Yahoo Tuning Groups Ultimate Backup tuning-math Attn: Gene 2

All right, folks . . . I'm not sure if I missed anything important
since I last posted, but before I catch up . . .

In the 3-limit, there's only one kind of regular TOP temperament:
equal TOP temperament. For any instance of it, the complexity can be
assessed by either

() Measuring the Tenney harmonic distance of the commatic unison
vector

5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988

() Calculating the number of notes per pure octave or 'tritave':

5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
.........TOP tritave = 1901 -> 19.01 notes per pure tritave.

The latter results are precisely the former divided by 2: in
particular, the base-2 Tenney harmonic distance gives 2 times the
number of notes per tritave, and the base-3 Tenney harmonic distance
gives 2 times the number of notes per octave. A funny 'switch' but
agreement (up to a factor of exactly 2) nonetheless. In some way,
both of these methods of course have to correspond to the same
mathematical formula . . .

In the 5-limit, there are both 'linear' and equal TOP temperaments.
For the 'linear' case, we can use the first method above (Tenney
harmonic distance) to calculate complexity. For the equal case, two
commas are involved; if we delete the entries for prime p in the
monzos for each of the commatic unison vectors and calculate the
determinant of the remaining 2-by-2 matrix, we get the number of
notes per tempered p; then we can use the usual TOP formula to get
tempered p in terms of pure p and thus finally, the number of notes
per pure p. Note that there was no need to calculate the angle
or 'straightness' of the commas; change the angles in your lattice
and the number of notes the commas define remains the same, so angles
can't really be relevant here. As I understand it, the determinant
measures *area* not only in Euclidean geometry, but also in 'affine'
geometry, where angles are left undefined . . . Anyhow, since both of
these methods could be used to address a 3-limit TOP temperament, in
5-limit could they be still both be expressible in a single form in a
general enough framework, say exterior algebra?

In the 7-limit, the two methods give us, respectively, the complexity
of a 'planar' temperament as a distance, and the cardinality of a 7-
limit equal temperament as a volume. But 7-limit 'linear'
temperaments get left out in the cold. The appropriate measure would
seem to have to be an *area* of some sort -- from what I understand
from exterior algebra, this is the area of the *bivector* formed by
taking the *wedge product* of any two linearly independent commatic
unison vectors (barring torsion). If the generalization I referred to
above is attainable, all three of the 7-limit cases could be
expressed in a single way. Anyhow, if this is all correct, I want
details, details, details. The goal, of course, is to produce
complexity vs. TOP error graphs for 7-limit linear temperaments,
something I currently don't know how to do. If someone can fill in
the missing links on the above, preferably showing the rigorous
collapse to a single formula in the 3-limit and with some intuitive
guidance on how to visualize the bivector area in affine geometry (or
whatever), I'd be extremely grateful.

🔗Carl Lumma <ekin@lumma.org>

This is the 3rd version of this post I've recieved, this time with
a different subject. Which version should I refer to?

-Carl

>All right, folks . . . I'm not sure if I missed anything important
>since I last posted, but before I catch up . . .
>
>In the 3-limit, there's only one kind of regular TOP temperament:
>equal TOP temperament. For any instance of it, the complexity can be
>assessed by either
>
>() Measuring the Tenney harmonic distance of the commatic unison
>vector
>
>5-equal: log2(256*243) = 15.925, log3(256*243) = 10.047
>12-equal: log2(531441*524288) = 38.02, log3(531441*524288) = 23.988
>
>() Calculating the number of notes per pure octave or 'tritave':
>
>5-equal: TOP octave = 1194.3 -> 5.0237 notes per pure octave;
>.........TOP tritave = 1910.9 -> 7.9624 notes per pure tritave.
>12-equal: TOP octave = 1200.6 -> 11.994 notes per pure octave;
>.........TOP tritave = 1901 -> 19.01 notes per pure tritave.
>
>The latter results are precisely the former divided by 2: in
>particular, the base-2 Tenney harmonic distance gives 2 times the
>number of notes per tritave, and the base-3 Tenney harmonic distance
>gives 2 times the number of notes per octave. A funny 'switch' but
>agreement (up to a factor of exactly 2) nonetheless. In some way,
>both of these methods of course have to correspond to the same
>mathematical formula . . .
>
>In the 5-limit, there are both 'linear' and equal TOP temperaments.
>For the 'linear' case, we can use the first method above (Tenney
>harmonic distance) to calculate complexity. For the equal case, two
>commas are involved; if we delete the entries for prime p in the
>monzos for each of the commatic unison vectors and calculate the
>determinant of the remaining 2-by-2 matrix, we get the number of
>notes per tempered p; then we can use the usual TOP formula to get
>tempered p in terms of pure p and thus finally, the number of notes
>per pure p. Note that there was no need to calculate the angle
>or 'straightness' of the commas; change the angles in your lattice
>and the number of notes the commas define remains the same, so angles
>can't really be relevant here. As I understand it, the determinant
>measures *area* not only in Euclidean geometry, but also in 'affine'
>geometry, where angles are left undefined . . . Anyhow, since both of
>these methods could be used to address a 3-limit TOP temperament, in
>5-limit could they be still both be expressible in a single form in a
>general enough framework, say exterior algebra?
>
>In the 7-limit, the two methods give us, respectively, the complexity
>of a 'planar' temperament as a distance, and the cardinality of a 7-
>limit equal temperament as a volume. But 7-limit 'linear'
>temperaments get left out in the cold. The appropriate measure would
>seem to have to be an *area* of some sort -- from what I understand
>from exterior algebra, this is the area of the *bivector* formed by
>taking the *wedge product* of any two linearly independent commatic
>unison vectors (barring torsion). If the generalization I referred to
>above is attainable, all three of the 7-limit cases could be
>expressed in a single way. Anyhow, if this is all correct, I want
>details, details, details. The goal, of course, is to produce
>complexity vs. TOP error graphs for 7-limit linear temperaments,
>something I currently don't know how to do. If someone can fill in
>the missing links on the above, preferably showing the rigorous
>collapse to a single formula in the 3-limit and with some intuitive
>guidance on how to visualize the bivector area in affine geometry (or
>whatever), I'd be extremely grateful.
>
>
>
>
>
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🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

'
> geometry, where angles are left undefined . . . Anyhow, since both of
> these methods could be used to address a 3-limit TOP temperament, in
> 5-limit could they be still both be expressible in a single form in a
> general enough framework, say exterior algebra?

That was my suggestion. You normalize by dividing each coordinate by
log2(p) and take the wedge product up to the (normalized) multival,
and then measure complexity by taking the max of the absolute values
of the coefficients. If you start from the monzo side, you get the
same normalized coefficients up to a constant factor, but now you
might rather take the sum of the absolute values (L1 vs L infinity.)

The goal, of course, is to produce
> complexity vs. TOP error graphs for 7-limit linear temperaments,
> something I currently don't know how to do.

Why not use the formula I gave before:

|| <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7,
|w4|/p3p5, |w5|/p3p7, |w6|/p5p7)

to measure complexity? Here p3=log2(3), etc. The corresponding
log-flat badness would be

BAD = ||TOP - JIP|| * ||Wedgie||^2

Here ||Wedgie|| is as above, and ||TOP-JIP|| is the maximum weighted
error, or distance to the JIP. If <t2 t3 t5 t7| is the top tuning for
the temperament given by Wedgie, then this is

||TOP - JIP|| = Max(|t2-1|, |t3/p3-1|, |t5/p5-1|, |t7/p7-1|)

Your choice whether to do everything in log2 or cents terms, of course.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> '
> > geometry, where angles are left undefined . . . Anyhow, since
both of
> > these methods could be used to address a 3-limit TOP temperament,
in
> > 5-limit could they be still both be expressible in a single form
in a
> > general enough framework, say exterior algebra?
>
> That was my suggestion. You normalize by dividing each coordinate by
> log2(p) and take the wedge product up to the (normalized) multival,
> and then measure complexity by taking the max of the absolute values
> of the coefficients. If you start from the monzo side, you get the
> same normalized coefficients up to a constant factor, but now you
> might rather take the sum of the absolute values (L1 vs L infinity.)

Sounds like there are still alternatives here (L1 vs L infinity),
while the complexity definition I was sketching should not have this
ambiguity. So I'm not sure if I'll end up agreeing with Gene's
complexity calculations. Again, it would really help if he went
through my original post and showed, step by step, how all the
results there generalize to a single formula.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> '
> > geometry, where angles are left undefined . . . Anyhow, since both of
> > these methods could be used to address a 3-limit TOP temperament, in
> > 5-limit could they be still both be expressible in a single form in a
> > general enough framework, say exterior algebra?
>
> That was my suggestion. You normalize by dividing each coordinate by
> log2(p) and take the wedge product up to the (normalized) multival,
> and then measure complexity by taking the max of the absolute values
> of the coefficients. If you start from the monzo side, you get the
> same normalized coefficients up to a constant factor, but now you
> might rather take the sum of the absolute values (L1 vs L infinity.)
>
> The goal, of course, is to produce
> > complexity vs. TOP error graphs for 7-limit linear temperaments,
> > something I currently don't know how to do.
>
> Why not use the formula I gave before:
>
> || <<w1 w2 w3 w4 w5 w6|| || = max(|w1|/p3, |w2|/p5, |w3|/p7,
> |w4|/p3p5, |w5|/p3p7, |w6|/p5p7)

This formula looks like a good argument for using square brackets
<< ]] instead of vertical bars << || for multivectors. I though we
had agreed to use square brackets at one stage.