The Origin and Well-Formedness of Tonal Pitch Structures
Aline Honingh

Abstract:
The research as described in this thesis is centered around the origin
of pitch structures such as scales and chords. It is often unclear how
these tonal pitch structures have developed and where they come
from. We could think of the following questions. Why does the Western
major diatonic scale (do, re, mi, fa, sol, la, si) consist of 7 tones
instead of for example 6 or 8 tones? And why does the Japanese
pentatonic scale consist of 5 tones? In other words, are these numbers
arbitrary and developed in different cultures, or are these numbers
related and perhaps arisen from one and the same origin? A lot of
research is in favor of the latter claim. This research can be divided
in several areas. For example, there exist the area of evolutionary
musicology where research on the evolution of music is done from
biological and cultural perspective. Furthermore, studies exist that
suggest that the scales from different cultures are linked to the
instruments that are used in a specific culture. Equal temperament (a
term described below) could also play a role. Specific equal tempered
scales can be preferable for several reasons, for example because they
approximate just intonation very well and at the same time they have
the possibility for modulation. Finally, there is the research in the
area of the so-called well-formed scales. A scale can be called
well-formed for several reasons, for example because it has a
symmetrical shape when it is being displayed on a tone-lattice or
circle of fifths.
This thesis focuses on the latter two research areas (equal
temperament and well-formedness) to possibly explain the origin of
scales. Besides a possible common origin, these studies could also
serve as evaluation of existing scales (some are perhaps better than
others, or more suitable for a specific purpose?). Finally, scales
that are developed by these theories, can serve as suggestions for new
scales which are interesting for composers, scientist and music
theoretics to study. In this thesis, a part is written about the study
and evaluation of equal tempered scales. Furthermore, a new notion of
well-formedness has been introduced, of which applications are
explained in the last two chapters.


Equal temperament

Already since Pythagoras it has been known that a musical interval of
ratio 2:1 is a beautiful sounding or `just' interval: the octave. The
fifth with ratio 3:2 is also a just interval. These (and more)
intervals appear to be impossible to combine in a musical
instrument. If, for example, you want to be able to play a pure fifth
or octave above every note on a piano, you would need an infinite
number of keys. As a solution of this problem, the equal temperament
has been invented in the 19th century, where the octave is divided in
12 equal parts. In this temperament, some intervals from just
intonation are approximated well. Still, several researchers and
composers wondered whether it is possible to divide the octave into
another number than 12, such that possibly more intervals from just
intonation can be approximated, or perhaps some intervals can be
approximated better. Therefore, several studies have been performed in
order to create an n-tone equal tempered scale, with n as optimal as
possible. The question of course is, what is optimal? Just intonation
describes an infinite number of intervals. Which intervals should be
approximated in a (finite) equal temperament? To come to an optimal
choice for n, a set of intervals from just intonation should be
selected that serve as the set of intervals that are to be
approximated. The next question is then: within this set of intervals,
which interval should be approximated best, and which interval second
best, and so on? Or is every interval equally important in this
respect? In this study I tried to formalize these questions to arrive
at a model that predicts the optimal choices for n. The values for n
that have been found, are: 12, 15, 19, 27, 31, 34, 41, 46, 53. The
12-tone equal temperament, the one used nowadays, appears to be a good
temperament according to this model, which is what we expected. A
number of the other temperaments has also been used and studied in the
literature, but to a lesser extent than the 12-tone temperament.

If the resulting equal temperaments are being used to play Western
music with, the temperament should be consistent with the Western
notational system. With this, we mean that an element from the equal
tempered scale can indicate more than one note name (like A, C#), but
one note name should be linked to no more than one element from the
scale. If one note name could refer to more than one element from the
scale, it would not be clear which key to press on a piano when
somebody for example asks to play an A. The fact that one element can
refer to more than one note name is called enharmonic equivalence. For
example, on a (12-tone equal tempered) piano, the key that refers to a
C#, refers also to a Db. These conditions form restrictions on the
number of possibilities for n, in a n-tone equal temperament. This
means that some n-tone temperaments are not suitable for playing music
that has been notated using the Western notation system. To make it
possible to do play in these `forbidden' n-tone systems given the
rules above, another notation system should be developed for this
purpose. Chapter 3 of this thesis deals with these questions and makes
a prediction of the possibilities for n, using the imposed
restrictions caused by the notational system. Combined with the values
for n that are listed above (that were predicted for the good
approximation of just intonation ratios), the Western notational
system implies that systems with n equal to 12, 19 or 31, would be
suitable choices. Instruments in these equal temperaments have indeed
been developed.



Well-formedness

In this part of the study the focus is on scales and chords in just
intonation. The central question is: when should a set of notes be
called a scale or a chord and what makes a good scale or chord? Until
now, no unique answer on this question exists. Therefore, we consider
a large number of scales in a tone space and look at aspects they have
in common. It turns out that almost all scales describe a convex shape
in this space. A convex shape is a shape without any wholes or inlets
(for example, circles or squares are convex shapes, but stars and
donuts are not convex). For Western chords it was found that all
diatonic chords have a convex shape. This thesis argues that the
convexity of scales and chords is connected to consonance. The more
tones are linked to each other on the tone-lattice, the easier one can
go from one tone to another via consonant intervals. With this
finding, an evaluation model for scales is made as well: if the scale
is convex, then it can be called well-formed. Convexity turns out to
be independent of the chosen basis of the tone-lattice, something that
makes this property even more special: it is not an artifact of the
lattice. Since the convexity of a lot of scales has been shown, among
which also non-Western scales, this gives reason to believe that
convexity can embody unifying properties of tonal pitch structures. A
property that is related to convexity is compactness, it is the
gradation to which elements are close together in the
tone-space. Contrary to convexity, compactness is dependent on the
choice of the basis of the tone-lattice. However, if the basis is
chosen in agreement with the most consonant intervals in music,
compactness can be interpreted as a measure for consonance as well:
the more compact a set of notes, the more consonant it is. Now we can
look at some direct applications of the properties of convexity and
compactness.



Application 1: Preferred intonation of chords

When we talk about chords, this is usually in terms of note names or
an indication of the place of a chord in a scale. For example, one can
speak about the tonic chord, the chord that is built on the tonic of
the scale. Or one could speak about the dominant seventh chord, the
chord that describes the tones G,B,D,F, in the key of C. However,
seldom, one speaks about chords in terms of frequency ratios (like the
chord 1,5/4,3/2) when one wants to indicate a chord in a piece of
music. The reason for this is, that for most chords it is not clear
what their intonation is, and therefore how to describe it in terms of
frequency ratios. Of course, many opinions exist, but no unique theory
exists that everybody follows. If we say that a chord should sound as
consonant as possible, the measures of convexity and compactness can
be used to discover which intonation (which frequency ratios) are
preferred for a number of chords. As a model of evaluation, another
measure of consonance is used, the Gradus function by Euler. It turns
out that compactness is a better indicator of consonance than
convexity.


Application 2: Pitch spelling

In many computer applications tones are encoded as MIDI pitch
numbers. In the MIDI system, the central C is encoded as the number
60; the tone which is a semi-tone higher (C#/Db) is encoded as number
61, and so on. This MIDI notation is similar to the 12-tone equal
temperament. Both notations do not distinguish between enharmonically
equivalent notes like the C# and Db. However, it is this notation that
contains a lot of information about the key, the harmony, melody and
intonation, which are important for a musician to know. For this
reason, it would be useful if a model would exist that transforms MIDI
numbers to note names. This is a difficult problem, because at some
instance a specific MIDI number can for example represent an A#, while
at another instance the same MIDI number represents a Bb, depending on
the musical context. In the literature, some models can be found, that
try to `spell the pitches', given a piece of music in MIDI
notation. No model works for 100 percent correct, which means that
none of these models can spell all notes of the piece of music in the
right way. The right spelling in this case, is the notation that was
used by the composer of the piece. In this thesis, I present a new
model for pitch spelling based on the notion of compactness. The key
of a piece of music makes an important contribution to the musical
context that causes a tone to be spelled in a particular way. It turns
out that, by choosing the most compact shape of a set of notes in the
used tone-space, the notes are often within a specific key that leads
to the correct spelling. In the resulting compactness-model, the
number of notes in such a set can be varied, and it turns out that,
the more notes are included in a set, the better the model works. This
compactness pitch-spelling model has been tested on all preludes and
fugues of the Well-tempered Clavier by J.S. Bach which contains 41544
notes. A result of 99.21 percent correctly spelled notes is gained if
the pieces of music are divided in sets of 7 notes. Although this
percentage of correctness is comparable with the correctness of other
models, it is very special to get these results from a model that is
based on only one principle.


Finally

The problems that have been studied in this thesis can be seen as
projections between the different representations of pitches. For
example, in the study on equal temperaments, a projection has been
made from the frequency ratios to the elements of equal temperament,
and from the note names to the elements of equal
temperament. Furthermore, in the study of intonation of chords, we
tried to make a suitable projection from the note names to the
frequency ratios. Finally, the problem of pitch spelling has to do
with the projection from the equal tempered pitch numbers to the note
names. We found a number of regularities in tonal pitch structures, on
basis of which the latter two projections have been established.

Coming back to the questions from the beginning, it can be said that a
number of aspects have possibly contributed to the origin of several
scales. The n-tone equal tempered scales that have been found in this
study by searching for a good approximation of just intonation
intervals and a suitable notation system, have also been found in
literature and practice. This supports the hypothesis that `just
intonation' and `suitable notation' have been underlying principles
for the development of these scales. Furthermore, the notion of
convexity has been found as a unifying property of a large number of
just intonation scales. On the one hand, this suggests that the
principle of convexity is an underlying principle that could have
played a role in the origin of scales. On the other hand, convexity
can be used as an evaluation model as described above. Finally, the
property of convexity can be used for further exploration and
development of new scales.


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