- --- In tuning@y..., jpehrson@r... wrote:
>

it

> 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

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. --- In tuning@y..., paul@s... wrote:

http://groups.yahoo.com/group/tuning/message/21768

> 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

to you.

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...

_________ ______ _____ _

Joseph Pehrson