Totally new concept!

I've been working recently on verbalizing another concept I've developed
in my contemplations of various ETs, and I think perhaps it's time I
shared it with y'all to see what you think of it. I have added the
following text to my personal webpage:

*** BEGIN QUOTED-WEBPAGE ***
[The chart at the following URL] contains data on another concept which I call
completeness. An ET is complete at a given harmonic limit if the basic
intervals within that limit form a basis which spans that ET, if you
think of the ET as a space or group. Example: 24TET is incomplete at
the 5-limit because 5/4 is approximated by 8 steps and 3/2 by 14,
which means no matter what combination of 5/4s and 3/2s (or 6/5s) you
use, you can never generate those intervals which contain an odd
number of steps.

http://library.wustl.edu/~manynote/complete.txt

So what do the numbers mean? Only those combinations of ET and (odd)
limit have entries which are both consistent and complete. (I stopped,
somwhat arbitrarily, at 300TET.) x/y means that the ET is x-level
consistent at that limit (as in the above charts), and y is the
diameter at which the ET is completed.

So what, in turn, does diameter mean? If n-ET has diameter y at the
m-limit, that means that there is at least one interval which requires
combining y m-limit (primary) intervals to derive it, but no intervals
which require more. Example: the primary intervals (consonances)
within the 5-limit (the senario) are represented in 12TET by 3, 4, 5,
7, 8, and 9 steps. 1 can be expressed as 4-3 (or 5-4, etc.), 2 by 5-3
etc, and 6 by 3+3. (The derivations of 10 and 11 are analogous to
those of their complements 2 and 1.) That completes the gamut of
12TET, therefore the diameter of 12TET at the 5-limit is 2.

A contrasting example: in 19TET, combinations of exactly 2 of the
primary 5-limit consonances (5, 6, 8, 11, 13, 14) give you all the
rest except for 4 (and its complement 15). 4 does, however, have a
ternary derivation (4=5+5-6), so the diameter of 19TET at the 5-limit
is 3.
*** END QUOTED-WEBPAGE ***

The point, in case it isn't clear, of introducing a parameter like
diameter is that it gives us another measure by which we can compare
ETs, i.e. an ET with a small diameter will be more "comprehensible" to
the ear/mind, whereas an ET with a large diameter will have intervals
which are interpreted as more complex and difficult to hear.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>

Several books I have read say that one of the largest reasons for the
popularity of Equal Temperament in western music is that the most
prominent piano manufacturers all decided to make pianos in ET, and that
constructing a JI piano creates some serious mechanical difficulties.
However, the books have never mentioned what these mechanical difficulties
are. Does anyone have any input on this question? It is one of the last
unanswered questions in my music perception thesis in which I compare
perception of ET with perception of JI in certain circumstances. I need
to finish by fri.

Thanks for the input,
Jesse Gay
Reed College

>Several books I have read say that one of the largest reasons for the
>popularity of Equal Temperament in western music is that the most
>prominent piano manufacturers all decided to make pianos in ET, and that
>constructing a JI piano creates some serious mechanical difficulties.
>However, the books have never mentioned what these mechanical difficulties
>are.

I have heard that too, from Ivor Darreg, who did a lot of piano tuning
in his day. If I remember correctly, he said that it would tend to go out
of tune more quickly in anything other than equal-temperament.