--- 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
it
> 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
notation!
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
otonalities.
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.
... http://groups.yahoo.com/group/tuning/message/21441 ... quite a bit of PRACTICAL literature out about it. That's a LOT ... systems, in MY opinion... ... so...
jpehrson@...
Apr 24, 2001 3:02 am
... it ... Yes -- and what's more, these inflections are immediately correlated with the prime-factors that you're using -- just as in the HEWM notation! This...
PERLICH@...
Apr 24, 2001 4:04 am
... http://groups.yahoo.com/group/tuning/message/21486 ... BOTH comma-based systems. However, 72-tET is, compared to the Johnston system, easy to play and...
jpehrson@...
Apr 24, 2001 4:22 am
... it ... produce ... It doesn't! In 72-tET, the Pythagorean chain is the same as in 12- tET -- it closes after 12 fifths. It takes 6 _separate_ Pythagorean ...
PERLICH@...
Apr 24, 2001 8:56 pm
... http://groups.yahoo.com/group/tuning/message/21533 ... Joe (and others who are still mystified as to why 72-EDO works so well in so many different...
monz
MONZ@...
Apr 24, 2001 9:41 pm
... http://groups.yahoo.com/group/tuning/message/21533 ... from any other interval. ... Well, this is pretty amazing, if I'm understanding it correctly... In...
jpehrson@...
Apr 25, 2001 4:14 am
Hi Joseph! The idea of 72-tET as a six-gear version of a 12-tET bike reminds me of something similar we both learned when talking about your 19-tone just ...
Paul H. Erlich
PERLICH@...
Apr 25, 2001 6:32 am
... http://groups.yahoo.com/group/tuning/message/21567 ... < <etc.> Hey Paul, this post gave a terrific simple way of thinking about some of these...
monz
MONZ@...
Apr 25, 2001 7:24 am
... 318-tET is six 53-tET near-Pythagorean circles. It's only consistent through the 5-limit, and you'd never leave the first 53-tET circle using any 5-limit...
PERLICH@...
Apr 25, 2001 8:39 am
... http://groups.yahoo.com/group/tuning/message/21574 ... Oops, my bad. I left a few letters out. http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm...
monz
MONZ@...
Apr 25, 2001 9:43 am
... Well it does look like you saw Aristoxenus's scales as encompassing six interlocking Pythagorean circles: "every 6th tick-mark [notated 'xx'] designates...
PERLICH@...
Apr 25, 2001 9:58 am
... You do say "144-tET - a useful approximation to Aristoxenus's system, much less sophisticated than 318-tET" which kind of suggests you'd compared the two...
Graham Breed
graham@...
Apr 25, 2001 2:47 pm
... http://groups.yahoo.com/group/tuning/message/21582 ... Yes, but that's in connection with older, more-established interpretations of Aristoxenus's theory....
monz
MONZ@...
Apr 25, 2001 5:30 pm
... http://groups.yahoo.com/group/tuning/message/21567 ... me of something similar we both learned when talking about your 19- ... gear version of 19-tET. Do...
jpehrson@...
Apr 25, 2001 7:40 pm
... Remember how you were finding that, when you compared various 7-limit JI scales to 19-tET, you kept seeing cent deviations that were multiples of 7 cents?...
PERLICH@...
Apr 25, 2001 9:34 pm
... http://groups.yahoo.com/group/tuning/message/21608 ... limit JI scales to 19-tET, you kept seeing cent deviations that were ... one step of 171-tET,...
jpehrson@...
Apr 26, 2001 3:38 am
... 7- ... Exactly . . . this was the 171-tET bike "in action"!...
PERLICH@...
Apr 26, 2001 5:06 pm
... http://groups.yahoo.com/group/tuning/message/21647 ... Just thought I'd point out that the prolific German tuning theorist Martin Vogel is a big advocate...
monz
joemonz@...
Apr 27, 2001 7:27 pm
Hi Paul, What do you think about 612 as a 51 gear of 12-teT. And 3125 as a 5 gear of 5 gear of 5 gear of 5 gear of 5-teT :-) Kees ... From: "Paul H. Erlich"...
Kees van Prooijen
kees@...
Apr 25, 2001 6:56 pm
... I don't -- none of the 12-tET intervals correspond to 612's best approximation to any consonant JI interval. ... I did notice 3125-tET but it also didn't...
PERLICH@...
Apr 25, 2001 9:21 pm
... http://groups.yahoo.com/group/tuning/message/21486 ... Thanks, Paul - this is really terrific! Joe [Pehrson], The post I just wrote (with the lattice...
monz
MONZ@...
Apr 24, 2001 6:46 am
... http://groups.yahoo.com/group/tuning/message/21486 ... tET. ... tET. ... HEWM notation you might have to notate a note with various 5-based, 7- ... in one...
jpehrson@...
Apr 24, 2001 4:30 pm
... limit...but ... Hi Joseph. Firstly, note that the 17-limit, which includes ratios of 3, 5, 7, 9, 11, 13, 15 and 17, is not _uniquely_ expressed in 72-tET....
PERLICH@...
Apr 24, 2001 10:00 pm
... http://groups.yahoo.com/group/tuning/message/21542 ... Thanks, Paul... Actually, this struck me as rather humorous... but it's late... Thanks for the...
jpehrson@...
Apr 25, 2001 4:21 am
... http://groups.yahoo.com/group/tuning/message/21542 Looking at the table at the bottom of Rick Tagawa's ... Thank you so much, Paul, for your assistance...
jpehrson@...
Apr 25, 2001 6:30 pm
... a ... book ... No you're right, Maneri & Co. don't mention JI, don't care about JI. To them, dissonance has already been emancipated, so there's no point ...
PERLICH@...
Apr 25, 2001 9:16 pm
... http://groups.yahoo.com/group/tuning/message/21605> No you're right, Maneri & Co. don't mention JI, don't care about JI. ... point trying to classify...
jpehrson@...
Apr 26, 2001 3:24 am
... 72- ... Well, you can map out the entire 72-tET system as a 5-limit system, which is what he's doing . . . but that doesn't preclude you from observing and...
PERLICH@...
Apr 26, 2001 5:05 pm
... http://groups.yahoo.com/group/tuning/message/21646 ... Joe, what Paul is saying here is that in Rick's diagram the small interval separating 75:64 from 7:6...
monz
joemonz@...
Apr 27, 2001 7:25 pm
... To summarize, my reaction was that I found it highly unlikely that your ear was really targeting a 75:64 _harmonic interval_ -- 75:64 can be built from two...