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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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🔗DFinnamore@aol.com

Hello, Graham,

Thanks for joining the fray! This is good stuff. Great sense of humor, too!

You wrote:

>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.
>...
>On a frequency scale, an nTET interval will be a ratio of 2^(1/n).
>No transcendental numbers and no logarithms.

Huh? I'm not sure Marion literally meant transcendental numbers; I took it
to be a bit of tongue-in-cheek overstatement. In any event, it seems evident
to me that the math of roots and exponents is more complex than that of
whole-number ratios. If you can solve for the nth root of x in your head,
and multiply it by the factor of the previous note of the scale in your head,
you really do have some mathematical skills to show off! And if I'm not
mistaken, that math is necessary for ETs whether or not you're trying to
approximate a JI tuning.

>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.

This sounds like interesting analysis but I'm not sure what it means. Do you
mean that the larger the number of equal divisions of a given interval, the
more musically useful the resulting scale? If so, why? If not, how small is
small? And why should a scale played in order be boring? "Joy to the world"
by G. F. Handel is a good example of musically interesting and effective use
of scale tones in their "natural" order. I must be missing your point.

>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.

This makes logical sense. It is true that human perception of audio
phenomena, pitch included, is essentially logarithmic. But why does it make
musical sense to simply quantize this logarithmic space? I'm not saying it
doesn't. But I don't yet know of any reason that it should, either
theoretically or experientially.

>>nature is the best basis of judging any tuning
>>system.

> doesn't hold unless you consider ETs to be less natural
>than JIs.

I do. :-) Sort of. At least at this point I suspect that they are. Part of
my purpose in lighting this fire was to learn what relationships exist, if
any, between ET tunings, musical acoustics, and psychoacoustics. Some seem
to be beginning to emerge on the list, in bits and pieces, and I'm compiling
and summarizing them as we go. I'll probably post my summary soon.

If you haven't yet seen Ray Tomes' web site on Cycles in the Universe and
Harmonics Theory - http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm - it
explains much better than I could what I mean by "natural." The whole cosmos
apparently can be explained (scientifically, that is, not philosophically) in
terms of harmonic overtones, and might even be thought of as a giant
vibrating string. This is traditional science, not metaphysics. Talk about
the music of the spheres! The Big Twang. I like his idea of AE0 Hz being
quite literally "in tune with the earth," and the idea of "tuning" tempi to
musical keys.

And finally,

>There's no reason why puns can't be implemented on a continuous
>pitch model. I'm thinking about software to do this.

That's only if you can get some semblance of agreement on what the term "pun"
means! :-D Good luck!

David J. Finnamore
Just tune it!

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🔗mr88cet@texas.net (Gary Morrison)

>Do you
>mean that the larger the number of equal divisions of a given interval, the
>more musically useful the resulting scale? If so, why?

I think it's fair to say that, with larger numbers of steps per octave,
there are fewer shades of chromaticism to choose from. I noticed that
effect when I first tried 10TET.

But I don't think that it's quite appropriate to take that idea too far.
I'm not sure that 53TET, for example, will prove all that much more
expressive than 41TET (although I'm largely speculating there). And of
course the more pitches you put into an octave's span, the more cumbersome
the tuning becomes to use.

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🔗mr88cet@texas.net (Gary Morrison)

🔗gbreed@cix.compulink.co.uk (Graham Breed)

Reply to David Finnamore:

>Thanks for joining the fray! This is good stuff. Great sense of humor, too!

Hmmm, worrying. Are you detecting my finely tuned wit, or did I
make some more laughable errors in my post? Hopefully the former.
The problem with, as you put it, "joining the fray," is that I
handle my e-mail offline so I end up replying to the digest before
the one I'm picking up. Anyway, here goes . . .

>In any event, it seems evident
>to me that the math of roots and exponents is more complex than that of
>whole-number ratios. If you can solve for the nth root of x in your head,
>and multiply it by the factor of the previous note of the scale in your head,
>you really do have some mathematical skills to show off! And if I'm not
>mistaken, that math is necessary for ETs whether or not you're trying to
>approximate a JI tuning.

If you're tuning a digital synth to an ET scale, all you need to
do is one division, and from then on it's just addition. This
is easier than all that multiplying of fractions you need to work
with JI -- unless you use ratio space axes. My point is that
there's no reason why you can't do all your work in a logarithmic
scale, and then an ET is the simplest scale to define. Most of
the work in microtonality now is done with electronic synthesis
rather than dividiing up strings.

Actually, I probably could calculate an nth root of 2 to a few of
decimal places (which is all you need for practical tuning) using
the Taylor series for (1+x)**n A first approx to the nth root
of two is 1+0.7/n for 10sensible nTETs. I cheated and got the 0.7 from my calculator, but
it can be estimated from the Taylor series. The true inverse
problem would be calculating the logarithms of whole number
ratios, which I could do with pencil and paper fairly easily if I
could remember the logarithms to base two of prime numbers. In
reality, of course, I would use a spreadsheet for both
calculations.

>>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.
>
>This sounds like interesting analysis but I'm not sure what it means. Do you
>mean that the larger the number of equal divisions of a given interval, the
>more musically useful the resulting scale? If so, why? If not, how small is
>small? And why should a scale played in order be boring? "Joy to the world"
>by G. F. Handel is a good example of musically interesting and effective use
>of scale tones in their "natural" order. I must be missing your point.

By the limited experience I've had of playing in alternative
tunings, I have found that ETs do have a common characteristic.
My ear can definitely tell that all the intervals are the same.
Melodically, this makes ET to me more natural than JI. But,
the effect is to make the scales sound "linear". Diatonic scales
fitted to 19tet sound great but, once I play all the notes within
a third, in any order, it starts to sound bland. This means that
the equal tempered nature of a scale is audible, but should be
hidden. Handel may have got away with it, but he was a better
composer than me!

>It is true that human perception of audio
>phenomena, pitch included, is essentially logarithmic. But why does it make
>musical sense to simply quantize this logarithmic space? I'm not saying it
>doesn't. But I don't yet know of any reason that it should, either
>theoretically or experientially.

Once you define a space, the simplest thing you can do is divide
it into equal steps. If that doesn't work, you can try something
more complicated, but there's no point in adding complexity for
the sake of it. JI comes from the harmonic series, which is a
quantization of linear frequency space.

>>>nature is the best basis of judging any tuning
>>>system.
>
>> doesn't hold unless you consider ETs to be less natural
>than JIs.
>
>I do. :-) Sort of. . . .

To correct my own grammar, I suppose something's either natural or
it isn't. My point is that ET scales _are_ natural. Of course,
the word "natural" could mean a lot of things. In this case, in
relation to mathematics, acoustics, human instinct or patterns
found in the natural world. I think it's the last one you mean,
which is fair enough.

More importantly, though, I disagree with your original statement
which, as you did say, is largely a question of personal taste. A
tuning should be judged primarily by its subjective musical
interest. By this criterion, just scales score very well. But,
with most music, they can be approximated quite adequately to,
say, 31tet. This is the usual, but not sole, reason for using
ETs. If you want to exploit JI with hair splitting accuracy, I
expect you'd have to work with slow, harmonic music with perfectly
harmonic timbres and no vibrato. Otherwise, go with whatever's
simplest. On my Wave Blaster, major triads actually sound better
in 31tet than JI. This may be because of sample looping (could
someone explain this, please?), phase shifting vibrato or a bug in
my program.

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🔗"David Madole" <madole@...>

🔗gbreed@cix.compulink.co.uk (Graham Breed)

🔗DFinnamore@aol.com

🔗mr88cet@texas.net (Gary Morrison)

🔗mr88cet@texas.net (Gary Morrison)

🔗gbreed@cix.compulink.co.uk (Graham Breed)