Tuning-Math Digests messages 8125 - 8149

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

8100 - 8125 -



top of page bottom of page down


Message: 8125

Date: Wed, 12 Nov 2003 21:58:19

Subject: Re: Definition of microtemperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...>
> > wrote:
> > > I would have said "would always be less than about 3 cents" 
> or "... 
> > > less than 3.5 cents" in order to include Miracle.  Or don't you 
> > > consider that a microtemperament, and if not, then what should 
we 
> > > call it?
> > 
> > I've always considered miracle to be a microtemperament at the 7-
> limit
> > (2.4 c) but not at the 9 or 11 limits (3.3 c).
> 
> I don't follow this.  The error of 4:5 in Miracle (with minimax 
> generator) is ~3.323c.

we were focusing on the 72-equal incarnation of miracle.

> If you're going to use 
> anything on the order of half the error of meantone as your cutoff, 
> then you should also extend this to half the error of 8:9 in 
meantone 
> for a 9 limit.

why? there's no analogy there. 1/4-comma meantone was not used for 
music where 8:9 is used as a consonance.

> The beating harmonics in a tempered 8:9 are much more difficult to 
> hear than for 2:3,

shouldn't that consideration lower the weight of 8:9 in the 
calculation, compensating this next point?

> hence that interval is more difficult to play in 
> tune with flexible-pitch instruments, hence the actual error for 
that 
> interval in a live performance is likely to be greater.

> > It's all pretty arbitrary, but I think we need to draw such a line
> > somewhere.
> 
> Yes.

noooooooooooooooooo! :)


top of page bottom of page up down


Message: 8126

Date: Wed, 12 Nov 2003 00:44:40

Subject: Re: Eponyms

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 about "mapping" instead of "val" with the implication 
> > (preferably stated along with n) that we are talking about ET.
> 
> I don't see the point. What about optimal et?

we're talking about how to optimally map primes to a given et, right?


top of page bottom of page up down


Message: 8127

Date: Wed, 12 Nov 2003 22:03:08

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > Here's what you've given us so far...
> 
> I've given way, way way more than that. I can't force anyone to 
read 
> it.
> 
> > ...It appears that in the case of the "standard 3-val for the 5-
> limit",
> > n=3.  Is that why you called it a 3-val? 
> 
> > Where did 3 come from?
> 
> A division of the octave into three parts, or in other words, a 
> mapping of 2 to 3.

excuse me, but i think the answer to carl's question is "the complete 
5-limit otonal chord has *3* notes". right?


top of page bottom of page up down


Message: 8129

Date: Wed, 12 Nov 2003 22:03:26

Subject: Re: 7-limit optimal et 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:
> > what is the optimality criterion?
> 
> Minimax error in the 7-limit.

any differences if you use rms?


top of page bottom of page up down


Message: 8130

Date: Wed, 12 Nov 2003 01:23:42

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > I don't see the point. What about optimal et?
> > 
> > we're talking about how to optimally map primes to a given et, 
> right?
> 
> I don't count it as an et unless it has a mapping; anyway "optimal 
> val" is shorter and sweeter.

ok.


top of page bottom of page up down


Message: 8133

Date: Wed, 12 Nov 2003 14:16:14

Subject: Re: Vals?

From: Carl Lumma

>> Here's what you've given us so far...
>
>I've given way, way way more than that. I can't force anyone to
>read it.

I've read everything you've ever posted to this list, much of it
more than once, and much of it I've saved locally.

-Carl


top of page bottom of page up down


Message: 8134

Date: Wed, 12 Nov 2003 03:18:51

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:
> 
> > Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping
> > in exactly the same way? 
> 
> Go ahead and do so, however a val is not necessarily an et-mapping of
> any kind.

In this message
Yahoo groups: /tuning-math/message/7528 *
you said the two could be identified (except for a question about
finiteness)?

Here is a possible Monz dictionary definition of an ET-prime-mapping
(improved from the one I already gave in the above message, that you
seem to have missed):
----------------------------------------------------------------------
ET-prime-mapping

A list of the (whole) numbers of steps of some equal temperament (ET)
(not necessarily octave based) used to approximate each prime number
(considered as a frequency ratio). An "n-limit" ET-prime-mapping
(where n is a whole number) only lists numbers of steps for primes no
greater than n.

The "standard" prime mapping for an ET is the one that gives the best
approximation for each prime, but note that this is not guaranteed to
give the best approximation for all ratios, and other mappings may be
more useful in some cases.

To find the number of steps approximating some ratio in some ET,
express the ratio as a prime-exponent-vector and multiply its elements
by the corresponding elements in the chosen ET-prime-mapping and sum
the products. This is called the dot product or inner product or
scalar product of the two vectors.

For example the standard 7-limit ET-prime-mapping for 12-EDO is [12 19
28 34]. These numbers can be calculated for any EDO as
Round(N*ln(p)/ln(2)) where N is the number of divisons per octave and
p is the prime number.

To find how many steps of 12-EDO approximate a 7/5 frequency ratio,
first express the ratio as a prime-exponent-vector.
7/5 = 2^0 * 3^0 * 5^-1 * 7^1 = [0 0 -1 1]
now find its dot product with the prime-mapping
[0 0 -1 1].[12 19 28 34] = 0*12 + 0*19 + -1*28 + 1*34 = 6
So 7/5 is approximated by 6 steps, a tritone.
----------------------------------------------------------------------
 
> And then a "p-limit ET-mapping" would be a
> > restricted one.
> > 
> > So they're exactly the same!!!!
> 
> I havn't seen your definition.

I suspect you just didn't recognise it as such, because it was in
plain English.

> > So why have we been calling them "vals" all this time? 
> 
> Because there wasn't a good word for "finitely generated homomorphic
> mapping from Q+ to Z" or "Z-linear combination of padic valuations"
> already in existence.
> 
> A mathworld
> > search on the term finds nothing. Did you invent the term? 
> 
> You bet. We needed a term for it, and there wasn't one.

_You_ might have needed a term for "finitely generated homomorphic
mapping from Q+ to Z" or "Z-linear combination of padic valuations",
but I don't think anyone else on this list did.

> Is it
> > merely an obscure synonym for "homomorphism", or "group homomorphism"?
> > The fact that it _is_ a group homomorphism is far from being its most
> > important characteristic as far as microtonality is concerned. The
> > fact that it maps ratios to steps of ETs is of far more interest.
> 
> A val *does not necessarily* map ratios to steps of an ET, but it *is*
> always a homomorphism. 

So what would you estimate is the percentage of the vals posted to
tuning-math that could not be read as mapping ratios to steps of an
ET. Please give some examples of these and explain what they _do_ mean
in tuning terms.

> If you insist, you could replace the term with
> "finitely generated homomorphic mapping from Q+ to Z", I suppose, but
> I imagine "a p-limit homomorphic mapping to the integers" or something
> like that would suit you better.

Those certainly don't suit me. By all means use the word "val" to
stand for this abstract mathematical category. But this is the
_tuning_ math list. We want names that indicate their meaning as
applied to _tuning_.

For example, in the application area of electrical theory we use
vectors to represent the magnitude and phase of sinusoidal voltages
and currents. But we don't just call them all vectors. We want names
that tell us what they _mean_. We call them "voltage phasors" and
"current phasors".

I assume that the val, strictly speaking, is the operation of taking
the dot product with a vector of step numbers, not the actual vector
of step numbers itself. That's ok. The term "mapping" is used
similarly ambiguously. It doesn't usually cause any misunderstanding.

> It most certainly would not have served me better. You do what you
> like, but please don't expect me to follow your lead.

So are you saying that you don't really care if only two or three
people on this list understand how the things you write about apply to
tuning?


top of page bottom of page up down


Message: 8136

Date: Wed, 12 Nov 2003 03:51:39

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...>
wrote:
> I would have said "would always be less than about 3 cents" or "... 
> less than 3.5 cents" in order to include Miracle.  Or don't you 
> consider that a microtemperament, and if not, then what should we 
> call it?

I've always considered miracle to be a microtemperament at the 7-limit
(2.4 c) but not at the 9 or 11 limits (3.3 c).

I originally said "less than half the 5-limit error of 1/4-comma
meantone", i.e. less than 2.7 c.

I let it creep up already so a couple of temperaments with 2.8 c
errors could scrape in, and I went up to 3 for this definition just
because it seemed silly to be as precise as 2.8 c, so I definitely
wouldn't want it to creep _past_ 3 cents.

Gene would like the limit set at 1 c, although I haven't read why.
However I believe this definition caters for that, by allowing the ear
to arbitrate, and mentionaing the context dependence. In some contexts
a temperament with an error between 1 and 3 cents may not be a
microtemperament.

All I'm saying with the 3 cent thing is that there is no context in
which an error _greater_ than 3 cents would be considered a
microtemperament, ear or no ear.

It's all pretty arbitrary, but I think we need to draw such a line
somewhere.


top of page bottom of page up down


Message: 8137

Date: Wed, 12 Nov 2003 22:43:16

Subject: Re: Definition of microtemperament

From: Paul Erlich

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

> > why? there's no analogy there. 1/4-comma meantone was not used 
for 
> > music where 8:9 is used as a consonance.
> 
> No, it wasn't historically, but that doesn't mean that someone 
> *couldn't* use an extended meantone temperament for 9-limit 
harmony.  

right, but then they'd be more likely to use something like 1/5-comma 
or 1/6-comma meantone.

> > > > It's all pretty arbitrary, but I think we need to draw such a 
> line
> > > > somewhere.
> > > 
> > > Yes.
> > 
> > noooooooooooooooooo! :)
> 
> Do you mean noooooooooooooooooo categories or noooooooooooooooooot 
> arbitrary, or booooooooooooooooooth?  :-)
> 
> --George

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.


top of page bottom of page up down


Message: 8141

Date: Wed, 12 Nov 2003 07:40:16

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:
> 
> > In this message
> > Yahoo groups: /tuning-math/message/7528 *
> > you said the two could be identified (except for a question about
> > finiteness)?
> 
> I understood you to mean any kind of prime mapping, whether it could
> be called an et or not.

I wrote:
> > How is a val different from an ET-mapping? i.e. a list of the
> > numbers of steps approximating each prime in some ET.

So I'm rather surprised you didn't know I was talking about ETs?

But that's good, because now it looks like "val", as applied to
tuning, can be replaced by "prime-mapping", which is even simpler than
"ET-prime-mapping".

> <0 1 4 10| should be familiar from the meantone temperament. 

This looks like one row of the 7-limit prime-mapping for the meantone
linear temperament using a fifth as the generator, in particular the
row giving the mapping to fifth generators. Isn't it somewhat
incomplete without the other row that gives the mapping to octave
generators (periods)?

Why do we want to give the same name to something which in one case is
the complete mapping for an ET (a 1D temperament), and in the other
case only a part of the mapping for an LT (a 2D temperament)?

But assuming that there's a good reason, I'd simply call them
"prime-mappings" or "1D-prime mappings".

But I'd prefer to be more specific and call one an ET-mapping and the
other an LT-generator-mapping. I'd call the missing row the
LT-period-mapping. Together the LT-generator-mapping and the
LT-period-mapping make up the LT-mapping. The word "prime" can be
inserted before the word "mapping" whenever this is not clear from the
context.

> > So are you saying that you don't really care if only two or three
> > people on this list understand how the things you write about
apply > > to tuning?
> 
> I've explained what a val is numerous times. I can't insist you pay
> attention to everything I say; these days you and George tend to lose
> me, after all, which is fair enough.

If all your explanations were similar to this one
Definitions of tuning terms: val, (c) 2001 by Joe Monzo *
I'm afraid it wouldn't have made any difference if I'd read them all.
But I can't hold it against anyone that they are not good at
explaining things.

I'm pretty sure I did read an early one, and said to myself, "I have
no idea what that means. I guess I need a bit more mathematical
background. I'll look into it later."

But it appears that little or no explanation would have been necessary
if you had simply called them prime mappings.

So is a val, as applied to tuning theory, simply a prime-mapping, or a
1D-prime-mapping?


top of page bottom of page up down


Message: 8142

Date: Thu, 13 Nov 2003 02:16:33

Subject: Re: Vals?

From: Dave Keenan

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?


top of page bottom of page up down


Message: 8146

Date: Thu, 13 Nov 2003 23:24:11

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- 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, (c) 1998 by Joe Monzo *
> 
> 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.

This whole thread is hilarious. :-) I haven't had such a good laugh
from tuning-math in a long time, but I admit I've been taking myself
too seriously lately.

Back when it said "always", and it had 3 cents (because I thought, as
a lot of people apparently do) that 2 significant digits of cents was
a bit too precise, George said "if you mean 2.8 cents then say 2.8
cents or you'll just encourage further creepage" or words to that
effect. I thought he had a good point. But now that we've changed it
to "typically" (which I understand most people support) then even
George agrees 2.8 is too finicky. So "typically less than 3 cents" is
ok with me.

Paul, I assume you were merely arguing that "typically less than 2.8
cents" is as about as good as any other nearby number as a
just-noticeble-difference, and you wouldn't really mind if the
microtemperament definition was changed to "typically less than 3 cents".

Gene, I wanted an actual cutoff too - an "always" rather than a
"typically" - but it looks like we're outvoted. Or to put it another
way, I can live with a "typically" for the sake of consensus.


top of page bottom of page up down


Message: 8149

Date: Thu, 13 Nov 2003 23:30:01

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:
> 
> > 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.

It's a deal. :-)


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

8100 - 8125 -

top of page