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Message: 8150 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 05:53:12

Subject: Re: Vals?

From: Gene Ward Smith

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


> But the fact is, that was not a pure-math question. 


It sounded to me like you wanted to tell me I should use matricies,
and forget about all the other issues I have in mind, such as for
instance wedge products. In other words, very much telling me not only
how to name things, but how to do my math.


> I don't don't see why it is like asking that. You _could_ just try
> answering the original question.


I want to talk about homomorphims to Z, because that is dual to
intevals, which has various implications. Concretely we see the
usefulness of that, to give one example, in the whole multilinear
algebra approach.


> I understand that. But isn't it the _only_ basis that we are using
> them with for _tuning_ purposes?


It's obviously the gold standard.
 

> I think we _can_ hold it
> against you if you insist on continuing to use an obscure term when
> you've been presented with perfectly transparent alternatives 
> _for_the_application_to_tuning_, 
> which is, after all, what this list is about.


Well, I did for my own mathematical reasons define vals as a finite
Z-linear combination of padic valuations, and that isn't *precisely*
like any of your proposed alternatives. More or less, but not 100%, as
I said.

Vals are an important concept and deserve a name. Why is this so
painful? I admit my names are not always terrific (eg "standard val")
and some of them (eg "icon") I haven't even attempted to inflict on
people here, while others (eg "notation") have generated no support,
but I really am not interested in using an inferior name for an
inferior definition. Why insist that everything must be done your way?

Message: 8151 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 18:55:40

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

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


> there's no way you could *hear* the point at which the line is 
drawn 
> (nor should it necessarily be drawn according to minimax), so i'd 
> prefer to use 'microtemperament' in a looser way -- if anyone cares 
> to check on the 'microtemperedness' of a particular temperament, 
the 
> exact numbers should be readily available. maybe 2.8 to 3.1 can be 
> considered a 'gray zone', where *context* will determine whether 
the 
> effect is one of microtemperament or not.


I don't see how you can draw the line at 3 cents, though. You *can* 
hear the difference between that and JI pretty clearly.

Message: 8152 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 23:40:26

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

From: Dave Keenan

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

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

>> can you *please* give a very detailed explanation of what
>> you're saying?  ... with lots and lots of 11-limit examples
>> that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?
>> 
>> thanks.
> 

> Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala 
> scale files. If you run "data" on them, you will find that 5, 7, 9 
> and 13 give Constant Structure scales, and 11 does not.
...


Bravo! Gene, this is an excellent minimal-math explanation!

It certainly is a curious fact. And I'm looking forward to hearing
from George, why he thinks it matters so much musically that he
wouldn't consider using an 11-limit tuning.

Message: 8153 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 06:23:45

Subject: Re: Vals?

From: Dave Keenan

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

>> Actually, if you need a shorter term than "prime-mapping", it seems
>> like "mapping" would do. What other kinds of mappings do we use in
>> tuning-math?
> 

> A "mapping", as it has been used, is sufficient to define a
> linear temperament.  A val is not.  

Agreed.

> But choo got me as to the
> exact relationship/difference between the two.


Then a val is just a mapping-row, which is itself still a mapping. A
vector can also be considered as an nx1 matrix (as it apparently has
been in the ET case).

In the case of an ET the complete mapping has 1 row, for an LT it has
2 rows, for a planar temperament (PT) it has 3 rows, etc.

But we can still refer to a single row of an LT or PT mapping as "the
mapping for the <something> generator". Or in the LT case, the
generator-mapping and the period-mapping. If you call them vals,
you're still going to have to say "the val for the <something>
generator", or the period val and the generator val. I don't see how
calling them vals adds anything to this. In fact I think it just
obscures things.

Message: 8154 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:00:58

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

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

> Microtemperament
> 
> A microtemperament is a temperament where the consonances sound 
justly
> intoned to most listeners in ordinary musical use. The allowed 
errors
> in the approximated ratios are therefore somewhat context-dependent
> but would typically be less than 2.8 cents.


I would change "typically be less than 2.8 cents" to "at minimum be 
less than three cents". I also wonder, if we adopt this definition, 
what we would call something like ennealimmal or octoid.

Message: 8155 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 06:39:14

Subject: Re: 7-limit optimal et vals

From: Dave Keenan

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

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

>>>>> what is the optimality criterion?
>>>> 

>>>> Minimax error in the 7-limit.
>>> 

>>> any differences if you use rms?
>> 

>> and are you allowing the octaves to be tempered? i.e. Do they apply
>> strictly to EDOs or to ET's generally?
> 

> EDOs only, but I didn't know you were calling non=integer mappings ets.


Aha. It seems Graham might be missing that too.

We've always had tET's and cET's. The step of n-tET is 1/n of an
octave, while the step of n-cET is n cents. Nowadays we seem to be
using EDO more for what used to be tET. And we have ED3 for the BP
tunings.

But I've certainly been guilty in the past, of being sloppy about this
and calling things ET's when I should have been more specific and said
tET's or EDO's.

Monz's definition agrees.
Definitions of tuning terms: equal temperament... * [with cont.]  (Wayb.)

But with non-EDO ETs the mapping still contains only integers, it's
just that the optimum step size (generator) is allowed to be something
other than an integer fraction of an octave. I assume that's what you
meant too.

Message: 8156 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 11:15:15

Subject: Re: Definition of microtemperament

From: Carl Lumma

>I don't see how you can draw the line at 3 cents, though. You *can* 
>hear the difference between that and JI pretty clearly.


It depends on the instrument.

-Carl

Message: 8157 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:15:58

Subject: Re: Vals?

From: Gene Ward Smith

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


> What confuses the hell out of me is that Gene keeps using
> the word "column" re. vals, but they don't give successive
> approximations to the same prime, they give a single
> approx. to various primes.


I don't understand this. The column business is to tell you they are 
dual to intervals. Calling vals "bra vectors" and monzos "ket 
vectors" does this also, and comes complete with a notational 
convention easier to deal with that row vs column vectors, so I 
suggest shifting to that.


> At the very least, I'd hope understand what vals are good
> for before trying to rename them. 


To start with, I wanted intervals and vals to work in precisely the 
same way in terms of the wedge product, and hence my particular 
definition. But, obviously, we are always talking about these things 
implicitly, and so why not name them? And if we name them, why not 
give a precise, mathematically correct defintion at some point, even 
if we don't normally need to sweat the details?

 Or maybe you understand

> why the 11-limit has no standard val, and can explain it
> to the rest of us.


Paul seems to have figured that out. 11-limit complete otonal and 
utonal chords just don't work in a consistent way.

Message: 8158 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 11:17:45

Subject: Re: Definition of microtemperament

From: Carl Lumma

>I would change "typically be less than 2.8 cents" to "at minimum be 
>less than three cents". I also wonder, if we adopt this definition, 
>what we would call something like ennealimmal or octoid.


Howabout "typically less than 2 cents" (the error of a 12-equal
fifth)?  Since this is meant to be applied by musicians, .1 cent
resolution should not be offerred.

-Carl

Message: 8159 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 17:04:10

Subject: Re: Vals?

From: Carl Lumma

>> >ene, since you won't say what's desirable about being a
>> standard val...
>

>Purely a matter of being easy to calculate.


Adding our birthdays together is easy to calculate.  There
must be some other reason.  Dave's 'the best approx. to each
element of a chord in n-tET' is better, but why n should equal
the number of notes in a chord is still a mystery.

-Carl

Message: 8160 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 07:09:36

Subject: Re: Vals?

From: Dave Keenan

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

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

>> But the fact is, that was not a pure-math question. 
> 

> It sounded to me like you wanted to tell me I should use matricies,
> and forget about all the other issues I have in mind, such as for
> instance wedge products. In other words, very much telling me not only
> how to name things, but how to do my math.


I assure you this is not the case.

I think we have quite complementary skills. You come up with the the
math tools and methods and I may _eventually_ be able to understand
them enough to put them into terms that others on this list can more
easily understand. But I don't think you should worry too much if my
explanations or recasting of terminology misses some of the more
subtle points as far as the pure mathematician is concerned, at least
on a first pass.


>> I don't don't see why it is like asking that. You _could_ just try
>> answering the original question.
> 

> I want to talk about homomorphims to Z, because that is dual to
> intevals, which has various implications. Concretely we see the
> usefulness of that, to give one example, in the whole multilinear
> algebra approach.


OK. Well I hope we can put those implications in tuning terms
eventually, but I'd prefer to get the basics translated first. The
sort of stuff people can do themselves using nothing more
sophisticated than Excel, and without needing to know the meaning of
the terms homomorphism, Z, dual or multilinear algebra.


> Vals are an important concept and deserve a name.

I agree.

> Why is this so painful? 


That isn't painful per se. But it's a term that belongs to the
pure-math side of things and isn't specific enough to our applications
of it.


> I admit my names are not always terrific (eg "standard val")
> and some of them (eg "icon") I haven't even attempted to inflict on
> people here, while others (eg "notation") have generated no support,
> but I really am not interested in using an inferior name for an
> inferior definition. Why insist that everything must be done your way?


Now you're exaggerating. I don't insist on that. How could I anyway?
I'm just expressing my opinion like anyone else.

All I'm saying is, if you want people on this list to understand what
you are on about, it's a good idea to invest in names that are
descriptive of their specific application to tuning, or even ones that
are more like terms i everyday use. By all means tell us "in
mathematics we call this a <whatever>", but when someone tells you,
"Oh I think I understand what you're talking about. That's a <whatever
tuning thing>.", and no-one seriously disagrees, then I think it would
be a good idea to try to construct a descriptive name from that for
use in future discourse on this list, even it's ony 99% the same
concept to start with.

I'm just disappointed we got this far with "val" without someone
figuring out a more tuning-specific or everyday cognate (or very near
cognate). But I don't blame you for that.

Message: 8161 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:19:15

Subject: Re: Vals?

From: Gene Ward Smith

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


> I think we have quite complementary skills. You come up with the the
> math tools and methods and I may _eventually_ be able to understand
> them enough to put them into terms that others on this list can more
> easily understand. But I don't think you should worry too much if my
> explanations or recasting of terminology misses some of the more
> subtle points as far as the pure mathematician is concerned, at 
least
> on a first pass.


Sounds reasonable, but I don't think you should worry to much if I 
want to make precise mathematical definitions for things, or make the 
definitions the way they are for reasons not immediately apparent to 
you.

Message: 8162 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 07:15:20

Subject: Re: 7-limit optimal et vals

From: Dave Keenan

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

>> And we have ED3 for the BP tunings.
> 

> Who's we?  I, for one, reject any and all EDx terminology
> with the Iron Fist of Discountenance...


That's fine, but we still _have_ EDO and ED3 whether we want to use
them or not. Or are you able to erase them from your memory? :-)

If so, sorry to remind you of them again, and don't ever look at the
index to Monz's dictionary. At least keep away from the E's, OK. ;-)

Message: 8163 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 17:15:21

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

From: Carl Lumma

>Here are the 5, 7, 9, 11 and 13 limit complete otonal chords as Scala 
>scale files. If you run "data" on them, you will find that 5, 7, 9 
>and 13 give Constant Structure scales, and 11 does not. You will also 
>find stuff about "JI epimorphic", but I don't understand what Manuel 
>is up to; it isn't what I expected.


So there's no val that sends all 11-limit intervals to integers
without collisions?

-Carl

Message: 8164 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:25:33

Subject: Re: Vals?

From: Gene Ward Smith

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

>> I'm just disappointed we got this far with "val"
> 

> ...with only 1 -- two if we're lucky -- persons who know
> how to use them for what they're capable of.


This hardly is the case.


> Gene, since you won't say what's desirable about being a
> standard val...


Purely a matter of being easy to calculate. 

and you haven't said what the lack of a

> standard 11-limit val means about the 11-limit...


It's not a *standard* 11-limit val, but one associated to 11-limit 
complete harmony. I haven't given it a name. At this point, I don't 
know if I should even consider doing such a thing.

What it means for the 11-limit is that a systematic way of looking at 
harmony by reducing it to an approximation plus a chord description 
does not work in the 11-limit, but does work in the 3, 5, 7, 9 and 13 
limits.

Message: 8165 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 17:31:37

Subject: Re: Vals?

From: Carl Lumma

>A prime-mapping (or val with log-prime basis) simply maps each prime
>number (or strictly-speaking the logarithm of each prime number) to an
>integer multiple of some interval (log of frequency ratio) that we
>call a generator.
>
>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.


Yes, I know this.  But why integers?  And why can't there be collisions?
And in what sense could the order in which the identities of a chord
are considered have any bearing on things?


>> But Gene's talking about finding vals for limits!!!
>

>He's just abbreviating excessively and assuming the meaning will be
>clear from your readings of his previous postings in the same thread.
>He's really talking about finding vals-with-log-prime-basis
>(prime-mappings) that map the complete chord of each limit to a tET
>with the same cardinality.


I finally got that.  Why the same card.?  


>Try the 6 possible possible voicings of the 11-limit otonality, that
>fit within an octave, and you'll see that none of them are very even.


It's proper and for that matter seems to fit to 6-tET reasonably
well.


>> Note that I have no idea what the bra ket notation stuff is about.
>

>It's just a way of distinguishing prime-mappings (vals) from
>prime-exponent-vectors (monzos) without having to say it in words
>every time. It only makes sense to multiply mappings by
>exponent-vectors, not any other combination and these brackets try to
>make that clear because ] and [ fit together, but > and <, > and [, ]
>and < do not.


So are monzos are now kets written [ ... > ?
and vals are bras written < ... ] ?

-Carl

Message: 8166 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:35:02

Subject: Re: Vals?

From: Gene Ward Smith

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

>> And the claim is not limited to standard mappings,
>> but any mappings at all.
> 

> According to Gene, Gram and other vals may get around 
this 'problem'.


No, you get inconsistency no matter what you try. The way around it 
would be to not use complete 11-limit chords, or to follow up on your 
suggestion and scramble the ordering, but that means abandoning the 
problem as stated as impossible.


>> I believe the claim is that there is no prime mapping that will map
>> the pitches of the 11-limit complete otonality, in any voicing, to
>> consecutive degrees of 6-tET.
>> 
>> Why 6-ET? Because that's how many pitches are in the chord.
> 
> Why consecutive?


Your going to get a dog's breakfast otherwise, but we could try the 
idea, I suppose.


>> and there's one that maps the 5-limit otonality to 3-ET,
> 
> Consecutive?

5/4-3/2-2


>> You can calculate the coefficient for prime p in the "standard"
>> mapping for n-tET as round(n*ln(p)/ln(2)).
> 

> Gene's already given that.


It's always helpful to repeat things anyway, considering the 
communication problems I seem to engender. I thought Dave's article 
was nicely clear.

Message: 8167 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:37:48

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

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

> i changed it a couple of days ago when you proposed the
> earlier version of the part i snipped here.  now it's as
> per your latest definition:
> 
> Definitions of tuning terms: microtemperament,... * [with cont.]  (Wayb.)


Could you change this back to "always less than three cents"? 2.8 
cents seems an absurd line to draw, and "usually" means it isn't even 
a line.

Message: 8168 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 08:06:25

Subject: Re: Vals?

From: Dave Keenan

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

>> Then a val is just a mapping-row,
> 

> What confuses the hell out of me is that Gene keeps using
> the word "column" re. vals, but they don't give successive
> approximations to the same prime, they give a single
> approx. to various primes.
> 

>> I don't see how
>> calling them vals adds anything to this. In fact I think it
>> just obscures things.
> 

> By now it should be no surprise that I'm utterly confused
> by your obsession over this word.  Considered a career in
> postmodern critical theory?


So are you telling me that things didn't become a lot clearer for you
when you figured out that a val was in fact a prime mapping (for
purposes of tuning theory)? When, incidentally, did you figure that out?


> At the very least, I'd hope understand what vals are good
> for before trying to rename them.


I believe I do understand what they are good for
_in_relation_to_tuning_, which is surely what matters for this list?


>  Or maybe you understand
> why the 11-limit has no standard val, and can explain it
> to the rest of us.


I don't thing anyone is saying the 11-limit has no standard prime
mapping. That doesn't make sense.

I believe the discussion you're referring to is about the 11-limit
complete otonality. And the claim is not limited to standard mappings,
but any mappings at all.

I believe the claim is that there is no prime mapping that will map
the pitches of the 11-limit complete otonality, in any voicing, to
consecutive degrees of 6-tET. 

Why 6-ET? Because that's how many pitches are in the chord.

Why is this interesting? Because there _is_ a mapping that maps some
voicing of the 3-limit complete otonality to consecutive degrees of
2-ET, and there's one that maps the 5-limit otonality to 3-ET, 7-limit
otonality to 4-ET, 9-limit to 5-tET and 13-limit to 7-tET. And in each
case it happens to be the "standard" mapping that does it.

The "standard" mapping for a tET is the one that gives the best
approximation to each prime number (and its octave equivalents). It
doesn't guarantee the best approximations to other ratios with
_combinations_ of primes. e.g. At the 5-limit, if some tET is
inconsistent, the "standard" mapping will give the best approximation
to 5/4 and 3/2 but not 5/3.

You can calculate the coefficient for prime p in the "standard"
mapping for n-tET as round(n*ln(p)/ln(2)).

Message: 8169 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 17:37:05

Subject: Re: Vals?

From: Carl Lumma

>> >ut Gene's talking about finding vals for limits!!!
>

>Come on Carl, this is no more true than that my Hypothesis concerned 
>temperaments.


Gene did use those words, apparently abbreviating excessively.  I'm
closer to what he meant now, but I have no idea what you're referring
to re. the Hypothesis.  It clearly concerns temperaments, since it
states things about what happens when you temper uvs out of a PB.


>>> I didn't say anything about restricting ourselves to one octave.
>> 

>> Then the standard 5-limit 3-val that Gene gave isn't consecutive.
>
>whaaaa???


Gene's going from smallest to largest interval, though as I just
Confessed, I have no idea what order has to do with anything.

-Carl

Message: 8170 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:50:27

Subject: Re: Vals?

From: Gene Ward Smith

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


> I think it may relate to the complete otonality being a constant
> structure. But we'll have to wait until Gene tells me if I got it
> right. Then he can tell you why consecutive.


Right; or that it is epimorphic. Put it into Scala and it will tell 
you both.


> Alternatively we could decide to say that there is no such thing as 
an
> odd-limit mapping - that the limit of a mapping is always considered
> to be its largest prime.


Which can map to 0.

Message: 8171 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 08:08:56

Subject: Re: 7-limit optimal et vals

From: Dave Keenan

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

> There are ways of attacking a terminology.  Publishing papers with
> similar but subtly different terminology, for example.


How about rational argument?


>  I don't think
> this will be necessary, though, as the worthlessness of "EDO" should
> be readily apparent to most onlookers.


I'm afraid it's worthlessness isn't apparent to me. I'd appreciate it
if you could take the time to explain.

Message: 8172 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:55:32

Subject: Re: Vals?

From: monz

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


>>> [Dave Keenan:]
>>> You can calculate the coefficient for prime p in
>>> the "standard" mapping for n-tET as round(n*ln(p)/ln(2)).
>> 
>> [Carl Lumma:]

>> Gene's already given that.
> 

> It's always helpful to repeat things anyway, considering
> the communication problems I seem to engender. I thought
> Dave's article was nicely clear.




amen!  by all means, *please* repeat stuff in as many
different ways as possible!  it sure helps me to understand.



-monz

Message: 8173 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 00:22:06

Subject: Re: Vals?

From: Carl Lumma

>So are you telling me that things didn't become a lot clearer
>for you when you figured out that a val was in fact a prime
>mapping (for purposes of tuning theory)? When, incidentally,
>did you figure that out?


Months ago, when Gene showed me how to use his Maple routines
to find linear temperaments from a pair of vals.


>I don't thing anyone is saying the 11-limit has no standard prime
>mapping. That doesn't make sense.
//
>I believe the discussion you're referring to is about the 11-limit
>complete otonality.


Right, "limit" means odd-limit unless it's "prime-limit", as
established by Partch and Erlich.


>And the claim is not limited to standard mappings,
>but any mappings at all.


According to Gene, Gram and other vals may get around this 'problem'.


>I believe the claim is that there is no prime mapping that will map
>the pitches of the 11-limit complete otonality, in any voicing, to
>consecutive degrees of 6-tET.
>
>Why 6-ET? Because that's how many pitches are in the chord.

Why consecutive?

>Why is this interesting? Because there _is_ a mapping that maps some
>voicing of the 3-limit complete otonality to consecutive degrees of
>2-ET,


It would have to be consecutive.


>and there's one that maps the 5-limit otonality to 3-ET,

Consecutive?

>The "standard" mapping for a tET is the one that gives the best
>approximation to each prime number (and its octave equivalents).


2 is a prime, so octaves are included.  But this doesn't mention
anything about consecuity (or ordering of any kind).  And it
doesn't include why we care that the number of notes in an octave
equals the number of notes in the chord.  And it only defines vals
for prime limits, not for odd limits.


>It
>doesn't guarantee the best approximations to other ratios with
>_combinations_ of primes. e.g. At the 5-limit, if some tET is
>inconsistent, the "standard" mapping will give the best approximation
>to 5/4 and 3/2 but not 5/3.
>
>You can calculate the coefficient for prime p in the "standard"
>mapping for n-tET as round(n*ln(p)/ln(2)).


Gene's already given that.

-Carl

Message: 8174 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 19:59:56

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

From: monz

hi Gene,

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


>> and you haven't said what the lack of a
>> standard 11-limit val means about the 11-limit...
> 

> It's not a *standard* 11-limit val, but one associated
> to 11-limit complete harmony. I haven't given it a name.
> At this point, I don't know if I should even consider
> doing such a thing.
> 
> What it means for the 11-limit is that a systematic way
> of looking at harmony by reducing it to an approximation
> plus a chord description does not work in the 11-limit,
> but does work in the 3, 5, 7, 9 and 13 limits.




i'm getting hopelessly confused about this, but it seems
like something i'd really like to understand.  

can you *please* give a very detailed explanation of what
you're saying?  ... with lots and lots of 11-limit examples
that don't work and 3-, 5-, 7-, 9-, 13-limit examples that do?

thanks.


i've renamed the subject header in anticipation that this
will become a big thread in its own right, but unfortunately
since you haven't named this phenomenon it's an ugly header.
feel free to rename the subject line if you define a good name.



-monz

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