Tuning-Math Digests messages 8200 - 8224

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 9

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8200 - 8225 -



top of page bottom of page down


Message: 8200

Date: Fri, 14 Nov 2003 16:20:11

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

From: Manuel Op de Coul

Gene wrote:
>You will also find stuff about "JI epimorphic", but I don't understand
>what Manuel is up to; it isn't what I expected.

Can you give an example of what you expected to be different?
I thought I implemented the epimorphism you discussed on this list.

Manuel


top of page bottom of page up down


Message: 8201

Date: Fri, 14 Nov 2003 22:17:59

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I am not picking nits. This is very important to the math. I don't 
> understand what your objection is anyway--we have a lot of defintions 
> of tuning and scale terms tossed around which are neither clear nor 
> elegant sounding. "Val" is short, and I gave it a precise definition, 
> thereby doing what I wish other people would more often do, judging 
> by the definitions in Joe's dictionary.

I thought we just agreed that you wouldn't worry too much if my
explanations didn't capture the precise pure-math meaning, and in
return I wouldn't worry that you give definitions that are
incomprehensible to most tuning-math readers

But if I've made a serious mistake I really need to know:
What do the integers [the val's coefficients] represent in tuning
terms? What are they counting?


top of page bottom of page up down


Message: 8202

Date: Fri, 14 Nov 2003 23:59:44

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > That's Gene's proposal (that we should write them that way, not 
> that
> > > we should call them bras and kets) and it seems like a reasonable 
> > one
> > > to me.
> > 
> > it'll help me, since i'm used to them.
> 
> Sounds like we are achieving consensus on something!
> 
> > > Except I wonder how we should write a complete mapping matrix for 
> a 
> > more-than-1D temperament.
> > 
> > a matrix is a matrix, not a bra or a ket. i never understood 
> > covariant vs. contravariant, though . . .
> 
> Let's leave matricies alone. 

But mappings for more-than-1D temperaments are naturally expressed as
matrices that map prime-exponent vectors to generator-count vectors.

In general a temperament has a prime-mapping matrix "M" that has a
column for each prime and a row for each generator. And a ratio has a
prime-exponent-vector (monzo) "a" of the same width. If we want to
know how many of each generator correspond to that ratio in that
temperament we simply calculate the matrix product
transpose(M*transpose(a)), or simply M*a if the monzo is already a
column vector and the result can be a column vector.

> As for covariant vs contravariant, if 
> you change the basis for monzos to something other than primes, you 
> have to make a complimentary change in basis for the val basis. The 
> standard basis is that monzos have a basis e_2, e_3 etc. 
> corresponding to primes, and vals v_2, v_3 corresponding to (whether 
> we want to call them that or not) padic valuations.

Sure. BWDIMAATT? :-) Never mind. I don't think we need to worry about
it. We just need to remember that mappings and monzos are different
kinds of things. They have different "units" as it were.


top of page bottom of page up down


Message: 8203

Date: Fri, 14 Nov 2003 02:17:37

Subject: Re: Vals?

From: Carl Lumma

>> Voicing shouldn't matter, since the voicing of the thing you're
>> mapping to (an ET) doesn't matter.  If I set...
>> 
>> 1= 9/8
>> 2= 5/4
>> 3= 11/8
>> 4= 3/2
>> 5= 7/4
>> 6= 2/1
>> 
>> ...can you show me the problem? 
>
>3/2 is 4 steps, so 9/4 is 8 steps, so 9/8 is 2 steps--except it is 
>also 1 step, contradiction.

You may be wondering why I ask a question and then answer it in
the same message.  If I'm wrong it saves a volley because the
reader knows what I was trying to do.

-Carl


top of page bottom of page up down


Message: 8204

Date: Fri, 14 Nov 2003 17:28:34

Subject: Re: Vals?

From: monz

hey paul,

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

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

> > p.s. would it be OK for me to attempt a modification of your page
> > 
> > Definitions of tuning terms: EDO prime error, (c) 2003 by Joe Monzo *
> > 
> > ? i realized that you *do* have the signed errors of the
> > primes in the text, despite your use of absolute values in
> > the graph. so if i just added the signed errors for the
> > odds that you omitted, i could then quickly locate any
> > inconsistency, since inconsistency occurs if and only if
> > the signed relative error of one odd differs by over 50% 
> > from the signed relative error of another odd. for example,
> > in 43-equal, the error on 7 is, as you show, +28%; the error
> > on 9 is double that on 3, so about -30%; the difference
> > between these two signed percentages (and thus the implied
> > error on 9:7) is 58%; so 43-equal is inconsistent in the
> > 9-limit. what do you think? i think the page would be
> > sorely misleading, and much less useful, without this 
> > information.
> 
> 
> 
> paul, you already know that i think that the information
> given on my "EDO prime error" page is useful as it is.
> 
> but if you envision a better version of the page, sure,
> send me the modification.  you know i trust your judgment
> on tuning matters!  :)



take a look at this:

Definitions of tuning terms: EDO 11-odd-limit error, (c) 2003 by Joe Monzo *




-monz


top of page bottom of page up down


Message: 8205

Date: Fri, 14 Nov 2003 22:20:09

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:
> 
> > > That's Gene's proposal (that we should write them that way, not 
> that
> > > we should call them bras and kets) and it seems like a 
reasonable 
> > one
> > > to me.
> > 
> > it'll help me, since i'm used to them.
> 
> Sounds like we are achieving consensus on something!
> 
> > > Except I wonder how we should write a complete mapping matrix 
for 
> a 
> > more-than-1D temperament.
> > 
> > a matrix is a matrix, not a bra or a ket. i never understood 
> > covariant vs. contravariant, though . . .
> 
> Let's leave matricies alone. As for covariant vs contravariant, if 
> you change the basis for monzos to something other than primes, you 
> have to make a complimentary change in basis for the val basis. The 
> standard basis is that monzos have a basis e_2, e_3 etc. 
> corresponding to primes, and vals v_2, v_3 corresponding to 
(whether 
> we want to call them that or not) padic valuations.

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


top of page bottom of page up down


Message: 8208

Date: Fri, 14 Nov 2003 22:39:33

Subject: Re: Definition of microtemperament

From: monz

hi paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > hi paul,
> > 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > wrote:
> > 
> > > You mean 11-odd limit? Well, meantone contained
> > > excellent approximations to ratios of 7, but practically
> > > no one considered them consonant historically. 
> > 
> > 
> > that's not true, and you know it: meantone gave good
> > approximations to a 4:5:7 triad in its "augmented-6th"
> > chord, which was used a *lot* in the "common-practice"
> > era.
> 
> but not as a consonance -- so what i was saying is true.


OK, you're right.

 
> > true, no-one at the time analyzed these chords as
> > consonant 4:5:7 chords,
> 
> but huygens *did* find these ratios in augmented sixth chord.



oops ... i *knew* you'd catch me on that one!

 
> > but in meantone, that's what
> > they were, and they were perfectly acceptable in
> > both theory and practice.
> 
> i didn't say they were unacceptable -- plenty of 
> not-so-easy-to-ratio-analyse sonorities were acceptable
> as dissonances as well -- just not considered consonant,
> that is, it was not used as a chord to resolve a dissonant
> chord to, but rather it was used as a chord that would 
> resolve *to* a consonant chord.


OK, now i understand perfectly what you were saying.

the augmented-6th chord was always used as a "dissonant"
chord which had to resolve, as you say.



-monz


top of page bottom of page up down


Message: 8211

Date: Fri, 14 Nov 2003 22:42:19

Subject: Re: Definition of microtemperament

From: Paul Erlich

but the appearance of this chord so easily in the meantone series 
certainly makes one wonder what direction western music might have 
progressed in had the movement for closure at 12 notes not won out.

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi paul,
> 
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > hi paul,
> > > 
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
> > > wrote:
> > > 
> > > > You mean 11-odd limit? Well, meantone contained
> > > > excellent approximations to ratios of 7, but practically
> > > > no one considered them consonant historically. 
> > > 
> > > 
> > > that's not true, and you know it: meantone gave good
> > > approximations to a 4:5:7 triad in its "augmented-6th"
> > > chord, which was used a *lot* in the "common-practice"
> > > era.
> > 
> > but not as a consonance -- so what i was saying is true.
> 
> 
> OK, you're right.
> 
>  
> > > true, no-one at the time analyzed these chords as
> > > consonant 4:5:7 chords,
> > 
> > but huygens *did* find these ratios in augmented sixth chord.
> 
> 
> 
> oops ... i *knew* you'd catch me on that one!
> 
>  
> > > but in meantone, that's what
> > > they were, and they were perfectly acceptable in
> > > both theory and practice.
> > 
> > i didn't say they were unacceptable -- plenty of 
> > not-so-easy-to-ratio-analyse sonorities were acceptable
> > as dissonances as well -- just not considered consonant,
> > that is, it was not used as a chord to resolve a dissonant
> > chord to, but rather it was used as a chord that would 
> > resolve *to* a consonant chord.
> 
> 
> OK, now i understand perfectly what you were saying.
> 
> the augmented-6th chord was always used as a "dissonant"
> chord which had to resolve, as you say.
> 
> 
> 
> -monz


top of page bottom of page up down


Message: 8213

Date: Fri, 14 Nov 2003 10:43:06

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:
> > 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.
> 
> This is absolutely not what I mean by a val, which maps to integers.

I think we're picking nits here. What do the integers represent in
tuning terms? What are they counting?


top of page bottom of page up down


Message: 8214

Date: Fri, 14 Nov 2003 14:40:16

Subject: Re: Integrating the Riemann-Siegel Zeta

From: Carl Lumma

Roger that!  I was going to post something to this effect, but I
still haven't studied the mathworld entry.

-Carl

>this is really hot, and i wish i understood it . . . maybe if manfred 
>schroeder wrote a book on it . . .
>
//
>> 
>> The point of this business is to give what you might call a generic
>> goodness measure for ets; meaning one not attached to any particular
>> prime limit. The result seems better than what we get for maximal
>> values of |Z(x)|, and much better than what we can glean from gaps
>> between the zeros.


top of page bottom of page up down


Message: 8217

Date: Fri, 14 Nov 2003 03:08:25

Subject: Re: Vals?

From: Carl Lumma

>I think we're picking nits here. What do the integers represent in
>tuning terms? What are they counting?

Not necessarily the most musically-obvious generator, if you're
coming at vals in the context of linear temperaments.  I don't
think integers is what confuses me.

I'm still wondering about 6.  In 22, the 11-prime-limit val
consistently maps the 9/8, and the resulting hexad taken as a
scale is a Constant Structure.  22 is even generally 11-limit
consistent.  Why not use 22?

-Carl


top of page bottom of page up down


Message: 8218

Date: Fri, 14 Nov 2003 12:11:13

Subject: Re: Vals?

From: Carl Lumma

>Speaking of which, I should probably have used < ... | for the val
>and | ... > for the monzo, as being the actual bra-ket notation.

The only reason I put it the other way was so that bra-ket would
tell you the position of the square brackets.  But you're using
a pipe instead of a square bracket, so this way the word still
tells you the location of the most bracket-like things, the angle
bracket.

-Carl


top of page bottom of page up down


Message: 8219

Date: Fri, 14 Nov 2003 14:43:21

Subject: Re: Vals?

From: Carl Lumma

>> Ok, now we're on the right track, but I'm still not grokking
>> you.  I started with six rationals and ended up with 6 integers.
>> What's the problem?
>
>Are your integers consecutive?

No, and that's part of the def. of standard val, but what
motivates it?

-Carl


top of page bottom of page up down


Message: 8220

Date: Fri, 14 Nov 2003 01:00:34

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> [Dave Keenan:]
> >So did it become a lot clearer what a val was, when you figured
> >out that it was a prime mapping?
> 
> Everything I've ever figured out about vals made them clearer,
> obviously.  The words "prime mapping" wouldn't have helped a bit.

This seems like a logical contradiction. You say that figuring out
that vals were prime mappings made them clearer. Wouldn't calling them
"prime mappings" have helped you figure out that they were prime
mappings? And wouldn't this have happened sooner if they had been
called prime mappings instead of vals?

I guess what you're saying is that this wasn't a very important factor
in your understanding of the concept.

> >Did you understand what they were mapping the primes _to_ at that
> >time?
> 
> I still don't think I do.

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.

If we were told it was for an ET3 then we could read off what ET3 it
was for as the second coefficient, and the generator is that fraction
of the tritave.

If however we are told that the mapping is for an arbitrary equal
temperament, a cET, then we would have to solve for the generator that
minimises some error function such as max-absolute (minimax) or rms
(sum of squares). It's fairly simple to do this numerically in Excel
using the Solver add-in. Let me know if you want more details on that.

In an ET, the generator is the step. But not necessarily so in higher
D temperaments

If we were told the mapping was one row (Gene says we can forget that
"column" stuff) of a linear temperament mapping, then to solve for the
generator that this row maps to, we would either need to know what the
other generator was, or what its mapping was, e.g. maybe the other
generator is the period and we are told it is an exact octave.
 
> 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. It's all about how evenly-spaced the chords
are. 

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.

Why such an apparently melodic property should be considered important
when applied to a vertical harmony, I don't know.

> 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.
 
> Then I don't know why the standard 11-limit mapping wouldn't be
> identical to the standard 11-prime-limit mapping.

It is.

> Anyway, saying mapping instead of val is already confusing me here.

Sorry. I've given both in several places above.

As far as tuning is concerned the only important difference I can see
is that in the case of temperaments with more than one generator, "the
 mapping" (unqualified) refers to the whole matrix (all the rows, one
per generator). There's no such thing as "the val" for such a
temperament. In this case a val is apparently only one row. But even
there, "the val for generator x" is the same as "the mapping for
generator x". So the term val isn't actually needed.


top of page bottom of page up down


Message: 8221

Date: Fri, 14 Nov 2003 14:42:38

Subject: Re: Vals?

From: Carl Lumma

>> Yeah well, the choice of 6 here still hasn't been accounted for.
>
>i explained why it was 3 in the case of 5-limit, and here it's the 
>same -- 6 is the number of notes in the 11-limit complete otonality! 
>this has nothing to do with the definition of vals, it's just one 
>particular problem that gene and george happen to be interested in.

Right, got that, just don't see why it's a "problem".

>> >So it's proper, but not a constant structure. I was under the
>> >misapprehension that proper always implied constant structure, i.e
>> >that propriety was a stronger condition. Hmm.
//
>> >However, the Enharmonic of Archytas is. Translate the scale 28/27 x
>> >36/35 x 5/4 x 9/8 x 28/27 x 36/35 x 5/4 into cents and generate the
>> >D-matrix.
>> >
>> >63  49   386  204 63  49   386
>> >112 435  590  267 112 435  449
>> >498 639  653  316 498 498  498
>> >702 702  702  702 561 547  884
>> >765 751  1088 765 610 933  1088
>> >814 1137 1151 814 996 1137 1151
>> >1200 .......
>> 
>> -Carl
>
>this is an example of . . . ?

A (wildly) improper constant structure.

-C.


top of page bottom of page up down


Message: 8223

Date: Fri, 14 Nov 2003 12:25:40

Subject: Re: Vals?

From: Carl Lumma

I wrote...
>>So it's proper, but not a constant structure. I was under the
>>misapprehension that proper always implied constant structure, i.e
>>that propriety was a stronger condition. Hmm.
>
>Nope.  The set of all non-CS scales is equivalennt to the set of
>all non-strictly-proper scales.

Sorry, it was too late.  There improper non-CS scales.  There's
no convenient way to express CS with Rothenberg's language that
I know of.

-Carl


top of page bottom of page up down


Message: 8224

Date: Fri, 14 Nov 2003 22:47:39

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Ok, now we're on the right track, but I'm still not grokking
> >> you.  I started with six rationals and ended up with 6 integers.
> >> What's the problem?
> >
> >Are your integers consecutive?
> 
> No, and that's part of the def. of standard val, but what
> motivates it?
> 
> -Carl

i can't make heads or tails of this question. the standard val puts 
the primes in order because it's easy to remember them that way. you 
could put them in a different order but you would have to remember 
which entry refers to which prime. so i don't see what there is to 
motivate.


top of page bottom of page up

Previous Next

8000 8050 8100 8150 8200 8250 8300 8350 8400 8450 8500 8550 8600 8650 8700 8750 8800 8850 8900 8950

8200 - 8225 -

top of page