Tuning-Math Digests messages 11250 - 11274

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Message: 11250

Date: Sun, 04 Jul 2004 08:53:30

Subject: Re: NOT tuning

From: Graham Breed

Carl Lumma wrote:
>>>>except where TOP already had pure octaves, in 
>>>>which case it would actually change!
>>>
>>>That's impossible given the criterion of NOT.
>>>
>>>Maybe I don't comprehend you.
>>
>>Some examples of this method of tuning would be nice, and
>>a definition even better.
> 
> 
> Which method?  Graham's?  I think he gave examples.
> 
> Graham, what's a good word to search for?  I know I have that
> post.  I think I replied to it.

I searched my local folder for "Kees metric" and found a post on 2nd Feb 
that you can work back from.  I didn't originally know I was using a 
Kees metric, so you won't find that post.

I think it must be different to NOT, partly because Gene mentioned some 
problems that I'm sure I'd already solved.


                   Graham


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Message: 11252

Date: Sun, 04 Jul 2004 09:07:47

Subject: Re: NOT tuning

From: Graham Breed

Paul Erlich wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> 
>>>Meantone
>>>
>>>5-limit: 698.0187 (43, 55, 98, 153, 251, 404)
>>>
>>>7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43)
>>
>>Hmm... I dunno, this seems a bit far from the old-style rms
>>optimum.
>>
>>-Carl
> 
> 
> Carl, when Graham investigated this same question here a few months 
> ago, he concluded that pure-octaves TOP would be a uniform stretching 
> or compression of TOP, except where TOP already had pure octaves, in 
> which case it would actually change!

You can always define the method to give the same answer for pure-odd 
ratios.  But yes, for the 5-limit it should give quarter comma meantone, 
because the 81:80 is shared between the four factors of 3 in the 
numerator.  It's clearly doing something different.

I haven't defined the 7-limit result because I don't generally know how 
to do 7-limit linear TOP.  What I do have is:

Minimax           696.58
RMS (7)           696.65
RMS (9)           696.44
PORMSWE           697.22

The last one you may recall is my alternative to TOP.  Here, the octave 
is stretched by 1.24 cents.  I can't generalize it to the 
octave-equivalent case (which is why I switched to odd limits in the 
first place).  But you can always unstretch the octave, which here gives 
a fifth of 696.49 cents.

Either there's a systematic error in all my calculations, or Gene's 
result is perverse.


                   Graham


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Message: 11253

Date: Sun, 04 Jul 2004 19:49:10

Subject: Re: A chart of syntonic comma temperaments

From: monz

hi Herman,

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:

> http://www.io.com/~hmiller/png/syntonic.png *
> 
> This is a chart of 7-limit temperaments that temper out
> the syntonic comma 81;80. The horizontal axis is deviation
> from 3:1 and the vertical axis is the deviation from 7:1.
> This time I limited the list to 7-limit consistent ET's.



so according to the criteria in this chart, the temperament
with the lowest error for both 3 and 7 is 36-ET gawel?

i've been missing a lot on the tuning lists until lately,
so i don't even know about the gawel family.



-monz


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Message: 11255

Date: Sun, 04 Jul 2004 14:31:51

Subject: Re: dual, and inner product space (was: Gene's mail server)

From: monz

hi Gene,


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > What do you mean? In quantum mechanics, the bracket
> > or "inner product" is simply the application of a 
> > linear functional acting on a vector.
> 
> In QM state vectors are unit vectors in a complex 
> Hilbert space, meaning there is a Hermitian inner product
> on the space. An eigenvector for a Hermitian linear operator
> (ie, an "observable") with discrete spectrum will *also*
> be a state vector if normalized to a unit vector. They
> live in the same space, so the eigenvector will be a
> wave function like the state vector, if those are wave
> functions. If you take the absolute value of the inner
> product and square it, you get the probability of a
> measurement coming up with with corresponding eigenvalue
> as a result of the measurement of the observable 
> (= Hermitian operator.) If you don't have a discrete spectrum
> you need to resort to spectral theory, but it's basically
> similar. The upshot is that the eigenvectors are bounded
> linear functionals on the states, but since we are in an
> inner product space we can identify these with states. 
> It's like identifying a row vector with a column vector.
> 
> Applications to music? I dunno; but the discrete spectrum
> business is intriging. Someone should tune up a hydrogen atom
> when taken down enough octaves, I guess.



i understand very little of what you wrote here, but ever
since i came up with the idea of finity and bridging in 1998 ...

http://tonalsoft.com/enc/finity.htm *
http://tonalsoft.com/enc/bridging.htm *

... what i *do* understand about QM has had me believing
that it might have some application to music.  i.e.,
there's a sort of "uncertainty principle" with regard
to our perception and comprehension of pitch / tuning / harmony.



-monz






 






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Message: 11256

Date: Sun, 04 Jul 2004 19:51:29

Subject: Re: A chart of syntonic comma temperaments

From: monz

oops ... of course, i see that 36-ET is also catler and
mothra, as well as gawel.  i don't know about either of
those two, either.


-monz


--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Herman,
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
> wrote:
> 
> > http://www.io.com/~hmiller/png/syntonic.png *
> > 
> > This is a chart of 7-limit temperaments that temper out
> > the syntonic comma 81;80. The horizontal axis is deviation
> > from 3:1 and the vertical axis is the deviation from 7:1.
> > This time I limited the list to 7-limit consistent ET's.
> 
> 
> 
> so according to the criteria in this chart, the temperament
> with the lowest error for both 3 and 7 is 36-ET gawel?
> 
> i've been missing a lot on the tuning lists until lately,
> so i don't even know about the gawel family.
> 
> 
> 
> -monz


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Message: 11257

Date: Sun, 04 Jul 2004 00:12:47

Subject: Re: Gene's mail server

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > It says that the bra is a covariant 1-vector, and the ket is a 
> > contravariant one-form. It also says that the combination of the 
two, 
> > as in <v|w>, is an inner product.
> 
> I know people talk that way. I also know it is very confusing to
> people trying to learn this stuff. But probably not worth worrying
> about in connection with your paper. I'd also flush all of that 
stuff
> about 1-forms from your brain immediately.

Well, I remember trying to understand differential forms when 
cramming for a big freshman math final, but really there's nothing to 
flush.

I just looked at Robert Griffiths' book "Consistent Quantum 
Mechanics", and it introduces kets explitictly as linear functionals, 
and then the bracket product as the "inner product". Luckily (or 
perhaps unluckily?), the music theory doesn't have to deal with 
taking complex conjugates.


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Message: 11259

Date: Sun, 04 Jul 2004 00:15:53

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:
> Hi,
> 
> Linear temperaments (or 2-dimensional tunings) are infinitely 
> extendable. Once you extend a linear temperament eonugh you'll 
start 
> getting different pitches that nevertheless are more or less 
> indistinguishable from each other. Even before that you'll get 
> approximations that are better than those the linear temperament is 
> supposed to give. 
> 
> So what would be a good place to close the circle and go from 
linear 
> to equal?
> 
> For TOP tempered linear temperaments I suggest closing the circle 
> when you start getting better approximations to the primes for 
which 
> the tuning is optimized.

Not a bad idea. I don't think any of my horagrams go further than 
this, although 5:4 is slightly better in TOP Catler, and maybe 
there's another similar example somewhere.

You'd have to make your criterion a little more precise -- are you 
assuming that the scales grow in one direction, or in both 
directions, as you apply the generator more and more times?


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Message: 11261

Date: Sun, 04 Jul 2004 00:18:29

Subject: Re: dual, and inner product space (was: Gene's mail server)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> > what is the dual?  if you could explain it along
> > the lines of my "prime-space" definition, that
> > would help me.
> 
> Paul and I have been tossing this back and forth at each other:
> 
> Dual Vector Space -- from MathWorld *
> 
> Linear Function -- from MathWorld *
> 
> > Definitions of tuning terms: prime-space, (c) 2003 by Joe Monzo *
> > 
> > 
> > > That is how you ended up with a bracket product
> > > despite the fact that no inner product space is
> > > being discussed, or would make any sense in the context.
> > 
> > 
> > what's an "inner product space"?
> 
> http://mathworld.wolfram.com/VectorSpace.html *
> Vector space - Wikipedia, the free encyclopedia *
> 
> Inner Product Space -- from MathWorld *
> Inner product space - Wikipedia, the free encyclopedia *
> 
> However, since we aren't using inner product spaces it is only 
vector
> spaces which need concern us. Other relevant encyclopedia pages are
> for abelian group
> 
> Abelian group - Wikipedia, the free encyclopedia *
> 
> You could also look up bra-ket notation, but because it assumes we 
are
> in an inner product space and we are not, it may not be that great.
> This is my problem with Paul wanting to use "inner product" for the
> bracket--we don't actually have an inner product, whereas in quantum
> mechanics we do. This isn't QM, thank heavens.

What do you mean? In quantum mechanics, the bracket or "inner 
product" is simply the application of a linear functional acting on a 
vector.


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Message: 11262

Date: Sun, 04 Jul 2004 20:34:41

Subject: Re: A chart of syntonic comma temperaments

From: monz

hi Herman,

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:

> http://www.io.com/~hmiller/png/syntonic.png *
> 
> This is a chart of 7-limit temperaments that temper out
> the syntonic comma 81;80. The horizontal axis is deviation
> from 3:1 and the vertical axis is the deviation from 7:1.
> This time I limited the list to 7-limit consistent ET's.



are the numbers associated with each temperament family
on this chart wedgies?  if not, then what are they?

on the meantone one, <<1, 4, 10, 4, 13, 12|| ,
the "1, 4, 10" part at least looks familiar as the 
generator mapping for primes 3, 5, and 7.  am i on
the right track?

please explain one, using the meantone one as an example.
if your triangular arrangement is appropriate, please
show that as well.  thanks.



-monz


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Message: 11263

Date: Sun, 04 Jul 2004 00:21:49

Subject: Re: from linear to equal

From: Paul Erlich

9-limit should also be considered when you're going "poptimal".

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
wrote:
> 
> > So what would be a good place to close the circle and go from 
linear 
> > to equal?
> > 
> > For TOP tempered linear temperaments I suggest closing the circle 
> > when you start getting better approximations to the primes for 
which 
> > the tuning is optimized. 
> > 
> > What are your thoughts about this?
> 
> For pure octave tunings, a system I sometimes use is to close at a
> "poptimal" generator. A generator is "poptimal" for a certain set of
> octave-eqivalent consonances if there is some exponent p, 2 <= p <=
> infinity, such that the sum of the pth powers of the absolute value 
of
> the errors over the set of consonances is minimal. This is 
convenient
> for Scala score files, since the notes are now represented by
> (reasonably small) integers. I also sometimes use it when cooking 
up a
> Scala scl file (just did, in fact, over on the tuning list) though 
in
> that case it makes little difference.
> 
> If you follow this system, 5-limit meantone closes for 81, 7-limit
> meantone for 31, and 11-limit meantone for 31. 5 and 7 taken 
together
> are 1/4-comma exactly, which doesn't close; 5 and 11 taken together
> closes at 112, and 7 and 11 of course also at 31. One rarely
> encounters problems; even a microtemperament like ennealimmal closes
> at 1053, which is perfectly reasonable for Scala applications; one
> does, however, need to ensure the division is divisible by 9.
> 
> A different naming convention than using TOP tuning would be to give
> the same name iff the poptimal ranges intersect. This isn't very
> convenient in practice, due to the difficulty of computing the
> poptimal range, but clearly it leads to quite different results.
> Miracle, for instance, has the same TOP tuning in the 5, 7 and 11
> limits, but while the 5 and 7 limit poptimal ranges intersect, the 5
> and 11 or 7 and 11 ranges apparently do not, though as I say 
computing
> these is a pain, so I may have the range too small. In any case,
> miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11-
limit.


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Message: 11265

Date: Sun, 04 Jul 2004 00:34:55

Subject: Re: A map of starling space

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> > I made a chart of ET's with the comma 126;125, and drew lines on 
it to 
> > illustrate some of the landmark starling temperaments.
> 
> Great! This is something I've suggested Paul do from time to time;

I must have missed those; I remember a request to plot according to 
the generator mapping.

> Here's a list of 21 starling temperaments, ordered in terms of
> increasing Graham-TOP badness:
>
> [4, 4, 4, -3, -5, -2]
> [[4, 6, 9, 11], [0, 1, 1, 1]]
> 4 5.871540 93.944647
Dimisept in my paper

> [6, 5, 3, -6, -12, -7]
> [[1, 0, 1, 2], [0, 6, 5, 3]]
> 6 3.187309 114.743119
Keenan in my paper

> [10, 9, 7, -9, -17, -9]
> [[1, -1, 0, 1], [0, 10, 9, 7]]
> 10 1.171542 117.154200
Myna in my paper

> [9, 5, -3, -13, -30, -21]
> [[1, 1, 2, 3], [0, 9, 5, -3]]
> 12 1.049791 151.169891

> [1, 4, 10, 4, 13, 12]
> [[1, 2, 4, 7], [0, -1, -4, -10]]
> 10 1.698521 169.852100
Meantone in my paper

> [3, 0, -6, -7, -18, -14]
> [[3, 5, 7, 8], [0, -1, 0, 2]]
> 9 2.939961 238.136875
Augene in my paper

> [7, 9, 13, -2, 1, 5]
> [[1, -1, -1, -2], [0, 7, 9, 13]]
> 13 1.610469 272.169318
Sensisept in my paper


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Message: 11269

Date: Sun, 04 Jul 2004 01:16:39

Subject: Re: Temperaments with a 7/5 generator

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith <genewardsmith@j...> 
wrote:
> Linear temperaments with a generator which is itself a consonant 
interval
> seem to me of particular interest, so I thought I would explore 
what is
> out there for generators of about 7/5. While nothing dramatic 
turned up,
> these systems might be of interest.
> 
> 
> [3, -5, -6, -1, -15, -18, -12, 0, 15, 18]
> <56/55, 64/63, 77/75>
> badness = 326 rms = 13.78
> 
> [3, -5, -6, 0, 18, -15]
> <64/63, 392/375>
> badness = 532 rms = 14.78
> 
> 7/15 < 15/32 < 8/17
> 
> 15/32 is a nearly exact 11-limit generator; 11/8 is closer than 7/5 
to
> this generator, which is convenient.
> 
> 
> 
> [3, 12, 11, -1, 12, 9, -12, -8, -44, -41]
> <56/55, 81/80, 540/539>
> badness = 404 rms = 12.62
> 
> [3, 12, 11, -8, -9, 12]
> <81/80, 686/675>
> badness = 634 rms = 9.05
> 
> 8/17 < 17/36 < 9/19
> 
> 17/36 is nearly exact 11-generator; again, 11/8 is closer. These 
two are
> the same in the 17-et.

17/36 or 19/36. Gawel.


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Message: 11270

Date: Sun, 04 Jul 2004 22:56:43

Subject: Re: NOT tuning

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >Meantone
> >> >
> >> >5-limit: 698.0187 (43, 55, 98, 153, 251, 404)
> >> >
> >> >7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 
43)
> >> 
> >> Hmm... I dunno, this seems a bit far from the old-style rms
> >> optimum.
> >> 
> >> -Carl
> >
> >Carl, when Graham investigated this same question here a few 
months 
> >ago, he concluded that pure-octaves TOP would be a uniform 
stretching 
> >or compression of TOP,
> 
> That seems obvious for ETs....

But it's not what Gene's definition gives you.

> 
> >except where TOP already had pure octaves, in 
> >which case it would actually change!
> 
> That's impossible given the criterion of NOT.
> 
> Maybe I don't comprehend you.

I didn't say NOT, I said "Graham" and "pure-octaves TOP".


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Message: 11271

Date: Sun, 04 Jul 2004 01:19:09

Subject: Re: NOT tuning

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >Meantone
> >
> >5-limit: 698.0187 (43, 55, 98, 153, 251, 404)
> >
> >7-limit: 697.6458 (31, 43, 934, 977; 0.0286 cents flatter than 43)
> 
> Hmm... I dunno, this seems a bit far from the old-style rms
> optimum.
> 
> -Carl

Carl, when Graham investigated this same question here a few months 
ago, he concluded that pure-octaves TOP would be a uniform stretching 
or compression of TOP, except where TOP already had pure octaves, in 
which case it would actually change!


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Message: 11272

Date: Sun, 04 Jul 2004 22:57:57

Subject: Re: NOT tuning

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > >except where TOP already had pure octaves, in 
> > >which case it would actually change!
> > 
> > That's impossible given the criterion of NOT.
> > 
> > Maybe I don't comprehend you.
> 
> Some examples of this method of tuning would be nice, and a 
definition
> even better.

I looked again and Graham said Feb 2nd.


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Message: 11273

Date: Sun, 04 Jul 2004 23:01:05

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >> 9-limit should also be considered when you're going "poptimal".
> > >
> > >True enough. Alas, even though we have the same wedgie, commas 
and
> > >tuning map, the poptimal range need not even overlap. Orwell is a
> > >typical example--there seems to be no overlap from 7 to 9, and 
none
> > >between 11 and 9, but the others are OK. So, 5 and 9 overlap, and
> > >have 43/190 as a common generator, but 7 and 9, no.
> > 
> > This is AWESOME.  Seriously, if you had come to me in a past
> > life and asked me to imagine the most heinously interesting
> > thing ever, for torturing curious folks in purgatory or
> > something, I wouldn't have come up with the half of this
> > temperaments thing.
> 
> Har. You think that is bad, try this: two different 11-limit linear
> temperaments are the meantone variants meantone or meanpop (sharing
> the same TOP tuning with the 7-limit temperament) and huygens 
(sharing
> the same NOT tuning with the 7-limit temperament.)

Isn't that a ridiculous name for an 11-limit temperament?


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Message: 11274

Date: Sun, 04 Jul 2004 23:06:04

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> 
> > For pure octave tunings, a system I sometimes use is to close at a
> > "poptimal" generator. A generator is "poptimal" for a certain set 
of
> > octave-eqivalent consonances if there is some exponent p, 2 <= p 
<=
> > infinity, such that the sum of the pth powers of the absolute 
value 
> of
> > the errors over the set of consonances is minimal. 
> 
> This is quite an interesting approach. What makes poptimal 
>generators 
> good?

Not much, IMHO -- the "true" value of p in any situation will be some 
number, not an infinite range of numbers.

> And why can't p be 1?

My graphs show p going even slightly below 1, and I think this is 
more than appropriate when you look at the kinds of discordance 
curves Bill Sethares predicts and George Secor prefers. Very sharp 
spikes at the simple ratios.

> These results are interesting. Do these poptimal generators make 
> these linear temperaments close exactly at these ETs?!

"Poptimal" doesn't imply uniqueness the way "optimal" does. Any 
generator within a certain finite range will be poptimal for a given 
situation. So you have to "feed in" ET generators at the beginning if 
you want the circle(s) to close.


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