Tuning-Math Digests messages 5851 - 5875

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Message: 5851

Date: Mon, 06 Jan 2003 08:04:30

Subject: Re: Scale theory resources

From: Carl Lumma

>By the way, Carl, did you see my question about scale theory
resources?

I had, but I don't really know any good sources for this type
of stuff.  Might be a good question to ask on specmus.

>Does anyone know of a good source for these, especially
>on-line? I was browsing around and found that Clampitt
>has been using Ramsey theory, which sounds as if people
>are wading out past the shallow end of the pond, and
>I'd like to catch up.

If you can get a hold of any of these...

Rothenberg, David. "A Mathematical Model for Perception Applied
to the Perception of Pitch", Lecture Notes in Computer Science:
Formal Aspects of the Cognitive Process, G. Goos and
J. Harmanis (eds.), chapter 22, 1975, pp. 127-141.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications. Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.

(from Manuel's Tuning & Temperament Bib.)

-Carl


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Message: 5857

Date: Mon, 06 Jan 2003 21:12:09

Subject: Re: Poptimal generators

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus 
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > "Poptimal" is short for "p-optimal". The p here is a real variable
> > p>=2, which is what analysts normally use when discussing these 
> Holder
> > type normed linear spaces.
> > 
> > A pair of generators [1/n, x] for a linear temperament is 
> >*poptimal* if there is some p, 2 <= p <= infinity,
> 
> why not go all the way to 1? MAD, or p=1, error certainly seems 
most 
> appropriate for dissonance curves such as vos's or secor's -- which 
> are in fact even pointier at the local minima (resembling exp
> (|error|)) . . .

Possibly because no one in the history of this endeavour has ever 
before now suggested that mean-absolute error corresponds in any way 
to the human perception of these things. I think everyone agrees that 
the worst errors dominate; either totally as in max-absolute (p=oo) 
or partially as in RMS (p=2).

However, I strongly object to this statement of Gene's in the tuning 
list:

"A "poptimal" generator can lay claim to being absolutely and ideally
perfect as a generator for a given temperament ..."

When we're talking about human perception, as we are, it should be 
obvious that nothing can be absolutely and ideally perfect for 
everyone. Even a single person might prefer slightly different 
generators for different purposes. To validate such a claim 
of "perfection" you would at least need to produce statistics on the 
opinions of many listeners.

Gene, you seem to be confusing beautiful mathematics with accurate 
modelling of a psychoacoustic property (yet to be established).

> > "Muggles" [5, 1, -7, -19, 25, -10] [1, 62/197]

I think the names "Muggles" and "Wizard" could be swapped for these 
two temperaments. The one above, also called "Narrow major thirds", 
is significantly better than the Twin major thirds temperament by 
anyone's badness. The twin major thirds temperament is so bad (due 
mostly to its complexity) that it's barely worth mentioning (at least 
at the 7-limit).

> > "Beatles" [2, -9, -4, 16, 12, -19] [1, 19/64]

Previously called "Neutral thirds with complex 5s". I guess the 
name's based of the "64" denominator?


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Message: 5858

Date: Mon, 06 Jan 2003 21:20:25

Subject: Re: 31, 112 and 11-limit Meantone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> We get a good version of 11-limit Meantone by using the mapping the 
31-et gives us. This has
> 
> [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
> 
> for a wedgie and
> 
> [[1, 1, 0, -3, 11], [0, 1, 4, 10, -13]]
> 
> for a mapping to primes. I *still* cannot get the 31-et to be 
poptimal using this! In fact, the best poptimal for it seems to be 
our old pal, 112-equal Meantone, a hitherto unknown star in the 
Meantone firmament.

What this says to me is that p-optimality is an interesting guide in 
many cases but we shouldn't let it take over from commonsense. In the 
case of meantone it says to me that the range of generators is too 
wide and we may need two ETs to cover it.


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Message: 5861

Date: Tue, 07 Jan 2003 11:46:51

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>I also looked at beta and gamma, but I didn't find much. Beta
>can be related to a cheeseball system obtainable as h75&h94,
>and it is possible to regard gamma as 5/171. Maybe Graham or
>Dave can point out something I am missing here.

Would you expect that tempering the "octave" instead of the
"generator" at a given limit would lead to a different list
of optimal temperaments or optimal generator for a temperament?

Perhaps we should, since our lists tend to keep 2:1 on both
axes of the map (or at least with a very short period of ie's)
while not considering its temperament.  That is, we often
discuss temperaments with 2:1 "octaves" and irrational or
dissonant "generators".  Perhaps we should instead allow the
"octave" to be any size while applying a single weighted error
function to all consonances, including the 2:1.

Most weighting schemes would probably attract the good
generators of good temperaments to consonant intervals.  It
might even be possible to always force one of the generators
to a pure 2:1 with a steep enough weighting.  Rather than
guessing, it seems like a good place to plug in harmonic
entropy.

But why average (max, rms, etc.) complexity and error across
a map before weighting and calculating badness?  Why not
weight per harmonic identity, then just sum to find badness at
the given limit?  If you weight the error right, you shouldn't
have to weight the complexity.

Let g(x) be the graham complexity of identity x, and e(x) be
the weighted error of that identity.  Then minimize

Sum [g(r) * e(r)]

where r goes over all the identities in the given limit.  If a
minimum could be found, we would know the optimal generator of
the optimal temperament.  Assuming a perfect error function,
which can't exist outside of perfect, deterministic neuroscience.

Given the way the error and graham complexity of identities
compound when covering a limit (some sort of consistency is
required by Gene's def. for linear temperaments, IIRC), per-
identity consideration should be enough.  But one could imagine
weighting per-dyad, or even per-chord.

Or maybe I'm missing something?

-Carl


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Message: 5862

Date: Tue, 07 Jan 2003 19:41:10

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>Let g(x) be the graham complexity of identity x, and e(x) be
>the weighted error of that identity. Then minimize
>
>Sum [g(r) * e(r)]
>
>where r goes over all the identities in the given limit.

That's supposed to be...


Let g(x) be the graham complexity of identity x, and e(x) be
the weighted error of that identity. Then minimize

Sum [g(r) * e(r)]

over all maps, where r goes over the identities of the given map.


-Carl


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Message: 5865

Date: Tue, 7 Jan 2003 12:57:28

Subject: Fw: FW: PLEEEEEASE READ!!!! It was on the news!

From: monz

----- Original Message -----
From: "Can Akkoc" <can193849@xxxxx.xxx>
To: "Gonca Tokuz" <gensek1@xxxxxx.xxx.xx>; "Mustafa Tokyay"
<mtokyay@xx.xxxx.xxx.xx>; "Oktay Yagiz" <obyagiz@xxx.xxx>; "Ali Yuksel
Selcukoglu" <selcukya@xxx.xxx>; "Ayhan Sicimoglu" <ayhan@xxxxxxxxxxxx.xxx>;
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Erbahar" <doganerbahar@xxxxxxxxxxx.xxx>; "Cenap Erenben"
<erenben@xxxxxx.xxx>; "Yener Erguven" <erguven@xxxxxxx.xxxxxxx.xxx.xx>;
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"Hasan Gokpinar" <hasan_gokpinar@xxxxxxx.xxx>; "Metin Guney"
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"Ergun Cagatay" <tetragon@xxxx.xxx>; "Gunsan Cetin" <gunsan_cetin@xxx.xxx>;
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"Nevra Akkoc" <nakkoc@xxxxx.xxx>; "Joseph Albree" <joe2@xxxxxxx.xxx.xxx>;
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<addemira@xxxxxxxx.xxx>; "Selim Akkoc" <selim@xxxxxxxxxx.xxx>; "Ali Askin"
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<ayberk@xxxxx.xxx>; "Sehvar Besiroglu" <besir@xxx.xxx.xx>; "Hakan Cevher"
<cevher@xxxxxxx.xxx.xxx.xx>; "Cullen Duke" <dukecul@xxxxxx.xxx>; "Ibrahim
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<yoktay@xxx.xxx.xx>; "Ceylan Orhun" <herana@xxxxxxxxxxx.xxx>; "Gamze Sisman"
<gamzesisman@xxxxx.xxx.xx>; "Erol Tasdemiroglu"
<siberasertas@xxxxxxxxxxx.xxx>; "Ufuk Tezer" <ufuk_tezer@xxxxxxx.xxx>;
"Hasan Toy" <hasantoy@xxxxxxx.xxx>; "Tacettin Yuksel" <t_yuksel@xxxxxxx.xxx>
Sent: Tuesday, January 07, 2003 12:24 PM
Subject: Fwd: FW: PLEEEEEASE READ!!!! It was on the news!


>
>
>  Sehvar Besiroglu <besir@xxx.xxx.xx> wrote:Date: Wed, 08 Jan 2003
05:53:23 -0800
> Subject: FW: PLEEEEEASE READ!!!! It was on the news!
> From: "Sehvar Besiroglu"
> To: "Aykanat, Acar" ,
> Can Akkoc , Cihat Askin ,
> "Murat AYDEMĞR" ,
> "n.serhan aytan" ,
> "Münir Nurettin Beken" ,
> Sibel Bozdogan ,
> ezgi CARDAKTAN ,
> hedia chaffai ,
> "Derya TÜRKAN"
> ,
> nilgun dogrusoz ,
> Sedat Eren ,
> "Tolga Gülen" ,
> Kamran Hooshmand ,
> "Ciğdem Kafescioğlu" ,
> Sevgi KANTAR ,
> Belma Kurtisoglu ,
> kurtisoglu ,
> oozbilen , Siir Ozbilge ,
> "Hadass Pal-Yarden"
>
>
>
> ----------
> From: "Feridun Ozgoren" <feridun.ozgoren@xxxxxxx.xxx>
> To: <VTunaligil@xxx.xxx>, "Semin Cagin" <gzaimler@xxxxxxx.xxx>, "Selis
Onel" <selis@xxx.xxx.xxx>, "Selim Alptekin" <alptekin@xxxxxxx.xxx>, "Sehvar
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<muzafferkanaan@xxxxxxx.xxx>, "Moshahida Sultana" <sultana.m@xxx.xxx>,
"Metin Sezgin" <mtsezgin@xxx.xxx>, "Ibrahim Kalin" <ikalin@xxxxxxxxx.xxx>,
"ibrahim hakki yigit" <ibrahimhakkiyigit@xxxxxxx.xxx>, Hüseyin Adalar
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"Hazal Selcuk" <hazalselcuk@xxxxxxx.xxx>, "Harun" <harunmusic@xxxxxxx.xxx>,
"Hamza Zeytinoglu" <hamza_zeytinoglu@xxx.xxxxxxx.xxx>, "Hakan Talu"
<refikhakan@xxxxxxxxxxx.xxx>, "Hakan Partal" <hpartal@xxxxx.xxx>, "Gunduz
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Ayyildiz" <gulhangayyildiz@xxxxxxx.xxx>, "gulden ayboga" <ayboga@xxxxx.xxx>,
"Ferhan Ozgoren" <grafik@xxxxxx.xxx.xx>, "Fatma" <fazomers@xxx.xxx>, "Emre
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Pervane" <bora@xxxxxxx.xxx>, "Birol Yesilada" <BYesilada@xxx.xxx>, "Barihuda
Tanrikorur" <barihudatanrikorur@xxxxxxx.xxx>, "Ahmet,Nurten,Zafer Sahin"
<ab_ce_43@xxxxx.xxx>, "Ahmet Ersoy" <ersoya@xxxx.xxx.xx>, "Ahmet Erdogdular"
<ahmeterdogdular@xxxxxxx.xxx>, "Ahmed Husrev Isbilir"
<ahmedhusrevi@xxxxx.xxx>
> Subject: Fw: PLEEEEEASE READ!!!! It was on the news!
> Date: Mon, Jan 6, 2003, 7:22 AM
>
>
>
> ----- Original Message -----
> From: Muzaffer Kanaan <mailto:muzaffer.kanaan@xxxxxxx.xxx>
> To: feridun.ozgoren@xxxxxxx.xxx <mailto:feridun.ozgoren@xxxxxxx.xxx>  ;
gupam@xxx.xxx <mailto:gupam@xxx.xxx>
> Sent: Tuesday, January 07, 2003 9:58 AM
> Subject: FW: PLEEEEEASE READ!!!! It was on the news!
>
> Boyle seylere de hic inanmam ama.,.....bir deneyelim bakalim.
>
> Muzaffer
>
> -----Original Message-----
> From: Burak KALAC [mailto:burak.kalac@xxxxxxx.xxx.xxx
> Sent: Tuesday, January 07, 2003 7:44 AM
> Subject: PLEEEEEASE READ!!!! It was on the news!
>
> BLL ABMLE ANLATIK DA...........
>
>
>
> Subject: FW: PLEEEEEASE READ!!!! It was on the news!
>
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>
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>
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>
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>
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>
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>
> Can Akkoc
>
> can193849@xxxxx.xxx
>
>
> ---------------------------------
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Message: 5867

Date: Tue, 07 Jan 2003 21:17:42

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>The problem with this is that it makes the question of what the
>consonances of the temperament are murky--we can't simply use
>everything in the odd limit for some odd n.  Of course, we could
>simply create a set of intervals and then temper, or use n-limit
>including evens.

Perhaps I'm not seeing it, but I don't think we need to change
our concept of limit.

>>But why average (max, rms, etc.) complexity and error across
>>a map before weighting and calculating badness?  Why not
>>weight per harmonic identity, then just sum to find badness at
>>the given limit?  If you weight the error right, you shouldn't
>>have to weight the complexity.
>
>I'll return to this after I've had my breakfast coffee. :)

That was a seriously late-night post, and hopefully it made
sense.  For all I know you could already be calculating badness
this way.

For linear temperaments, we have a 2-D lattice of generators.
The map turns points on this lattice into points on the
(weighted, if you like) harmonic lattice, and back again.  The
complexity of a pair of such points is the taxicab distance on
the lattice of generators, and the error is the taxicab distance
on the harmonic lattice.

You can define as many mapping points as you like.  But if you
stick to consistent maps of the identities only, you should get
reasonable results for all the members of what we normally call
a limit.

-Carl


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Message: 5869

Date: Tue, 07 Jan 2003 21:51:17

Subject: Re: Temperament notation

From: Carl Lumma

>>Since learning septimal notation gives access to 300 years of
>>Western music, the potential payoff for each new one shouldn't
>>be grounds for complaint.
> 
>What, exactly, do you mean by "septimal notation"?

A notation with 7 nominals!

>But in either case my conclusion holds:
> 
>>>so yes, it can be done without
>>>extraordinary effort.
>>
>>I'd say there's nothing far worse about such a setup than
>>in using the Halberstadt for 12-equal.
> 
>I'd be the first to say that it's far better than the
>Halberstadt for 12-ET, because like intervals occur in like
>patterns, so you need only a single fingering pattern for a
>given interval, chord, scale, etc., regardless of the key.

We agree, then.

>(However, like patterns do not guarantee like intervals, as
>would be the case with a decimal keyboard, so you must take
>care to observe in which keys this will work, just as you
>would need to avoid the wolf of meantone temperament on a
>Halberstadt keyboard.)

I don't follow this.  It is possible to make a transpositionally
invariant decimal keyboard, so that like patterns do guarantee
like intervals.

>>>(Or suppose that a piece is heptatonic in one place and
>>>decatonic in another.  Do we switch notations in the middle
>>>of the page?)
>>
>>Absolutely!  Just like switching clefs or key signatures.
> 
>Hey, you were supposed to say "no, of course not!"  Suppose it
>goes back and forth from heptatonic to decatonic every measure,
>or the right hand does something decatonic, while the left hand
>is heptatonic, etc.

It's just a judgement call on the part of the composer, as it
is with clefs and key signatures.

>>You mention Partch's music, which doesn't really use any fixed
>>melodic scale.  I would think transpositionally invariant
>>notation would be optimal, for the scores at least.
>> 
>>But Partch had the right idea... since his instruments played
>>different scales (tonality diamond, microchromatic scales,
>>ancient melodic scales, etc.), he notated differently for each
>>of them.
> 
>Help!  That's what I'm trying to avoid -- insofar as possible, I
>want the pitch notation to be independent of the instrument.  How
>can you understand vertical harmony in a score if the notation is
>different in different lines?

Oh, you have to have unified notation for a score, as I say,
"for scores at least".  But parts are a different matter.

>>I am genuinely interested to see how it looks.  You should debut
>>it with sample music, both original and classical.
> 
>My paper has a couple of samples, plus examples of the 17-limit 
>consonances in the key of C.  These at least give you an idea of
>how things look on paper.  This, along with the explanation of
>the notation, would be enough to get you started.  (I'll have to
>get part of that posted soon, after I go through it and make a
>few small changes.)

Great!  Please let me know with XH 18 (or is it 19?) comes out.

>>>... 6:7:8:9:10:11 isn't a constant structure).
>> 
>>The diatonic scale in 12-equal isn't a constant structure either.
> 
>That's true only as a technicality -- the augmented fourth and
>diminished fifth in the diatonic scale just happen to be the same
>size in 12-ET, but since they are functionally different, this
>isn't a liability.

And in harmonics 6-12, the aug 3rd and dim 4th don't function
differently?

>Partch's hexads make more sense if you consider them 
>heptatonically (with a vacanct position between 6 and 7).

I'd imagine I'd mainly stick to triads if I were to write
'diatonic' music with harmonics 6-12.

>>Did you see any of the virtual keyboard projector posts I've
>>made to the main list over the past year?
> 
>No, I haven't.  I've been spending so much time on the notation 
>project that I usually only have time to read quickly (or scan) 
>through postings on the main list, so there are a lot of things
>I have spent less time on than I would have liked.  Would you
>refer me to an example or two?

Msg. #s 35809 and 41680.

-Carl


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Message: 5871

Date: Tue, 07 Jan 2003 22:01:01

Subject: Quick summary of thoughts on notation

From: Carl Lumma

() I would like to distinguish between two types of notation.

....() Systems in which a given acoustic interval always
covers the same distance on the staff, in terms of lines
and spaces. ("trans. invariant notations")

....() Systems in which a given scale interval (2nd, 3rd,
etc.) always covers the same distance on the staff.
("diatonic" notations)

() For non-diatonic music, the former type of notation
is preferable.

() Regarding the former type, the number of nominals is
more-or-less irrelevant.

() For diatonic music, the latter type of notation is
preferable.

() Regarding the latter type, the number of nominals is
crucial.

() A single notation of the latter type has occupied
generations of musicians in our culture.

-Carl


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Message: 5872

Date: Tue, 07 Jan 2003 00:32:30

Subject: Re: 31, 112 and 11-limit Meantone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> Too wide? The problem has been that it is too damned narrow, being
> stuck in the neighborhood of 1.53 to 1.58 or thereabouts in terms 
of Blackwood's constant. We can't seem to get down to 1.5, for 31, 
and only for 5-limit do we get up to 1.6, for 81.

I guess I don't really understand this poptimal stuff after all. 
Isn't there rather a big difference between the RMS and the max-
absolute optimum generators for meantone, at least at the 5 limit. 
I'm sorry I don't know what Blackwood's constant is.


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