Tuning-Math Digests messages 8777 - 8801

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Message: 8777

Date: Thu, 11 Dec 2003 09:36:27

Subject: Re: Enumerating pitch class sets algebraically

From: monz

hi paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

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

> > i think the main reason harmonic analysis would be
> > characterised as an "art" is precisely *because* of
> > the ambiguity available to a composer like Brahms,
> > whether his intended tuning is 12edo or a meantone
> > (the only two likely possibilities for Brahms IMO).
> > 
> > my point: that *temperament* allows composers to play
> > the kinds of games ("punning") that aren't possible
> > in JI.
> 
> Irrelevant -- Dante and I were talking about 'conventional' tonal 
> harmonic analysis, which never distinguishes any 81:80s anyway.




OK, my bad.  i didn't follow the thread from the beginning
and probably should have just stayed out of it.

... in fact, my eyes have glazed over with nearly every
post i've seen here over the last week, except for this one.


 
> > and of course JI is the tuning which offers
> > the straightforward "scientific" approach to harmonic
> > analysis.
> 
> BS.


i guess i didn't express myself clearly enough.  you and
i both already know each other's viewpoints on this.

anyway, since i did miss so much here in the last week,
it's probably not worth it for me to try to clarify now ...



-monz


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Message: 8779

Date: Thu, 11 Dec 2003 00:24:42

Subject: Re: Digest Number 862

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> When I said that some 
> physicists would agree with me and some with you, I was talking 
> about agreement on the *real* issue at hand which is the 
> _mathematical _verses _the _musical _definition _of _the 
> _harmonic _series, which - I'm sorry - does not involve 
> acoustics!

So you think some physicists would agree with you and say what, 
exactly? I honestly want to understand you better.


> 
> Sheesh, this has become a royal waste of time...
> 
> Aaron


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Message: 8780

Date: Thu, 11 Dec 2003 01:58:02

Subject: Re: An 11-limit linear temperament top 100 list

From: Paul Erlich

Dave, I have a different interpretation of what's going on here than 
you. Gene has gone about the process of finding 11-limit linear 
temperaments in several ways. Proceeding directly from a prescribed 
set of commas was never one that anyone thought would capture _the_ 
top 100, or top N, according to any badness function (of complexity 
and error). That's why Gene wrote "An" and not "The" in the title. 
Proceeding directly from commas is probably something that I've put 
more weight on than anyone else, but when there is more than one 
comma being tempered out, *straightness* enters the picture and some 
combinations of the "best" commas will be worse than some 
combinations that include a "non-best" comma. Gene has always seemed 
fully cognizant of this fact. Still, I think Gene should reply for 
himself!

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> ...
> > The extra commas I suggested were all that was needed in the 7-
limit
> all had epimericity less than .46. I suggested .5 as a cutoff for
> > the 7-limit and .3 for the 11-limit; I boosted this to .35, with a
> 50 cent cutoff for size. This gave me the following list of 51 
commas,
> > in order of badness of the corresponding planar temperament:
> > 
> > [9801/9800, 3025/3024, 3294225/3294172, 151263/151250, 441/440,
> 385/384, 225/224, 2401/2400, 56/55, 176/175, 4375/4374, 540/539,
> 64/63, 100/99, 250047/250000, 5632/5625, 36/35, 1375/1372, 126/125,
> 45/44, 99/98, 43923/43904, 896/891, 81/80, 49/48, 50/49, 121/120,
> 117649/117612, 55/54, 41503/41472, 1771561/1771470, 77/75, 
4000/3993,
> 6250/6237, 8019/8000, 6144/6125, 1029/1024,5120/5103, 3388/3375,
> 3136/3125, 32805/32768, 245/242, 243/242, 128/125, 12005/11979,
> 245/243, 1728/1715, 19712/19683, 625/616, 1331/1323, 2200/2187]
> > 
> > Wedging these three at a time led to 6135 wedgies. Taking the best
> 100 of these by geometric badness gave me my list. 
> ... 
> 
> Hi Gene,
> 
> I was looking for names for linear temperaments I had found using
> Graham's online finder, and I noticed this 11-limit one wasn't in 
your
> list:
> 
> Complex aug fourths
> generator mapping [[1, ?, ?, ?, ?], [0, -7, -26, -25,  3]]
> minimax generators [1200., 585.14]
> minimax error 4.1 c
> 
> Does this mean there is another 11-limit comma that should be added 
to
> your list above?
> 
> I called it "complex" in deference to this one in your list:
> 
> > Tritonic
> > [5, -11, -12, -3, -29, -33, -22, 3, 31, 33] [[1, 4, -3, -3, 2], 
[0,
> -5, 11, 12, 3]]
> > 
> > generators [1200., 580.274408364]
> > bad 6158.168745 rms 5.154394 comp 70.204409


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Message: 8781

Date: Thu, 11 Dec 2003 00:26:06

Subject: Re: Digest Number 862

From: Paul Erlich

Aaron, hopefully this post will make it up soon, because my posts 
have not appeared in order. One crucial post I made before this one 
has not yet appeared, and it seems we are on bad footing as a result. 
Please be patient until that post appears.

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
> <perlich@a...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" 
> <cfaah@e...> wrote:
> > 
> > > imaginary because you invoked an army of like-minded 
> > > physicists in order to rebut my point of view, and this seemed 
> > > rather fantastical and unnecessary to me.
> > 
> > Fantastical and imaginary as in untrue?
> > 
> > > By the way, Paul, it wasn't necessary to go to the trouble.
> > 
> > If it seemed untrue/imaginary/fantastical before, but now 
> seems true, 
> > then certainly *something* changed your mind . . .
> 
> 
> Ah, I see the problem now.
> 
> Paul, you have not told me anything I do not already know. You 
> started bringing in these legions of physicists to support your 
> argument from a physicists point of view and I told you that you 
> were missing my point completely. When I said that some 
> physicists would agree with me and some with you, I was talking 
> about agreement on the *real* issue at hand which is the 
> _mathematical _verses _the _musical _definition _of _the 
> _harmonic _series, which - I'm sorry - does not involve 
> acoustics!
> 
> Sheesh, this has become a royal waste of time...
> 
> Aaron


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Message: 8782

Date: Fri, 12 Dec 2003 17:32:14

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

George wrote:
>(For example, why are 64/63 and
>63/32 being replaced by 45/44 and 88/45?)

Because they have a lower van Prooijen harmonic distance
value. Also a lower Erlich complexity, which is easier:
log2( max( num and den without factors 2 ) ).

Manuel


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Message: 8786

Date: Fri, 12 Dec 2003 22:37:36

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

>However useful those criteria may be, I consider 64/63 and 63/32
>simpler because:
>1) The prime numbers in the factors are lower; and
>2) The range of numbers in the ratios (32 to 64) is lower (than 44 to
>88).

Still there are more consonant chords in the scale with the original
pitches.

Manuel


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Message: 8789

Date: Fri, 12 Dec 2003 14:55:53

Subject: Re: Question for Manuel, Gene, Kees

From: Carl Lumma

>After fixing my program, here is what I am getting for Prooijen and
>geometric 11-limit reductions:

Thanks for the follow-up, Gene.  I wonder what you and Manuel are
doing differently?

-Carl


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Message: 8790

Date: Fri, 12 Dec 2003 23:13:35

Subject: Re: An 11-limit linear temperament top 100 list

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I later added 12 more to my list of 100, but still don't find it on my
> top 112 list. The reason seems to be that it is above the badness
cutoff. 
> 
> Here's some information on Complex Augmented Fourths:
> 
> Wedgie: [7, 26, 25, -3, 25, 20, -29, -15, -97, -95]
> 
> Mapping: [[1, 5, 15, 15, 2], [0, -7, -26, -25, 3]]
> 
> MT basis: <540/539, 896/891, 1375/1372>
> 
> ets: 41, 80, 121
> 
> rms error: 2.583842867
> geometric complexity (natural log style): 124.3706717
> badness: 8006.869167

OK. Thanks. While it isn't anything to write home about, it doesn't
seem as bad to me as the above complexity figure makes it. Can someone
please explain what geometric complexity is, and how the badness
figure is obtained?


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Message: 8791

Date: Sat, 13 Dec 2003 00:10:17

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

George wrote:
>Another question is: why 15/14 and 15/8 (when 16/15 would have been
>the inversion of 15/8)?

Then it wouldn't be epimorphic anymore, nor a constant structure.
The alternatives are limited to changes by the unison vectors of
the PB.

Manuel


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Message: 8792

Date: Sat, 13 Dec 2003 00:11:05

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

Gene, your geometric reduced scale isn't epimorphic. Is that
a mistake?

Manuel


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Message: 8793

Date: Sat, 13 Dec 2003 00:12:52

Subject: Re: Question for Manuel, Gene, Kees

From: Manuel Op de Coul

Carl wrote:
>Thanks for the follow-up, Gene.  I wonder what you and Manuel are
>doing differently?

We used different periodicity blocks to optimise.
At least that's what I think.

Manuel


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Message: 8796

Date: Mon, 15 Dec 2003 17:59:11

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

George wrote:
>If 15/14 were changed to 16/15, the tuning would still be a constant
>structure.

Sorry, yes. I must have made a typo when I tried it.
Those changes are ok indeed, since 225/224 is one of the unison vectors,
the others are 3025/3024, 1375/1372 and 4375/4374.
I'll change it in the archive too.

>Is there now a definition that does not require a
>degree in mathematics to comprehend?

See Definitions of tuning terms: epimorphic, (c) 2002 by Joe Monzo *

Considering all the higher than 11-limit ratios, I can imagine
it would take a lot of time to change the diagram.

>(I'm presently trying to finish up the rest of the
>sagittal graphics for Scala.)

Splendid, by the way I now don't use xpm files anymore, but png,
but that doesn't matter to you.

Manuel


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Message: 8799

Date: Mon, 15 Dec 2003 19:11:19

Subject: Re: Question for Manuel, Gene, Kees, or whomever . . .

From: Manuel Op de Coul

> Can you tell me
>where you think there is a problem?

I checked and there's a numerical problem in Scala which was silently
ignored. Thanks, I'll see if I can fix it.

Manuel


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