Re: question on Ben Johnston notation
- --- In tuning@y..., jpehrson@r... wrote:
> Well, actually, I didn't know that! So, they are, technically BOTH
> comma-based systems. However, 72-tET is, compared to the Johnston
> system, easy to play and notatate... since 12-tET is contained in
> and there are simply 6 inflections for each basic step, correct??Yes -- and what's more, these inflections are immediately correlated
with the prime-factors that you're using -- just as in the HEWM
This amazing fact stems from the fact that the unaltered pitches in
72-tET notation, 12-tET, play the role of the Pythagorean tuning in
HEWM. And of course, 12-tET is very close to Pythagorean tuning.
The entire 72-tET notation is simply 6 interlocking 12-tET systems.
Every factor of 5 you put in the numerator (or move up in the
lattice), you have to use the 12-tET system 1/72 octave lower (or
1/12 tone lower) than standard 12-tET. And the reverse for every
factor of 5 you put in the denominator. So the 1/12 tone down
indicators play the role of the -, and 1/12 tone up indicators play
the role of the +, in 72-tET notation.
Every factor of 7 you put in the numerator, you have to use the 12-
tET system 2/72 octave lower (or 1/6 tone lower) than standard 12-tET.
Every factor of 11 you put in the numerator, you have to use the 12-
tET system 3/72 octave lower (or 1/4 tone lower) than standard 12-tET.
What's really great is that these accidentals don't pile up. In HEWM
notation you might have to notate a note with various 5-based, 7-
based, and 11-based accidentals. In 72-tET notation, you're always in
one of the 6 12-tET systems . . . no sweat!
Accuracy: The entire 11-limit Tonality Diamond (Partch's 29 "primary
ratios" in the 2/1) is represented with a maximum error of 4 cents.
Well within the "subconscious adjustment range" within which players
can seek to eliminate beats, if so instructed.
Uniqueness: Every 11-limit interval (I mean odd limit, of course) is
represented by a _different_ 72-tET interval.
Consistency: You have consistency through the 17-limit -- so there
will be no possible confusion when notating, say, big 17-limit
Hence I think _most_ JI music can be adequately, and very easily,
notated in 72-tET. The exceptions would be
1) If you had a progression which "pumped" one of the commas that
vanish in 72-tET, such as the 224:225. This progression would not
drift in 72-tET, even though it would in JI (by about 7 cents each
time you repeated it).
2) If you wanted to use 19-limit or higher sonorities in an intricate
manner with accurate intonation . . . the notational inconsistencies
could cause practical difficulties here.
--- In tuning@y..., paul@s... wrote:
> The real point of my message that should be of interest to you, in
considering 72-tET, was well-illustrated in the meantone I-vi-ii-V
analogy I was making there . . . if you're still interested, feel
free to go back and read that analogy and tell me if it makes sense
Thanks, Paul... I believe I posted that that "reprise" was very
valuable for me, and I saved it... I can see how eliminating the
comma can make for a greater multiplicity of consonant intervals in
ANY of the systems...
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