Tuning-Math Digests messages 9325 - 9349

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Message: 9325

Date: Mon, 19 Jan 2004 02:26:24

Subject: Re: Question for Dave Keenan

From: Carl Lumma

>> It might, depending on the value of 's' or hearing resolution assumed 
>> (this is essentially the only free parameter in harmonic entropy, 
>> which subsumes considerations of timbre, register,

Not register, I thought.  The neigborhood around 5/2 probably looks
a lot like that around 5/4 but the actual entropy values will be
different, right?

-Carl


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Message: 9326

Date: Mon, 19 Jan 2004 05:49:06

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > wrote:
> > 
> > > But remember that the disagreement on things like "beep" 
> > and "father"
> > > is not whether they contain consonances but whether those 
> > consonances
> > > have anything to do with their supposed 5-limit mappings.
> > 
> > What would you call a virtual-pitch-based phenomenon where chord 
> > tones are assigned by the brain to specific partials within a 
> > subsuming 'harmony' -- to take typical examples, 3:4:5, 4:5:6, 
5:6:8, 
> > or hypothetically, 10:12:15 (or still more hypothetically -- 
remember 
> > George Kahrimanis? -- 1/6:1/5:1/4)?
> 
> I suppose you'd like to call it 5-limit consonance, and I would have
> no great objection to that. But what I want to know is, how can we
> tell whether it's happening or not?

A psychological experiment concerning 'roots', perhaps? Ultimately it 
will come down to perception and the reporting of perception, which 
as we know, are not amenable to the exact sciences.

> Let's assume we agree on what
> sounds consonant. How would we tell if the consonance is due to one 
of
> these approximate 5-limit alignments or some other more complex but
> more accurate alignment.

Well the perceived position of the 'root' would determine that.

> e.g. with TOP Beep, you were asking us to believe that the 260 c
> generator could be "experienced as" an approximate 5:6 even though 
it
> is only 7 cents away from 6:7, and 55 cents away from 5:6.

Not really, because though the optimality of TOP can be seen as 
concerning *all intervals*, most of which are even more complex than 
5:6 and have worse errors, its optimization property still holds for 
any, say, "product limit" n*d not smaller than the largest prime. So 
you could use a "product limit" of 5 or even 20 (thus allowing 
voicings -- favored anyway -- like 2:3:4:5) without running into this 
difficulty.

Also, I'm not convinced what you say would necessarily be impossible 
in the right context, such as a full 1:2:3:4:5:6:8 chord, or even, 
perhaps, a 4:5:6:8 chord. After all, Gene and others would have us 
believe that meantone dominant seventh chords were "experienced as" 
4:5:6:7 chords, even though the 6:7 interval would typically be tuned 
far closer to 5:6.

Anyway, when I yielded to you on 'father', the same was pretty much 
implied for 'beep' . . .


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Message: 9327

Date: Mon, 19 Jan 2004 05:57:04

Subject: Re: Annotated Dave Keenan file

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've taken this file
> 
> Good 7-limit generators *
> 
> and annotated it by adding names when available, and descriptions when
> not, of the 7-limit linear temperaments it discusses. While I think
> the value judgments are a little eccentric, it surely is worth looking
> at the temperaments here which don't have names. Maybe Dave wants to
> suggest something?

I'd like them to be named "", "", and "" respectively (without the
quotes). :-)


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Message: 9328

Date: Mon, 19 Jan 2004 10:33:16

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> It might, depending on the value of 's' or hearing resolution 
assumed 
> >> (this is essentially the only free parameter in harmonic 
entropy, 
> >> which subsumes considerations of timbre, register,
> 
> Not register, I thought.

Yes register. S is clearly larger lower down.

> The neigborhood around 5/2 probably looks
> a lot like that around 5/4 but the actual entropy values will be
> different, right?

What does that have to do with it? The right comparison would be a 
given interval with that same interval transposed lower down, not 
between two different intervals.

For octave-equivalent harmonic entropy, s still depends on register, 
but *voicing* and *inversion* of course become irrelevant to the 
actual entropy values.


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Message: 9329

Date: Mon, 19 Jan 2004 02:59:05

Subject: Re: Question for Dave Keenan

From: Carl Lumma

>What does that have to do with it? The right comparison would be a 
>given interval with that same interval transposed lower down, not 
>between two different intervals.

You're right  -C.



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Message: 9330

Date: Mon, 19 Jan 2004 06:11:51

Subject: to whoever put up the links

From: Paul Erlich

Please,

User John Starrett on math.cudenver.edu *

should be replaced with

User John Starrett on rainbow.nmt.edu *

Thanks!



________________________________________________________________________
________________________________________________________________________




------------------------------------------------------------------------
Yahoo! Groups Links

To visit your group on the web, go to:
 Yahoo groups: /tuning-math/ *

To unsubscribe from this group, send an email to:
 tuning-math-unsubscribe@xxxxxxxxxxx.xxx

Your use of Yahoo! Groups is subject to:
 Yahoo! Terms of Service *


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Message: 9331

Date: Tue, 20 Jan 2004 19:00:31

Subject: Q: The missing link -- affine geometry & exterior algebra?

From: Paul Erlich

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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Message: 9332

Date: Tue, 20 Jan 2004 19:01:30

Subject: Q: The missing link -- affine geometry & exterior algebra?

From: Paul Erlich

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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Message: 9333

Date: Tue, 20 Jan 2004 19:15:16

Subject: Re: A potentially informative property of tunings

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
> wrote:
> > Take a generator of 260.76 cents and a period of 1206.55 cents. 
This
> > defines a linear tuning which belongs to a family of related 
linear
> > temperaments. The simplest mapping is the "beep" mapping, which 
> distributes
> > the 27;25 interval:
> > 
> > [(1, 0), (2, -2), (3, -3)]
> > 
> > but after 6 iterations of the generator, there's a better 5:1 at 
> (1, 6),
> > about 15 cents flat (compared with the 51 cent sharp "beep" 
version 
> of the
> > interval). That means this particular tuning is consistent 
> with "beep"
> > temperament only up to a range of 5 generators -- or to coin a 
> phrase, its
> > "consistency range" with respect to "beep" is 5. In comparison, 
top
> > meantone has a "consistency range" of 34: its (17, -35) version 
of 
> 5:1 is
> > only 2 cents flat, compared with the 4-cent sharp (4, -4). 
Quarter-
> comma
> > meantone has a "consistency range" of 29, since it has a better 
3:1 
> at
> > (-11, 30).
> > 
> > First of all, I don't like the term "consistency range", but I 
> couldn't
> > think of anything better. I'd appreciate ideas for what to call 
this
> > property. 
> 
> Since you're describing a relationship or comparison between two 
> temperaments, I would suggest "compatibility range".  The 
> term "consistency" is usually used to describe only relationships 
> within a single temperament.

I don't see any comparison between two temperaments in what Herman is 
proposing! It all looks "within temperament" to me.


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Message: 9334

Date: Tue, 20 Jan 2004 19:17:54

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >There's also the posssibility that the dominant seventh chord
> >functions best when its harmonic entropy (or maybe only the HE of 
one
> >of its dyads) is locally _maximised_.
> 
> What would this look like?  The dominant seventh chord is defined
> as a local minimum of entropy.

Defined?? By whom?


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Message: 9335

Date: Tue, 20 Jan 2004 19:20:28

Subject: Re: Harmonic Entropy and Minkowski's ? function

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> It seems to me a Stieltjes integral with respect to the Minkowski ? 
> function is worth exploring.
> 
> Stieltjes Integral -- from MathWorld *
> 
> Minkowski's Question Mark Function -- from MathWorld *
> 
> The ? function has the property that
> 
> ?((p1+p2)/(q1+q2)) = (p1/q1 + p2/q2)/2
> 
> for adjacent Farey fractions.
> 
> We can ask for Integral exp(-((x-c)/s)^2) d?
> 
> for various values of s, and use it as one definition for the 
> harmonic entropy of c. ?(x) is continuous and it doesn't look 
> terribly difficult to compute, though I've never tried.

I'll eagerly await further details on the harmonic entropy list.


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Message: 9336

Date: Tue, 20 Jan 2004 19:23:48

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >There's also the posssibility that the dominant seventh chord
> > >functions best when its harmonic entropy (or maybe only the HE 
of one
> > >of its dyads) is locally _maximised_.
> > 
> > What would this look like?
> 
> Pretty much like a dominant seventh chord in 12-tET.
> 
> Using noble-mediants to estimate them, a max entropy minor seventh
> should be around 1002 cents, a max entropy diminished fifth should 
be
> around 607 cents, and a max entropy minor third should be around 284
> cents.

Why look at these intervals and not the major third, etc.?


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Message: 9337

Date: Tue, 20 Jan 2004 19:25:01

Subject: Re: Octacot?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Given dicot and tetracot,

Don't forget tricot.

 shoudn't octafifths really be octacot?
> 
> "Octacot" can be described as nonoctave 88tet with octaves, or as
> tetracot sliced in half. If we are using 5/34 as a generator for
> tetracot, then we get octacot by using 5/68 instead. Octacot has TM
> basis {245/243, 2400

/2401?


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Message: 9338

Date: Tue, 20 Jan 2004 19:26:26

Subject: Re: A potentially informative property of tunings

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> Herman,
> 
> I quite agree that it would be very useful to know, for any n-limit
> temperament, the max number of generators before we obtain a better
> approximation for some n-limit consonance, than that given by the
> temperament's mapping.
> 
> I'd love to see this figure for all our old favourites. The only 
thing
> that bothers me is that I assume it will vary according to which
> particular optimum generator we use, and if so, then it isn't 
entirely
> a property of the temperament (i.e. the map).
> 
> But otherwise, it works fine for me to say that temperament X has a
> _consistency_limit_ of Y generators.

I'd avoid the term 'limit' since it's already so loaded. But I agree 
with Dave, and disagree with George, about the appropriateness 
of 'consistency' here.


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Message: 9344

Date: Tue, 20 Jan 2004 19:47:02

Subject: Re: Harmonic Entropy and Minkowski's ? function

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > I'll eagerly await further details on the harmonic entropy list.
> 
> Must it be moved there? That list has been incredibly 
> counterproductive, at least to me, since it's served to keep me out 
> of the conversation.

I don't understand.

> Why not just repost relevant articles?

I could do that if necessary.


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Message: 9347

Date: Tue, 20 Jan 2004 07:41:35

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >There's also the posssibility that the dominant seventh chord
> >functions best when its harmonic entropy (or maybe only the HE of one
> >of its dyads) is locally _maximised_.
> 
> What would this look like?

Pretty much like a dominant seventh chord in 12-tET.

Using noble-mediants to estimate them, a max entropy minor seventh
should be around 1002 cents, a max entropy diminished fifth should be
around 607 cents, and a max entropy minor third should be around 284
cents.

>  The dominant seventh chord is defined
> as a local minimum of entropy.

Who defined it as such, when, and why?

As far as I know, the only widely accepted definition of the dominant
seventh chord is a chord containing the diatonic scale degrees V, VII,
II, IV. It may or may not be a local harmonic entropy minimum
depending how the scale is tuned. It seems to me that the more
dissonant it is, the more relief is likely to be felt when it
"resolves" to a consonance.


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Message: 9348

Date: Tue, 20 Jan 2004 19:51:08

Subject: Re: Octacot?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...> 
> > wrote:
> > > Given dicot and tetracot,
> > 
> > Don't forget tricot.
> 
> Too late, I already did. Not only that, googling restricted to 
Yahoo 
> groups turns up mostly French language groups, plus Bisexual Pagan 
> Teens. Can you brief me?

As usual, you need only look here:

Yahoo groups: /tuning/database? *
method=reportRows&tbl=10&sortBy=6

or here:

Tonalsoft Encyclopaedia of Tuning - equal-temperament, (c) 2004 Tonalsoft Inc. *


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