On 10 Sep 2009, at 10:26, Torsten Anders wrote:
It seems to be that that staff indicates the Pythagorean tuning,
with accidentals to indicate offsets relative that. Right?
Exactly: nominals (c, d, e...) and the "common
accientals" (natural, #, b, x, bb) denote a spiral of Pythagorean
fifths. Other accidentals detune this Pythagorean by commas etc.
Multiple comma-accidentals can be freely combined for notating
arbitrary just intonation pitches. The Sagittal notation (http://users.bigpond.net.au/d.keenan/sagittal/
) follows exactly the same idea.
Yes, I thought so.
This is in contrast, for example, to the older just intonation
notation by Ben Johnston (see David B. Doty (2002). The Just
Intonation Primer. Just Intonation Network), where some intervals
between nominals are Pythagorean (e.g., C G) and others are a just
third etc (e.g., C E). Accidentals again denotes various comma
shifts exactly. However, as the notation is less uniform music not
notated in C is harder to read. I assume this experience led to the
development of the Pythagorean-based approach of the Helmholtz-
Ellis and Sagittal notation.
The Sagittal notation allows for an even more fine-grained tuning
(e.g., even comma fractions for adaptive just intonation), and also
provides a single sign for each comma combination. However, I find
the Helmholtz-Ellis notation more easy to read (signs differ more,
less signs).
The Western musical notation system is limited to what I call a
diatonic pitch system (as "extended meantone" suggest certain
closeness to the major third).
For a major second M and minor second m, this is the system of
pitches generated by p m + q M, where p, q are integers. The case
(p, q) = (0,0) could be taken to be the tuning frequency. Sharps and
flats alter with the interval M - m.
I have implemented it into ChuCK, so that it can easily be played in
various tunings. The Pythagorean and quarter-comma meantone are of
course special cases. But also others, like the Bohlen-Pierce scale
in which the diapason is not the octave.
Now, inspired by Hormoz Farhat's thesis on Persian music, I extended
it by adding neutral seconds. For each neutral seconds n between M &
m, one needs accidentals to go from m to n, and from M to n. This
suffices in Farhat's description of Persian music (sori and koron).
For Turkish music, one needs the "dual" neutral n' := M - n; the
reason is that different division of the perfect fourth leads to
negative n coefficients. So then one needs to more accidentals to go
from m to n', and from M to n'.
In this kind of music notation, one just tries to extend the
Pythagorean tuning with 5-limit intervals. So one neutral n is
sufficient in this description. For higher limits, one needs more
neutrals, and for notation, a way to sort out preferred choice and
order.
Now, one advantage of this model is that, like the Western notation
system, one does not need to have explicit values for these symbols,
though one can do so.
Basically just a FYI.
Hans