Tuning-Math Digests messages 5581 - 5605

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Message: 5581

Date: Fri, 15 Nov 2002 10:40:19

Subject: Re: 43edo 7-limit periodicity-block

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:

>kees van prooijen's page, which i just
>referred you to in a private e-mail, presents an impressive attempt
>to incorporate the smaller-numbers-ratio->smaller-distance idea onto
>an octave-equivalent lattice (at least for the small, comma-like
>intervals), the octave-equivalence being necessary for representing
>periodicity blocks with a finite number of points. i tried very hard
>to get members of this list interested in kees' idea, and to help
>figure out what was going on with this metric, but i found it akin to
>beating my head against a wall.

I haven't told yet that this metric is implemented in Scala:
SET ATTRIBUTE PROOIJEN
and there's a little text in tips.par.

>we therefore know that 12288:12005 is one possible choice for forming
>a complete basis for 43 along with 81:80 and 225:224.

This PB doesn't look like what Joe plotted on his page.
Is it really a PB? I'm not sure.

Manuel


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Message: 5582

Date: Sat, 16 Nov 2002 13:29:41

Subject: Re: 43edo 7-limit periodicity-block

From: monz

> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, November 15, 2002 11:47 AM
> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
>
>
> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > Paul wrote:
> > 
> > > we therefore know that 12288:12005 is one possible
> > > choice for forming a complete basis for 43 along
> > > with 81:80 and 225:224.
> > 
> > This PB doesn't look like what Joe plotted on his page.
> 
> i said a complete basis, not necessarily a set of edges
> for a fokker parallelepiped.


i don't know the difference between a "complete basis" and
"a set of edges for a fokker parallelepiped", and would
greatly welcome an explanation.


 
> > Is it really a PB? I'm not sure.
> 
> is what really a PB? Joe's block? you have to resolve
> the ambiguous positions he indicated for different
> occurences of the same note at the same distance from
> 1/1, but once you've done that, it most certainly is
> a PB, since it contains one and only one instance of 
> each of the 43 tones of 43-equal.


paul's explanation here is exactly right.  i left in the
doubled and tripled instances of pitches which are an
equal-number of taxicab steps away from 1/1 in any of
the six directions (+/- 3/5/7), and connected them with
lines showing their equivalence.


but in any case, Manuel is absolutely correct that 
12288:12005 is *not* one of the unison-vectors defining
the periodicity-block in my graphic.

the matrix i published on the webpage to represent
Gene's "7-limit MT reduced bases" for 43-edo is:

  2  3  5  7

[-4  4 -1  0]  =     81/80
[ 1  2 -3  1]  =    126/125
[12  1 -1 -4]  =  12288/12005


the periodicity-block in my graphic only uses 7^(-1,0,+1).


so my question still has not been answered:  in addition
to 81:80 and 225:224, which i can easily see in my
diagram, what is the third necessary unison-vector which
defines my periodicity-block?  and how does one figure
that out?




-monz


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Message: 5585

Date: Sun, 17 Nov 2002 01:06:28

Subject: Re: 43edo 7-limit periodicity-block

From: monz

> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, November 17, 2002 12:19 AM
> Subject: [tuning-math] Re: 43edo 7-limit periodicity-block
>
>
> anyway, 12288:12005 is certainly going to work for this purpose. 
> i don't see why you're denying it above. if you're worried that just 
> because its 7s exponent is -4, while you're only using 3 different 
> levels along the 7 direction in your plot, it isn't going to work, 
> worry no further. it will work just fine. if you prefer to keep the 7s 
> exponent within the plus-or-minus 3 range, you can always add 
> or subtract (the exponents of ) any of the other unison vectors. for 
> example, if you add the exponents of 126:125 ([2 -3 1]) to those 
> of 12288:12005 ([1 -1 -4]), you get [3 -4 -3] . . . now how about 
> adding the exponents of 224:225 ([-2 -2 1]), resulting in [1 -6 -2] . 
> . . all of these are valid choices for the third unison vector as well.


ah, yes -- of course!  duh, my bad.  i just didn't see it at first.
got it.



-monz


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Message: 5586

Date: Wed, 20 Nov 2002 11:25 +0

Subject: Re: A common notation for JI and ETs

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <5.1.1.6.1.20021120144249.01b20b38@xx.xxx.xx>
David C Keenan wrote:

> We might propose a standard format for that. I imagine something like 
> the tempo specification at the start of some scores that says "crotchet 
> = 120" or some such. e.g. "C:G = 700 c" or "~2:3 = 700 c" or "P5 = 700 
> c". Additional words might say things like "7-limit JI" or "Miracle 
> temperament", or "22-ET", or "Blackjack tuning", but if the reader has 
> never heard of Blackjack at least they have the size of the fifth, and 
> can proceed to play it correctly.

"~2:3" wouldn't be appropriate.  You're saying something that's written a 
certain way is heard as a certain interval.  What you're writing is C:G, 
not ~2:3.  For JI, you could say C:G = 2:3.

For generality, you could specify the octave as well.  Or any other 
interval that would be helpful.


                   Graham


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Message: 5587

Date: Wed, 20 Nov 2002 14:43:27

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 07:32 AM 15/11/2002 -0800, you wrote:
>From:  George Secor, 11/15/2002 (#5015)
>Subject: A common notation for JI and ETs
>
>I'd like to finalize the notation for the two remaining multiples of
>12:
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4662]:
> > At 06:19 PM 17/09/2002 -0700, George Secor wrote:
> > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > > > At 10:24 AM 13/09/2002 -0700, George Secor wrote:
> > > > >132a:  ~|(  /|  |)  |\  (|~  ~||(  /||  ||)  ||\  (||~  /||\
>(MS)
> > > > >132b:  ~|(  /|  |)  |\  (|~  /|\  /||  ||)  ||\  (||~  /||\
>(MS)
> > > >
> > > > I prefer 132b, but why not |( as 5:7-comma for 1deg132?
> > >
> > > I try to choose symbols that are as valid in as many roles as
>possible.
> > >  |( is valid only as the 5:7 comma and not as the 11:13 or 17'-17
> > > commas (1 out of 3), whereas ~|( needs to be valid only as the 17'
> > > comma (1 out of 1).  This is another one that I don't have strong
> > > feelings about, and in the course of working on the spreadsheet I
>might
> > > change my mind.  Even if we don't get any final agreement at this
>point
> > > about some of these less common divisions, at least our discussion
>of
> > > these will provide some examples from which I can arrive at general
> > > principles for choosing symbols.
> >
> > OK
>
>Here I've taken the single-shaft symbols of 132b and used their
>rational complements:
>
>132c:  ~|(  /|  |)  |\  (|~  /|\  /||  ||)  ||\  (||(  /||\    (RC)
>
>But I'm beginning to wonder if we should allow /|\ to exceed (|), which
>would give us a more meaningful 5deg symbol:
>
>132d:  ~|(  /|  |)  |\  (|)  /|\  /||  ||)  ||\  (||(  /||\    (RC)
>
>This might be justified on the same basis that we have allowed /| to
>exceed |) and even |\ in a few instances.  After all, we are already
>used to seeing either sharps or flats higher in pitch in different
>octave divisions.

Yes. I approve of allowing (|) to be smaller than /|\ in the larger 
multiples of 12-ET. It's what I had earlier but only starting with 204-ET. 
However, I think I was using /|) in its place as the 5+7 comma for the 
smaller multiples, which I've now agreed we should only do if it's also the 
13-comma.

I can accept either 132c or 132d. You (or your spreadsheet) should decide. 
I'm mentally too distant from such details at present.

>Now for 144.  I would really like to have its notation in the XH
>article, because it's been mentioned quite a bit on the tuning list.
>
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4654]:
> > At 10:24 AM 13/09/2002 -0700, George Secor wrote:
>
> > >144:  ~|(  /|  )|)  |\  /|)  /|\  (|\  /||  )||)  ||\  /||)  /||\
>
> > Agreed.
>
>This is what we get if we use the above with rational complements:
>
>144b:  ~|(  /|  )|)  |\  /|)  /|\  (|\  /||  )/||  ||\  (||(  /||\
>(RC)
>
>I've now think that I wouldn't want to use )|) if I didn't have to --
>it's a more unusual symbol (and therefore less memorable) than the
>23-comma:
>
>144c:  ~|(  /|  |~  |\  /|)  /|\  (|\  /||  ~||)  ||\  (||(  /||\
>(RC)
>
>In the past you have used prime limit as a measure of simplicity, but I
>would justify using a 23-comma symbol on the basis of product
>complexity.

I used prime limit for much of our discussion, but with the introduction of 
5:7 commas etc, I started using product complexity, perhaps inconsistently 
and without making a point about it. So I agree that product complexity is 
more meaningful.

>   This would also enable us to keep the notation for all the
>multiples of 12 up to 144 without going beyond the 18 single-shaft
>symbols that I am presenting in the article:
>
>)|  |(  ~|  ~|(  |~  )|~  /|  |)  |\  (|  ~|)  (|(  //|  /|)  (|~  /|\
>(|)  (|\
>
>These symbols are sufficient to notate all 17-limit consonances and all
>harmonics and subharmonics through 29, relative to the natural notes.
>Also, their rational complements collectively have the same
>combinations of flags as in the single-shaft set:
>
>(||~  /||)  //||  (||(  ~||)  (||  ||\  ||)  /||  )||~  ||~  ~||(  ~||
>(|\  )||
>
>So I think that these 18 symbols could be a useful set for the
>moderately sophisticated user, just as the "starter set" of 7 symbols
>that I have in Table 3 of my article would be for the simpler ETs
>(including all multiples of 12 through 96):
>
>/|  |)  |\  /|)  /|\  (|)  (|\  and rational complements ||\  ||)  /||

Yes. That sounds very sensible to me.

I agree with 144c.

Just a few more thoughts before going "public".

We must point out that the saggital notation on a score, by itself is not 
enough. The score must also have something to tell the reader what tuning 
it is in. In fact the minimum piece of information required is what size 
the notational fifths are (e.g. in cents).

We might propose a standard format for that. I imagine something like the 
tempo specification at the start of some scores that says "crotchet = 120" 
or some such. e.g. "C:G = 700 c" or "~2:3 = 700 c" or "P5 = 700 c". 
Additional words might say things like "7-limit JI" or "Miracle 
temperament", or "22-ET", or "Blackjack tuning", but if the reader has 
never heard of Blackjack at least they have the size of the fifth, and can 
proceed to play it correctly.

Something else that may need standardising is how one pronounces the 
saggital symbols when reading a score out loud. It seems to me that one 
should say "5-comma up", "11-diesis down" etc., but these are a bit of a 
mouthful (compared to e.g. "sharp" and "flat") and I can see them being a 
problem with composers who just want to use an ET without having to know 
anything about JI. Note that Sims-notation users say "twelfth up, sixth 
down, quarter up, etc., referring to that fraction of a 12-ET whole tone.

We shouldn't make extreme claims about the universality of this notation. 
We can do a lot of JI and ETs (and linears by mapping to ETs), but how do 
we deal, for example, with non-octave tunings such as Bolen-Pierce or 
88-cET or planar temperaments, or randomly chosen pitches. Can we give an 
algorithm, that Manuel might implement in Scala, to give a notation for say 
90% of the tunings in the Scala archive, that will be accurate to within 
+-0.5 c? Assume it is allowed to consult a table of all our agreed ET 
notations.

Just some thorts.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5588

Date: Wed, 20 Nov 2002 23:47:16

Subject: Re: A common notation for JI and ETs

From: monz

hi Dave (and Graham)

> From: "Dave Keenan" <d.keenan@xx.xxx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, November 20, 2002 6:51 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@y..., graham@m... wrote:
> >
> > For generality, you could specify the octave as well.  Or any other 
> > interval that would be helpful.
> 
> Another excellent idea. A full spec might look like this.
> 
> A = 440 Hz, A:A = 1:2, D:A = 2:3
> 
> or
> 
> A = 440 Hz, A:A = 1200 c, D:A = 700 c
> 
> or perhaps more usefully 
> 
> A = 440 Hz, A:A = 1:2, D:A = 2:3 - 2 c



terrific!



-monz


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Message: 5589

Date: Thu, 21 Nov 2002 02:51:28

Subject: Re: A common notation for JI and ETs

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> "~2:3" wouldn't be appropriate.  You're saying something that's
written a 
> certain way is heard as a certain interval.  What you're writing is
C:G, 
> not ~2:3.  For JI, you could say C:G = 2:3.

Good point. And P5 is probably not appropriate in those extreme
tunings where it would seem odd to refer to the notational fifth as
"perfect".

But should it be C:G? Some tunings will not contain the notes C or G
natural. Why not D:A since D is the natural centre of a chain of
fifths and A is the pitch standard.

> For generality, you could specify the octave as well.  Or any other 
> interval that would be helpful.

Another excellent idea. A full spec might look like this.

A = 440 Hz, A:A = 1:2, D:A = 2:3

or

A = 440 Hz, A:A = 1200 c, D:A = 700 c

or perhaps more usefully 

A = 440 Hz, A:A = 1:2, D:A = 2:3 - 2 c


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Message: 5597

Date: Sat, 23 Nov 2002 02:15:35

Subject: Re: Adaptive JI notated on staff

From: monz

hmmm ... the adaptive-JI example reminds me very much of
another musical notation which is nearly-unique in the
literature: that "Daseian" notation used in the _musica
enchiriadis_ and _scolia enchiriadis_ treatises of c. 800 AD.

(i was writing a paper about my speculations on the possible
intonational meanings of that notation back around 1997,
but never finished it.)


-monz


----- Original Message -----
From: "Dave Keenan" <d.keenan@xx.xxx.xx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Friday, November 22, 2002 8:01 PM
Subject: [tuning-math] Re: Adaptive JI notated on staff


> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > > > right, but i'd like to see this actually notated, on a staff.
> > >
> > > Here it is.
> > > Yahoo groups: /tuning-math/files/Dave/AdaptiveJI.bmp *
> >
> > this notation . . . personally, it doesn't do much for me -- for
> > example, looking at this 217-equal example,
> >
> > Yahoo groups: /tuning-math/files/Dave/AdaptiveJI.bmp *
> >
> > only a few of the pure thirds are immediately recognizable from the
> > notation, unless you've memorized all the symbols and the order in
> > which they occur in 217-equal. the symbol for a syntonic comma
> > alteration will quickly be learned by any user of the system, but all
> > the sets of symbols whose difference is a syntonic comma in a given
> > tuning?
>
> Hi Paul. Thanks for your belated response. I totally agree with you re
> the adaptive JI example. But surely you're not rejecting all possible
> uses of the notation on the basis of that?
>
> I gave that example, not because I thought it was a particularly good
> use of the notation, but in response to your request in message 3993:
>
> > > i think it would be cool if someone notated the adaptive-ji
> version
> > > of the chord progression
> > >
> > > Cmajor -> A minor -> D minor -> G major -> C major
> > >
> > > in 217-equal. then we could all look at it and see if we have any
> > > major problems with it.
>
> Can you tell us what you expect of a notation for 217-ET? How might it
> be done better so the pure thirds could all be immediately
> recognisable? Surely any notation for something as large as 217-ET
> will require a significant learning curve?
>
> Why not tell us instead how you feel about the way the notation would
> work in your old favourite, 22-ET. It only needs one pair of new
> symbols /| (for the 5-comma), and its semantics are the same as the
> standard Scala one I've been promoting for ages, and I think it has
> the same semantics as the one Alison Monteith uses. Or in 31-ET, where
> there is also only one new pair of symbols /|\ which are
> simultaneously the 7-comma and the 11-comma (a semi-sharp in this
> case). Or in 72-ET where its semantics are identical to the Sims
> notation. Only the symbols change. /|  |)  /|\
>
>
> 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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