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Message: 8325 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 19:22:57

Subject: Re: "does not work in the 11-limit" (was:: Vals?)

From: George D. Secor

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:
> 

>> Is it important that a musical scale be 
>> a constant structure, and if so, why?
> 

> but we're talking about a chord, not a whole scale . . .


Aha!  That's something that wasn't clear to me.  If someone starts 
talking about 5 to 7 tones in some sort of combination, I start 
thinking of them in terms of a "scale" rather than a "chord".  Sure, 
I would say that it's not necessary or even important for a chord to 
be a constant structure.

But we could take something that I said a step further: just as a 
major or minor scale is a subset of a pythagorean, meantone, or 12-ET 
tuning, major and minor triads are subsets of major and minor 
scales.  Then we could ask the question: Is it important that new 
musical scales be organized in such a manner that they can be 
described as a set of tones that is the union of several subsets of 
tones (chords) that are alike in structure, differing only in that 
they built on different tones (i.e., have different roots), just as a 
major scale could be described as the union of the tones in a tonic, 
dominant, and subdominant triads?  Certainly this would be a very 
desirable characteristic in a scale -- hmmm, it seems to me that you 
wrote a very brilliant paper on the possibilities of doing this in a 
22-tone octave that would serve as a prime example.  ;-).  I also 
found a couple of scale subsets of a 17-tone octave that have this 
property, and I have a composition in progress in 17-WT that will 
illustrate one of these using non-5 harmony.  (The process of 
composing is very slow going; I try lots of things, and some of them 
work, but some don't.)

But I would say that this property is not essential in a new scale -- 
just that it's a very nice thing to have working in your favor.  (But 
I digress.)


>> (By "scale", I am referring to a set of tones that may be used to 
>> write a simple melody.
> 

> exactly . . . the two champions would have to be the diatonic 
> pentatonic and heptatonic scales . . .
> 

>> If I'm using a pentatonic scale made from a 9-limit otonal chord:
>> 8 : 9 : 10 : 12 : 14 : 16
>> then I have two intervals each of 2:3 (both pentatonic "4ths") 
and 
>> 3:4 (both pentatonic "3rds").
> 

> personally, i'm not fond of this as a scale or melodic entity at 
all -
> - when i improvise over a dominant ninth chord, simply using its 
> notes is about the worst way to come up with a melody . . .


I understand, and I wouldn't have much to say about the harmonic 
possibilities either.  But I think that we've been spoiled by the 
harmonic sophistication of the major-minor system to such an extent 
that it's difficult to appreciate the resources of a simple scale.  
We would have to immerse ourselves in gamelan music (particularly 
slendro) to get in the proper frame of mind to be able to even begin 
to create something decent with such limited tonal resources.  
(Again, we're off on another topic.)


>> So far, so good.
>> 
>> But if I try to use hexatonic scale made from an 11-limit otonal 
>> chord:
>> 8 : 9 : 10 : 11 : 12 : 14 : 16
>> then one of my 2:3s is a hexatonic "5th" and the other is a 
>> hexatonic "4th", and likewise one of my 3:4s is a hexatonic "4th" 
and 
>> the other is a hexatonic "3rd".  Most attempts to transfer a 
melodic 
>> figure beginning on a certain scale degree to another scale 
degree 
>> (such as is required in the musical device called a "sequence") 
will 
>> tend to produce undesirable consequences (such as listener 
>> disorientation) due to the fact that the 2:3 and/or 3:4 must 
switch 
>> degree-roles in the process.
> 

> yet even an algorithmic composition program, such as those written 
by 
> prent rodgers, can produce lovely music by simply using a single 
such 
> hexad at a time, for both harmony and melody (or for polyphony). 
i'm 
> not disputing your constant structure argument too vehemently, 
> especially when it concerns such an important interval as 3:2, but 
> note especially that the diatonic scale in 12-equal is not CS, and 
> yet doesn't cause any more listener disorientation than the 
diatonic 
> scale in, say, 19-equal or 17-equal, where it it CS.


I agree.  I thought about this over the weekend, and I've figured out 
how this differs from the 11-limit otonal scale.  Suppose that we 
were to modify this 11-limit otonal hexatonic scale:
8 : 9 : 10 : 11 : 12 : 14 : 16
so that the 8:12 and 9:12 are each tempered wide by one cent, so that:
8:12 = 703c, 4 scale degrees
12:18 = 701c, 3 deg
9:12 = 499c, 3 deg
12:16 = 497c, 2deg
thereby giving us a constant structure.

Would this eliminate the problem of the potential for 
disorientation?  Of course not, because it still remains that tones 
having the same functional relationship are separated by different 
numbers of degrees in the scale.  We still perceive slightly tempered 
fifths as fifths (i.e., members of the same interval-class), and 
tempering does not change their harmonic identity or function within 
a scale one bit.

Carl mentioned (in msg. #7670) that:

> Paul E. has suggested that we only care about collisions if they
> occur to a consonant interval.  That allows the diatonic scale
> in 12-equal to pass.


That's a good hypothesis, but I think that there's more to it than 
that.  Observe that there is a collision in a harmonic minor scale 
between the augmented 2nd (a dissonance between the 6th and 7th scale 
degrees) and the minor 3rds in the scale (which are *consonant*), but 
I wouldn't say that this results in any disorientation.

Also consider this:  If we were to make each of the 12-ET fifths in a 
major scale narrower by 0.1 cents (so as to make the augmented 4th 
slightly different in size from the diminished 5th), then the scale 
would be CS, even though we would be hard pressed to hear any 
difference from 12-ET.  So constant structure (taken alone) is not 
the whole issue.

I believe that the potential for _functional scale disorientation_ 
(if I may attempt to coin a term) is caused by a particular 
combination of circumstances:

1) If there are two *functionally* different intervals (i.e., aug4 & 
dim5, or aug2 & min3) in a scale that only *happen* to be the exact 
same size (because their ratios, which may be defined either as 
rational or not, happen to be conflated in the tuning in which the 
scale is being used) then there is *no* potential for functional 
scale disorientation.

2) But if there are two intervals in a scale that are *not 
functionally different* (such as the two 2:3s or 3:4s in our 11-limit 
hexatonic otonality), but which span different numbers of steps in 
the scale, then the possibility for functional scale disorientation 
exists.  Since it would be very difficult to perceive any interval 
anywhere in the ballpark of 2:3 or 3:4 as having some other identity 
or functional role, even tempering to make the scale CS would not 
address the problem of functional scale disorientation.  So we see 
that consonance and harmonic entropy are involved here, but it is 
*functional equality* in combination with a *non-CS* condition for 
two intervals that are the conditions for disorientation.

To summarize:  Two intervals with *different* functionality that are 
the same size will not cause disorientation so long as each one spans 
the proper number of steps in the scale.  It is only when two 
intervals with the *same* functionality span differing numbers of 
steps that the problem arises (regardless of whether they are the 
same size or not).

I'm not familiar with any of the 11-limit music that Prent Rodgers 
has produced, so I can only make a guess about why it works.  Suppose 
that I happen to write a composition using a major scale with the 6th 
degree omitted.  Is it hexatonic or heptatonic?  I think that we 
would hear it as heptatonic (due in no small part to the fact that we 
are so heptatonic-oriented, but on the other hand, I don't know how 
someone coming from a culture that uses only a pentatonic scale would 
interpret it).  So I think that it's possible to avoid functional 
scale disorientation with an 11-limit otonal scale by interpreting it 
as an incomplete heptatonic scale.


>> ...
>> So it would not have been possible for the methods of 
conventional (5-
>> limit) harmony to have reached such sophistication if the major 
and 
>> minor scales were not constant structures, because our whole 
method 
>> of building chords (by 3rds) has depended on the fact that the 
simple 
>> ratios of 3 would always be heptatonic 4ths and 5ths and that the 
>> simple ratios of 5 would always be 3rds and 6ths.
> 

> ah, but you're depending on the heptatonic scale here! if we used 
> some sort of heptatonic or other scalar framework to understand 11-
> limit harmony, the same property might hold, despite the fact that 
> the hexad itself is not CS. so the latter fact seems irrelevant.


So I guess I'm agreeing with you by what I said above.


>> ...
>> If you want 11-limit otonal harmony in a conherent scale, then I 
>> think it will have to be at least heptatonic and that you're 
going to 
>> have to fill that extra position with something or other, such as:
>> 8:9:10:11:12:27/2:14:16.
>> Hmmm, that's really not a bad choice, if you'll notice that 
22:27:32 
>> is an isoharmonic triad.  I remember that this scale works very 
>> nicely in 31-ET, since the 27/2:16:20 ends up as an ordinary 
minor 
>> triad.
>> 
>> Likewise, you can have constant-structure 17 and 19-limit otonal 
scales:
>> 16:17:18:20:21:22:24:26:28:30:32 (with chords built in 
decatonic "4ths")
>> and
>> 16:17:18:19:20:21:22:24:25:26:28:30:32
> 

> ok, but not much harmonic movement possible here.


I also find that the dodecatonic scale has more tones than I would 
like to see in anything that I would consider a scale.  While it's 
not quite the same challenge as trying to writing a successful melody 
using all the tones of 12-ET (e.g., problems with identifying a tonal 
center), it's still a bit much for a listener to take in.

I would guess that your statement about limitations on harmonic 
movement comes from looking at this as a single scale, but I find 
that if I look at it as a just tuning, then it possesses considerable 
harmonic richness by virtue of the great variety of heptatonic scales 
that can be found in it -- a direct result of the large number of 
intervals present in the tuning (especially compared to 12-ET).  
There's almost nothing that you can transpose (in the strict sense of 
the term), but one thing that you *can* do is transpose in a looser 
sense: just as a theme may be transposed from a major to its relative 
minor key, so may a pentatonic or heptatonic theme be transposed in 
this 19-limit tuning into another "key", and *always* with a change 
of mode.  And if the term "modulation" doesn't apply, then where does 
it?

--George

Message: 8326 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 19:23:50

Subject: Re: "does not work in the 11-limit" (was:: Vals?)

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> George,
> 
> I was already convinced that Constant Structure is a valuable 
melodic
> property of a scale. But what's wrong with using complete 11-limit
> hexads as vertical harmony within a larger CS scale? Why should we
> care that the hexads _themselves_ are not CS?


Nothing wrong at all with that, just as long as your larger scale 
doesn't get so large that there are too many tones for it to be 
comprehended as a scale.  See my reply to Paul.

--George

Message: 8327 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 19:24:55

Subject: Re: "does not work in the 11-limit"

From: George D. Secor

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

>> I've never really thought very much about this, because for me 
this 
>> was something that seemed to be fairly obvious: that a musical 
scale 
>> that is not a constant structure will tend to result in confusion 
or 
>> disorientation by an inherent contradiction between the acoustical 
>> properties of certain intervals and their identity (or ability to 
>> function) as members (i.e., degrees or steps) of that scale.
> 

> Does that include the diatonic scale in 12-equal?


No.  See my reply to Paul.


> ...
> But incidentally, I'd love a musical example of a hexatonic 11-limit
> melody where the non-CS "collision" causes a problem with 
constructing
> a musical sequence.  With all the ink I've spilled on this subject,
> I'm probably more guilty than anyone of not having come up with
> musical examples to demonstrate propriety...


I'd have to compose something to illustrate that and produce a midi 
or mp3 file that we could listen to.  Then we would have to decide 
that, if we both agreed that it didn't work, that it was due to the 
non-CS property and not because I had a bad composing day.  Or worse 
yet, maybe I might have an extraordinarily good composing day and end 
up turning out something that did work, in spite of the non-CS scale 
property.

Unfortunately, I haven't found much time lately to compose things 
that exemplify ideas that I believe *will* work.  :-(

--George

Message: 8328 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 00:29:22

Subject: Re: Vals?

From: Dave Keenan

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

> If we fix a prime limit, we have duality between multivals and 
> multimonzos; in the 7-limit, a bival and a bimonzo can be identified.
> In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, can be 
> identified, as can trivals with trimonzos. This involves changing the 
> basis of the n-monzos to make them numerically correspond to the
> (pi(p)-n)-vals. My approach to all this has been to swap the 
> n-monzo for the corresponding (pi(p)-n)-val, and use that, but this 
> does require we fix a prime limit.


Don't we always fix the prime limit anyway? Why might this be a problem?



>> e.g. Why is a 5-limit bi-val a monzo?
> 

> The above duality. It isn't, except if you make the identification.


But why is that identification even possible, when, as I understand
it, the bases are incommensurable? There's something I'm not getting here.

Gene, can you please post your code for calculating the wedge-product
for arbitrary dimensions and arbitrary combinations of grades.

And Graham, if yours uses the same ordering of coefficients, please do
the same. (I know it's in the Python code on your website somewhere,
but I'm lazy).

Any accompanying comments will be gratefully received.

For those who haven't followed John Browne's intro, the "grade" of the
multivector is what we're now indicating by the number of nested
brackets and the "bi-" "tri-" etc.

In the Pascal's-triangle of multivector types, dimension increases
downward and grade increases from left to right.

Message: 8329 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 13:05:18

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>I also 
>found a couple of scale subsets of a 17-tone octave that have this 
>property, and I have a composition in progress in 17-WT that will 
>illustrate one of these using non-5 harmony.


Sweet!  Can't wait to hear this...


>Carl mentioned (in msg. #7670) that:

>> Paul E. has suggested that we only care about collisions if they
>> occur to a consonant interval.  That allows the diatonic scale
>> in 12-equal to pass.
>

>That's a good hypothesis, but I think that there's more to it than 
>that.  Observe that there is a collision in a harmonic minor scale 
>between the augmented 2nd (a dissonance between the 6th and 7th scale 
>degrees) and the minor 3rds in the scale (which are *consonant*), but 
>I wouldn't say that this results in any disorientation.

Hmm...

>Also consider this:  If we were to make each of the 12-ET fifths in a 
>major scale narrower by 0.1 cents (so as to make the augmented 4th 
>slightly different in size from the diminished 5th), then the scale 
>would be CS, even though we would be hard pressed to hear any 
>difference from 12-ET.  So constant structure (taken alone) is not 
>the whole issue.


It's a bit of a red herring, to use tolerance in this way.  Things
like CS can be assumed to operate through a blur filter.  It's always
better to explicitly spec the filter, as I did for some of Rothenberg's
measures, but anyway....


>2) But if there are two intervals in a scale that are *not 
>functionally different* (such as the two 2:3s or 3:4s in our 11-limit 
>hexatonic otonality), but which span different numbers of steps in 
>the scale, then the possibility for functional scale disorientation 
>exists.  Since it would be very difficult to perceive any interval 
>anywhere in the ballpark of 2:3 or 3:4 as having some other identity 
>or functional role, even tempering to make the scale CS would not 
>address the problem of functional scale disorientation.  So we see 
>that consonance and harmonic entropy are involved here, but it is 
>*functional equality* in combination with a *non-CS* condition for 
>two intervals that are the conditions for disorientation.


Can you write a melody in the 11-limit 'scale' that sounds wrong
because of functional scale disorientation?


>To summarize:  Two intervals with *different* functionality that are 
>the same size will not cause disorientation so long as each one spans 
>the proper number of steps in the scale.  It is only when two 
>intervals with the *same* functionality span differing numbers of 
>steps that the problem arises (regardless of whether they are the 
>same size or not).


[I'm quoting this here so I only have to save this message.]


>I'm not familiar with any of the 11-limit music that Prent Rodgers 
>has produced, so I can only make a guess about why it works.  Suppose 
>that I happen to write a composition using a major scale with the 6th 
>degree omitted.  Is it hexatonic or heptatonic?  I think that we 
>would hear it as heptatonic (due in no small part to the fact that we 
>are so heptatonic-oriented, but on the other hand, I don't know how 
>someone coming from a culture that uses only a pentatonic scale would 
>interpret it).  So I think that it's possible to avoid functional 
>scale disorientation with an 11-limit otonal scale by interpreting it 
>as an incomplete heptatonic scale.


Of all the scales that might sound like a diatonic scale, I think
the 11-limit otonal scale is probably least.  Anyway, you should
definitely download some of Prent's music, even though I don't think
it's a very good example of melodic writing with harmonic series
segments (Prent's music isn't very melodic) -- Denny Genovese or
Jules Siegel have provided much better examples (though not
downloadable at this point).


>I also find that the dodecatonic scale has more tones than I would 
>like to see in anything that I would consider a scale.


We agree on that.

-Carl

Message: 8330 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 09:44:01

Subject: Re: Vals?

From: monz

hi Gene and Graham (and probably paul too)

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


> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
wrote:
> 


>> I don't know if anybody's following this, but I think
>> the multi-bra notation makes it clearer than anything
>> we've had before.
> 

> I think you and I are probably the only ones without
> headaches, but I do agree. We seem to have passed rapidly
> from disappointing incomprehension to far more
> understanding of this stuff than I've learned to expect,
> and I think the notation thing is a big help. I'm going
> to revise my web pages, and I think revised dictionary
> entries are in order.




yes, absolutely.  i totally agree.  would you guys
*please* rewrite them for me?  thanks.



-monz

Message: 8331 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 13:07:50

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>> >..
>> But incidentally, I'd love a musical example of a hexatonic 11-limit
>> melody where the non-CS "collision" causes a problem with 
>> constructing a musical sequence.
>

>I'd have to compose something to illustrate that and produce a midi 
>or mp3 file that we could listen to.  Then we would have to decide 
>that, if we both agreed that it didn't work, that it was due to the 
>non-CS property and not because I had a bad composing day.


And would it be so much harder than writing all these messages,
full of so much speculation?

-Carl

Message: 8332 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 22:38:11

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

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

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:

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

>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" 
<d.keenan@b...> 
>>>> wrote:
>>>> 

>>>>> If we are told that the mapping is for a tET then _which_ 
tET 
>> it is

>>>>> for can be read straight out of the mapping, as the 
coefficient 
>> for

>>>>> the prime 2 (the first coefficient). And the generator is 
>> simply one

>>>>> step of that tET.
>>>> 

>>>> just wondering why you keep saying "tET" -- 'If we are told 
that 
>> the 

>>>> mapping is for a tone equal temperament then . . .' ??
>>> 

>>> I agree it's awkward. Carl objected so vehemently to EDO and I 
>> wanted

>>> to reserve ET for the most general term (including EDOs ED3s 
cETs).
>>> Perhaps this would be a misuse of ET. Do we have some other 
term for
>>> the most general category of 1D temperaments, i.e. any single
>>> generator temperament whether or not it is an integer fraction 
of 
>> any

>>> ratio? I guess "1D-temperament" will do.
>>> 

>>>> actually, > and < fit together and create a X (as in times) !
>>> 

>>> Oops. Well we could interpret that as the matrix-product as 
opposed 
>> to

>>> the scalar-product (dot-product), but I don't know of any 
meaning 
>> for

>>> that in tuning.
>> 

>> the symbol normally indicates the cross-product, which is 
extremely 
>> useful in tuning: for example, if i take the monzo for the 
diaschisma
>> 
>> [-4 4 -1>
>> 
>> and cross it with the (transpose of the?) monzo for the syntonic 
comma
>> 
>> <-11 4 2]
> 

> Should have been [-11 4 2>


no, the point is that you transpose it so that the angle bracket is 
at the beginning.


>> i get the val for the et where they both vanish:
>> 
>> [12    19    28]
> 

> Now you could write <12 19 28]

maybe.

> That's magic! I never knew that! But of course if someone ever said 
it
> before I wouldn't have understood it since I didn't have a clue 
what a
> val was.
> 
> So [-4 4 -1> (x) [-11 4 2> = <12 19 28]


the idea, though, is that the if the second vector has the angle 
bracket at the beginning, you end up with the symbology ><, which 
already looks like a "x".

Message: 8333 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 22:43:15

Subject: Re: Vals?

From: Paul Erlich

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

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

>> so how can i tell which one is covariant and which one is 
>> contravariant?
> 

> Which one do you regard as the vectors you start from 
(contravariant 
> vector) and which as linear functions on the space of such vectors 
> (covariant vector?) Obviously, in our case the monzos are the 
> objects, and the vals are the mappings, and not the other way 
around. 
> However, we *can* consider linear mappings of vals, which can be 
> identifified via unique isomorphim with monzos.
> 
> Anyway, we have this:
> 
> monzo = ket = contravariant
> 
> val = bra = covariant


so it's pretty much a matter of convention which ones you consider 
covariant and which ones you consider contravariant, but you're ok as 
long as you keep the two categories straight? a little math wouldn't 
hurt :)

Message: 8334 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 22:45:18

Subject: Re: Vals?

From: Paul Erlich

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 

>> I agree it's awkward. Carl objected so vehemently to EDO and I 
> wanted

>> to reserve ET for the most general term (including EDOs ED3s 
cETs).
>> Perhaps this would be a misuse of ET. Do we have some other term 
for
>> the most general category of 1D temperaments, i.e. any single
>> generator temperament whether or not it is an integer fraction of 
> any

>> ratio? I guess "1D-temperament" will do.
> 

> Not 1D. These are 0-dimensional temperaments, I'm afraid.


if linear temperaments are 2-dimesional as you always stress, why 
would these be 0-dimensional and not 1-dimensional? for example, 
88cET has a single generator of 88 cents . . . seems 1 dimensional to 
me!

Message: 8335 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 22:56:57

Subject: Re: Vals?

From: Paul Erlich

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

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

>> so how can i tell which one is covariant and which one is 
>> contravariant?
> 

> Which one do you regard as the vectors you start from 
(contravariant 
> vector) and which as linear functions on the space of such vectors 
> (covariant vector?) Obviously, in our case the monzos are the 
> objects, and the vals are the mappings, and not the other way 
around. 
> However, we *can* consider linear mappings of vals, which can be 
> identifified via unique isomorphim with monzos.
> 
> Anyway, we have this:
> 
> monzo = ket = contravariant
> 
> val = bra = covariant


in quantum mechanics though, a state can be represented as either a 
bra or a ket, depending on the mathematical operation being 
performed . . .

??

Message: 8336 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 22:58:04

Subject: Re: Vals?

From: Paul Erlich

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

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

>> right, but i still want to understand it, since it was in my 
>> relativity textbooks . . .
> 

> It's more complicated in relativity. There you have tangent spaces 
> and cotangent spaces *at every point*, which have to connect 
> together, plus you have a non-positive inner product which changes 
> from point to point. We've got it easy and should enjoy ourseves.


aren't you talking about general relativity? i was only taking 
special relativity . . .

Message: 8337 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 23:06:18

Subject: Re: Vals?

From: Graham Breed

Dave Keenan wrote:


> And since a 7-limit monzo has coefficients [e2 e3 e5 e7> then a
> 7-limit trimonzo will have coefficients ordered [[[e357 e572 e723 e235>>>.
> 
> Is this how your software does it too Graham?


The wedgies are stored in a dictionary, indexed by the bases.  So the 
order only becomes important for some display functions.  I order them 
by increasing index.  And everything uses increasing numbers left to 
right.  So it'd be [[[e235 e237 e257 e357>>>.


> But how do you order the coefficents of a 7-limit bimonzo or bimap
> (bival) so it's its own complement???


Gene does it so you reverse the order to do the complement.  But he's 
never given the general case, and I haven't worked it out.  If I could, 
I might be able to go on to write an efficient implementation in C.


                 Graham

Message: 8338 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 23:34:17

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

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

>> If we fix a prime limit, we have duality between multivals and 
>> multimonzos; in the 7-limit, a bival and a bimonzo can be 
identified.
>> In the 11-limit, bivals and 4-monzos, and bimonzos and 4-vals, 
can be 
>> identified, as can trivals with trimonzos. This involves changing 
the 
>> basis of the n-monzos to make them numerically correspond to the
>> (pi(p)-n)-vals. My approach to all this has been to swap the 
>> n-monzo for the corresponding (pi(p)-n)-val, and use that, but 
this 
>> does require we fix a prime limit.
> 

> Don't we always fix the prime limit anyway? Why might this be a 
problem?


sometimes you want to use a set of nonconsecutive primes, as you've 
mentioned yourself, dave.

Message: 8339 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 23:45:13

Subject: Re: Vals?

From: Dave Keenan

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

> the idea, though, is that the if the second vector has the angle 
> bracket at the beginning, you end up with the symbology ><, which 
> already looks like a "x".


I'm afraid I don't like that at all. It would only work for two
arguments, not 3 or more, and in any case we don't need to use the
cross-product operator since we're using the more general exterior
product (wedge product) ^. And the < ... ] is meant to tell us we're
looking at a map not a monzo. And it is beneficial to know that you
can't wedge maps with monzos. You have to convert them to the same
kind first.

Message: 8340 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 23:51:12

Subject: Re: Vals?

From: Dave Keenan

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

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

>> Anyway, we have this:
>> 
>> monzo = ket = contravariant
>> 
>> val = bra = covariant
> 

> in quantum mechanics though, a state can be represented as either a 
> bra or a ket, depending on the mathematical operation being 
> performed . . .
> 
> ??


I have to admit I'm not too concerned if analogies between quantum
mechanics and tuning theory don't pan out. :-)

I'm only concerned with whether (our extension of) the notation makes
the Grassman Algebra clearer for us. And it certainly seems to be
doing so.

Message: 8341 - Contents - Hide Contents

Date: Mon, 17 Nov 2003 23:54:01

Subject: Re: "does not work in the 11-limit" (was:: Vals?)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:


>> exactly . . . the two champions would have to be the diatonic 
>> pentatonic and heptatonic scales . . .
>> 

>>> If I'm using a pentatonic scale made from a 9-limit otonal 
chord:
>>> 8 : 9 : 10 : 12 : 14 : 16
>>> then I have two intervals each of 2:3 (both pentatonic "4ths") 
> and 

>>> 3:4 (both pentatonic "3rds").
>> 

>> personally, i'm not fond of this as a scale or melodic entity at 
> all -

>> - when i improvise over a dominant ninth chord, simply using its 
>> notes is about the worst way to come up with a melody . . .
> 

> I understand, and I wouldn't have much to say about the harmonic 
> possibilities either.  But I think that we've been spoiled by the 
> harmonic sophistication of the major-minor system to such an extent 
> that it's difficult to appreciate the resources of a simple scale.  
> We would have to immerse ourselves in gamelan music (particularly 
> slendro) to get in the proper frame of mind to be able to even 
begin 
> to create something decent with such limited tonal resources.  
> (Again, we're off on another topic.)


i don't know . . . i mentioned the diatonic pentatonic scale above. 
that's an equally simple scale, isn't it, and yet i could probably 
live a happy life with no other melodic resources. so it seems you 
missed my point entirely.

i realized, since i made my original post, that the "dominant 
pentatonic" is not CS in 12-equal. perhaps that's one source of my 
difficulty?


> Carl mentioned (in msg. #7670) that:

>> Paul E. has suggested that we only care about collisions if they
>> occur to a consonant interval.  That allows the diatonic scale
>> in 12-equal to pass.
> 

> That's a good hypothesis, but I think that there's more to it than 
> that.  Observe that there is a collision in a harmonic minor scale 
> between the augmented 2nd (a dissonance between the 6th and 7th 
scale 
> degrees) and the minor 3rds in the scale (which are *consonant*), 
but 
> I wouldn't say that this results in any disorientation.


the suggestion of mine that carl was referring to . . . did i ever 
make it quite clear? i don't remember :(


> Also consider this:  If we were to make each of the 12-ET fifths in 
a 
> major scale narrower by 0.1 cents (so as to make the augmented 4th 
> slightly different in size from the diminished 5th), then the scale 
> would be CS, even though we would be hard pressed to hear any 
> difference from 12-ET.  So constant structure (taken alone) is not 
> the whole issue.


well yes, that's exactly the point i was trying to make in my 
original post.


> I believe that the potential for _functional scale disorientation_ 
> (if I may attempt to coin a term) is caused by a particular 
> combination of circumstances:
> 
> 1) If there are two *functionally* different intervals (i.e., aug4 
& 
> dim5, or aug2 & min3) in a scale that only *happen* to be the exact 
> same size (because their ratios, which may be defined either as 
> rational or not, happen to be conflated in the tuning in which the 
> scale is being used) then there is *no* potential for functional 
> scale disorientation.
> 
> 2) But if there are two intervals in a scale that are *not 
> functionally different* (such as the two 2:3s or 3:4s in our 11-
limit 
> hexatonic otonality),


why aren't they functionally different? because we don't have a well-
defined sense of hexatonic musical function, while we know all too 
much about the history and theory of the diatonic scale? i don't 
think that the "happen to" above can be defined in any precise or 
perceptually relevant sense -- though it would be nice . . .


> To summarize:  Two intervals with *different* functionality that 
are 
> the same size will not cause disorientation so long as each one 
spans 
> the proper number of steps in the scale.  It is only when two 
> intervals with the *same* functionality span differing numbers of 
> steps


i look forward to a definition of "functionality" . . .


> I'm not familiar with any of the 11-limit music that Prent Rodgers 
> has produced,


please do listen to it as soon as possible . . .

Microtonal Music by Prent Rodgers * [with cont.]  (Wayb.)


> so I can only make a guess about why it works.  Suppose 
> that I happen to write a composition using a major scale with the 
6th 
> degree omitted.  Is it hexatonic or heptatonic?  I think that we 
> would hear it as heptatonic (due in no small part to the fact that 
we 
> are so heptatonic-oriented, but on the other hand, I don't know how 
> someone coming from a culture that uses only a pentatonic scale 
would 
> interpret it).  So I think that it's possible to avoid functional 
> scale disorientation with an 11-limit otonal scale by interpreting 
it 
> as an incomplete heptatonic scale.


does the composer or algorithm have to so interpret it in some way 
for this to work?

Message: 8342 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 18:56:51

Subject: Re: "does not work in the 11-limit"

From: George D. Secor

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

>> I also 
>> found a couple of scale subsets of a 17-tone octave that have this 
>> property, and I have a composition in progress in 17-WT that will 
>> illustrate one of these using non-5 harmony.
> 

> Sweet!  Can't wait to hear this...


Me too.  I conceived the idea for it over 25 years ago and started 
working on it in Cakewalk a little over a year ago.  It's in a 9-tone 
MOS scale structure -- very different from anything else I've ever 
tried.  It's been very slow going to figure out how to make 
everything work to my satisfaction.  I have only about 20 bars of 
music done, but I'm absolutely delighted with it so far.  At this 
rate, maybe I'll have it done by the end of this decade.  ;-)


> ...
> Can you write a melody in the 11-limit 'scale' that sounds wrong
> because of functional scale disorientation?


I actually tried to run a couple of ideas through my head last night 
about how to do it, and it's not as easy as I thought.  
Paradoxically, it takes a certain lack of talent or lack or 
familiarity with microtonal resources to write something 
intentionally bad enough that it sounds wrong. -- it's only when I'm 
not trying to do it that I seem to be able to succeed.  Hmmm, perhaps 
I've stumbled on a new method of composition -- guaranteed to give 
good results.  ;-)

--George

Message: 8343 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 18:58:02

Subject: Re: "does not work in the 11-limit"

From: George D. Secor

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

>>> ...
>>> But incidentally, I'd love a musical example of a hexatonic 11-
limit
>>> melody where the non-CS "collision" causes a problem with 
>>> constructing a musical sequence.
>> 

>> I'd have to compose something to illustrate that and produce a 
midi 
>> or mp3 file that we could listen to.  Then we would have to decide 
>> that, if we both agreed that it didn't work, that it was due to 
the 
>> non-CS property and not because I had a bad composing day.
> 

> And would it be so much harder than writing all these messages,
> full of so much speculation?
> 
> -Carl


Yes, I know from experience that it would be both harder and more 
time-consuming, because I would need to make a lot of decisions on 
how I would go about it.  It's much quicker and easier just to rattle 
thoughts out of my head into a keyboard.  Speculation is cheap, but 
creating music (even bad music) requires a bigger investment -- it 
takes a bit of work.

That said, I think I've convinced myself that I need to take a break 
from the tuning lists for a while and use the time to do some 
composing.  That's the way to discover first-hand which techniques 
and ideas will work (at least for me) and which ones won't.  (And 
then perhaps I'll also save excerpts of some of the bad ideas I try, 
just for illustration when discussions like these come up.)

--George

Message: 8344 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 18:58:57

Subject: Re: "does not work in the 11-limit" (was:: Vals?)

From: George D. Secor

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:
> 

>>> exactly . . . the two champions would have to be the diatonic 
>>> pentatonic and heptatonic scales . . .
>>> 

>>>> If I'm using a pentatonic scale made from a 9-limit otonal 
> chord:

>>>> 8 : 9 : 10 : 12 : 14 : 16
>>>> then I have two intervals each of 2:3 (both 
pentatonic "4ths") 
>> and 

>>>> 3:4 (both pentatonic "3rds").
>>> 

>>> personally, i'm not fond of this as a scale or melodic entity 

at all -

>>> - when i improvise over a dominant ninth chord, simply using 
its 
>>> notes is about the worst way to come up with a melody . . .
>> 

>> I understand, and I wouldn't have much to say about the harmonic 
>> possibilities either.  But I think that we've been spoiled by the 
>> harmonic sophistication of the major-minor system to such an 
extent 
>> that it's difficult to appreciate the resources of a simple 
scale.  
>> We would have to immerse ourselves in gamelan music (particularly 
>> slendro) to get in the proper frame of mind to be able to even 
begin 
>> to create something decent with such limited tonal resources.  
>> (Again, we're off on another topic.)
> 

> i don't know . . . i mentioned the diatonic pentatonic scale above. 
> that's an equally simple scale, isn't it, and yet i could probably 
> live a happy life with no other melodic resources. so it seems you 
> missed my point entirely.


No, I don't think I did.  I recognize that a scale that is 
essentially a just dominant 9th chord makes it a little more 
difficult to imply any changes in the harmonic element other than 
alternation between 4:5:6 to 6:7:9 triads in an accompaniment.  An 
unaccompanied melody using a diatonic pentatonic scale, on the other 
hand (such as _Auld Lang Syne_), can easily evoke a heptatonic 
chordal accompaniment in our imaginations -- it takes only the 
presence of the 5/3 in the scale to imply that there should be a 4/3 
(subdominant) somewhere in context.

My motive in suggesting "slendro therapy" was to experience that 
there is a feeling of satisfaction that can be achieved with music 
that has little or no harmonic motion.


> i realized, since i made my original post, that the "dominant 
> pentatonic" is not CS in 12-equal. perhaps that's one source of my 
> difficulty?


I wouldn't think so.


>> ...
>> 2) But if there are two intervals in a scale that are *not 
>> functionally different* (such as the two 2:3s or 3:4s in our 11-
limit 
>> hexatonic otonality),
> 

> why aren't they functionally different? because we don't have a 
well-
> defined sense of hexatonic musical function, while we know all too 
> much about the history and theory of the diatonic scale? i don't 
> think that the "happen to" above can be defined in any precise or 
> perceptually relevant sense -- though it would be nice . . .


As I see it, interval function is independent of the number of tones 
in the scale, but instead has to do with the (just) *ratio* that is 
either directly expressed (in JI) or implied (in a temperament) by 
that interval.  So two tempered intervals that (in a given context) 
are implying the same just interval are functionally the same, even 
if they are not exactly the same size (such as in a well-
temperament).  But two tempered intervals that (by context) imply 
different just intervals are functionally different, even if they are 
exactly the same size in a particular tuning.

In the context of a diatonic scale the tones are all assumed to be in 
a chain of fifths.  If one member of that chain is taken to represent 
1/1, then each of the other members can be assigned at least one 
(rational) ratio that is unique to that member.  An augmented 4th and 
diminished 5th (or a minor 3rd and augmented 2nd, etc.) will 
therefore be considered to be serving different harmonic functions, 
since they represent different ratios.

In an 8:9:10:11:12:14:16 scale there is no question that the two 2:3s 
(or the two 3:4s) are for all intents and purposes identical (since 
this is JI), so on a *harmonic* level they are functionally 
equivalent.  But since these pairs of intervals subtend different 
steps in the scale, the potential for _functional scale 
disorientation_ (if you don't like the term, then please suggest 
something else) exists.


> 

>> To summarize:  Two intervals with *different* functionality that 
are 
>> the same size will not cause disorientation so long as each one 
spans 
>> the proper number of steps in the scale.  It is only when two 
>> intervals with the *same* functionality span differing numbers of 
>> steps
> 

> i look forward to a definition of "functionality" . . .


I think that I've given enough information above to arrive at one.  
Perhaps I haven't chosen the right term -- would something containing 
the words "harmonic identity" or "interval identity" be better?


>> I'm not familiar with any of the 11-limit music that Prent 
Rodgers 
>> has produced,
> 

> please do listen to it as soon as possible . . .
> 
> Microtonal Music by Prent Rodgers * [with cont.]  (Wayb.)
> 

>> so I can only make a guess about why it works.  Suppose 
>> that I happen to write a composition using a major scale with the 
6th 
>> degree omitted.  Is it hexatonic or heptatonic?  I think that we 
>> would hear it as heptatonic (due in no small part to the fact 
that we 
>> are so heptatonic-oriented, but on the other hand, I don't know 
how 
>> someone coming from a culture that uses only a pentatonic scale 
would 
>> interpret it).  So I think that it's possible to avoid functional 
>> scale disorientation with an 11-limit otonal scale by 
interpreting it 
>> as an incomplete heptatonic scale.
> 

> does the composer or algorithm have to so interpret it in some way 
> for this to work?


It's hard to say.  It's possible for a composition to be so free-form 
that we (as listeners) just take it in without attempting to perceive 
a particular scale structure in our minds.

--George

Message: 8345 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 11:08:23

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>Me too.  I conceived the idea for it over 25 years ago and started 
>working on it in Cakewalk a little over a year ago.  It's in a 9-tone 
>MOS scale structure -- very different from anything else I've ever 
>tried.  It's been very slow going to figure out how to make 
>everything work to my satisfaction.  I have only about 20 bars of 
>music done, but I'm absolutely delighted with it so far.  At this 
>rate, maybe I'll have it done by the end of this decade.  ;-)


Are you writing it in Sagittal?  ;-)


>> Can you write a melody in the 11-limit 'scale' that sounds wrong
>> because of functional scale disorientation?
>

>I actually tried to run a couple of ideas through my head last night 
>about how to do it, and it's not as easy as I thought.  
>Paradoxically, it takes a certain lack of talent or lack or 
>familiarity with microtonal resources to write something 
>intentionally bad enough that it sounds wrong. -- it's only when I'm 
>not trying to do it that I seem to be able to succeed.  Hmmm, perhaps 
>I've stumbled on a new method of composition -- guaranteed to give 
>good results.  ;-)


Remember, we're trying to ignore composition as a factor.  We don't
care if it's bad, just if it has functional scale disorientation.
If we try and try but can't ever hear functional scale disorientation,
it's important -- it means it probably doesn't exist.

-Carl

Message: 8346 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 19:59:38

Subject: Re: "does not work in the 11-limit"

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:


> ... I think I've convinced myself that I need to take a break 
> from the tuning lists for a while and use the time to do some 
> composing.  That's the way to discover first-hand which 
> techniques and ideas will work (at least for me) and which
> ones won't.  (And then perhaps I'll also save excerpts of
> some of the bad ideas I try, just for illustration when
> discussions like these come up.)



this is interesting to me.

i have one piece which i wrote in 12edo, and later converted
to JI, but i was never really happy with the JI version.

_In A Minute_:
program notes for In A Minute, (c)1993, 1999 b... * [with cont.]  (Wayb.)

this page is supposed to open with the mp3, but in case
there's a problem, here it is:
ÿ À * [with cont.]  (Wayb.)


i'm expecting that once my software has music composition
capability (which will be within a month), i'll be able
to do a better job of this.

but anyway, even tho i'm not happy with it, i thought it
was interesting to document my "justification" of this piece.



-monz

Message: 8347 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 21:12:15

Subject: Re: "does not work in the 11-limit"

From: George D. Secor

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

>> Me too.  I conceived the idea for it over 25 years ago and started 
>> working on it in Cakewalk a little over a year ago.  It's in a 9-
tone 
>> MOS scale structure -- very different from anything else I've ever 
>> tried.  It's been very slow going to figure out how to make 
>> everything work to my satisfaction.  I have only about 20 bars of 
>> music done, but I'm absolutely delighted with it so far.  At this 
>> rate, maybe I'll have it done by the end of this decade.  ;-)
> 

> Are you writing it in Sagittal?  ;-)


No, because it's not actually "written", and if I were to print out 
anything from Cakewalk, it would only show the nearest 12-ET 
pitches.  I'll have to wait for Monz's software to come out before I 
can do any better than that.


>>> Can you write a melody in the 11-limit 'scale' that sounds wrong
>>> because of functional scale disorientation?
>> 

>> I actually tried to run a couple of ideas through my head last 
night 
>> about how to do it, and it's not as easy as I thought.  
>> Paradoxically, it takes a certain lack of talent or lack or 
>> familiarity with microtonal resources to write something 
>> intentionally bad enough that it sounds wrong. -- it's only when 
I'm 
>> not trying to do it that I seem to be able to succeed.  Hmmm, 
perhaps 
>> I've stumbled on a new method of composition -- guaranteed to give 
>> good results.  ;-)
> 

> Remember, we're trying to ignore composition as a factor.  We don't
> care if it's bad, just if it has functional scale disorientation.
> If we try and try but can't ever hear functional scale 
disorientation,
> it's important -- it means it probably doesn't exist.
> 
> -Carl


What I was concerned about was that disorientation might occur, but 
the result wouldn't be perceived as a problem, but rather as 
something that, though unexpected, ends up sounding new and exciting 
(rather like a deceptive cadence).  (Oh boy, I think I'm just digging 
myself into a hole.  Maybe I'll get a chance to try this out 
sometime, but don't hold your breath.)

--George

Message: 8348 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 13:28:01

Subject: Re: "does not work in the 11-limit"

From: Carl Lumma

>No, because it's not actually "written",


Are you entering notes from a keyboard?

-C.

Message: 8349 - Contents - Hide Contents

Date: Tue, 18 Nov 2003 21:46:55

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

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

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:

>>> Don't we always fix the prime limit anyway? Why might this be a 
>> problem?
>> 

>> sometimes you want to use a set of nonconsecutive primes, as 
you've 
>> mentioned yourself, dave.
> 
> Good point.
> 

> There must be a convenient way of dealing with these. Does it 
actually
> matter if you use non-consecutive primes, as long as you do it
> consistently throughout the calculations. Isn't it really just the
> _dimension_ of the multi-vectors that must be fixed for any given 
set
> of calculations?


of course. it's just that you might not be dealing with a "prime 
limit"!

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