Download Chapter 1: The Basics Microtonal Notation

Chapter 1: The Basics
Microtonal Notation
Basics 1
There have been a number of attempts to
devise logical signs for microtones but there
is still limited agreement between
publishers, musicologists or composers.
This situation is partly due to there being
a number of ways to organise microtones
in composition. The main ones are listed
below:
– to divide the tone into more than 2 parts
(e.g. third-tone; quarter-tone; fifth-tone; sixthtone, etc.)
– to divide the octave into more or less than
12 parts (11-div; 13-div; 19-div, etc.)
– to use structured ‘pure’ intervals based
on the harmonic series (referred to as ‘Just’
Intonation)
The above represent the principal
organisational categories used by
microtonal composers over the past fifty
years or so.1 It will be noted that all of
those in the first category could also be
described using the second (though not visa
versa). In such cases, it is not uncommon
for systems to be identified by reference to
either the tone or the octave. For example,
third-tone music may be described as 18-div
and fifth-tone music as 30-div, and so on.
While there is no categorical reason why
the word ‘division’ should be associated
with a division of the octave as opposed to
a division of a tone (or any other interval)
this terminology is widely accepted. The
terms ‘tet’, ‘et’, ‘edo’ or ‘equal’ are also used,
as in ‘19tet’ (19 tone equal temperament);
‘19et’ (19 equal temperament); 19edo (19
equal divisions of the octave) or ‘19 equal’.
All these terms refer specifically to equal
divisions of the octave. The description
n-tone, in the sense that it is sometimes
used to describe n-divisions of the
octave (as in 19-tone), is best avoided in
For two concise summaries of past and present microtonal composers and their work, see: Bob Gilmore, ‘The
Climate Since Harry Partch’, Contemporary Music Review, ‘Microtones and Microtonalities’ 2003, Vol. 22, Parts
1 and 2, pp. 15-34, and Daniel James Wolf, ‘Alternative Tunings, Alternative Tonalities’, Contemporary Music
Review, ‘Microtones and Microtonalities’ 2003, Vol. 22, Parts 1 and 2, pp. 3-14. Although there are many other
ways to organise pitch in microtonal compositions, for example, with unequal divisions of an interval, with nonoctave divisions and so on, the three approaches listed have all been regularly applied to acoustic music which is
the main focus of this book.
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the authors’ view because of the possibility
of confusing a division of the octave with a
division of a tone.2 Furthermore, there is
the suggestion that there are 19 ‘tones’ to
the octave when in fact the unit size is c.63
cents, less than a semitone.3
For clarification, here is a concise definition
of the use of the term ‘division’ as used in
this book:
‘Division’ (n-div) is preferred in this publication firstly because it doesn’t have any
other pitch-related connotations (as ‘tone’
does) and, secondly, because it doesn’t refer
specifically to equal divisions. This second
point is important in relation to performance on acoustic instruments where, just
as in 12-div music, there is a natural tendency to alter intervals by subtle degrees
depending on context.4
An octave division of theoretically equal
steps which, in performance on acoustic
instruments, will undergo many small inflections from the theoretical pitch depending
on context.
n-division
(where ‘n’ is any number)
Understanding how these microtonal systems
influence notational approaches is
fundamental, and in the next section we will
explain why.
In cases where it is necessary to describe a
division specifically as ‘equal tempered’
(perhaps to define it theoretically as equal
steps) the term ‘et’ is used, as in 19et.
the size of the number will probably imply the correct meaning but there could be ambiguity, for example, with
11-tone, which would be feasible as a division of the tone or the octave.
2
This usage has clearly developed from the double meaning of ‘tone’ in music as an interval and as a sound, or
pitch, so ‘19-tone’ would generally be interpreted as meaning ‘19 pitches per octave’.
3
For further information see: Donald Bousted, ‘Tuning In: Intonation in Performance’, The Recorder Magazine,
Summer 2002, pp. 28-61; Donald Bousted, ‘An Empirical Study of Quarter-Tone Intonation’, Contemporary Music Review, ‘Microtones and Microtonalities’ 2003, Vol. 22, Parts 1 and 2, pp. 53-85; and Mieko Kanno, ‘Thoughts
on How to Play in Tune: Pitch and Intonation’, Contemporary Music Review, ‘Microtones and Microtonalities’
2003, Vol. 22, Parts 1 and 2, pp. 35-52
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Music in the First Category
(divisions of the tone)
In music which is structured by subdividing the tone, the predominant notational
method has been to design a different sign
for each subdivision. This is logical enough:
the known and established signs (the sharp,
flat and natural) remain unchanged and new
signs are devised which indicate (and usually suggest visually) a deviation from these
established standard-bearers.
This can be seen clearly in the quarter- and
eighth-tone signs below. In the sequence
of quarter-sharp, sharp and three-quarter
sharp, for example, the addition of vertical
lines suggest a sequence getting sharper, in
increments of a quarter-tone. The flats are
less logical and there are probably a number
of as-good equivalents. The choices below
take Tartini’s quarter flat as standard, with a
three-quarter flat which is one symbol (as
opposed to two, which is a common variant) and not a filled-in flat (which is also
common but is too easily read as a pitch).
Example 1: Quarter-Tone signs used in this book
quarter-sharps
& nœ
µœ
1/4
mœ
1/2
˜œ
nœ
3/4
nœ
Bœ
bœ
1/4
1/2
quarter-flats
bœ
nœ
3/4
All the signs used are available in standard
music engraving software programmes,
which was one reason for their selection.
signs, which tends to alienate performers
whose excursions into microtonality are
usually complicated enough.
The reason why no absolute consensus has
developed has been because the aspirations
and objectives of composers have been
different. A system devised mainly for quarter-tones may not expand naturally to thirdtones, twelfth-tones or sixteenth-tones for
example – the logic of the symbols may
break down, or some signs may be too similar to others. Conversely, systems which
have as part of their objective to normalise
smaller intervals, such as twelfth-tones have
often utilised non-standard quarter-tone
Other issues which may affect the choice of
sign are whether or not the microtones are
‘structural’ or ‘ornamental’. ‘Ornamental’
suggests that microtones are for ‘colour’
and are ‘inexact’; while ‘structural’ suggests
that each pitch is equal with any other. The
authors have chosen to use arrows, added
to the standard sharp, flat and natural signs,
to indicate ‘exact’ eighth-tones even though
these signs have been used previously to
represent inexact inflections, for example
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in Brian Ferneyhough’s solo flute piece
Cassandra’s Dream Song.5 The suggestion
of flexibility created by the arrow (which
could be interpreted as an ‘inflection in the
direction of’) perhaps does not sit
perfectly with the concept of structural
microtonal writing. However, there is a
satisfying graphical logic to the eighth-tone
chromatic scale and it is also believed that
the logic of the system can realistically be
extended to twelfth-tones.6 Given what we
have already suggested about flexibility of
pitch, this does not seem to be taking compromise too far and Ferneyhough’s piece is,
after all, just a single pieces as opposed to a
genre.
Example 2: 48-div signs used in this book
eighth-sharps
& n œ k œ µ œ L œ m œ l œ ˜ œ Kœ n œ
1/8
1/4
3/8
1/2
5/8 6/8
7/8
Arrowheads could clearly be attached to
quarter-tone signs too. They are not in this
volume in order to keep things as clear
and simple as possible. The signs chosen
correspond to the suggested names of the
intervals given in Basics 2.
Music in the Second Category
(divisions of the octave)
Pitches in the second category are generally
notated using a different approach which
follows the Pythagorean note-naming
system (based on a cycle of fifths). The
principal derives from meantone tuning sys-
eighth-flats
n œ Kœ B œ jœ b œ Jœ b œ k œ n œ
1/8
1/4
3/8 1/2
5/8
6/8 7/8
tems. Historically, such systems sought to
overcome the problem associated with the
non-closure of a cycle of pure fifths.7 The
premise is that the closest mathematical approximation to the fifth in the given division
(a compromised or tempered fifth) forms
the basis of a cycle of those intervals (written as a fifth). This is sometimes known as
‘faux’ or ‘enharmonic’ notation.
One of the consequences of this approach
is that sharps and flats, which we normally
recognize as being enharmonic equivalents
of the same pitch (for example, D sharp
Brian Ferneyhough, Cassandra’s Dream Song (London, 1975)
Donald Bousted, ‘More Microtonality – ATS: ZNS’, The Recorder Magazine, Winter 2002, pp. 136-138
7
This is fully explained in the sections on 19-div although the same approach has been applied to
other systems.
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and E flat), equate to different pitches. With
a tuning such as 17-div this approach to
notation leads to rather odd consequences
in that D flat is lower than C sharp which
looks very odd in an ascending sequence
and would be extremely counter-intuitive to
most performers.
For this reason, it is normal to use
replacement signs as is common practice in
another popular alternative tuning, 31-div.8
In the case of 19-div, one of the subjects
of this study, enharmonic notation behaves
logically without substitute signs and, with
practice, is relatively intuitive for the performer.
If this all sounds complicated and confusing,
don’t worry – we will discuss faux notation
in much more detail in the sections on 19div and we believe that the patient reader
will be rewarded not only with a clear understanding of the microtonal implications
of 19-div but also with an appreciation of
ideas which have key in the development of
western music.
Music in the Third Category
(’pure’ intervals)
Music in this category is called ‘Just Intonation’. Just Intonation has, as its basis,
the pure intervals of the harmonic series
(although the organisation of such intervals
is usually much more sophisticated than this
may imply). Although some of the pitches
at the beginning of the series correspond
more or less to equal tempered pitches,
there are substantial differences from the
seventh harmonic upwards. These intervals
have mathematical relationships with each
other which can be described by mathmatical ratios. Historically, music composed in
just intonation has often been notated using
the rather musician-unfriendly language
of ratios rather than stave-based notation
although Ben Johnston’s notation, which is
fully explained and used in the musical examples throughout David Doty’s introduction, The Just Intonation Primer9 and 10 is a
convincing alternative. David Doty’s book is
thoroughly recommended to reader’s who
wish to pursue this area.
8
see Erv Wilson, ‘A Classification of Tonal Systems, and a Proposed Standardisation of Signatures’, Xenharmonikôn, No. 2 (Autumn, 1974), unpaged
9
David B. Doty, The Just Intonation Primer (San Francisco, 1994)
10
Despite the sometimes off-putting notational complexity of just intonation, musicias should be encouraged
to pursue this area because it, essentially, uses intervals which they use all the time: the beat-free, pure sounds
which characterize good tuning. The techniques described in this book will encourage an open-minded and
flexible approach to tuning which will enable players to find solutions to music in just intonation: this book does
not concern itself with structured just intonation but it does discuss the influence of just intervals in harmonic
music in relation to the 3 tuning systems covered.
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This project is concerned with quarter- and
eighth-tones, which fall under the first and
second category (so the music is described
as quarter- or eighth-tone or 24- or 48div) and 19-div, which belongs only to the
second.
In Basics 2, terminology is presented for
describing intervals in quarter- and eighthtones, thus fulfilling the need for a
clear and comprehensive language with
which to describe these new intervals.
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