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Re: question on Ben Johnston notation

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  • PERLICH@ACADIAN-ASSET.COM
    ... 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
    Message 1 of 93 , Apr 23, 2001
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      --- 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.
    • jpehrson@rcn.com
      ... http://groups.yahoo.com/group/tuning/message/21768 ... considering 72-tET, was well-illustrated in the meantone I-vi-ii-V analogy I was making there . . .
      Message 93 of 93 , Apr 28, 2001
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        --- 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
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