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Message: 4400 - Contents - Hide Contents

Date: Wed, 27 Mar 2002 22:10:34

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> 

>> The best 16 would be enough for me; I don't see AMT as 
particularly
>> interesting as a 5-limit temperament, and the last three just 
don't 
> seem

>> all that useful for musical purposes as far as I can tell. I 
> haven't done

>> as much playing around with these scales as I'd like, but it 
> generally

>> seems to be the case that the less complex scales are also the 
ones 
> that

>> are most musically interesting to me.
> 

> I think we should leave room for various preferences in this 
> department, and don't see why we can't have a best 20.


that would be fine with me (though it's not really the 'best 20' by 
any complete criterion, just the 'first 20' in complexity to pass a 
certain condition).

Message: 4401 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 03:27:34

Subject: Re: Pitch Class and Generators

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> yes, but to claim (as balzano did) that the fundamental importance 
of 
> the diatonic scale hinges on this fact is to pull the wool over the 
> eyes of the numerically inclined reader. 


It's an incorrect assessment, I don't suppose it was intentionally 
misleading.

the important properties of the diatonic scale must, i 

> feel, be found in the scale itself, in whatever tuning it may be 
> found (with reasonable allowances for the ear's ability to accept 
> small errors) -- any 'chromatic totality' considerations should 
wait 
> until, and be completely dependent upon, the establishment of the 
> fundamental 'diatonic' entity upon which the music is to be based. 


Is it fundamental to the diatonic scale that 81/80~1, in your view?

Message: 4402 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 17:28:45

Subject: Re: Decatonics

From: Carl Lumma

>arranged--that's a valid insight.  The number ... * [with cont.]  (Wayb.)
>
>he didn't really get wilson's 'constant structures' right so maybe 
>he's mistaken on 'Rothenberg's "propriety"' as well?


Well, let me reply to his post:


>	The problem as I saw it was that so many not very interesting
>looking scales were being generated that stronger criteria were
>necessary. Just ensuing that all the notes were parts of major or
>minor (or septimal) chords was not a fine enough filter. Melodic
>characteristics are also important.


So he has already an embarrassment of riches as far as he's
concerned, so erring on the side of fewer scales may not be
his worst fear.


>	The criterion I've used the most is Rothenberg's "propriety,"
>which is equivalent essentially to Balzano's "coherence,"


So he agrees with us.


>and Wilson's "constant structure."


This isn't right.  Both John and I independently made the mistake
that CS was equivalent to *strict* propriety in 1999, forgetting
that there are improper scales which are CS.


>A proper scale is one which holds together as a scale, rather than
>being perceived as a set of principal and ornamental tones.


This is a gloss, but seems okay.  Rothenberg says that in practice,
especially without a drone, for scales with very low stability,
composition will tend to trace out proper subset(s) of the scale,
if they exist, and treat the propriety-breaking scale degrees as
ornamental.  Playing with the harmonic series segment from 8-16,
I found that I tended to use 13 this way, and I later looked, and
indeed the scale was proper without it.

Yes, he should have said stability.


>	A more important question is how are the scales going to be
>used. Are they  collections of tones (gamuts, like the 12-tone set)
>or are they going to be projected as scales and are scalar passages
>going to be used thematically.

Ah-ha!

>	I have considerable doubt about JI in nature. Most sounds in
>nature are nonharmonic, if only because the oscillators and resonators
>are complex 3-D structures instead of ideal 1-D strings and air columns.
>It takes considerable effort to train the voice to make harmonic timbres,
>and the vocal quality is decidedly un-natural, whether in relation to
>speech or the usual singing voice. Most animal cries don't strike me
>as especially musical, except perhaps song birds. I suspect that our
>perception of JI as something special is an epiphenomenon of our
>auditory system. A system evolved to recognize the sounds of predators,
>human voices, and especially to decode speech, may find JI easy to
>process because of acoustic and/or neurological laws. Hence, we find
>it exceptional and special.


He's right.  Many authors use the former (natural sounds) as a
justification for inharmonic/nonharmonic timbres/music, forgetting
the latter (biology).
 

>	It may also be partly a learned phenomenon; JI may not be so 
>easily perceived or appreciated in cultures whose musical instruments 
>have mostly nonharmonic timbres (Indonesia, S.E. Asia). There is
>some speculation that in pre-columbian Mesoamerican music, melodic 
>contour was more important than interval size. Perhaps we learn
>to perceive JI because it is part of our unnatural environment,
>even in its approximated tempered form. It's hard to escape harmonic 
>timbres from muzak, radio, stereos, etc. But, this is speculation.


Well, I tend to think cultural conditioning is not significant here.
I don't buy the predator noises thing -- we have excellent spatial
stuff for that (not only timing between ears, but apparently also
the the direction sounds enter the outer ear -- somebody's even got
an algorithm to simulate this over headphones now).  The key thing
I think is vowels and inflection in speech, which exist in all
cultures, and the biology here is so strong and exposed that we're
already beginning to discover it!

-Carl

Message: 4403 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 03:28:58

Subject: Re: Hermite normal form version of "25 best"

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> Lets move on to the full 7-limit. When you've got time Gene, could 
you 
> set your limits somewhat wide to start with. Say 35 cents rms 
error. 
> Weighted complexity of 30 gens, badness to give about 40 
temperaments?


I've been thinking of doing 7-limit planars first.

Message: 4404 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 19:15:16

Subject: Re: Decatonics

From: Carl Lumma

Sorry to all of you who have already gotten this from the other
list.  I've deleted it there and posted it here.


>> and Wilson's "constant structure."
>

>This isn't right. Both John and I independently made the mistake
>that CS was equivalent to *strict* propriety in 1999, forgetting
>that there are improper scales which are CS.


It's worth noting that CS can be interpreted in the same light
as propriety, though. Both can be viewed as measures of interval
recognizability -- linking acoustic intervals to scalar ones (ie
3:2 -> 5th). If you assume that the ear tracks acoustic intervals
by *relative* size, you get propriety/stablity. If you assume the
ear can recognize *particular* *absolute* intervals, you get CS
instead. Since I think the latter idea is complete nonsense (with
the possible exception of a few strong consonances), I think CS
should not be applied to the generalized diatonics problem. However,
you, Paul, have shown (and it seems that Wilson intuitively
understands) that CS is important with respect to periodicity
blocks. I can only remember one facet of this -- that all PBs with
unison vectors smaller than their smallest 2nd are CS (was there
ever a proof?).

-Carl

Message: 4405 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 09:36 +0

Subject: Re: Decatonics

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a7tcea+f5v4@xxxxxxx.xxx>
Mark:

>> This is inconsistent with my rule ii: it contains segments of 3 and 
> more
>> adjacent PCs
>> 
>> 0 1 2, 4 5 6 7, 9 10 11 12, etc
>> 
>> So it will show up as intervallically inchoerent (as defined by 
> Balzano).

Paul:

> this is what we call 'rothenberg improper'. but i don't think that's 
> a good reason to throw it out. the diatonic scale in pythagorean 
> tuning is rothenberg improper!


It doesn't look like impropriety to me.  The diatonic scale would only 
fail in the degenerate case of 7-equal.  The classic pentatonic would be 
incoherent in 7-equal, because it's a "pentatonic".


> i sure hope my other responses to you show up, mark. but basically, i 
> wish there was some text accompanying your decatonic diagram. i have 
> no idea why you are using 4 (218¢) and 5 (273¢) as your 'generators', 
> for example.


Yes, I'm looking forward to these.  How come the only messages that come 
through from you are ones where you say your messages aren't coming 
through?


                 Graham

Message: 4406 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 18:54:36

Subject: Re: A common notation for JI and ETs

From: gdsecor

i--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

>> I thought that, in the event somebody *absolutely must* have 23, 
one 
>> could allow a little bit of slack if the following were taken 
into 
>> account:
> ...

>> and, in addition, 21, 25, and 27 are all 
>> consistent with that.
> 

> You're right. I'd even like to see if we can push it to 31-limit, 
> consistent with 311-ET, since we're so close, but this would use 
> additional flags (and/or additional schismas like 4095:4096 and 
> 3519:3520) and not affect the existing 23-limit, 217-ET 
> correspondence.


Considering that the semantics of the notation have already put us 
past the 19 limit, that 217 is not 23-limit consistent, and that 311 
is such an excellent division, I'd say let's go for it!


> You know I went thru the prime factorisation of all the 
> superparticulars in John Chalmer's list, and you've found the only 
two 
> that are useful for this purpose.


I guess I just have a knack for finding useful commas (even before I 
start looking for them).  Are you ready for the next one?  It's a 
honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents).  And it turns 
out that we don't need any new flags to get the 29 factor: its 
defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the 
convex left flag that we already have (715:729) is ~33.571 cents.


> ...  I was considering putting a blob on the end of the straight 7 
> flag, but no. I agree with you now. Keep the curved flag for the 
> 7-comma. It is most important to get the 11-limit right. The rest 
is 
> just icing on the cake, and a little lateral confusability there 
can 
> be tolerated.


As long as we are going for a higher prime limit that will almost 
certainly require an additional kind of flag, perhaps that will 
present an opportunity to de-confuse the situation a bit, but that 
remains to be seen.

Here's something to keep in mind as we raise the prime limit.  I am 
sure that there are quite a few people who would think that making a 
notation as versatile as this one promises to get is overkill.  I 
think that such a criticism is valid only if its complexity makes it 
more difficult to do the simpler things.  Let's try to keep it simple 
for the ET's under 100 (as I believe we have been able to do so far), 
keeping the advanced features in reserve for the power-JI composer 
who wants a lot of prime numbers.  If we build everything in from the 
start and do it right, then there will be no need to revise it later 
and upset a few people in the process.


>> By the way, something else I figured out over the weekend is how 
to 
>> notate 13 through 20 degrees of 217 with single symbols, i.e., 
how to 
>> subtract the 1 through 8-degree symbols from the sagittal apotome 
>> (/||\).  The symbol subtraction for notation of apotome 
complements 
>> works like this:
>> 
>> For a symbol consisting of:
>> 1) a left flag (or blank)
>> 2) a single (or triple) stem, and
>> 3) a right flag (or blank):
>> 4) convert the single stem to a double (or triple to an X);
>> 5) replace the left and right flags with their opposites 
according to 
>> the following:
>> a) a straight flag is the opposite of a blank (and vice versa);
>> b) a convex flag is the opposite of a concave flag (and vice 
versa).
> 

> You gotta admit this isn't exactly intuitive (particularly 5a). I'm 
> more interested in the single-stem saggitals used with the standard 
> sharp-flat symbols, but it's nice that you can do that.


Believe it or not, the logic behind 5a) is pretty solid, while it is 
5b) that is a bit contrived.  The above is an expansion of what I 
originally did for the 72-ET notation before any curved flags were 
introduced.  Allow me to elaborate on this.  Consider the following:

81:80 upward is a left flag:  /|
33:32 upward is both flags:  /|\
so 55:54 upward is 33:32 *less* a left flag:  |\
Since an apotome upward is two stems with both flags:  /||\
then an apotome *minus 81:80* is the apotome symbol *less a left 
flag*:  ||\
which illustrates how we arrive at a symbol for the apotome's 
complement of 81/80 by changing /| to ||\ according to 4) and 5a) 
above.

Using curved flags in the 72-ET native notation to alleviate lateral 
confusibility complicates this a little when we wish to notate the 
apotome's complement (4deg72) of 64/63 (2deg72), a single *convex 
right* flag.  I was doing it with two stems plus a *convex left* 
flag, but the above rules dictate two stems with *straight left* and 
*concave right* flags.  As it turns out, the symbol having a single 
stem with *concave left* and *straight right* flags is also 2deg72, 
and its apotome complement is two stems plus a *convex left* flag 
(4deg72), which gives me what I was using before for 4 degrees.  So 
with a little bit of creativity I can still get what I had (and 
really want) in 72; the same thing can be done in 43-ET.  This is the 
only bit of trickery that I have found any need for in divisions 
below 100.

As you noted, it is nice that, given the way that we are developing 
the symbols, this notation will allow the composer to make the 
decision whether to use a single-symbol approach or a single-symbols-
with-sharp-and-flat approach.  And the musical marketplace could 
eventually make a final decision between the two.  So while we can 
continue to debate this point, we are under no pressure or obligation 
to come to an agreement on it.


>> I will prepare a diagram illustrating the progression of symbols 
for 
>> JI and for various ET's so we can see how all of this is going to 
>> look.
>> 
>> Stay tuned!
> 

> Sure. This is fun.


More fun (if more complicated) than I had ever expected!

--George

Message: 4407 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 20:08:56

Subject: Re: Decatonics

From: Carl Lumma

>>> >he criterion I've used the most is Rothenberg's "propriety,"
>>> which is equivalent essentially to Balzano's "coherence,"
>> 

>> So he agrees with us.
>

>i thought you objected to calling it *rothenberg* propriety . . . 
>isn't that the whole point?


No, Rothenberg did coin the term propriety, and a scale is
either proper or not, and IIRC is is the same as Balzano
coherence.  It's just that R. doesn't use it to eliminate
scales, and talks most often of stability.

I suppose I've been guilty here too.  From now on, I'm saying
"Rothenberg stability".

-Carl

Message: 4408 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 12:44:35

Subject: Re: Decatonics

From: Carl Lumma

>> >his is what we call 'rothenberg improper'. but i don't think that's 
>> a good reason to throw it out. the diatonic scale in pythagorean 
>> tuning is rothenberg improper!
>

>It doesn't look like impropriety to me.  The diatonic scale would only 
>fail in the degenerate case of 7-equal.  The classic pentatonic would be 
>incoherent in 7-equal, because it's a "pentatonic".


When I read Balzano's paper, incoherency was indeed equivalent to
propriety.  I don't remember anything about "pentatonic" and
"diatonic" being a majority or minority of s vs. L, but maybe I'm
just repressing it cause it's such a poor idea.

-Carl

Message: 4409 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 20:54:57

Subject: Re: Decatonics

From: Carl Lumma

>> >t's just that R. doesn't use it to eliminate scales,
>

>so what? i don't get what you were yelling at me about! i was just 
>saying exactly the same thing you're saying here -- that it's the 
>same as balzano coherence. balzano (and some of his followers) *do* 
>use it to eliminate scales, and *that's* what i was responding to.


Ok, sorry.  I did ask...


>> Who has been throwing out scales for being improper in the way
>> that the Pythagorean diatonic is improper?


...and in the past, you've used this argument to say that
'propriety doesn't matter', which is true but incomplete,
since it assumes the strictest possible definition for the
term "propriety".

-Carl

Message: 4410 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 21:28:27

Subject: Euler and harmonic entropy

From: genewardsmith

How does Euler's ranking of chords, where chord a:b:c:d would be given
a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy
rankings of chords? Euler's method gives the same value to otonal as
to utonal chords, so it must have some differences. How great are
they?

Message: 4411 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:35:29

Subject: Re: Euler and harmonic entropy

From: Carl Lumma

>How does Euler's ranking of chords, where chord a:b:c:d would be
>given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy
>rankings of chords? Euler's method gives the same value to otonal as
>to utonal chords, so it must have some differences. How great are
>they?


I guess we won't really know until we have chordal harmonic entropy.

I remember the totient function, which didn't work at all, even for
dyads, but the above measure looks different.

-Carl

Message: 4412 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:41:39

Subject: Re: Decatonics

From: Carl Lumma

>this is what we call 'rothenberg improper'. but i don't think that's 
>a good reason to throw it out. the diatonic scale in pythagorean 
>tuning is rothenberg improper!


Paul, how long are you going to continue using this fallacious
application of Rothenberg?  Who has been throwing out scales
for being improper in the way that the Pythagorean diatonic is
improper?  Certainly not Rothenberg!

-Carl

Message: 4413 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:56:03

Subject: Re: Euler and harmonic entropy

From: Carl Lumma

Doesn't have harmonic entropy, but it does have totient and
gradus suavatatis, and n*d, which *is* similar to harmonic
entropy, at least for dyads:

http://www.uq.net.au/~zzdkeena/Music/HarmonicComplexity.zip - Type Ok * [with cont.]  (Wayb.)

Dave, it looks like all of the pointers are broken.  Maybe it has
to do with me running Excel 2000 now?

-Carl

Message: 4414 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:00:51

Subject: Re: Pitch Class and Generators

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 

>> yes, but to claim (as balzano did) that the fundamental 
importance 
> of 

>> the diatonic scale hinges on this fact is to pull the wool over 
the 
>> eyes of the numerically inclined reader. 
> 

> It's an incorrect assessment, I don't suppose it was intentionally 
> misleading.


well, no, i think balzano managed to fool himself, and much of the 
academic community in the process.


>> the important properties of the diatonic scale must, i 
>> feel, be found in the scale itself, in whatever tuning it may be 
>> found (with reasonable allowances for the ear's ability to accept 
>> small errors) -- any 'chromatic totality' considerations should 
> wait 

>> until, and be completely dependent upon, the establishment of the 
>> fundamental 'diatonic' entity upon which the music is to be 
based. 
> 

> Is it fundamental to the diatonic scale that 81/80~1, in your view?


well, the diatonic scale is pretty strong already in the 3-limit, 
where 80 of course doesn't even come into play. but yes, in the 5-
limit, whether the diatonic scale is in ji or tempered, 81/80 is one 
of its defining unison vectors. as joe monzo might put it, the 
smallness of 81/80 helps determine the 'finity' of the diatonic 
scale -- if 81/80 were large (pardon the arithmetical counterfactual, 
and don't take it too seriously), you'd tend to keep adding more 
notes until you did hit up against a small unison vector.

Message: 4415 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:01:44

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 

>> Lets move on to the full 7-limit. When you've got time Gene, 
could 
> you 

>> set your limits somewhat wide to start with. Say 35 cents rms 
> error. 

>> Weighted complexity of 30 gens, badness to give about 40 
> temperaments?
> 

> I've been thinking of doing 7-limit planars first.


wow -- that could be a long list. the nice thing is that my heuristic 
should work well for these, since there's only one unison vector.

Message: 4416 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:06:02

Subject: Re: Decatonics

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>>> this is what we call 'rothenberg improper'. but i don't think 
that's 
>>> a good reason to throw it out. the diatonic scale in pythagorean 
>>> tuning is rothenberg improper!
>> 

>> It doesn't look like impropriety to me.  The diatonic scale would 
only 
>> fail in the degenerate case of 7-equal.  The classic pentatonic 
would be 
>> incoherent in 7-equal, because it's a "pentatonic".
> 

> When I read Balzano's paper, incoherency was indeed equivalent to
> propriety.


me too. so graham, what gives?

Message: 4417 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 00:03:09

Subject: Re: Hermite normal form version of "25 best"

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>> I think we should leave room for various preferences in this 
>> department, and don't see why we can't have a best 20.
> 

> that would be fine with me (though it's not really the 'best 20' by 
> any complete criterion, just the 'first 20' in complexity to pass a 
> certain condition).


Sure. If any one of us wants the 20, I'm happy to go with it. 

Lets move on to the full 7-limit. When you've got time Gene, could you 
set your limits somewhat wide to start with. Say 35 cents rms error. 
Weighted complexity of 30 gens, badness to give about 40 temperaments?

Message: 4418 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:08:22

Subject: Re: Decatonics

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>> this is what we call 'rothenberg improper'. but i don't think 
that's 
>> a good reason to throw it out. the diatonic scale in pythagorean 
>> tuning is rothenberg improper!
> 

> Paul, how long are you going to continue using this fallacious
> application of Rothenberg?  Who has been throwing out scales
> for being improper in the way that the Pythagorean diatonic is
> improper?  Certainly not Rothenberg!


sorry -- just force of habit (acquired from john chalmers, i 
believe). so who did introduce the terms 'proper' and 'improper' in 
this context, if not rothenberg?

Message: 4419 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:12:15

Subject: Re: Euler and harmonic entropy

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


>How does Euler's ranking of chords, where chord a:b:c:d would be 
>given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy 
>rankings of chords? Euler's method gives the same value to otonal as 
>to utonal chords, so it must have some differences. How great are 
>they?


they could be very great, in many cases. for example, harmonic 
entropy is a *continuous* function of the input intervals, defined 
for irrational as well as rational intervals.

i wish this were euler's actual ranking method, but of course he went 
even further (off the deep end) with his final formulae for GS.

Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)

Euler and music, by Patrice Bailhache, transla... * [with cont.]  (Wayb.)

Message: 4420 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:55:43

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

let's see how the heuristic for complexity matches up with the g_w 
measure. i'll leave out the heuristic for error, since we appear to 
have settled on an unweighted error measure.



> 135/128   (3)^3*(5)/(2)^7
> 
>  g_w   2.558772839


log(135)/2.558772839 = 1.9170



> 256/243   (2)^8/(3)^5
> 
>  g_w   3.472662942


log(243)/3.472662942 = 1.5818



> 25/24   (5)^2/(2)^3/(3)
>
>   g_w   1.597771402


log(25)/1.597771402 =  2.0146



> 648/625   (2)^3*(3)^4/(5)^4
>
>  g_w   3.484393186


log(625)/3.484393186 = 1.8476



> 16875/16384   (3)^3*(5)^4/(2)^14
>
>  g_w   4.719203505


log(16875)/4.719203505 = 2.0625



> 250/243   (2)*(5)^3/(3)^5
>
>  g_w   3.413658644


log(250)/3.413658644 = 1.6175



> 128/125   (2)^7/(5)^3
>
>  g_w   2.613294890


log(125)/2.613294890 = 1.8476



> 3125/3072   (5)^5/(2)^10/(3)
>
>  g_w   4.128050871


log(3125)/4.128050871 = 1.9494



> 20000/19683   (2)^5*(5)^4/(3)^9
> 
>  g_w   5.817894303


log(19683)/5.817894303 = 1.6995



> 81/80   (3)^4/(2)^4/(5)
>
>  g_w   2.558772839


log(81)/2.558772839 = 1.7174



> 2048/2025   (2)^11/(3)^4/(5)^2
>
>  g_w   3.822598772


log(2025)/3.822598772 = 1.9917



> 78732/78125   (2)^2*(3)^9/(5)^7
>
>  g_w   6.772337791


log(78125)/6.772337791 = 1.6635



> 393216/390625   (2)^17*(3)/(5)^8
>
>  g_w   6.722154036


log(390625)/6.722154036 = 1.9154



> 2109375/2097152   (3)^3*(5)^7/(2)^21
> 
>  g_w   7.187006703


log(2109375)/7.187006703 = 2.0261



> 4294967296/4271484375   (2)^32/(3)^7/(5)^9
>
>  g_w   10.74662038


log(4271484375)/10.74662038 = 2.0635



> 15625/15552   (5)^6/(2)^6/(3)^5
> 
>  g_w   4.990527341


log(15625)/4.990527341 = 1.9350



> 1600000/1594323   (2)^9*(5)^5/(3)^13
> 
>  g_w   8.314887839


log(1594323)/8.314887839 = 1.7176



> 1224440064/1220703125   (2)^8*(3)^14/(5)^13
> 
>  g_w   11.61862841


log(1220703125)/11.61862841 = 1.8008



> 10485760000/10460353203   (2)^24*(5)^4/(3)^21
> 
>  g_w   13.57752022


log(10460353203)/13.57752022 = 1.6992



> 6115295232/6103515625   (2)^23*(3)^6/(5)^14
> 
>  g_w   11.20594372


log(6103515625)/11.20594372 = 2.0107



> 19073486328125/19042491875328   (5)^19/(2)^14/(3)^19
> 
>  g_w   16.55086763


log(19073486328125)/16.55086763 = 1.8476



> 32805/32768   (3)^8*(5)/(2)^15
> 
>  g_w   5.957335766


log(32805)/5.957335766 = 1.7455



> 274877906944/274658203125   (2)^38/(3)^2/(5)^15
> 
>  g_w   13.67967551


log(274658203125)/13.67967551 = 1.9254



> 7629394531250/7625597484987   (2)*(5)^18/(3)^27
> 
>   g_w   19.05445924


log(7625597484987)/19.05445924 = 1.5567



> 9010162353515625/9007199254740992   (3)^10*(5)^16/(2)^53
> 
>   g_w   17.87941745


log(9010162353515625)/17.87941745 = 2.0547


the ratios fall between 1.5567 and 2.0635. so it's good as a rough 
guesstimate . . .

Message: 4421 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 14:58:30

Subject: Re: Decatonics

From: Carl Lumma

>> >aul, how long are you going to continue using this fallacious
>> application of Rothenberg?  Who has been throwing out scales
>> for being improper in the way that the Pythagorean diatonic is
>> improper?  Certainly not Rothenberg!
>

>sorry -- just force of habit (acquired from john chalmers, i 
>believe). so who did introduce the terms 'proper' and 'improper'
>in this context, if not rothenberg?


I'm not sure.  I've seen it around the list(s) on occasion, and
complained bitterly every time.  :)

To recap: Rothenberg defines his "ideal measure" as one that works
on all the subsets of a scale.  However, due mainly to computational
constraints, he uses stability, without a cutoff -- he lists all
the scales in 12-et and ranks them by stability.  I've proposed a
varient of stability based on the actual log-frequency amount of
overlap, and this is available in Scala.  Even for low stability
scales, Rothenberg does not throw them out -- he simply predicts
that a different kind of composition suits them better.

I don't endorse all of Rothenberg's model -- in fact, I only
understand a small part of it.  I do agree that he tries to go too
far without considering harmony -- I suspect his derrivation of
the diatonic, pentatonic scale, etc, using only stability and
efficiency blows up in a tuning other than 12 where almost
everything is consonant.  He may also have jumped to conclusions
with the ethno-musical data, though I've found that trying to
transpose themes over the modes of low-stability scales doesn't
seem to work.

-Carl

Message: 4422 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 16:47:29

Subject: generalizing diatonicity

From: Carl Lumma

Hello Mark, Jeff, All,

I've just updated my diatonicity shopping list:

 * [with cont.]  (Wayb.)

Does it make any more sense now?

-Carl

Message: 4423 - Contents - Hide Contents

Date: Fri, 29 Mar 2002 09:03:14

Subject: Re: Digest Number 331

From: John Chalmers

Manuel: Thanks for the counter-example to CS equalling strict propriety.
I stand corrected

As for harmonic and inharmonic vocal timbres. I was apparently mistaken.
What confused me was the fact that outside of the European culture area,
vocal timbres are usually nasal and/or strident and their use may be
correlated with non-JI (or close approximations) tunings and intervals.
For example, how harmonic is the spectrum of the Indonesian singing
voice or that of American Indians? For that matter, how harmonically
related are the formants of speech in many languages (Khoisan, North
Caucasian, etc.). It seemed to me that to produce the clear harmonic
tone of European singing (primarily Church and Italianate styles) takes
a lot of training. Untrained voices often sound less harmonic to me, but
I could be wrong. 

How in tune are the harmonics and are the usual pitches of the vowel
formants for most speakers actually close to harmonics? I don't know. Is
this information in the literature (Sundberg, perhaps?)? 

--John

Message: 4424 - Contents - Hide Contents

Date: Fri, 29 Mar 2002 09:48:39

Subject: Re: Decatonics

From: Carl Lumma

>> >n 1/3-comma meantone is strictly proper and CS
>> in 12-et is proper but not strictly so, not CS
>> in Pythagorean tuning is improper and CS
>

>All of the above are epimorphic.
>
>I think of scales in terms of the properties epimorphic,
>convex, and connected.


If I'm right about the terms convex and connected, they
only apply to periodicity blocks.  What about non-just
scales?  Rothenberg claims some of the stuff of melody
doesn't have anything to do with harmony.

-Carl

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