source file: mills2.txt
Date: Sat, 10 May 1997 16:27:35 +0200

Subject: RE: Why x-ET?

From: gbreed@cix.compulink.co.uk (Graham Breed)

David J. Finnamore wrote:
>This is one of those subjects that never
>seems to end.  Maybe it's too large for any one person to comprehend fully.

Or maybe it's something people can argue the toss about without
actually changing their minds.  Well, I'm quite happy to join in!
I missed the end of TD1064 through carelessness, so apologies if
I'm repeating someone in this message.

Marion wrote:
>It seems that much discussion of ET is framed in JI terminology,
>so that the study of ET scales could just be viewed as a study of
>a limited subset of JI scales--those that approximate ET spacing.

It seems that a lot of people take it for granted that ETs will
be used to approximate JI intervals.  Hence this idea that ET
scales are the more mathematically complex.  From my own
experiments, I think that it is possible to recognise equal
intervals, and so ET scales do make sense in and of themselves.
Hence, David's idea:

>... my premise for suspecting that JI may be
>superior to ET is not that one tuning system is the basis of judging the
>value of others, but that nature is the best basis of judging any tuning
>system.

    doesn't hold unless you consider ETs to be less natural
than JIs.  I consider small number ETs to be the less valuable
because the natural order, which can be made audible by
playing equal intervals in series, sounds pretty boring.

On a frequency scale, an nTET interval will be a ratio of 2^(1/n).
No transcendental numbers and no logarithms.  If you're ignoring
JI, there's no reason why you can't take the logarithm of
frequency as your fundamental scale, because this is nearer to
human perception.  Then, ETs are a simple quantisation of this
space.  The logarithms only come in when you try to express
frequency ratios in this space, and therefore compare nTET and JI.
Which you consider more complicated depends upon your own
prejudices, and the instrument you're trying to tune.

I use ratio space and matrices to relate integer ratios to a
logarithmic scale.  I explained this a while ago on the list, and
no-one seemed to be interested.  Note, though, that it gives
a new intellectual respectability to JI, because it uses matrices
and these come a lot later on the maths curriculum than logs.
I can't work out any way of bringing differential equations into
tuning theory, unfortunately, so I may have to move into timbral
synthesis to show off my mathematical abilities.

Which brings me to the importance of tuning theory in practical
music making.  There have been some pretty starry eyed posts
lately with sentiments like this from Paul E:

>Here's my point of view: We have several hundred years of Western music in 
>12-tET, by, say, 100 great composers. Have the possibilities been exhausted? 
>Somewhat. Does the existing repertoire have enough diversity to provide a 
>lifetime of listening enjoyment of the most transcendent and sublime sort? 
>Many seem to feel that it does. Therefore, even a single new tuning system 
>should be enough for a composer to do a lifetime of work and, even if the 
>composer is of the first rank, the composer will not exhaust 1% of the 
>tuning's resources. Considering the range of expression contained in 12-tET 
>music, any single microtonal tuning will, by increasing pitch resources, 
>lead to an unimaginable new universe of moods and sensations. The technical 
>difficulty of mastering a single new tuning, even a new ET, in the current 
>educational environment, is plenty to expect of a composer whose main goal 
>is to express himself/herself in a new way.

Remember that the vast majority of great music written in 12TET 
works because of good melody, structure and whatever, not
_because_ it was written in 12TET. Alternative tuning systems may
be a useful aid to producing good, original music, but no more so
than a new synth.  I think "an unimaginable new universe of moods
and sensations" is going a bit over the top.

There's no reason why puns can't be implemented on a continuous
pitch model.  I'm thinking about software to do this.  I'll say
more if I ever get around to writing it.

Well, this seems to be getting quite long already, so I'll leave
2-D tunings and the like for another day.

                    Graham

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