source file: m1461.txt
Date: Sun, 28 Jun 1998 20:08:15 -0700

Subject: Re: TUNING digest 1457

From: Carl Lumma <clumma@nni.com>

>Apparantly, Bosanquet studied 22-tET extensively, and Ogolevets
>proclaimed it the future of music (along with 17-tET). The section of my
>paper entitled "History of 22" now seems really underinformed. But my
>chances of ever being able to get a hold of those writings seems very
>small -- anyone know if McLaren actually had the primary sources on
>hand?

I wouldn't say that Bosanquet studied 22 "extensively".  I am certain he
was unaware of the decatonic scales.  What he did know was that 22 was one
of a group of positive 2nd order systems that had good 3rds.  He knew such
systems wouldn't work on his generalized keyboard, which was designed for
1st order systems, so he developed a generalized mapping for use with 2nd
order systems.  This mapping was based on the diatonic scale, and no
keyboard has ever been built to play it to the best of anyone's knowledge.
His work with 22 was towards the end of his music theory investigations;
the last years of his career were devoted to the study of magnetism.

For those un-acquainted with Bosanquet's ideas and terminology, I offer
this primer:

1. The size of an interval can be measured in...
      a) "departure", the difference from the 12tET approximation
      b) "error", the difference from just

2. All tunings can be described by specifying the size of the 3/2, or
nearest approximation.

3. A tuning is "positive" if its best 3/2 has a positive departure.  A
tuning is "negetive" if its best 3/2 has a negetive departure.  [Although
note recent post from John Chalmer's regarding McLaren's XH17 article and
the possibility of revising this terminology to use error instead of
departure].

4. Pitches in a tuning are described by their location on the chain of
these 3/2's.

5. Significant things can be told about a tuning by how various commas are
represented.  Special terms are assigned to the two most important commas...
     a) "order", the number of steps of the tuning representing the
Pythagorean comma
     b) "class", the number of steps of the tuning representing the
Syntonic comma
   For example, 12tET has order zero and class zero.  22tET has order two
and class one.

Carl