"Notation - Part 1"
From: George Secor
January 22, 2002
Patience comes to those who wait for it, and I thank you all for your
patience. Here at last is Part One of my saggital notation
presentation, and I hope you agree that it was worth waiting for.
In this first part of the presentation I will illustrate the process
by which I arrived at the 72-EDO form of the notation. Subsequent
installments will address its multi-system application, both in
native (or EDO-specific) and transcendental (or trans-system generic)
forms, leaving most of the questions and comments that have been made
regarding the more controversial aspect of the subject for the final
It is perhaps a bit of a stretch to call this buried treasure,
inasmuch as this is so new that there has barely been enough time to
get any "dust" on the paperwork (most of which is virtual, in the
form of computer files; hmmm, I do seem to notice some dust on the
monitor screen). I reasoned that the presentation would be more
widely read, especially by future members of the Tuning List, if I
put it in my Buried Treasure column.
At the beginning of the year I made a new year's resolution to
complete the development and testing of my notation, and I am sharing
it with you to elicit your comments and suggestions to make this the
very best notation possible, one that will come closest to "doing it
all" and doing it well.
So as not to keep you in further suspension, let the resolution begin!
*A Challenge I Couldn't Resist!*
Please note: The figures for this presentation are in:
I have always believed that the best notation is that which is
simplest. A good example of this is the Tartini fractional sharps
(shown in the right half of the top row of Figure 1), which are so
clear that they require virtually no explanation. Although these
were employed by both Ivan Vyshnegradsky (for 24-EDO) and Adriaan
Fokker (for 31-EDO), it is rather surprising that quartertone
composers never adopted these as a standard notation. Instead, they
often preferred to place arrows in front of notes, which, in
combination with sharps or flats, tend to clutter a musical
manuscript, especially when chords are notated on a single staff.
Existing methods of notating 72-EDO have also used this approach, and
the diversity of symbols used somewhat arbitrarily (and not always
logically) to designate three different amounts of alteration in
pitch strikes me as a conglomeration of add-ons or do-dads intended
to supplement traditional notational practice. However, I did not
see these 72-EDO notations (including the one devised by Ezra Sims)
until several months after I had produced the initial (expanded)
version of my saggital notation, so they had absolutely no influence
in its development. To be completely honest, once I did see them, I
was appalled. I later learned that the symbols that were proposed by
those on the Alternate Tuning List were ASCII versions for
theoretical use only, not practical notation intended for use on an
actual musical manuscript, and the goal was largely to emulate the
Sims notation. Inasmuch as my goal was to arrive at the very best
notation possible, it is understandable that, immediately upon seeing
it, I found that I had absolutely no desire to emulate the Sims
notation, and it should be evident by the end of this part of the
presentation that any similarity between the Sims and the saggital
symbols is purely coincidental.
It is not an easy matter to arrive at a simple notation that would
require only a single symbol to modify the pitch of the seven
naturals notes on the staff for 72-EDO. In the first place, 24
symbols would be needed for a complete range of alteration by a whole
tone, both upward and downward. In order for this approach to be
successful, the new symbols would need to have an intuitiveness that
would enable them to be quickly and easily understood. They would
also need to be similar enough that they could be easily remembered,
yet different enough that there would be no difficulty in
distinguishing them from one another. This was a challenge that I
The solution did not come quickly, however, as it soon became evident
that this is one situation where the desired result would not be
achieved without investing a considerable amount of time and effort.
I spent hours putting all sorts of symbols, both old and new, on a
piece of paper, seeking as many ideas as possible from which to
choose. In the end I found that the best ideas were ones that had
already been successfully used in the past, and my saggital notation
integrates three of these into a unified set of symbols. These three
ideas are: 1) the use of arrows to indicate alterations in pitch up
and down, 2) the intuitiveness of the Tartini fractional sharps, and
3) the slanted lines used by Bosanquet to indicate commatic
*Tartini Plus Arrows*
Up and down arrows can be employed to indicate clearly the direction
in which the pitch is to be altered, and it was immediately obvious
that it would be necessary to have only 12 different symbols if each
symbol of the new notation could be inverted or mirrored vertically
to symbolize equal-but-opposite amounts of alteration. This would
require discarding the traditional single and double sharp symbols
(as well as excluding the Tartini fractional sharps from
consideration), inasmuch as they look virtually the same when
inverted. A traditional flat symbol can be inverted and does
resemble a hand with a finger pointing; the problem is that it points
in the wrong direction, so I concluded that it would also need to be
discarded. Of the conventional symbols, only the "natural" symbol
would be retained.
In my first version of the sagittal notation of August 2001 (which I
now call the expanded saggital symbols), I used arrows as semisharp
and semiflat symbols, with multiple arrowheads for single, sesqui,
and double sharps and flats. These are shown in the second row of
Figure 1. The use of arrows to represent semisharps and semiflats
may seem somewhat arbitrary, inasmuch as they have been used in
different instances to represent various amounts of pitch alteration,
but I felt that their frequent use for notating quartertones was
In December I realized that these symbols could be simplified by
replacing the multiple arrowheads with single arrows that are
combined with one to three vertical strokes, as in the Tartini
fractional sharps, with an "X" for the double sharp and flat, as
shown in the third row of Figure 1. The single arrowheads not only
make the symbols more compact, but they also permit a bolder print
(or font) style to be employed, which improves legibility.
If the abandonment of the conventional sharp and flat symbols seems a
bit shocking, we need to realize that, although they have served us
well since they were devised in the Middle Ages, 21st-century
microtonality will be better served by something new and better, and
I think that it is safe to say it is about time for an upgrade. We
can continue to call these sharps and flats with semi, sesqui, and
double prefixes added as appropriate, inasmuch as it is only the
symbols that are changing, not their names or meanings.
This set of 9 symbols is sufficient to notate 17, 24, and 31-EDO.
However, more symbols would be needed for 72-EDO.
The third idea to find its way into my saggital notation was the
symbol for commatic alterations in 53-EDO that Bosanquet used around
1875. These are shown in the top row of Figure 2, which illustrates
a lateral grouping for multi-comma alterations. The single degree of
72-EDO is similar in size to that of 53-EDO, with the intervals
representing just (5:4) and Pythagorean (81:64) major thirds
differing in size by this amount in each system, so the use of this
sort of symbol would not be inappropriate to indicate an alteration
of a single degree in 72-EDO. I first added a stem to the Bosanquet
symbol to form a sort of half-arrow or flag. I then stacked several
of these flags to indicate multiple-degree alterations, as in the
second row of Figure 2.
I quickly realized that the symbol that I was already using to alter
by 3 degrees differed from the 1-degree symbol by only a right half-
arrow or flag, and that it would be quite logical to represent a 2-
degree alteration with a backward 1-degree symbol. The resulting
expanded saggital symbols are shown in the third row of Figure 2.
These were subsequently simplified into the compact saggital notation
shown in the fourth row of Figure 2. Observe that each new half-
arrow (or Bosanquet flag) symbol is adjacent to a full-arrow symbol,
with the slant of the Bosanquet flag corresponding to the direction
in which the pitch symbolized by the adjacent (full-arrow) symbol
must be altered to arrive at the pitch symbolized by the Bosanquet
flag symbol: upward slope signifies alteration one degree (or comma)
up, while downward slope signifies one degree (or comma) down.
The full range of symbols is shown in Figure 3, along with some
examples on a musical staff comparing other notations with the new
Both the compact and expanded versions of the saggital symbols may be
simulated with ASCII characters for e-mail messages, etc., using a
combination of the slash, backslash, pipe, and capital X characters.
One comma down is \|, semisharp is /|\, and doubleflat is \X/
(compact) or \\\\|//// (expanded). While this generally involves
more characters than with other proposed ASCII notation, it is more
intuitive, and it inconveniences the theorist rather than the
musician. (Please note that the combination of ASCII symbols has a
better appearance when a proportionally spaced font is used; my
choice is Ariel.)
The next part of this presentation will discuss how the notation may
be applied logically and consistently to other EDO's, beginning with
31 and 41, as well as the use of the 72-EDO symbols as a
transcendental notation for sets of just (or near-just) tones mapped
onto a lesser division of the octave.
Until next time, please stay tuned!
Love / joy / peace / patience ...