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       M U S I C          T H E O R Y         O N L I N E

                     A Publication of the
                   Society for Music Theory
          Copyright (c) 1995 Society for Music Theory
| Volume 1, Number 4        July, 1995     ISSN:  1067-3040   |

  All queries to: mto-editor@boethius.music.ucsb.edu or to
AUTHOR: Cuciurean, John D.
TITLE: Review of Mark Lindley and Ronald Turner-Smith.
*Mathematical Models of Musical Scales: A New Approach*.
Bonn: Verlag fuer systematische Musikwissenschaft, GmbH, 1993.
KEYWORDS: scale, interval, equal temperament, mean-tone
temperament, Pythagorean tuning, group theory, diatonic
scale, music cognition

John D. Cuciurean
State University of New York at Buffalo
Department of Music
222 Baird Hall
Buffalo, NY 14260-4700

ABSTRACT: In *Mathematical Models of Musical Scales*, 
Mark Lindley and Ronald Turner-Smith attempt to model scales 
by rejecting traditional Pythagorean ideas and applying
modern algebraic techniques of group theory. In a recent 
MTO collaboration, the same authors summarize their work 
with less emphasis on the mathematical apparatus. This review
complements that article, discussing sections of the book
the article ignores and examining unique aspects of their 

[1] From the earliest known music-theoretical writings of the
ancient Greeks, mathematics has played a crucial role in the
development of our understanding of the mechanics of music.
Mathematics not only proves useful as a tool for defining the
physical characteristics of sound, but abstractly underlies
many of the current methods of analysis. Following
Pythagorean models, theorists from the middle ages to the 
present day who are concerned with intonation and tuning
use proportions and ratios as the primary language in their
music-theoretic discourse. However, few theorists in dealing
with scales have incorporated abstract algebraic concepts in
as systematic a manner as the recent collaboration between
music scholar Mark Lindley and mathematician Ronald
Turner-Smith.(1) In their new treatise, *Mathematical Models
of Musical Scales: A New Approach*, the authors "reject the
ancient Pythagorean idea that music somehow 'is' number, and
. . . show how to design mathematical models for musical scales
and systems according to some more modern principles" (p.7).

1. A representative group of recent articles in this field
which do, in fact, incorporate abstract algebraic concepts
include Eyton Agmon, "A Mathematical Model of the Diatonic
System," *Journal of Music Theory* 33/1 (1989): 1-25; Gerald
J. Balzano, "The Group-theoretic Description of 12-Fold and
Microtonal Pitch Systems," *Computer Music Journal* 4/4
(1980):66-84; Norman Carey and David Clampitt, "Aspects of
Well-formed Scales," *Music Theory Spectrum* 11/2 (1989):
187-206; John Clough and Gerald Myerson, "Variety and
Multiplicity in Diatonic Systems," *Journal of Music Theory*
29/2 (1985): 249-270; and, John Clough and Jack Douthett,
"Maximally Even Sets," *Journal of Music Theory* 35 (1991):
93-173. Lindley and Turner-Smith cite all of these articles
with the exception of the Balzano and Clough and 
Douthett articles, although they do mention "Maximally
 Even Sets" (p.70).

[2] The authors "have in mind several kinds of reader,
including mathematicians interested in music, musicians with
a substantial but non-professional ability in mathematical
reasoning, and students wishing to learn about both areas at
once" (p.13). The book itself is presented in two main parts.
The first comprises the systematic and rigorous discussion of
their models of musical scales; the second consists of
several short sections, including one that introduces the
mathematical principles invoked in part I (for readers who
consider themselves mathematical novices) and another on the
rudiments of music (for readers with little or no musical
training). While it may be convenient in a multidisciplinary
treatise to include primers of this nature, both the surveys
of the algebraic background (section 1) and the introduction
to musical notation, diatonic scales and elementary harmony
(section 4) are poorly suited to the mathematically
illiterate musician and the musically illiterate 
mathematician, respectively. However, keeping in mind the 
intended audience, these sections will help give readers 
some knowledge of these disciplines.(2) 

2. I strongly urge readers who do not have access to the
book, or who struggle with the mathematical notations
contained in part I, to read Mark Lindley and Ronald Turner-
Smith, "An Algebraic Approach to Mathematical Models of
Scales," *Music Theory Online* 0.3 (1993). 

[3] The mathematical models of musical scales are presented
in their entirety in part I, although the reader is directed
to part II, section 2 for all of the mathematical proofs and
additional commentary on the theorems. The models are 
facilitated through the invocation of mathematical group 
theory as described in chapter one, where intervals
operate on sets of notes to create scales. The scales are
"guaranteed finite by two rules: that no two notes may
overlap, and that the scale has a top and a bottom note"
(p.7). Even though their models are abstractly generalized,
the authors relate the models to musical thought and practice
at every opportunity. The models, which derive from extant
evidence of pitch-class systems found throughout the history
of Western art music, summarize different conceptions of
scales that appear in the theoretical literature. 

[4] The authors are concerned with models for "musical
systems [which consist of] sets of pitch classes with [pitch
class relations] operating on them. ... Pitch class relations
... are equivalence classes of intervals differing by an
integral number of octaves, [which] amount to equivalence
classes of musical-interval numbers modulo 1, which we call
*flog*s" (p.7).(3) In an attempt to construct their model
from actual pitch-class relations, the authors state "we have
asked ourselves how pitch-class relations are built up in
musical reality, and have decided to provide for two ways in
our modelling: [equal division and harmonically generated]"
(p.24). In an equal division system, the pitch continuum is
divided into n equal parts where adjacent pitch-classes are
separated by an interval equal to 1/n-th of an octave. In a
system which uses an harmonic generator, the basic harmonic
relations consist of a set, B^n^, where each element of the
set, b^n^, represents a specific harmonic relation (indicated
in the book with Roman numerals). The identity element (or
unison or octave), b^0^, is represented by I, which equals
flog(2) (=0). The fifth generator, b^1^, is represented by V,
which equals flog(3); the third generator, b^2^, is
represented by III, which equals flog(5); and the seventh
generator, b^3^, is represented by VII, which equals flog(7).
Furthermore, for each individual harmonic relation, there
exists a tempering factor, t^n^. 

3. flog(k) = log^2^(k) (mod 1). {That is, log (base 2) of k.}

[5] The relationships between the author's models and
traditional theories of scales, pitch-systems, intervals, and
temperaments are made clear in the second part of the book. 
Sections 5 and 6 of part II (which are intended to be read 
together, as section 6 includes annotated examples for 
section 5) put each model from part I in historical 
perspective, relating contributing theorists with 
representative composers for each model developed, from 
medieval plainchant to the present day. Indeed, if a picture 
is really worth a thousand words, then the illustration shown 
in figure 31(p.134) provides a clear and concise synopsis of 
nearly one thousand years of scale theories, and summarizes 
the 85 pages of text and examples that follow. In fact, the 
authors consume a mere 61 pages from the introduction of their 
models to the final chapter of part I, while they devote 115 
pages to the historical outline of their models (part II, 
sections 5 and 6) along with the theories pertaining to 
Pythagoreanism, Euler's theory (both in appendix 1) and 
Rameau's "temperament ordinaire"
(appendix 2).(4) The 
inclusion of the historical material also addresses 
musicologists and history of theory scholars, thereby 
diversifying the appeal of this book and increasing 
its potential impact. 

4. The scholarly presentation of the historical portions of
the book, including copious references, are in large part
thanks to Mark Lindley's diligent research in the history of
tuning and tempering systems over the last three decades.
Lindley's contributions to this field include: "Early 16th-
Century Keyboard Temperaments," *Musica Disciplina* 28
(1974); "Fifteenth-century Evidence for Meantone
Temperament," *Proceedings of the Royal Association* 102
(1976); "Pythagorean Intonation and the Rise of the Triad,"
Royal Music Association *Research Chronicle* 16 (1980);
"Equal-temperament", "Interval", "Just [pure] intonation",
"Mean-tone", "Pythagorean intonation", "Temperaments", and
"Tuning", in *The New Grove Dictionary of Music and
Musicians* (London, 1980); "La 'pratica ben regolata' di
Francescantonio Vallotti," *Rivista italiana di musicologia*
16 (1980); "Leonhard Euler als Musiktheoretiker," in
*Kongressbericht Bayreuth* (Kassel, 1981);"Der Tartini-
Schuler Michele Stratico," in *Kongressbericht Bayreuth*
(Kassel, 1981); *Lutes, Viols and Temperaments* (Cambridge,
1984); "J.S. Bach's Tunings," *Musical Times* 126 (1985); and
"Stimmung und Temperatur," in F. Zaminer, ed., *Horen, messen
und rechnen in der fruhen Neuzeit*, vol. 6 of *Geschichte der
Musiktheorie* (Darmstadt, 1987).

[6] In reviewing for an electronic journal a book which relies 
heavily on specialized mathematical notations and conventions, 
the medium of the review poses certain technical
difficulties when replicating the symbols in the body of the
review. Moreover, the vast repertoire of symbols derived from
the Roman and Greek alphabets, including multiple meanings
for certain letters depending on the case and appearance of
the letter in question, as well as an assortment of
mathematical symbols (some standard and some devised by the
authors) make it virtually impossible to discuss the models
without a proverbial "score card." Toward this end, Lindley
and Turner-Smith aid the reader by providing definitions for
all of their terms in appendix 4 (pp. 262- 269) and a lexicon
of symbols in appendix 5 (pp.270-279). Both appendices are
cross referenced to the location in the main text where the
term or symbol is originally defined.(5)

5. I found it desirable to have a copy of these pages close
by for quick reference while reading the main text in part I.

[7] Rather than summarize and condense part I of this book 
chapter by chapter, I would like to focus on some unique aspects 
of the book. The remainder of the review addresses three 
points: "leeway"; limitations of the "diatonic" model with 
respect to existing diatonic theories; and cognitive implications 
of their historical models. 

[8] The first and most noticeable divergence the authors make
from traditional scale theory is the introduction of the
conceptual "note neighborhood" in an effort to account for
vibrato, inexact intonation, and the effects of timbre and
intensity. They allow for a certain tolerance around a note
center which they term "leeway" (p. 20). In trying to account
for the magnitude of this leeway, they posit an upper limit
of 1/(2*n) of an octave where n is the number of divisions in
the octave (p.30, Theorem 7) and a lower limit of 0.2
mil (where 1 mil = 1/1000 of an octave = 1.2 cents) based on
the premise that "1/10 millioctave [is] beneath the threshold
of perception, but ... [the leeway] must always be more than
the threshold of perception" (p.39).(6)

6. Theorem 7, part (1) (p.30) which reads |Q|<= 1/2 u, should
probably have an italicized u, indicating that |Q|<= 1/2 (the

[9] The reason for drawing attention to this feature of their
model is that both the leeway factor and the tempering factor
of harmonically generated systems are of the same order of
magnitude. To avoid confusion in later chapters, the leeway
amount is not incorporated into the model when measuring
intervals--intervals are taken between ideal pitch centers.
However, leeway does return in chapter 15 in the discussion
of approximate equivalence and quasi-systems. One common 
example of a quasi-system is the piano trio, where the string
players are required to balance the purer thirds available
when the strings play together with the equal-tempered tuning
of the piano. By blurring the pitch leeway with the tempering
factor, an acceptable compromise may be achieved. (Additional
examples of quasi-systems are discussed on pp.50-51.) In a
generalized model of a musical scale, it may have been
desirable to carry the leeway factor throughout the
calculations. Lindley and Turner-Smith respond to this
criticism when they write "Someone might work out a more
elaborate model, for a more intricate account of musical
realities. Our object is not to design a model for every
purpose, but only present a new approach to mathematical
modelling of scales" (p.51). While they manage to do an
admirable job based on their stated objectives, I believe
that the "more elaborate model" is within their grasp at this
stage of development of their model, and the inclusion of the
elaborate model would prove to be a valuable enhancement to
the book.(7)

7. It is beyond the scope of this review to develop all of
the models with the inclusion of "leeway." However, the 
rewards of this task would include a deeper understanding 
of actual music as a physical phenomenon. Mathematically, 
a more generalized model affords the study of dynamically 
changing pitch-systems which drift over time, and the effect 
a small change in leeway has on the characteristics of the 
overall system. Work of this nature has been the norm for 
many years in mathematical modelling of physical systems 
for control system engineers and physicists, but musicians 
tend casually to disregard the effects of minute perturbations 
to the system. In practice, it may sensible to ignore leeway, 
but in theory there is no logical reason to dismiss its effect.

[10] The second aspect of the book I would like to address
concerns their "diatonic" model. According to their model,
a strictly diatonic scale in an ideal-system is defined as a 
scale "which represents a coherent system with a span of 6" 
(p.35), i.e., a harmonic system with a chain of 6 fifth (V) 
relations running through the 7 pitch classes, or a ^7^H^1^ 
system. By this definition, a diatonic system is not possible 
in an equal-division system. Easley Blackwood, in his study of 
diatonic structures from a tuning perspective, concludes that 
in order to model a recognizable 7-note diatonic system in any 
chromatic system, the cardinality of the chromatic system must 
admit an integer solution for w and h to the equation 5w+2h=c, 
where w is the number of chromatic steps in a diatonic 
whole-step, h is the number of chromatic steps in a diatonic 
half-step, c is the cardinality of the chromatic universe, and 
0 < h < w.(8) This model will always produce a 7-note diatonic 
scale, irrespective of the cardinality of the chromatic universe 
or the tuning system used. Therefore, Blackwood's model admits 
diatonic scales in both harmonically-generated and equal-tempered 
pitch-systems while Lindley and Turner-Smith never relax their 
model to make provisions for this.

8. Easley Blackwood, *The Structure of Recognizable Diatonic
Tunings*, (Princeton, 1985): 204-208.

[11] Elsewhere, Eyton Agmon defines a family of "diatonic
systems" whose generated scales satisfy several conditions
that may be loosely summarized as: g.c.d.(c,d)=1 (that is, c
and d are co-prime) and c=2(d-1), where c is the cardinality
of the chromatic universe and d is the cardinality of the
diatonic scale embedded within the chromatic universe.(9)
Agmon considers the whole issue of intonation separable from
the development of "diatonic systems." The remarkable aspect
of Agmon's work is that introducing a constraint based on the
intonation of the interval of a fifth to the family of diatonic 
systems "reduces the infinitely large number of specific 
'diatonic systems' to unity, this unique 'diatonic system' being 
the familiar diatonic system" (p.2). That is, the diatonic 
system where c=12 and d=7 -- the same system Lindley and 
Turner-Smith arrive at with their ^7^H^1^ system. 

9. Eyton Agmon, "A Mathematical Model of the Diatonic
System," *Journal of Music Theory* 33/1 (1989): 11-13. I have
used variables (c,d), rather than Agmon's original (a,b), in
an effort to remain consistent within the context of this
review. While my summary of Agmon's "diatonic system" may 
be an oversimplification in some respects, a thorough
explication of the mathematics contained in his article is 
unwarranted here.

[12] John Clough and Jack Douthett define hyper-diatonic
scales in which d is maximally even with respect to c, 
g.c.d.(c,d)=1 and c=2(d-1).(10) Although the notion of a 
"generated scale" is not a necessary condition for 
hyper-diatonic scales, they are in fact generated.
This model for hyper-diatonic scales is valid when c=12 
and d=7 (the usual diatonic scale) but again, this definition
admits diatonic scales irrespective of the intonation employed 
for the pitch-system. Lindley and Turner-Smith would argue 
that the notion of a 7-note diatonic scale in a 12-note equal-
tempered system is analogous to forcing a square peg in a
round hole. They would argue further that the diatonic 
scale has harmonic origins, and that imposing this scale on 
some other tempering system is an artificial construct. It is 
at this juncture that the present treatise being reviewed 
diverges from most other theories of the diatonic scale. In 
other words, Lindley and Turner-Smith have not broadened 
the notion of the diatonic scale by presenting a generalized 
model. Instead they have chosen to restrict the definition to 
a harmonically generated coherent system with a span of 6. 
This imposed limitation is not necessarily objectionable, 
but it is important to acknowledge. 

10. John Clough and Jack Douthett, "Maximally Even Sets,"
*Journal of Music Theory* 35 (1991): 138-141. Hyper-diatonic
scales are presented as Theorem 2.2.

[13] The third aspect of the book I would like to address
deals with the cognitive implications of the historical
models. During the development of their harmonically
generated model, the authors discuss the relationship between
the cardinality of a system and the size of the neighborhood
within that system. They suggest that "cardinality is
important because musical composition is not only an art of
imagined sonorities, but also ... a cognitive game" (p.31).
They assert that the cardinality of the system must be large
enough to sustain the listener's interest, "but not so large
that the listener fails to sense intuitively the juggling
aspect of the game" (p.31). They "wonder if the size of the
neighborhood can sometimes guarantee a cardinality suited to
traditional composition" (p.31). The examples discussed in
sections 5 and 6 of part II provide an ideal starting point
for further research into cognitive issues surrounding the
perception of well-known pieces (including several
masterpieces) when performed in different yet similar scale
systems. Indeed, most of us can recognize the subtle
differences between an equal tempered instrument and say, a
mean-tone tempered instrument in performance, but to sit down
and analyze the subtle changes in the pitch-system in Bach's
Toccata in F# minor, mm.109-136 (pp.196-199, Ex.29b) as the
authors suggest requires resources that are unavailable to
most readers.(11) I would speculate that cognitive research
in this area would support the "cause-and-effect link" that
the authors claim may exist. The authors hope for as much in
the final sentences of part I (p.71). The frequent return to
issues of perception addresses yet another potential
audience, that of cognitive psychologists. And even though the
cognitive issues are presented briefly and speculatively and
are never really followed up, the diversification to a third
discipline of study further increases the potential impact of
this book. 

11. While appendix 3 of this book provides tuning
instructions for the various systems defined in part I of the
book, I am doubtful that the casual reader will be inclined
to experiment in this manner with his/her own instrument(s).
Perhaps future editions might include prepared recordings of
the examples, as is the case with Mark Lindley's previous
book, *Lutes, Viols and Temperaments* (Cambridge, 1985).
Furthermore, while it is true that some electronic
instruments can approximate many of the tuning systems
described in this book through the use of the MIDI tuning
standard, I believe that electronic simulation of natural
acoustical instruments provides an inadequate substitute for
the genuine article.

[14] As I have stated previously, *Mathematical Models of
Musical Scales* does not provide a conceptual framework for
building exotic scales, nor does it hypothesize abstract
scales in a generalized universe. Rather, it uses abstract
algebraic methods to model existing scales, which was its 
original intention. The book includes an extensive list of 
works cited, providing an excellent opportunity for future 
comparative study. The rigorous treatment of the mathematical 
modelling and the historical perspective deserve praise. And 
section 3 of part II holds particular interest for music 
theorists who wish to correlate harmonic and equal-division 
systems in greater depth. However, some aspects of the book, 
such as the cognitive issues discussed in the preceding 
paragraph and the treatment of "leeway", remain superficial 
or incomplete and require a more thorough investigation. 
Aside from this, there are very few other shortcomings save the 
occasional editorial oversight. In general, for those interested 
in mathematics as applied to music theory, particularly tuning 
and scale theory, this book is well worth the effort. Theorists, 
musicologists, tuning specialists, mathematicians and cognitive 
psychologists alike will find something to stimulate further 
thought and discussion.


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