subminor, supermajor etc

Hi Ozan,

I wonder if it may be worth saying a bit about why
some of us make these finer distinctions beyond
tones and semitones for exact pitches, rather than
just consider them inflections of the twelve tones
with the amount of the bend up to the taste of the
performer, as I gather is often the case with
Arabic music.

If you work with three limit ratios
like the pythagorean ones involving numbers
which can be expressed as a prodcut of 2 and 3
only, then one naturally works with tones and
semitones.

If you work with five limit ratios
- the fifth partial in the harmonic series
C G D A E
then you have syntonic comma inflections
of the pythagorean intervals and those
can be considered to be inflections
of the twelve tones (as can the extra
notes in the 17 tone Arabic scale).

The thing there is that e.g. a 5/4
or a 6/5 is a much simpler ratio than
81/64 or 32/27. So it is natural to think
of them as inflections to bend those notes
to make pure harmonic series type ratios.

However when you get to seven limit ratios
involving products of 2, 3, 5 and 7, then
you get new simple ratios such as 7/6
the septimal or subminor third and
9/7 for the supermajor third
- so that the septimal minor chord is
1/1 7/6 3/2
which is often referred to as a subminor
chord. Like a minor chord but the third
is flatter.

The corresponding major chord is
1/1 9/7 3/2
which is the supermajor chord.

Then you can use eleven limit ratios
which give notes mid-way between
the twelve tone notes (approximtely)
such as 11/8, 11/9 etc which can be
called undecimal or neutral intervals.

Some microtonal composers like working
with numbers even higher up the harmonic
series. The thing there is that
the notes in a harmonic series chord
of many notes, say a ninth or eleventh
or thirteenth, will sound more harmonious
if they are in exact harmonic series ratios
and indeed one can go up to high numbers
there and even say a 29/4 or whatever may
be desired by a composer on occasion.
They may want it to be as exactly a
29/4 as it can be to within the
limits of the performer's ability.

If the instrument is rich enough in
partials then a performer who is used
to listening out to such things may be
able to hear many of the higher
partials within the instrument
timbre of a single note, and tune to them directly.
One can certainly do that with
7/4, 11/4 and 13/8, 17/8 too
if the instrument is rich enough
in partials such as a typical
bowed string instrument sound for
instance.

Or if they are using a keyboard retuned
electronically, then it is no
problem for the note to be pitched
to an exact ratio even for pitches
high up the harmonic series, if they use it
with a synth or soft synth capable
of such fine distinctions (e.g.
with C-Sound).

So then also that explains the reason
for all those inflections and the
large variety of accidentals in
Sagittal - and the reason they
sometimes need to be notated
precisely rather than as just inflections
of the twelve tone notes with the intonation
up to the performer.

If one is working with three limit
or five limit harmonies then there
is no need for the more exotic
Sgittal accidentals. Sagittal
also accomodates users who
want less precision like that.

But they are needed for more ratios
that involve use of higher prime
numbers from 7 upwards, related
to the higher partials in the
harmonic series.

Just intonation Sagittal is a new notation
and there are many other notations used
by microtonalists.

72-equal is notated using variou accidental schemes
for instance. Also many of the smaller ets
such as 31 are easy to notate using
half sharps and flats, e.g. for 31-et
C C+ C# Db D- D D+ D# Eb E E+ F- F
..

Scala has many notations for different equal
temperaments.

Sagittal has a whole range of systems of
accidentals for various n-ets, and I gather
that the aim is to standardise these notations
so that composers working in different ets
can speak to eash other in a common language,
and to make it easier for composers to find
performers able to perform their works
because the performers too would have
only the one standard noation to learn
for each equal temperament.

There also with the equal temperaments,
many composers now may compose a piece
entirely within a particular equal temperament.
Particularly guitarists as fretted instruments
are easy to tune to equal temperaments so
you get guitars which are tuned to 19 equal
or 31 equal etc. Also nowadays it is
possible to tune keyboards to equal temperaments
electronically. Occasionally you get purpose
built acoustic keyboard instruments in
various ets such as 31-et or 72-et
or 96-et etc (something that I gather
goes back to the middle ages).

Obviously a player of such an instrument
will be most interested in a system of
accidentals that matches the equal temperament
that there instrument is tuned to.
Also each n-et has its own "flavour"
on fixed pitched instruments so composers
too can be interested in exploring the
flavour of a particular n-et as a set
of fixed pitches e.g. for guitar
or keyboard, rather than as a notation
system for free form just intonation
with inflections.

This isn't to say that one has to always
use these high precisino systems.
If the composer wants the performer to
use inflections according to the taste
of the performer as in Arabic music
then it would indeed be inappropriate
to notate the music using a higher
resolution system as the performer
wouldn't be expected to follow it
anyway. That could only perhaps be
done after the event as a way of
notating what the performer actually
played - or perhaps as a way to notate the
music for a performer not familiar with
the tradition to see how it might
be performed - somewhat in the way that
one might use a written out figured bass part
in Western early music. With the idea
that this is just one example of many
ways of performing the same piece.

It is too new yet to see how well this
will succeed, but I hope it does myself
as I can see the many advantages.

So then for instance if you are using
36-et then you could write a piece
using the Sagittal system of accidentals
for 36-et and anyone who is familiar
with that system would be able to read
your score, and if a performer, play it
without the need to learn a new system.
And a higher resolution system may
well be inappropriate for use in some
contexts such as Arabic music.

I hope this might help to put
your discussion of notations for
36-et into context and make the
connections with Sagittal notation
and the other microtonal notations
clearer.

Thanks,

Robert