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Message: 8175 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 08:40:09

Subject: Re: Definition of microtemperament

From: monz

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


> Monz,
> 
> You could just change every ocurrence of "microtemperament" in it to
> "planar temperament" and change its name to "planartemp.htm" and
> there's your definition for planar temperament.
> 
> I'm hoping that the following will make everyone happy. I've changed
> "would always be less than 3 cents" to "would typically be less than
> 2.8 cents". I've also changed "JI scale" to "JI tuning" throughout.
> 
> I'm sure this definition could be improved, but can we just get
> something in place of that bad definition in Monz's dictionary?




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.)



and i already had in the Dictionary a definition of 
"planar temperament" from Graham:

Definitions of tuning terms: planar temperamen... * [with cont.]  (Wayb.)



since i've already changed the definition of "microtemperament"
according to your revisions, can you post the old definition
which i should revamp into "planar temperament"?  

... the old HTML is still in the source code of 
"microtemperament" in a comment, in case you need it.



-monz

Message: 8176 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 17:58:53

Subject: Re: Vals?

From: Carl Lumma

>I suppose we should say "and it is also
>the best (i.e. most accurate) mapping when the ET is consistent.


Which fits with:

>> Dave's 'the best approx. to each
>> element of a chord in n-tET' is better,

>> Dave's 'the best approx. to each
>> element of a chord in n-tET' is better,
>

>Err. I don't remember writing that.


You wrote "primes" not 'elements of any chord'.


>Standard vals (standard mappings), or vals (mappings) of any kind,
>have absolutely nothing to do with chords. Except that you can apply
>vals (mappings) to rational pitches to see where they end up in the
>temperament that the val (mapping) corresponds to.


So far so good.


>> but why n should equal
>> the number of notes in a chord is still a mystery.
>

>It's pretty much of a mystery to me too. This is not a necessary
>property of vals or even of standard vals. It has nothing to do
>with them.

Ok....

>It's just that Gene and George found it interesting to look at how
>complete chords can be mapped to a single octave of the ET of the same
>cardinality as the chord. It turns out that the 11-limit otonality
>can't be. There is no mapping and no voicing of the chord that will do
>this.


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?  Lessee, would the val would be
(the parens pending clarification on the ketbra situation)...

(val 0 10 14 17 21)

and reversing the process to get the above I only need to worry
about 9/8, which is (monzo -3 2 0 0 0).  It looks like this gives
me a 20-18 = 2, which is supposed to be 5/4.  Is *this* the
problem?

-Carl

Message: 8177 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 20:00:02

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
And considering what I have to 

> say below, it might be best to forget about boundaries altogether 
and 
> express the magnitude to one decimal place (as with stellar 
magnitude 
> in astronomy).


Works for me.


> I have never thought of any tuning as anything less than 11-limit 
if 
> it contains 11-limit intervals. 


Miracle still makes a lot of sense if you don't use the 11-limit. Is 
there a rule we need the 7 and 11 limits in meantone, simply because 
they are there?

 Even from an historical perspective 

> miracle has always been an 11-limit tuning.


You don't need to use 11-limit harmony in miracle.

Message: 8178 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 00:49:16

Subject: Re: 7-limit optimal et vals

From: Carl Lumma

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

>How about rational argument?


I tried in msgs. 7372 and 7523 to the big list.


>> 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.


We already have very well-accepted terminology for EDO.  It seems
perverse to have a problem with it but not with the term "octave".
EDx isn't as flexible as -CET when x isn't a whole number.
Otherwise, I'd suggest using ET not ED and omitting the 2 instead
of additng an O, thus arriving at the standard usage for things
like 12 ET, and at 13 ET3 for things like Bohlen-Pierce.

-Carl

Message: 8179 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 20:21:38

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

From: Gene Ward Smith

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

! fivelim.scl
!
Five-limit otonal chord
3
!
5/4
3/2
2


! sevenlim.scl
!
Seven-limit otonal chord
4
!
5/4
3/2
7/4
2


! ninelim.scl
!
Nine-limit otonal chord
5
!
9/8
5/4
3/2
7/4
2


! elevenlim.scl
!
Eleven-limit otonal chord
6
!
9/8
5/4
11/8
3/2
7/4
2


! thirteenlim.scl
!
Thirteen-limit otonal chord
7
!
9/8
5/4
11/8
3/2
13/8
7/4
2

Message: 8180 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 08:50:53

Subject: Re: Vals?

From: monz

hi Dave and Gene,


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


> I notice that even Paul, who along with Graham, appears
> to understand your stuff better than any of us, 



that's the impression i have too.  

i think Paul Hjelmstad and Carl Lumma can grasp a bit
of Gene's stuff too.  more than me, anyway.




> was gently suggesting a similar thing in another thread
> (although I thought he caved in rather too easily).



Gene, i hope you don't feel like i too am joining the
bandwagon to deride you.  in any case, i always try hard
to temper what i write in an opposing debate with some
compassion for the person on the other end.

but i must agree that while i feel (intuitively) that your
contribution to tuning theory in the last two years has
been truly mighty, i actually understand very little of
what you're doing and describing, and i unfortunately also
have to agree that the reason for that lack of understanding
is primarily the *way* you describe your work.

i applaud your eagerness to post definitions of terms as
you need to coin them, and i lemming-like pop them into
the Tuning Dictionary ... but they really don't help me
(nor, apparently, many others among us).


i'm anticipating that working with the software i now
have under development, upon its first release, will
aid me in comprehending your (Gene's) ideas, and i look
forward to your criticisms and comments on it after
you try it.



-monz

Message: 8181 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:27:11

Subject: Re: Vals?

From: Paul Erlich

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


>> 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).


it does imply it in most cases, since the primes are already in 
order, so the best approximations to them will have to be a non-
decreasing sequence.


> And it
> doesn't include why we care that the number of notes in an octave
> equals the number of notes in the chord.


george apparently cares about this very much too -- his observation 
about the 11-limit hexad not being CS implies gene's observation.


> And it only defines vals
> for prime limits, not for odd limits.


all you need is the previous point.

Message: 8182 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 08:58:47

Subject: Re: Vals?

From: monz

hi Gene and Dave,

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


> 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?




Gene, Dave can't help it ... he's a systems designer.

Dave, i can go along with Gene to some extent.  that's
why the Tuning Dictionary is there.  i understand your
desire for standardization, and agree with that too.

but so what if there are two different terms which mean
*very nearly* the same thing, but not exactly.  i'll put
them both in the Dictionary and you guys provide me with
proper definitions that show exactly what they have in
common and exactly where they differ.


i've been thru this same sort of thing myself with
my term "xenharmonic bridge" and its similarity to
Fokker's "unison vector".  

but despite the extreme similarity between the two,
i perceive a distinction (and i grant the possibility
that i could be wrong about that too).

so, they both go into the Dictionary and their
definitions both grow as i learn more and more about
them.

what's the problem?  


i think both of you guys need to chill out a little.

:)




-monz

Message: 8183 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:36:31

Subject: Re: Vals?

From: Paul Erlich

monz, this is a very specific, and perhaps even unusual, application 
of the val or mapping concept. it may warrant mention in an 
Encyclopedia but probably not in a dictionary.

p.s. would it be OK for me to attempt a modification of your page

Definitions of tuning terms: EDO prime error, ... * [with cont.]  (Wayb.)

? 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.

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

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

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

> this is the first time this has been explained in a way
> that made sense to me.
> 
> and this is good enough to put in the Tuning Dictionary
> ... but under what definition?  should "standard mapping"
> be a Dictionary entry?  or is this an amendment to "val"?
> 
> 
> 
> -monz

Message: 8184 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 09:01:06

Subject: Re: 7-limit optimal et vals

From: monz

hi Carl,

--- 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...


why?




-monz

Message: 8185 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:42:47

Subject: Re: Vals?

From: Paul Erlich

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


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


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


>> 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???

Message: 8186 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 09:07:57

Subject: Re: Vals?

From: monz

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

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

> 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)).




this is the first time this has been explained in a way
that made sense to me.

and this is good enough to put in the Tuning Dictionary
... but under what definition?  should "standard mapping"
be a Dictionary entry?  or is this an amendment to "val"?



-monz

Message: 8187 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:46:26

Subject: Re: Definition of microtemperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:

> 
>>> Definitions of tuning terms: microtemperament,... * [with cont.]  (Wayb.)
> 
> 

> As a general definition, '2.8 cents' seems a bit finicky to me, 
> especially with the qualifier 'depending on context'.


aaron, 'depending on context' was a qualifier inserted so that we 
wouldn't seem so finicky as to insist on an exact figure of 2.8 
cents! how is it that you are seeing this so differenly?


> Based on 
> psychoacoustic experiments for the JND, you can safely change 
> '2.8 cents' to 3 cents.


there are of course many JNDs, all of which are a function of 
absolute frequency. which do you have in mind? of course johnny 
reinhard on the tuning list can distinguish *melodic* changes of 1 
cent (and most of us can distinguish *harmonic* changes of 1 cent 
from a simple ratio), and he claims to be able to accurately 
conceive, in his aural imagination, any interval to an accuracy of 1 
cent.

Message: 8188 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 00:13:51

Subject: Re: Definition of microtemperament

From: Dave Keenan

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

> Maximum error in cents
> 
> Magnitude 0: 0.25-0.5
> Magnitude 1: 0.5-1.0
> Magnitude 2: 1.0-2.0
> Magnitude 3: 2.0-4.0
> 
> -log2(4 * error) is the formula.


Well this looks like a good system to me. But I'm afraid it will make
too many people unhappy to try to align the definition of
"microtemperament" with it.


> Miracle is a third magnitude temperament by this.


Hopefully this will make George happy.

Paul,

I wasn't considering the 72-EDO incarnation. When I gave 2.4 c and 3.3
c I was considering the minimax optima for 7-limit and 11-limit
respectively (9-limit is the same as 11-limit).

George,

I think most of us find it quite natural to speak of "7-limit miracle"
even though some may consider the "true" miracle temperament to be
11-limit. We have enough trouble agreeing on names as it is. I'd hate
to have to find different names for different lower-limit subsets (or
whatever the right term is) of the same mapping.

The most important thing is for Monz to get rid of the current false
definition of microtemperament. 

Monz,

You could just change every ocurrence of "microtemperament" in it to
"planar temperament" and change its name to "planartemp.htm" and
there's your definition for planar temperament.

I'm hoping that the following will make everyone happy. I've changed
"would always be less than 3 cents" to "would typically be less than
2.8 cents". I've also changed "JI scale" to "JI tuning" throughout.

I'm sure this definition could be improved, but can we just get
something in place of that bad definition in Monz's dictionary?

----------------------------------------------------------------------
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.

A JI tuning might be microtempered to increase the number of available
consonances or to regularise the scale for some purpose such as
allowing more full-width continuous frets on a stringed instrument.
Microtemperament may also be used to introduce deliberate slight
mistunings to avoid phase-locking when a JI tuning is implemented on an
electronic instrument.

A microtemperament may be equal, linear, planar or of any dimension
less than that of the JI tuning being approximated.
----------------------------------------------------------------------

Message: 8189 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 10:07:38

Subject: Re: Vals?

From: Dave Keenan

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

>> 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.


So did it become a lot clearer what a val was, when you figured out
that it was a prime mapping?

Did you understand what they were mapping the primes _to_ at that time?


>> 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.


Maybe so, but that's not why it doesn't make sense. It doesn't make
sense because limits don't have mappings. Temperaments have mappings.
These mapings are usually limited to some maximum prime. So mappings
have prime limits, not the other way 'round.


>> 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 missed the definition of a "Gram" mapping.


>> 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?


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.


>> 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.


Why so? It should be easy to find a voicing of the 1:3 chord, or a
3-limit 2-tET mapping, that has an unused pitch between the two
pitches of the chord. A 1:3 voicing would be such a voicing, with the
standard mapping <2 3]. I didn't say anything about restricting
ourselves to one octave. However, if I did say we had to fit within
the octave then I wouldn't need to say consecutive for any of them. So
it may well have been the octave limitation that Gene had in mind,
rather than the consecutive steps, since they are equivalent restrictions.


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


Why not? The standard 5-limit mapping for 3-tET is <3 5 7]. i.e 
round(n*ln(p)/ln(2)), for n = 3 and p = 2, 3, and 5.

Consider a 4:5:6 voicing of the complete 5-limit otonality. Convert
each pitch to a prime exponent vector. That gives us [2 0 0]:[0 0
1]:[1 1 0]. Now take the dot product of the mapping with each pitch in
turn.

<3 5 7].[2 0 0> = 3*2 + 5*0 + 7*0 = 6
<3 5 7].[0 0 1> = 3*0 + 5*0 + 7*1 = 7
<3 5 7].[1 1 0> = 3*1 + 5*1 + 7*0 = 8

So we see we have consecutive numbers of steps. No gaps. But as I said
we could instead just say we must use every pitch-class exactly once.
It's the same game.


>> 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.


Sure. It's only because we've restricted ourselves to equal divisions
of the octave that octave-equivalents also get their best
approximations. I probably shouldn't have even mentioned it.


> 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.


You'll have to ask Gene about that. If I've got that right.


>  And it only defines vals
> for prime limits, not for odd limits.


A mapping with an odd-limit of n has a number of steps (or generators)
for every prime <= n. So the standard 9-limit mapping for some ET is
identical to its standard 7-limit mapping.

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.

There are of course odd-limit temperaments because that's talking
about what ratios we want to optimise the approximations of.

Message: 8190 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:53:32

Subject: Re: Definition of microtemperament

From: Paul Erlich

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


> I have never thought of any tuning as anything less than 11-limit 
if 
> it contains 11-limit intervals.


You mean 11-prime limit? Miracle is of course a temperament, and it 
can be derived easily and naturally from the 7-limit lattice. You 
simply temper our 225:224 and 2401:2400 (or 225:224 and 1029:1024).

You mean 11-odd limit? Well, meantone contained excellent 
approximations to ratios of 7, but practically no one considered them 
consonant historically. So i see no problem with considering miracle 
a 7-limit temperament if someone uses it in a style where ratios of 
11 or their approximation are used as dissonances. In fact miracle is 
one of the very best choices for a 7-limit temperament.


> Even from an historical perspective 
> miracle has always been an 11-limit tuning.


Not true at all!! I think the 11-limit was an unexpected bonus, that 
mostly Dave Keenan was keeping track of at the time.

Message: 8191 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 00:17:11

Subject: Re: Gaps between Zeta function zeros and ets

From: Dave Keenan

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

> I took a list of the first 10000 zeros of the Riemann zeta function,
> rescaled so that they read in terms of equal divisions of the octave.
> The top zero is then 1089.695, so we are looking at ets from 1 to 1089. 
> The gaps between these zeros are on average 1/log2(n) in the vicinity
> of n, so I took the gaps between successive zeros and multiplied by
> log2 of the average of the two successive zeros. As I expected, for
> the largest gaps these are centered around equal divisions of the
> octave. The first fifty of these, in order, are as follows:
> 
> 954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882, 
> 1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075,
> 692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997, 441,
> 643, 597, 373, 1046, 795, 449, 224, 513, 328, 966
> 
> Some tendency for the ets in question to be the kind approximating a
> lot of primes (311) instead of a relatively few (411) seems to be in
> evidence, but it isn't really clear what is going on.

I agree.

Message: 8192 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 10:18:01

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- 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.)


Thanks Monz. I just needed to hit the reeload button.

I hope it's ok with everyone else.


> and i already had in the Dictionary a definition of 
> "planar temperament" from Graham:
> 
> Definitions of tuning terms: planar temperamen... * [with cont.]  (Wayb.)


It's fine. You can leave it how it is, as far as I'm concerned.

Message: 8193 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 21:54:45

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:

>>>> 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?
> 

> Maybe I misunderstood the question.


the question was "why 3"? your answer was not an answer at all, it 
just assumed the number three all over again, without saying where it 
came from.

Message: 8194 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 01:02:02

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:
> The octaves would be another val.
> 

>> 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)?
> 

> Is it just barely possible I might know something about mathematics?


There is no doubt about that. We're all very grateful for the new
tools you've given us. If only we could figure out what they actually
are in tuning terms, and how to use them.

But the fact is, that was not a pure-math question. It was a
tuning-math question. And it was not intended as a jibe, it was a
sincere question. I had hoped that would be clear from my following
sentence where I assumed that there _was_ a good reason.
 

> To me this is a bit like asking why we would care about a single 
> comma, when we need more than one of them for a Fokker block, BTW.


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


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

> Call them what you like, but clearly your names are clumsier.


I really don't think 3 syllables is excessive (nor 5 if we need to add
the "1D"). We can always make up a brand new shorter word for
something, but it isn't always a good idea. We'd be in a sorry state
now if we invented a new (initially meaningless) term every time a
descriptive term went over two syllables.


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

> They aren't prime mappings per se; that's just a basis.


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


> I *did* 
> explain they were homomorphic mappings, and give examples with column 
> vectors, etc etc.


OK. Sorry I missed them. But like I said, if you'd used an obvious
name that related to the application area, instead of a newly invented
abstract pure-math term, the explanations would barely have even been
necessary. Talk about secret decoder rings. :-)


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

> More or less.


OK. So now I feel like the boy who cried "the emperor has no clothes".
While we can't hold it against you that you think better in pure-math
terms and are not very good at explaining the relationships to tuning
in a way the rest of us can understand, 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.

I notice that even Paul, who along with Graham, appears to understand
your stuff better than any of us, was gently suggesting a similar
thing in another thread (although I thought he caved in rather too
easily).

Regards,
-- Dave Keenan

Message: 8195 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 03:06:49

Subject: Re: Vals?

From: Carl Lumma

>>> >o 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.
>

>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.

Terminology has absolutely nothing to do with why Gene's writing
is utterly incomprehensible to me.  I think it has more to do
with:

() Assumes too much prior math skill.  This doesn't mean jargon,
which I can look up.  It's something else.  I can often perform
text substitution on his writing and still be either utterly
lost or left with something that is utterly obvious (but useful
in some aspect I'm not seeing).
() Unable or unwilling to explain the same thing with multiple
tools (ie math, natural language, diagrams).
() Is unable or unwilling to explain the same thing in multiple
ways.
() Unable or unwilling to give reasoning/motivation behind
definitions.


>Did you understand what they were mapping the primes _to_ at that
>time?


I still don't think I do.


>>> 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.
>

>Maybe so, but that's not why it doesn't make sense. It doesn't make
>sense because limits don't have mappings. Temperaments have mappings.
>These mapings are usually limited to some maximum prime. So mappings
>have prime limits, not the other way 'round.


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


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

>I missed the definition of a "Gram" mapping.


You're not the only one.


>>> 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.
>
>Why so?


Because there are only 2 elements!


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


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


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

>Why not? The standard 5-limit mapping for 3-tET is <3 5 7]. i.e 
>round(n*ln(p)/ln(2)), for n = 3 and p = 2, 3, and 5.


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


>> 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.
>

>You'll have to ask Gene about that. If I've got that right.
>

>> And it only defines vals
>> for prime limits, not for odd limits.
>

>A mapping with an odd-limit of n has a number of steps (or generators)
>for every prime <= n. So the standard 9-limit mapping for some ET is
>identical to its standard 7-limit mapping.


Then I don't know why the standard 11-limit mapping wouldn't be
identical to the standard 11-prime-limit mapping.  Anyway, saying
mapping instead of val is already confusing me here.

-Carl

Message: 8196 - Contents - Hide Contents

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

Subject: Re: Definition of microtemperament

From: George D. Secor

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

>> 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


And it depends on the interval(s) involved.  Error would be most 
evident with the most consonant intervals, particularly fourths and 
fifths.  The language in question says "typically less than 3 cents", 
which leaves quite a bit open to interpretation.  I would expect that 
the most consonant intervals would be quite a bit "less than 3 
cents", while more dissonant "consonances" might even be acceptable 
if allowed to stray a little over 3 cents.  As soon as you are 
considering 9-limit consonances you have automatically restricted the 
allowable error for 2:3 and 3:4 to half of what will be allowed for 
8:9 (assuming that we are dealing with regular temperaments).

--George

Message: 8197 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 01:08:25

Subject: Re: 7-limit optimal et 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:

>> --- 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?

Message: 8198 - Contents - Hide Contents

Date: Thu, 13 Nov 2003 22:00:35

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

From: Paul Erlich

maybe george would be better equipped to explain why this is 
musically so significant.

--- 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. You will 
also 
> find stuff about "JI epimorphic", but I don't understand what 
Manuel 
> is up to; it isn't what I expected.
> 
> ! fivelim.scl
> !
> Five-limit otonal chord
> 3
> !
> 5/4
> 3/2
> 2
> 
> 
> ! sevenlim.scl
> !
> Seven-limit otonal chord
> 4
> !
> 5/4
> 3/2
> 7/4
> 2
> 
> 
> ! ninelim.scl
> !
> Nine-limit otonal chord
> 5
> !
> 9/8
> 5/4
> 3/2
> 7/4
> 2
> 
> 
> ! elevenlim.scl
> !
> Eleven-limit otonal chord
> 6
> !
> 9/8
> 5/4
> 11/8
> 3/2
> 7/4
> 2
> 
> 
> ! thirteenlim.scl
> !
> Thirteen-limit otonal chord
> 7
> !
> 9/8
> 5/4
> 11/8
> 3/2
> 13/8
> 7/4
> 2

Message: 8199 - Contents - Hide Contents

Date: Fri, 14 Nov 2003 10:11:52

Subject: Re: Vals?

From: Gene Ward Smith

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


> 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.

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