RE: [tuning] Hello and a Question

>My name is Keenan Pepper and I love JI! It's the only musical system that
>makes sense

Don't expect everyone on the list to agree with you. I don't. For example,
how would you play a I-vi-ii-V-I progrssion in JI?

>and I am overjoyed that there's a lyst on it.

>Now to the question part. An obvious (to me) consequence of spacial arrays
>of ratios (lattices, ratio spaces & such) is the concept of a numerical
>"musical distance" between two pitches. I've already made my own musical
>distance formula based on the Pythagorian theorem, but I was wondering if
>there was an existing one to use instead. I haven't heard of one yet, but
>I'm not yet well acquainted with the Just Intonation world.

There are several such distance measures, e.g., Euler's G.S., Barlow's
digestibility, Wilson's harmonic complexity, and Tenney's harmonic distance.
Tenney's is the only one of these that seems to satisfy some realistic
acoustical requirements about reflecting the consonance of the interval
itself, as I've discussed (see the archives), while the others may be seen
as taking into account possible contexts for the interval. See
http://www.ixpres.com/interval/dict/harmonicdistance.htm (the "4/b" should
be "a/b"). It's equivalent to the "city-block" distance between the two
pitches on an appropriately constructed rectangular lattice. It is crucial
not to "octave-reduce" any ratios when computing Tenney's distance function.
For octave-equivalant scenarios, the archives contain discussions about the
appropriate distance measures, which are computed on a triangular instead of
a rectangular lattice. Or equivalently, you can use a more complex distance
metric on the rectangular lattice, as in
http://www.kees.cc/tuning/perbl.html.

I'd be happy to discuss all this in greater depth, but there is one
important problem with these lattice-based measures. Eventually you will
find very distant intervals that are so close to consonant ones that one
cannot audibly distinguish the difference. In such a scenario, what meaning
is there to the vast difference between the "musical distances" of the two
intervals?

Keenan Pepper wrote,
>>>[JI is] the only musical system that makes sense

I wrote,

>>Don't expect everyone on the list to agree with you. I don't. For example,

>>how would you play a I-vi-ii-V-I progrssion in JI

Kraig Grady wrote,

>Any way one wishes or is possible in JI. They are all musically useful and
expressive in different ways. >I wouldn't want to give any of them up.

I wouldn't want to give up the melodic integrity of the meantone version, or
of an adaptive version such as Vicentino's.

>I would like to once again call attention to my own which like Helmholtz
recognizes that inversions are different.

To clarify for others, here you are referring to mirror inversions rather
than inversions as in "first inversion, second inversion". The poster was
asking about distances between two notes, and a two-note interval is the
same as its mirror inversion.

>Formula=The sum of the pitches and the first order difference tone but
omitting any unison duplication.

This will simply be twice the larger number in the ratio, since a + b + (a -
b) = 2*a, unless the difference tone coincides with the lower tone, which it
only does for the ratio 2/1, or with both tones as in 1/1. So, except for
the unison and octave, Kraig's measure is the same as the "integer limit" of
the ratio.

>>I'd be happy to discuss all this in greater depth, but there is one
>>important problem with these lattice-based measures. Eventually you will
>>find very distant intervals that are so close to consonant ones that one
>>cannot audibly distinguish the difference. In such a scenario, what
meaning
>>is there to the vast difference between the "musical distances" of the two

>>intervals?

>This is true, but it is playing with numbers. If one was working in a
tuning that contained two such intervals, the >musical context in which they
would both occur at the same time is almost non existent. One can superpose
any >interval up to an extreme till they equal any interval one wants, but
who is going to modulate that many places up >till you get there.

One only has to go up two 3/2s and two 5/4s to find an interval (with the
1/1) so close to 7/4 that the consonance of 7/4 greatly outweighs the
largeness of the numbers in the actual interval, 225/128. That is only one
of myriad examples.

I wrote,

>>This will simply be twice the larger number in the ratio, since a + b + (a
-
>>b) = 2*a, unless the difference tone coincides with the lower tone, which
it
>>only does for the ratio 2/1, or with both tones as in 1/1. So, except for
>>the unison and octave, Kraig's measure is the same as the "integer limit"
of
>>the ratio.

Kraig wrote,

>not quite. If we take my example of one of the inversions of the 6-7-8-9
and space it D=9 A#=14 G=24 C=32. in this >inversion A# occurs as a 14 not a
7. Sorry if this was not clear.

Those are tetrads. For dyads, which is what the poster was talking about,
isn't the above correct?

I wrote,

>>Those are tetrads. For dyads, which is what the poster was talking about,
>>isn't the above correct?

Kraig wrote,

>D A# would still be interpreted as D=9 A#=14

Yes, and the ratio 14/9 would still get a rating two times the higher
number, since 14 + 9 + (14 - 9) = 28.

For those of you interested, here are two examples of my distance metric on
a septimal major sixth (12/7) and a regular major sixth (5/3):

First, prime factor the ratio, excluding all powers of two because of the
Octave Rule:

12/7 = 3^1 * 7^(-1); 5/3 = 5^1 * 3^(-1)

Then, multiply each number by its exponent:

3, -7; 5, -3

Then take the sum of the squares of those numbers:

3^2 + (-7)^2 = 9 + 49 = 58; 5^2 + (-3)^2 = 25 + 9 = 34

And that's the distance (squared) from 1/1 (more for the septimal, which is
what one would expect).

This is exactly the same as if we had used the Pythagorean metric in prime
ratio space (where a move of N along the X axis multiplies by 3^N, 5 for the
Y axis, 7 for the Z axis... so there are infinte dimensions (one for each
prime number)) and multiplied the distance along each axis by the prime it
represents.

This means prime every overtone and undertone will be simply that distance
(not squared), i.e. 7/4 is 7 and 37/32 is 37.

Both the major and harmonic minor scales fit in a circle of radius six
centered about the origin, which gives an idea of the relative sizes of
things in "music space".

That's enough confusing from me for a while.

Stay Tuned,
Keenan

Kraig Grady wrote,

>I love it when you play Plato!

Not familiar with the reference.

>Yes and this is the same rating as 14/11 and 14/13. These really don't
sound so markedly different to my ear.

Nor to mine.

>The problem I see with my own system (it includes most of the rest) is that
it is tied into the numbers of the >harmonics in a way that might not be
proportional to their perceived dissonance. For example is 2 twice as
dissonant >as 1, is 13 13 times more dissonance than 1, I don't think so.
How to measure them I have not managed to do so.

Clearly perceived dissonance does not go to infinity along with the harmonic
series. One can use an arbirtary function to limit it, say define consonance
using 1/n and dissonance as 1-consonance. But again, getting too fancy with
all this is not recommended because beyond ratios of 17 proximity to simpler
ratios becomes as or more important than the numbers in the ratio itself.