From: mclaren
Subject: French spectral composers - 2
--
Franck Jedrezjewski mentioned that the
French spectral composers use 24-TET
and "not 1/6 tones, not 1/8 tones" when
producing their spectral compositions.
On the surface of it, this would appear to
make no sense whatever. Going from 12 to
24 tones per octave yields only a superior
approximation of harmonic 11.
The approximation of harmonic 5 is not
improved at all: it's still 13.6 cents off,
an error easily audible.
The approximation of harmonics 7, 13, 14,
21 and 23 is not improved in the slightest
by going to 24 tones per octave.
So let's do some back-of-the-envelope math
and see how many of the first 24 harmonics
are improved by going from 12 to 24 tones
and see how many of the first 24 harmonics
are improved by going from 12 to 24 tones
per octave:
FOR 12 TONES PER OCTAVE (+ indicates
a good approximation, within about 7 cents,
while - indicates a poor approximation)
+ - + - + - - + - - -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
+ + + - - - - +
17 18 19 20 21 22 23 24
Since harmonics 1, 2, 4, 8 and 16 will always be
well approximated in *any* equal tempered division
of the octave, they've been left out of the calculation.
FOR 24 TONES PER OCTAVE (+ indicates
a good approximation, within about 7 cents,
while - indicates a poor approximation)
a good approximation, within about 7 cents,
while - indicates a poor approximation)
+ - + - + - + + - - -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
+ + + - - + - +
17 18 19 20 21 22 23 24
--
Thus the total improvement in going from 12 to
24 tones per octave is: 8 good approximations (12-TET)
to 10 good approximations (24-TET). This is a mere 25%
increase in the number of harmonics which fall
within 7 cents of the scale degrees (7 cents is
roughly the audible difference limen in the central 500 hz
frequency region.) 8 -->10 well-approximated harmonics
is a remarkably *poor* improvement in recompense for
going to all the trouble of doubling the number of pitches
per octave.
going to all the trouble of doubling the number of pitches
per octave.
By going from 12 to 48 tones per octave, however,
the number of good approximations increases from
8 to 14, a 75% increase:
FOR 48 TONES PER OCTAVE:
+ - + + + - + - + + -
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
+ + + - + + + +
17 18 19 20 21 22 23 24
--
Thus it's incomprehensible why the French spectral
composers would consider 24 tones per octave
even *remotely* adequate to approximate the
harmonic series.
Even the most naive observer must conclude that
PINE 3.91 MESSAGE TEXT Folder: INBOX Message 1 of 1 END
harmonic series.
Even the most naive observer must conclude that
since 24-TET yields an improvement in only the 11th
harmonic and the 22nd, while 48-TET yields
audible improvements in the 7th, 11th, 13th, 14th,
21st, 22nd and 23rd harmonics, 48-TET is obviously
the minimum multiple of 12 tones per octave which
produces a improvement commensurate with the
effort in subdividing the whole tone.
Perhaps Franck Jedrezjewski can explain this
puzzle?
--mclaren
