On 8 Dec 2008, at 12:28, Graham Breed wrote:
Are you on board with the regular mapping paradigm? I may as well
promote it while I'm here.
http://x31eq.com/paradigm.html
I looked a bit at it, the section "The Core Paradigm". The model I
indicated also chooses some generators, but in addition reflects the
notion of scale degrees that the Western musical notation system
brings out, which in its turn relies on an underlying empiric
principle, also in the case when augmented with intermediate pitches
as in Persian and Arab music:
Given intervals x (resp. y) in a two (resp. three) generator model x
= p m + q M (resp. y = p m + q M + r n), define a scale degree deg x
= p + q (resp. deg y = p + q + r). Empirically, melodic development
normally takes place between different scale degrees, also on say
chromatically altered ornaments. This is also true in the Persian
dastgahs, using Farhat's description. If one alters scale degrees,
melodic development still normally takes place between different
scale degrees, as on a parallel, altered scale. In the Western
notation system, one achieves this by simply minimizing the amount of
temporary accidentals, if the notes are not too chromatically dense,
which is very intuitive.
Now, the construction on the link above seems to ignore those scale
degrees. If one defines a scale from say an octave P8 and a perfect
fifth P5, then scale degrees can be defined by setting deg P8 = 7,
deg P5 = 4, and then work it out for other combinations. This gives
deg M = deg (2 P5 - P8) = 2 deg P5 - deg P8 = 1, deg m = deg (P4 - 2
M) = deg (P8 - P5 - 2M) = 7 - 4 - 2 = 1.
Then, in addition, to get an (extended) Western notation system, one
must define pitches A B C D E F G ... of scale degrees 0, 1, ..,
(n-1), where n is the number of scale degrees in what is called an
"octave".
Hans
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