _harmonia perfetta_ with the
5:4 major third below and the 6:5 minor third above.

Our mean-frequency divisions produce these favored arrangements -- but
the application of frequency-ratios to music goes back in the Western
European tradition only to around the early 17th century. After
considering a convenient convention of modern notation, we take a more
"musicologically correct" approach to deriving these favored
sonorities using only string-ratios.

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1.3. Multi-voice ratios: A mirrorlike notation
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When discussing the ratios of simple intervals, convention generally
places the larger number first: thus a 2:1 octave, 3:2 fifth, etc. In
a medieval or Renaissance setting before the widespread introduction
of frequency-ratios in the 17th century, this ordering fits a
preference for beginning with the lower note (longer string or pipe).

Another 17th-century development, the application of logarithms to
music (most familiar today in the cents measurement), may also
motivate this ordering. Ratios smaller than unity, such as 1:2 or 2:3,
would seem to imply negative logarithmic measures, for example "an
octave of 1:2, or -1200 cents."

When stating frequency-ratios or string-ratios for sonorities with
three or more notes, however, I find it a useful convention always to
list notes in ascending order: for example, a mean-frequency trine of
2:3:4 but a mean-string trine of 4:3:2.

One great advantage of this convention is that we can tell immediately
that 2:3:4 is a frequency-ratio while 4:3:2 is a string-ratio -- and
likewise 4:5:6 and 6:5:4 for the mean-frequency and mean-string
flavors of 5-limit triads.

Also, this convention makes graphically clear both the symmetrical
nature of these related divisions, sharing identical intervals in
converse arrangements, and the sequence of intervals in each
arrangement:

2:3:4 "a 2:3 fifth followed by a 3:4 fourth"
4:3:2 "a 4:3 fourth followed by a 3:2 fifth"

4:5:6 "a 4:5 major third followed by a 5:6 minor third"
6:5:4 "a 6:5 minor third followed by a 5:4 major third"

These last examples suggest a possible exception to the rule of
stating two-voice interval ratios with the higher number first. When
reading the intervals in a multi-voice sonority defined by a
frequency-ratio such as 2:3:4, it seems natural to follow the order of
the notation: "a 2:3 fifth followed by a 3:4 fourth," etc.

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2. With string-ratios attached: harmonic/arithmetic contrasts
-------------------------------------------------------------

Although medieval theorists of the Pythagorean tradition such as
Boethius (c. 480-524) describe higher notes as having more "intense"
or rapid vibrations than lower notes, the concrete basis of medieval
and Renaissance music theory focuses on string-ratios. Thus the use of
frequency-ratios in Section 1, while convenient, was somewhat
anachronistic -- beware medievalists bearing synthesizers!

Relying on string-ratios exclusively to generate our paired flavors of
3-limit trines and 5-limit triads, we discover another approach to
these sonorities: the "harmonic/arithmetic" contrast.

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2.1. Gothic perfection
----------------------

We have already used string-ratios to construct one variety of trine,
the "mean-string" or 4:3:2 division of the octave with the 4:3 fourth
below and the 3:2 fifth above:

| 2
| 3:2 (5)
2:1 (8) | 3
| 4:3 (4)
| 4

In medieval and Renaissance terms, this is an "arithmetic" division of
the octave, in which the two pairs of adjacent terms have equal
differences. That is, the difference in length between the lower pair
of strings forming the 4:3 fourth (4 - 3 = 1) is equal to that between
the upper pair of strings forming the 3:2 fifth (3 - 2 = 1).

A diagram may make this equality of differences between the pairs of
adjacent terms more clear:

4 : 3 : 2
(4 - 3) (3 - 2)
1 = 1

Arithmetic division of 2:1 octave

To obtain the ideally euphonious form of Gothic trine with the fifth
below and fourth above, we need to find a different kind of division.
Here it may help to choose a larger number of units for the lowest
string, avoiding the complication of fractional string lengths and
also following the medieval tradition that musical ratios should be
expressed in whole integers or "multitudes" rather than possibly
fractional "magnitudes."

Taking the length of our lowest string as 12 -- evenly divisible by 2,
3, and 4 -- we begin with a 12:6 octave and seek a division which will
yield our trine with fifth below and fourth above.

Our middle string must form a 3:2 fifth with the lowest string, and
thus be equal to 2/3 of its length of 12 units. We find that a string
of 8 units forms the required 12:8 or 3:2 fifth, and also forms our
desired 8:6 or 4:3 fourth with the upper string:

| 6
| 4:3 (4)
2:1 (8) | 8
| 3:2 (5)
| 12

Interestingly, Jacobus observes that the ratio of 12:8:6 suggests the
structure of a cube, which has 12 edges, 8 corners, and 6 faces. More
generally, this ratio -- more simply expressed as 6:4:3 -- exemplifies
a kind of division which can be applied to various intervals: the
_harmonic_ division.

In an arithmetic division such as 4:3:2, the two pairs of adjacent
terms have equal differences. Our 12:8:6 division has a different
property capturing the admiration of medieval and Renaissance
musicians: the differences between these pairs of adjacent terms have
a proportion equal to that between the outer terms.

A diagram may help in appreciating this somewhat subtle point:

12 : 8 : 6
(12 - 8) (8 - 6)
4 : 2

Harmonic division of 2:1 octave

Here the two strings forming the lower 12:8 fifth have a difference in
length of (12 - 8) or 4, while the two strings of the upper 8:6 fourth
have a difference of (8 - 6) or 2. These differences yield a ratio of
4:2 or 2:1 -- identical to that of the outer 12:6 or 2:1 octave.

For theorists such as Jacobus, following ancient Greek tradition,
these mathematical and musical divisions can have political overtones,
with the arithmetic division representing the indiscriminate equality
of a democracy, and the harmonic division the justly distributed
inequality of an ideal aristocracy. A philosopher of a different bent
might compare the arithmetic division to a flat tax, and the harmonic
division to an equitably distributed progressive income tax.

In any case, the harmonic and arithmetic divisions of the octave -- in
simplest form 6:4:3 and 4:3:2, respectively -- yield our two flavors
of trine. While both trines are used to superb musical effect in the
grand style of a composer such as Perotin (c. 1200), the harmonic
division with the fifth below and fourth above is the ideally
euphonious form in 13th-century practice and theory.

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2.2. Zarlino and the 5-limit: _harmonia perfetta_ in two flavors
----------------------------------------------------------------

We have already met the "mean-string" division of the fifth yielding a
complete 5-limit sonority with a 6:5 minor third below and a 5:4 major
third above -- or, in 16th-century terms, the arithmetic division:

| 4
| 5:4 (M3)
3:2 (5) | 5
| 6:5 (m3)
| 6

We can confirm that this is an arithmetic division by noting that the
lower pair of strings forming the 6:5 minor third have a difference of
length (6 - 5 = 1) equal to that of the upper pair of strings forming
the 5:4 major third (5 - 4 = 1). Or, in diagrammatic terms:

6 : 5 : 4
(6 - 5) (5 - 4)
1 = 1

Arithmetic division of 3:2 fifth

As Zarlino informs us, this division can be very expressive but is
less "natural" and "joyful" than that of the ideal harmonic division
of the fifth with the major third below and minor third above. Let us
now find the ratio for this ideal 16th-century arrangement.

In seeking the harmonic division of any unequal integer ratio, there
is a helpful rule for choosing numbers large enough to avoid the need
for fractional string lengths. To select a length for our lowest
string, we add the two numbers of the ratio to be divided, and then
multiply by the larger number of the ratio. More formally stated:

In finding the harmonic division of an interval n:m
where n > m, we may conveniently start with a lowest
string having a length of n (n + m).

Here we are dividing a pure fifth, with a ratio of 3:2. Thus we add
the two terms 3 + 2, getting 5, and multiply by the larger term 3,
getting 3 x 5 or 15 -- a convenient length for the lowest string of
our division.

If the lowest string is 15, then the highest string forming with it an
outer fifth must have a length equal to 2/3 of 15, or 10.

Our problem is now to add a third string forming a 5:4 major third
with the lower string -- and thus having a length equal to 4/5 of 15.
A string 12 units long neatly solves the problem, and also obligingly
forms our desired 6:5 minor third with the upper string (12:10):

| 10
| 6:5 (m3)
3:2 (5) | 12
| 5:4 (M3)
| 15

This is indeed a harmonic division: the lower pair of strings forming
the 5:4 major third have a difference of (15 - 12) or 3, while the
upper pair of strings forming the 6:5 minor third have a difference of
(12 - 10) or 2. These differences for a ratio of 3:2, identical to the
15:10 or 3:2 ratio of the strings forming the outer fifth.

In diagrammatic terms:

15 : 12 : 10
(15 - 12) (12 - 10)
3 : 2

Harmonic division of 3:2 fifth

Thus with Zarlino's divisions of the fifth to form 5-limit sonorities
later dubbed "triads" by Lippius, as with the medieval divisions of
the octave to form 3-limit trines, the harmonic and arithmetic ratios
produce symmetrical "mirrorlike" sonorities here called conversities.
In each case, while both divisions are musically important, the
harmonic division produces the arrangement deemed more blending and
conclusive in practice and theory.

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2.3. Finding divisions: formula for a 13-limit diversion
--------------------------------------------------------

Finding the arithmetic division of any unequal integer ratio -- note
that a 1:1 unison is indivisible -- is a matter simply of finding the
average of the ratio's two terms and adding a third term equal to this
average. In some cases, we may find it convenient to multiply both
terms by two in order to avoid fractions, but this is the only
complication.

For example, suppose we wish to find the arithmetic division of a pure
8:5 minor sixth. Since the sum of the terms, 13, is not evenly
divisible by two, we simply multiply both terms by two, making the
ratio 16:10, and find their average of 13 -- for an arithmetic
division of 16:13:10.

If this division represents a string-ratio -- as the present context,
and also the descending order 16:13:10, imply -- then we have a
sonority with a 16:13 below (a largish neutral third, ~359.5 cents)
and a 13:10 above (about midway between major third and fourth, ~454.2
cents).

The harmonic division is a somewhat more complicated matter. In
applying harmonic string-ratio divisions to a variety of intervals
with integer ratios, one might use an assortment of algebraic or
"intuitive" methods. Fortunately, it is also possible to use a rote
formula like the following.

For any musical interval with an integer ratio n:m,
where n > m, we may define a three-note harmonic
division as follows:

upper string = m (n + m)
middle string = 2 mn
lower string = n (n + m)

Thus to find the harmonic division of an 8:5 minor sixth, we calculate
the lower string at 8 (8 + 5) or 13 x 8, giving 104. Our upper string
is 5 (8 + 5), or 5 x 13, giving 65. Our middle string is 2 (8 x 5) or
2 x 40, giving 80 -- a division of 104:80:65.

We have a lower interval of 104:80 or 13:10, our 454-cent
"between-major-third-and-fourth"; and an upper interval of 80:65 or
16:13, our 359-cent neutral third. This division of 8:5 involves
intervals identical to those of our 16:13:10 arithmetic division --
but conversely arranged, as with the more historically familiar
3-limit and 5-limit divisions above.

Incidentally, returning briefly from our 13-limit excursion to the
routine 5-limit practice of the Renaissance, we may duly observe that
not all popular divisions need be either harmonic or arithmetic.

Composers and theorists of the 16th century get along happily with two
5-limit divisions of an 8:5 minor sixth. The 24:20:15 division
features a 6:5 minor third (24:20) below and a 4:3 fourth (20:15)
above, e.g. E3-G3-C4. The alternative 8:6:5 division has a 4:3 fourth
(8:6) below and a 6:5 minor third above, e.g. A3-D4-F4 -- the fourth
above the bass calling in this era for a somewhat cautious treatment,
but offering a mild sort of tension very attractively leading into a
cadence. These two sonorities are conversities with the same intervals
in mirrorlike arrangements, but neither is an arithmetic or harmonic
division.

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3. Strings and frequencies: connections and conventions
-------------------------------------------------------

For people oriented both to frequency-ratios and string-ratios, the
notation of "mean-frequency" and "mean-string" ratios as advocated in
Section 1.3 above -- the notes and intervals in either case listed in
ascending order -- may provide a convenient way of describing the
mirrorlike sonorities here termed conversities.

For example, one might say: "This composer finds 4:5:6 a bit obvious,
but often favors 6:5:4." Here it is understand that 4:5:6 is a
frequency-ratio (4:5 major third below 6:5 minor third), while 6:5:4
is the converse string-ratio division (6:5 below 5:4).

Likewise one might say, "I find a 6:7:9 quite harmonious, but a 9:7:6
rather unsettling unless the supermajor third expands to a fifth in
the Gothic fashion." Here, although the sonorities are not
mean-frequency or mean-string divisions, the notation nevertheless
conveys the converse ordering of intervals.

Since even those of us today who are medievalists tend to be familiar
with frequency-ratios as well as string-ratios, this style of notation
has the advantage of visual symmetry as well as a simple and concrete
basis: the frequencies of tones or the lengths of the strings or pipes
producing them.

In discussions involving the harmonic/arithmetic concept, historical
associations would suggest that these terms might conventionally imply
string-ratios rather than frequency-ratios, although strictly speaking
they may apply to either. Thus "a harmonic flavor of trine" suggests
6:4:3, while "an arithmetic flavor" suggests 4:3:2 -- and likewise
15:12:10 and 6:5:4 for harmonic and arithmetic 5-limit triads.

However, when desired, we may speak of harmonic/arithmetic
frequency-ratios as well as string-ratios -- with an understanding
that a harmonic string-ratio division is equivalent to a corresponding
arithmetic frequency-ratio, and vice versa. Thus 6:4:3 is equivalent
to 2:3:4, and 3:4:6 to 4:3:2 -- and likewise 15:12:10 to 4:5:6, and
6:5:4 to 10:12:15.

Here I should emphasize that the more traditional harmonic/arithmatic
contrast and the newer "mean-frequency/mean-string" contrast may
provide congenial approaches to various historical as well as current
musics, without in any way excluding other approaches which may better
fit some musical contexts and styles.

For example, a Partchian outlook based on "otonality/utonality" and
"undertones" might ideally fit not only Partch's own styles of music,
but other styles which share such related concepts as "major/minor
tonality." However, this outlook may make certain assumptions which
may not fit so well the early 5-limit music of around 1450-1650, let
alone the 3-limit trinic music of the 13th and 14th centuries.

An awareness of the musical assumptions implied by different systems
and conventions may promote a fuller appreciation not only of the
systems but of the music which the systems seek to explicate, an
attractive incentive for more dialogue on the Tuning List.

Most respectfully,

Margo Schulter
mschulter@value.net