Yahoo Tuning Groups Ultimate Backup tuning hilarious ri-post

Sorry about the length of this it's a quoting of the review
by
Richard Chrisman of Eric Regener's book, in its entirety. It was
originally published in _Journal of Music Theory_, volume 19.1
[Spring 1975], p. 161-171.

I had a question about the equations, but *after* going thru
the trouble of typing all of this, I found out that there was
an error in the original article, and that's why the math was
unclear to me. So even tho I understand it now, I'd still
like feedback from others on Regener's ideas.

Many subscribers here will probably find this of great interest.
I've only seen one copy of Regener's book, on a shelf at the
University of Pennsylvania library, covered with dust. As far as
I know, the book is out-of-print and appears to be rather hard to
obtain at this date.

In this reproduction of Chrisman's review, I indicate subscript
numbers by an underscore; for example, "F sub 0" appears as F_0.
The notes given in the `References' section at the end are
part
of Chrisman's review. Any comments I make in the body of the text
are enclosed in square brackets.

-monz

---------------------------------------------------------

Regener, Eric. 1973.
_Pitch notation and Equal Temperament: A Formal Study_
University of California Press
Berkeley, Los Angeles, and London.

Review by Richard Chrisman

This book provides a mathematical representation of conventional
pitch notation and of conditions for correspondence between
pitch notation and what Mr. Regener specifies as "regular", or
notationally consistent, intonation systems. Although Mr. Regener
hopes that his formalism will lead toward the development of a
system for the objective analysis of music, the more immediate
goal of establishing a mathematical model useful for expressing
any notated interval in relation to its associated sound (given
in terms of the frequency ratio between the two pitches forming
the interval) better represents the nature and value of this study.

In the mathematical approach live both the main advantages and
primary disadvantages of the book. On the one hand, the formalism
ensures a logical, systematic, and generalized description of pitch
notation and of particular intonation systems. On the other hand,
facility in reading a book of this sort depends to a great degree
on one's facility in reading mathematical notation and in grasping
the necessary mathematical concepts, even though every concept is
defined and important ideas illustrated by examples, tables, and
graphs. For a mathematician, the mathematics would be elementary
and probably uninteresting. For a musician interested in Mr.
Regener's ideas on notation theory, the mathematics might seem
overly complicated, actually detracting from the understanding
of the material.

Following introductory comments, certain mathematical background,
and a discussion of the author's long-range purposes for the book
(to be discussed below), the main body of the formalism begins in
Chapter 6, entitled "Notated Pitches". Here Regener establishes
a means for representing any notated pitch, apart from its rhythmic
value, in terms of two numbers. The first number, called the
"diatone" of a note, corresponds to the staff position of the note,
specifically its distance in diatonic steps from a reference pitch:
the F below middle C. Letting the diatone of this reference
pitch "F_0" be equal to 0, the diatone of G_0 (or Gb, G#, Gbb,
G##, etc. in the same register) a step above F_0 equals 1; A_0
two steps up equals 2; B_0 equals 3, and so on. Below F_0, the
same principle applies, thought the numbers are negative: the
diatone, for example, of E_-1 a step below equals -1; D_-1 two
steps below equals -2. The subscript for each letter refers to
the note's register in relation to F_0.

The second number identifying a pitch, called the "quint" of a
note, measures the distance from F (again equal to 0) in a sequency
of perfect fifths that Regener refers to as the "quint group":

... Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# ...
... -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 ...

For a given pitch, the quint thus fills in the information lacking
from the diatone, that is, the accidental given to a particular
letter-name: for example, here a pitch with a number between 0 and
6 has no accidental, those with numbers between 7 and 13 have one
sharp, those with numbers between -7 and 1 have one flat.

Defined as the "notated pitch" of a note, the two numbers form a
pair (d,q) where "d" equals the diatone of the note and "q" equals
its quint. *1 The "notated pitch" of middle C would be read as
(4,1): the C without an accidental that is 4 steps above F_0. The
C# a chromatic semitone above middle C is represented as (4,8):
the C whose staff position lies four steps (diatones) above F_0
but whose form with one sharp is a distance of 8 away from F in
the quint group. The note Cb in the same register has as its
identification number (4,-6); the Cb an octave higher would be
(11,-6). *2

The pair (d,q) thus uniquely defines every possible notated pitch
(even such unlikely notes as Abbb or F###, since the diatone and
quint series can be extended infinitely). Regener then shows how
these notated pitches can be plotted on a graph, with the horizontal
axis corresponding to the quint group and the vertical axis
representing the diatonic distances. (Example 1 - taken from
Fig. 3, p. 41, in the book - shows the notated pitches about the
two axes and on an enlarged treble and bass staff.) The point of
origin of the two axes if F_0, with coordinates (0,0). Any note
with coordinates (d,q) has its own point on the graph, such as
Ab_0 = (2, -3) which lies up 2 from F_0 on the vertical (diatone)
axis and -3 (3 points to the left from F_0) on the horizontal
(quint) axis.

[NOTE FROM MONZO: I have only reproduced part of this diagram in
ASCII format; it is greatly reduced and omits the clefs, but does
at least show the single-flats and the naturals for the octave
above and below the central F_0.]

[Chrisman's "Example 1"]

----- Fb_1 -------------------- F_1 --------------------- F#_1 7
Ebb_0 | Eb_0 | E_0 | 6
| Db_0 | D_0 | 5
| Cb_0 | C_0 | 4
Bbb_0| Bb_0| B_0| 3
| Ab_0 | A_0 | 2
| Gb_0 | G_0 | 1
Fb_0 -------------------- F_0 --------------------- F#_0 0
Ebb_-1 | Eb_-1 | E_-1 | -1
| Db_-1 | D_-1 | -2
| Cb_-1 | C_-1 | -3
Bbb_-1| Bb_-1| B_-1| -4
| Ab_-1 | A_-1 | -5
| Gb_-1 | G_-1 | -6
---- Fb_-1 ------------------- F_-1 ------------------- F#_-1 -7

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Vertical axis: diatone _d_, horizontal axis: quint _q_.
Subscript: register-number as defined.

Each notated pitch, therefore, is actually represented as an
interval from the basepoint F_0, since both the diatone and the
quint define each point in terms of distances from F_0 = (0,0). *3
This set of points, or intervals, and the particular rules for
associating form what Regener calls an "interval space",
which he
labels _I_ [NOTE FROM MONZO: This appears in Chrisman's text as a
capital I with a wavy line under it.] (the wavy underscore
symbolizing boldface type; I'll just refer to this as
"I").
The chapter on interval space, Chapter 8, is the most important
of the entire book, as everything from that point on depends on
concepts and mathematical expressions defined here. Unfortunately,
I also found this chapter the most difficult to understand. For
these reasons, I will go into more detail, while trying to simplify
and to clarify the essential ideas presented.

In Chapter 8, Regener establishes the means by which every notated
pitch with coordinates (d,q) in the interval space "I" can be
reinterpreted as a point with new coordinates (a,b) in relation to
a new set of vertical and horizontal axes, thus forming another
interval space. Regener shows how one of these new interval spaces
can be generated by two intervals that act as the "minimal
basis"
of that space, just as a diatone and a quint act as basic units of
measure and of vertical and horizontal direction "unit
vectors"
in "I". Regener uses the intervals (3,-1) (the upward perfect
fourth F_0 to Bb_0) and (1,2) (the upward major second
F_0 to
G_0) to illustrate one possible interval space which he calls
"K_0".
*4

The creation of "K_0" from the original space "I" can
be visualized
by representing with solid lines the two generating intervals (3,-1)
and (1,2). (See Example 2a, which is simply a portion of the figure
from Example 1.)

[Chrisman's "Example 2a"; the reader must bear with the
limitations
of ASCII and imagine that the two diagonal lines are dotted lines
and the tilted rectangle in the center has heavy solid lines on
the left and bottom edges, and light solid lines on the right and
top edges.]

d
|
` |5
` |
` _|4'` C_0 (4,1)
` _.-` | `
Bb_0\ |3 `
(3,-1)\ | ` ,
\ |2 ` ,
\ | ` ,
\ |1 _.-` G_0 (1,2)
F_0 (0,0) \| _.-`
-----------\-------------------- q
-2 -1 , 0|` 1 2 3 4
, -1| `
, | `
-2| `
| `
-3| `

When extended in either direction these lines form the new axes
(dotted lines in Ex 2a) of the interval space "K_0". Each of
the
two generating intervals also defines the unit of distance of each
point along its respective axis, whereas in "I" a unit of
distance
measured one diatone away from F_0 and one quint away from F_0.
In the new space, therefore, with F_0 again being the point of
origin (0,0), the Bb and the G each represent point "1" along
its
respective axis; point "2" would then be Eb along the F-Bb
axis
and A along the F-G axis. Thus the transformation from one space
to another, though using the same points, effects a change in
coordinates from (d,q) in "I" to (a,b) in "K_0".
Example 2b
illustrates this and also shows how all the points from "I"
can
be realigned by placing the new axes vertically and horizontally
and by equalizing the distances between the point along each axis.
Under realignment the parallelogram that connects four points in
"I" (drawn in Ex. 2a) becomes a square in "K_0" (Ex.
2b) yet still
connects the same four points. Like a huge lattice, all the points
in the old space are similarly affected.

[Chrisman's "Example 2b"; here, I use `x' to
represent the heavy
solid line.]

a
|
|2
|
|
Bb_0 (1,0)x1 . . .C_0 (1,1)
x .
x .
F_0 (0,0)x .G_0 (0,1)
--------------xxxxxxxx----------- b
-2 -1 |0 1 2
|
|-1
|
|
|-2
|

For any two intervals that generate a new interval space from
"I",
the following transformation equations relate the coordinates of
a point (d,q) in "I", whose unit vectors are (1,0) and (0,1),
with the new coordinates (a,b) in the generated space, whose
intervals i and j the new unit vectors serve as a minimal
basis for the new space:

d(1,0) + q(0,1) = ai + bj

In other words, the coordinates times the unit vectors in the old
space equals the new coordinates times the new unit vectors.

Interval space "K_0" was used only to illustrate the
transformation
from "I" to some other space. Regener now constructs another
space "K_1" based on the intervals (7,0) (the upward perfect
8ve)
and (4,1) (the upward perfect fifth), an interval space much more
useful and historically consistent in terms of intonation theory,
and one which also simplifies the mathematics. *5 The
transformation equations between coordinates in the two spaces
now become:

d(1,0) + q(0,1) = a(7,0) + b(4,1) or
(d,q) = (7a+4b,b) and
d = 7a + 4b
q = b

Solving for a and b (Equation 8.21, p. 82):

a = (d 4q)/7 (= k_0)
b = q (= q_0)

[NOTE FROM MONZO: this equation contained an error which was
corrected in the following issue of the Journal.]

In this new space "K_1", when the axes are realigned (as was
"K_0"
in Ex.2b), each successive point (a) along the vertical axis now
corresponds to an interval of an octave, while each successive
point (b) along the horizontal axis still represents the quint.
For an interval in "K_1" between two notated pitches (each
with
its own two coordinates), Regener therefore calls the distance
between the two respective points along the vertical axis the
"octave-span" of the interval (cf. "diatonic-span",
reference 3),
thus generalizing coordinate "a" above; he renames this
"k_0".
Similarly, the distance in quints between the respective "b"
coordinates of the two notes forming the interval is called
(as in "I") the "quint-span", labeled "q_0".

Regener is now able to define a correspondence between notation
and sound i.e., between any notated interval and a particular
frequency-ratio. Specifically, he is most concerned with what
he defines as "regular intonation systems", symbolized by the
function lf (log frequency-ratio), for which two conditions must
be satisfied: the perfect octave (7,0) is defined to have a
frequency ratio of 2:1; and the frequency ratio of two notated
intervals combined equals the product of their two individual
frequency ratios. This second condition implies that exactly
one frequency ratio will always correspond to any given interval
of the same name, thus ensuring that intonation is always
consistent with the notation. (It is possible, of course, that
more than one interval will have the same frequency ratio, as
in twelve-tone equal temperament.)

Regener then proves that in a regular intonation system all the
frequency ratios will be completely determined by assigning a
particular frequency ratio to just one notated interval other
than the perfect prime or its octave equivalents. It is here
(pp. 89-94) that the final link between notated pitches and
frequency ratios is made: the log frequency-ratio of any interval
i = (d,q) can be given in terms of the transformation coordinates
(k_0,q_0), which as shown above can be derived from (d,q), and in
terms of the log frequency-ratio of the perfect fifth. *6 The log
frequency-ratio of the perfect fifth thus acts as the "determining
constant" for the entire regular intonation system: its value
determines the value of every other interval (except the perfect
octave, which is already defined).

With the essential relationships between notation and frequency
ratios established, the second half of the book deals with
applications of the now generalized description. First Regener
distinguishes between "open" and "closed" regular
intonation
systems, borrowing Murray Barbour's term "closed" in the
sense
of a "regular temperament in which the initial note is eventually
reached again", as in equal temperament *7; by analogy its
counterpart is called an "open" system, as in Pythagorean
tuning.
Although Regener briefly discusses open systems, he is mainly
concerned with closed regular systems, for which he proves that
a system is closed if and only if it has a "non-trivial"
enharmonic
interval: an interval (from F_0) other than the perfect prime,
whose frequency-ratio is 1:1 an interval that closes the system
in the sense of which Barbour speaks. Mathematically, an
intonation system containing enharmonic intervals results in a
simplified form of the determining constant, one equal to k_0/q_0
(from the transformation coordinates) where q_0 corresponds to the
number of tones which divide the octave and where k_0 gives the
number of tones comprised by the perfect fifth in the system.
In twelve-tone equal temperament, for example, the determining
constant would be 7/12: twelve divisions to the octave with the
perfect fifth equal to seven of those divisions. It is at this
point that the mathematical approach really proves worthwhile,
resulting in a very simple, useful, and descriptive relationship.

Regener compares regular systems with "just" intonation
systems whose frequency ratios are rational numbers, i.e.,
expressible as a quotient of integers (p. 117). Here he also
presents criteria for "preferred" frequency ratios, criteria
which historically have sometimes governed decisions of choosing
certain frequency ratios over others. *8 When correlated with
criteria for notationally consistent closed regular systems,
these criteria for preferred ratios result in a number of systems
whose determining constant lie within the range 11/19 (nineteen
divisions of the octave) and 7/12. In these later chapters
familiar items keep emerging, though approached from a new
perspective.

Following a chapter that relates the conclusions reached thus far
to a generalization of pitch notation, Regener returns to his hope
that his system for representing notated pitches and intervals can
be used as a means for analyzing tonal music in a completely
objective manner. Considering a musical score as a "more
reliable"
representation of the music than any performance of that music
(a case could be made, I think, that supports the opposite
viewpoint), he speculates that it might be possible to find and
to codify the "internal syntactic arrangement of the score" by
treating it as a "detached, abstract symbolic entity, irrespective
of its possible realizations in sound and their conceivable effects
on potential auditors" (pp. 2-3). *9 I sense that underlying
Regener's thinking is the notion that for tonal music there is
some "natural law" that governs the patterns of symbols in a
musical score, irrespective of composer or of the evolution
of the composer's style. Although I am skeptical of such an
assumption, especially when one is concerned with the leading
tonal composers of the eighteenth and nineteenth centuries, I
am curious about what it might lead to, more in terms of process
rather than of end results. Very significant work has been
produced by persons seeking natural laws, even though these were
not always, if at all, found.

Also, I am not convinced that Regener's notational system would
be necessary for an objective study, since there already exists
a completely objective means for representing the musical score
the Ford-Columbia Music Representation for computer analysis.
I assume that Regener is also implying the application of
mathematics, beyond the notation itself, which might very well
prove to be important, as it has in other fields (quantum
mechanics, for example).

In general, I think that this book offers most to intonation
theorists. Unfortunately, the formal approach might discourage
interested readers from proceeding beyond the second chapter,
which is why I tried to "translate" some of the problematic
areas.
Unquestionably, mathematics and mathematical notation are useful
and often necessary, to a certain degree, for music theory, as in
the case of this book. But the apparatus itself can completely
negate whatever ideas one hopes to convey. I think that a
preferable approach would be to present, in the final discussion,
a minimum of mathematics. If proofs are essential, these can be
placed in appendices or, even more simply, can be referred to in
other sources, since nearly all of the math used for music can be
found in basic undergraduate texts. A formal approach may very
well lead to important conclusions, but there is usually no need
to present these conclusions in a form that may limit the
readership to what I suspect is a very small number.

REFERENCES

*1 Although I found that I would usually just refer to the charts
of these numbers that are provided in the book (Figures 1 and 2,
pp. 36-37), I also tried to find a simple way of remembering these
numbers, at least within the range of 0 to 6: F always equals 0;
the diatone of a note corresponds to the scale degrees in a G major
scale G=1, A=2, , E=6; the quints of notes F,G,A,B are in
the
even sequence 0,2,4,6 respectively and of notes C,D,E are odd,
1,3,5 respectively.

*2 Both the diatone and quint numbers are set up so that notes of
the same letter-name have numbers that are related by multiples of
7 that is, are equivalent mod7. In both cases if a number is
greater than 6 or less than 0, the number 7 can be subtracted or
added respectively one or more times until the number lies within
the range of 0 to 6. For the diatones, the numbers of times that
7 is subtracted or added tells the number of octaves the note lies
from the register above or below F_0, respectively. The number of
times 7 can be added to or subtracted from the quint number outside
the 0 to 6 range tells the number of sharps or flats, respectively,
that the letter-name carries. These relationships essentially
form the basis for the chapter on Modular Arithmetic (Chapter 5)
as well as in the chapter on Notated Pitches.

*3 Intervals between other notes besides those formed with F_0
can be ascertained by finding the distance between the diatones
of the two notes (the "diatonic span") and between the quints
of
the two notes (the "quint-span"); the resultant two numbers
give
the diatone and quint of the interval formed. Interval addition
works in the same way, though here the d and q coordinates of the
resultant interval equals respectively the sum of the diatones and
the sum of the two quints of the intervals.

*4 The conditions for two intervals being the generators of an
interval space are given on pages 65 through 75.

*5 Apparently, the reason that an interval space other than
"K_1"
was used to illustrate transformations from "I" is because the
quint axis in "K_1" is the same as that in "I".
Interval space
"K_0" necessitated a change in both the vertical and
horizontal
axes and coordinates, thus representing a more general example of
these transformations.

*6 As shown, any interval i = (d,q) in "I" can be
reinterpreted
in terms of coordinates (k_0, q_0) in interval space "K_1" as
given by the transformation equation: i = k_0(7,0) + q_0(4,1).
The frequency ratio of i, expressed as a logarithm to the base 2
and symbolized as the log frequency-ratio lf(i), is given by the
equation: lf(i) = k_0 * lf(7,0) + q_0 * lf(4,1). Since the value
of the perfect octave (7,0) was defined to be 2:1, its log
frequency-ratio to the base 2 equals 1 and the equation now reads:
lf(i) = k_0 + q_0 * lf(4,1).

*7 J. Murray Barbour, TUNING AND TEMPERAMENT: A HISTORICAL SURVEY,
2nd ed. (East Lansing, 1953), ix.

*8 The first of these relates to "consonance": that preferred
frequency-ratios are "those involving the lower numbers when in
lowest terms" (p. 127). Second, "preferred"
frequency-ratios
are "those which can be derived by linear combination from
known `preferred' values for other intervals, beginning with
the ratios 3/2 for a perfect fifth and 5/4 for a major third"
(pp. 127-8).

*9 I could not imagine how an analysis of the musical score
could be made on the basis of pitch alone, since the rhythmic
and metric context determines so many of the melodic and harmonic
considerations. Almost as though finishing an incomplete sentence,
the final chapter of the book does show the necessity for relating
pitch with the metric/rhythmic dimensions.

------------------------------------------

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

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