Tuning-Math Digests messages 6428 - 6452

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Message: 6428

Date: Tue, 11 Feb 2003 06:29:42

Subject: poking monz (was: Re: naming temperaments(

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
> <d.keenan@u...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
> > > <d.keenan@u...> wrote:
> > > > > Definitions of tuning terms: equal temperament, (c) 1998 by Joe Monzo *
> > > > 
> > > > This is an extraordinarily beautiful and informative graphic.
> > > 
> > > thanks -- i can't believe this is your first time seeing it.
> > 
> > I think I saw a very early version and hadn't looked at it since.
> 
> since you're interested in 217, 494 and the like, be sure to use the 
> mouse-over zoom feature.

Awesome! 

Except I found I wanted a 333&1/3 zoom so I could see 217 and 494 (and
possibly 612) on the same plot. Some people are never satisfied. ;-)

I couldn't find any explanation of why some numbers are in various
hues of red, orange and magenta. Only some stuff about red lines not
drawn.


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Message: 6430

Date: Tue, 11 Feb 2003 17:23:30

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 07:05 AM 11/02/2003 +0000, George Secor wrote:
> >  > ... unless there is something between 13.47 and 14.37
>cents
> >  > that we need to have in the comma category.
> >
> > I believe there is. Namely the 7:125-comma and the 43-comma.
> >
> > N      From C with cents   Popularity  Ocurrence
> >                             ranking
> > ------------------------------------------------
> > 7:125  Ebb-9.67  D+13.79    35          0.21%
> > 43     E#+9.99   F-13.473   58          0.10%
> > 143    Ebb-11.40 D+12.06    66          0.09%
> > 17:19  D+11.35   Ebb-12.11  72          0.08%
> >
> > The 143 (=11*13) and 17:19 cases above are not a problem because
>we'd be
> > forced to notate them all as ~)| anyway.
> >
> > The question really becomes: How far either side of the half
>Pythagorean
> > comma would a pair of "commas" have to be before we'd notate them
>using two
> > different symbols?
> >
> > In size order we have
> > ~)|
> > .~|(
> > '~)|
> > ~|(
> >
> > The 5:17-kleisma of 12.78 cents is notated exactly as .~|( and it
>needs to
> > be called a kleisma because there is also a 5:17-comma at 36.24
>cents
> > (unless we were going to pull the comma-carcinoma boundary down
>below
> > 36.24, which I don't recommend).
> >
> > I propose that if it's notated as ~)| or .~|( then it's a kleisma
>and if
> > its notated as ~|( or '~)| it's a comma.
> >
> > So in size order we have:
> > ~)|    primarily the 17:19-kleisma 11.35 c
> >                (but the 143-kleisma 12.06 c is more popular)
> > .~|(   primarily the  5:17-kleisma 12.78 c
> > '~)|                      43-comma 13.473 c
> >             or possibly 7:125 comma 13.79 c
> > ~|(    primarily the      17-comma 14.73 c
> >
> > The boundary then is most tightly defined between .~|( and '~)|. We
>already
> > have the 5:17-kleisma at 12.78 cents for .~|(. The most popular
>thing I can
> > find that _might_ be notated as '~)| is the 7:125-comma of 13.79
>cents. It
> > would otherwise be notated as ~|( so it would still be called a
>comma.
> > However the most popular that _needs_ to be notated as '~)| is the
>43-comma
> > of 13.473 cents.
> >
> > Similarly the comma-carcinoma boundary should be between
> > ~|)  primarily the 5:17-comma 36.24 c
> > /|~  primarily the 5:23-carcinoma 38.05 c
> >
> > These are less than a 5-schisma apart and so there are no
>combinations with
> > the 5-schisma flag to confuse the issue. Halfway is at 37.14 cents.
> >
> > Many commas come in pairs that differ by a Pythagorean comma, so it
>would
> > be an advantage to have the distance from the kleisma-comma
>boundary to the
> > comma-carcinoma boundary being exactly a Pythagorean comma. That
>way we are
> > guaranteed never to find such a pair falling into the comma
>category.
> >
> > A Pythag comma up from 13.47 is 36.93 cents, which will do nicely.
> >
> > To summarise:
> > 0
> > schismina
> > 0.98
> > schisma
> > 4.50
> > kleisma
> > 13.47
> > comma
> > 36.93
> > carcinoma
> > 45.11
> > diesis
> > 56.84
> > ediasis
> > 68.57
>
>The way you have it, the kleisma-comma boundary is right at the 43
>comma.

Just below it.

>  If we put the kleisma-comma boundary at ~13.125c, or halfway
>between the 5:17 kleisma (~12.777c) and the 43 comma (~13.473c), then
>a Pythagorean comma up from this would be ~36.585c.  But if we put
>the comma-diesis boundary at ~37.144c, or halfway between the 5:17
>comma (~36.237c) and the 5:23 comma (~38.051c), then a Pythagorean
>comma down from this would be ~13.684c.  Why not split the difference
>and make the boundaries ~13.404c and ~36.864c?

This would seem to make sense, but there's always a fly in the ointment. 
You may have missed where I later wrote:

>>Here's another data point relevant to the comma-name boundaries
>>discussion.
>>
>>49:125 E-13.469 Fb-36.929
>>
>>36.929 c must be notated as ~|) which should make it a comma.
>>Therefore 13.469 c ought to be a kleisma, as it would be with a 13.47
>>c boundary.

What it amounts to is that it is impossible to have the boundaries based on 
the change of symbols and at the same time satisfy the no-two-anomalies in 
the same category (for a given ratio) requirement. Although we can make it 
work for a fair way down the popularity list. Which is more important: 
no-two-anomalies in the same category, or categories correspond to sets of 
symbols?
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6432

Date: Tue, 11 Feb 2003 00:18:41

Subject: Re: naming temperaments

From: Carl Lumma

>{{As it stands, there's no good way to talk about the *blocks*
>behind popular temperaments.}}
>
>Why do you say blocks are behind temperaments?

What do you call meantone without the 81:80 tempered out?  Paul
tried to deny the existence of such beasts, but this hardly
seems possible in light of adaptive JI.

-Carl


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Message: 6434

Date: Tue, 11 Feb 2003 00:32:21

Subject: Re: poking monz (was: Re: naming temperaments(

From: monz

hi Dave,


> From: <d.keenan@xx.xxx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, February 10, 2003 10:29 PM
> Subject: [tuning-math] poking monz (was: Re: naming temperaments(
>
>
re: Definitions of tuning terms: equal temperament, (c) 1998 by Joe Monzo *

> I couldn't find any explanation of why some numbers
> are in various hues of red, orange and magenta. Only
> some stuff about red lines not drawn.



since you were sort-of offlist for a while,
you may not know that i too have been mostly
offlist since last March.  i really needed
a vacation over the summer (when i was enjoying
riding my new motorcycle a lot), and then around
September i got a lot busier with work (which
is real good, because i needed the money).


i slapped paul's graphics into this page and created
the javascript mouseover "zoom" feature, but
unfortunately haven't yet invested the time in
cleaning up the table and text that go along with
these graphics ... as paul has already mentioned
in this thread.

so ... be patient, i'll get around to it eventually.




-monz


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Message: 6435

Date: Tue, 11 Feb 2003 15:10:42

Subject: Re: naming temperaments

From: Carl Lumma

>> What do you call meantone without the 81:80 tempered out?
>
>i would *not* call it "untempered dicot"!

Oh, neither would I.  I switched the example.

>> Paul tried to deny the existence of such beasts,
>
>i did?

Well, you asked "dicot is a temperament, generated by neutral
thirds. in what sense does it make sense to speak of an untempered
temperament? 

>> but this hardly seems possible in light of adaptive JI.
>
>please fill in the blanks for us, carl.

Do you not argue in the Forms of Tonality that PBs are fundamental
musical structures?

Is there a difference between music written for meantone and
music written for 5-limit JI?  Does that difference go away
when the former is rendered in adaptive JI?

-Carl


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Message: 6436

Date: Tue, 11 Feb 2003 11:27:05

Subject: Re: naming temperaments

From: Graham Breed

Carl Lumma wrote:
> i've been dreaming of a huge website where scales are organized by 
> blocks and one can click on which unison vectors to 
> temper/detemper . . .
> 
> That would be truly awesome.  The culmination of years of work.

Then the sooner we start, the sooner it'll be ready.

What do you mean by "blocks"?  Planar temperaments?

I can see how it would be nice to have a dynamic version of Monz's 
diagram, where you could click on equal temeperaments or pairs of equal 
temperaments to get linear temperaments.  But what you want seems to be 
the other way round -- starting with commas rather than equal 
temperaments.  The problem with that is that it becomes multidimensional.

Planar temperaments combine with equal temperaments to give linear 
temperaments, but planar temperaments combining with other planar 
temperaments won't work in general.


>>>An alternative would be to name the important commas, and then
>>>name blocks and temperaments by concatenating the names of the
>>>commas involved, with prefixes to indicate vanishing.
>>
>>already there's the problem that the pythagorean comma doesn't vanish 
>>in pythagorean tuning. but i like the idea . . . nevertheless, what 
>>basis do you use? the TM basis for the 7-limit miracle kernel is 
>>{225:224, 1029:1024}, yet the breedsma does vanish too, which this 
>>wouldn't tell you by names alone . . .
> 
> Good point.  Maybe we need to name wedgies... does that solve the
> problem?

Wedgies or mappings will tell you if a given comma vanishes in a given 
temperament.  You can start with a list of known commas and keep all 
those that vanish in each temperament -- no need to restrict yourself to 
the TM basis.  What do you want them for?


                   Graham


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Message: 6438

Date: Tue, 11 Feb 2003 15:38:48

Subject: Re: naming temperaments

From: Carl Lumma

>ok, so what are we talking about exactly?

What terminology should be used when discussing
untempered PBs.

You said many different PBs might represent a
single untempered temperament.*  Would they all
still share some essential feature?

* I assumed this happens like: given a bunch of
commas, one can swap factors around between them
and come up with a different bunch of commas,
that generate a different block, but the temperament
would map both blocks to the same pitches.

>>>>Paul tried to deny the existence of such beasts,
>>> 
>>>i did?
>>
>>Well, you asked "dicot is a temperament, generated by neutral
>>thirds. in what sense does it make sense to speak of an untempered
>>temperament?"
>
>now we *are* talking about "untempered dicot" again? make up your 
>mind!

It was the "in what sense..." part that I was referring to.

>>Is there a difference between music written for meantone and
>>music written for 5-limit JI?
>
>there can be . . .

Ok, that's an answer.  Forget Western music.  I've always
believed in an affirmative answer here, too, though I realize
I'm at a loss as to how I'd test for it.  So I asked.

>> Does that difference go away
>> when the former is rendered in adaptive JI?
>
>it might not, depending on the music and how it's notated. the forms 
>of tonality argues that the diatonic model holds up quite well in all 
>these alternatives (i think i list 5).

Ok, so it does make sense to talk about an untempered temperament.
What's the appropriate way to talk about it?

-Carl


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Message: 6442

Date: Tue, 11 Feb 2003 15:52:19

Subject: Re: naming temperaments

From: Carl Lumma

>> What terminology should be used when discussing
>> untempered PBs.
>
>well, there are fokker periodicity blocks, other things 
>like "semiblocks" that gene's defined . . . and of course you might 
>have commatic unison vectors even in the untempered case . . .
//
>ok, so in a sense, my paper discusses, as one 
>possibility, "untempered meantone". in which comma drift can occur.
>
>how are we doing?

Good!  So how can one quickly refer to blocks that correspond
to temperaments?  Do I have to say "the 7-tone PB which has
commatic uv x and chromatic uv y"?  Does it make sense to say
"untempered [temperament-x]"?  etc.

-Carl


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Message: 6446

Date: Tue, 11 Feb 2003 22:17:08

Subject: Re: poking monz (was: Re: naming temperaments(

From: monz

hi paul,


> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, February 11, 2003 2:32 PM
> Subject: [tuning-math] poking monz (was: Re: naming temperaments(
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> > unfortunately haven't yet invested the time in
> > cleaning up the table
> 
> monz, the table you have there now was simply pasted in from a link i 
> gave you -- except that you fixed some of the scientifically notated 
> ratios so that they didn't look like unisons.
> 
> the table at the original url now includes the ratios in full decimal 
> representation, as well as important info like the generator of each. 
> so it would seem that you simply need to do the pasting again -- does 
> it take a lot of time? my impression was that it was an instantaneous 
> operation -- please correct me if i'm wrong.
> 
> sorry for poking,
> paul



no need to apologize -- i try to stay on top of my webpages
to make sure they're correct and as up-to-date as i can make
them, and i know that you've sent me (several times) some
stuff that i still need to incorporate.  i've just been real
busy lately ... sorry.

can you please post the link to the table again?
i'll try to get right on it.


-monz


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Message: 6447

Date: Wed, 12 Feb 2003 10:18:56

Subject: New from Springer

From: Gene W Smith

The Springer 2003 ad arrived a few days back, and had the following two
new books in it:

(1) Mathematics and Music, edited by Assayag, Feichtinger and Rodrigues

(2) Foundations of Diatonic Theory *A Mathematically Based Approach to
Music Fundamentals*,
by T.A. Johnson, Ithaca College. 

I'm having modem problems, so I can't respond yet to some things I read,
but Paul might like to know that my announcement of the Baton Rouge
meeting was a consequence of his email, which I did not mention because I
was not sure if it was to be kept private.


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