Tuning-Math Digests messages 11276 - 11300

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Message: 11276

Date: Sun, 04 Jul 2004 23:10:23

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > 
> > > For pure octave tunings, a system I sometimes use is to close 
at a
> > > "poptimal" generator. A generator is "poptimal" for a certain 
set of
> > > octave-eqivalent consonances if there is some exponent p, 2 <= 
p <=
> > > infinity, such that the sum of the pth powers of the absolute 
value 
> > of
> > > the errors over the set of consonances is minimal. 
> > 
> > This is quite an interesting approach. What makes poptimal 
generators 
> > good? And why can't p be 1?
> 
> It could be 1. In fact, Paul thinks it should be 1.

No I don't -- TOP actually uses p=inf, in a sense -- I just think 1 
should be included in the range if infinity is.

> What makes them good is that they approximate a
> given list of target consonances in an optimal way, for some sense 
of
> optimal.

It's not a sense which gives a unique answer, which seems to have 
tripped everyone up so far. All your logic is right there in your 
math, but you have to understand that this is essentially a foreign 
language for most of us.


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Message: 11278

Date: Sun, 04 Jul 2004 23:12:47

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> By the way, Gene, how does poptimal relate to TOP? 
> >
> >Not very well, apparently.
> >
> > If the
> >> commas dictate the TOP tuning, is there necessarily a
> >> generator/period pair that give it?
> >
> >You've lost me.
> 
> I meant, for a given TOP-tuned linear temperament, does it not
> stand to reason that there is at least one generator/period
> pair (in cents) that produces scales in said tuning?

Yes, and there's exactly one if you fix that the octave is multiple 
of the period, and the generator is within a prescribed 1/2-period 
range (say, between 0 or 1/2 period, or between 1/2 and 1 period).


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Message: 11279

Date: Sun, 04 Jul 2004 23:17:50

Subject: Re: 9&11 poptimal secor

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> The 9 and 11 poptimal range intersect only at the minimax for 9 and
> 11, which is the (18/5)^(1/19) secor. The continued fraction for 
this
> gives
> 10, 31, 41, 72, 329, 2046 ... as the et convergents. The 7/72 secor 
is
> between the poptimal range for 9 and and 11 and the range for 5 and 
7,
> which makes it an all-purpose utility choice, and it's actually
> possible that the 11-limit poptimal range includes it, since it at
> least gets quite close.
> 
> The 5 and 7 limit minimax tuning is (12/5)^(1/13), which defines the
> upper part of their range. Is either of these the official George
> Secor secor?

The first one -- (18/5)^(1/19) -- is. He chose it because it's the 11-
(odd-)limit minimax.


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Message: 11282

Date: Mon, 05 Jul 2004 22:09:04

Subject: Re: bimonzos, and naming tunings (was: Gene's mail server))

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Why harder? Can you show this?
> 
> From my explanation, you can see it is easy to read off the prime
> factors in a frontward way and figure out the signs in a bival. With
> the 7-limit bimonzo, you don't have the same regular pattern where 
two
> signs are the same, and one opposite; sometimes they all have the 
same
> sign, and sometimes they don't.

When don't they? I must have missed something.

> I think bivals are clearly easier.
> Moreover, for higher limit linear temperaments, they are still 
bivals
> and you can use the same rule, whereas trimonzos are what you get in
> the 11-limit, etc. A big fat mess by comparison.

I already answered this, so I won't repeat myself.

> > > and I
> > > don't think this makes much of a reason for using bimonzos. I 
really
> > > would like to know why Paul insists on this so stubbornly.
> > 
> > I welcome constructive suggestions for making the paper go bival, 
> > without adding to its math-heaviness.
> 
> It's really, really, really easy. Simply replace the bimonzos you 
list
> with the corresponding bivals, and you are done.

You've got to be kidding me.

> You need explain
> nothing, nor define anything.

I'd like to do better by my readers.

> > > If you have a bival <<a1 a2 a3 a4 a5 a6||, then
> > > 
> > > 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.
> > 
> > You don't need to remove common factors?
> 
> I said "gives the comma", not "is the comma".

But when you were talking about the bimonzo, you made it seem a whole 
lot more complicated by introducing b1, b2, b3.

> > > 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the 
power
> > > of the (3,7) coefficient, 3 to minus the power of the (2,7)
> > > coefficient, and 7 to the power of the (2,3) coefficient;
> > 
> > How is this easier than the bimonzo case?
> 
> Because there is a simple rule I can explain which works for any 
prime
> limit.

Show me how a bimonzo gets so bad in a higher prime limit.


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Message: 11283

Date: Mon, 05 Jul 2004 22:16:36

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
> > wrote:
> > > 
> > > > > This is quite an interesting approach. What makes poptimal 
> > > > >generators 
> > > > > good?
> > > > 
> > > > Not much, IMHO -- the "true" value of p in any situation will 
be 
> > some 
> > > > number, not an infinite range of numbers.
> > > 
> > > What in the world does this mean? What allegedly "true" value?
> > 
> > If you're using this p-norm model in the first place, it's 
probably 
> > because you think it's true for some value of p.
> 
> I have no idea how a norm can possibly be either true or false. My
> assumption is that it is a valid definition of optimum for any p in
> the range 2 to infinity, and that is simply because if you assume 
the
> endpoints define a valid sense of optimum, so should all the
> intermediate values.

OK. And a lot of other values may define a valid sense of optimum, as 
well, for example weightings, etc.

> > > As for 1, I think a lot of people would find the supposedly 
optimal
> > > tunings not really very optimal in some cases.
> > 
> > And yet there is a significant bunch of composers who refuse to 
> > temper, clinging to their JI scales. Might they be modelled too? 
(no 
> > offense to them, of course.)
> 
> If you refuse to temper at all, what in the world are you doing 
trying
> to decide which tuning is best, on the assumption that a given
> temperament will be used?

Exactly. This corresponds to the p<1 area, or at least to some of it.


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Message: 11284

Date: Mon, 05 Jul 2004 22:19:08

Subject: Re: NOT tuning

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >except where TOP already had pure octaves, in 
> >> >which case it would actually change!
> >> 
> >> That's impossible given the criterion of NOT.
> >> 
> >> Maybe I don't comprehend you.
> >
> >I didn't say NOT, I said "Graham" and "pure-octaves TOP".
> 
> Ok, it would seem to violate the definition of
> "pure-octaves TOP".

It doesn't. It still has pure octaves.


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Message: 11285

Date: Mon, 05 Jul 2004 22:22:57

Subject: Re: from linear to equal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi Gene and Paul,
> 
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> >  
> > > > You'd maybe prefer Fokker?
> > > 
> > > Did Fokker have a particular route along the circle of fifths 
that he 
> > > preferred to get 11?
> > 
> > I doubt it. These two temperaments should both probably be melted 
down
> > into 31 equal, however, which of course makes them the same; hence
> > huygens or fokker might be good names.
> 
> 
> huygens or fokker are indeed the two most appropriate names
> for 31edo.
> 
> but if your main criteria in naming is to honor someone
> who advocated 11-limit,

It would have to be a particular mapping of the 11-limit along a 
particular chain of fifths that would, for example, correspond to a 
particular path in the 31-equal circle of fifths. Huygens never went 
beyond 7-limit, and Fokker didn't prescribe one method of generating 
(by fifths) 31-equal's approximation of 11 over another.


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Message: 11286

Date: Mon, 05 Jul 2004 22:25:13

Subject: Re: 24 7-limit temperaments with 245/243 as a comma

From: Paul Erlich

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

> godzilla/hemifourths
> [2, 8, 1, 8, -4, -20]
> [[1, 2, 4, 3], [0, -2, -8, -1]]

Herman's chart calls this "mothra". My paper calls it "semaphore". 
Let's call the whole thing off :)


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Message: 11288

Date: Mon, 05 Jul 2004 17:46:15

Subject: Re: NOT tuning

From: Carl Lumma

>> >> >except where TOP already had pure octaves, in 
>> >> >which case it would actually change!
>> >> 
>> >> That's impossible given the criterion of NOT.
>> >> 
>> >> Maybe I don't comprehend you.
>> >
>> >I didn't say NOT, I said "Graham" and "pure-octaves TOP".
>> 
>> Ok, it would seem to violate the definition of
>> "pure-octaves TOP".
>
>It doesn't. It still has pure octaves.

Oh, I thought you were saying the octaves changed.  So in
fact I have no clue what you were trying to say.

-Carl


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Message: 11293

Date: Mon, 05 Jul 2004 20:13:00

Subject: Re: NOT tuning

From: Carl Lumma

>Graham's "pure-octaves TOP" is just a uniform stretching or 
>compression of normal TOP -- except in those cases where normal TOP's 
>already got pure octaves, in which case the change is not a mere 
>uniform stretching or compression.

Aha!  Got'cha.

That *is* interesting, and a bit unsettling I suppose.

-Carl


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Message: 11295

Date: Mon, 05 Jul 2004 00:25:37

Subject: Re: NOT tuning

From: Carl Lumma

>> >except where TOP already had pure octaves, in 
>> >which case it would actually change!
>> 
>> That's impossible given the criterion of NOT.
>> 
>> Maybe I don't comprehend you.
>
>I didn't say NOT, I said "Graham" and "pure-octaves TOP".

Ok, it would seem to violate the definition of
"pure-octaves TOP".

-Carl


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Message: 11296

Date: Mon, 05 Jul 2004 00:27:36

Subject: Re: from linear to equal

From: Carl Lumma

>> I meant, for a given TOP-tuned linear temperament, does it not
>> stand to reason that there is at least one generator/period
>> pair (in cents) that produces scales in said tuning?
>
>Yes, and there's exactly one if you fix that the octave is multiple 
>of the period, and the generator is within a prescribed 1/2-period 
>range (say, between 0 or 1/2 period, or between 1/2 and 1 period).

Great, thanks.  That's as I thought, then.

-Carl


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Message: 11298

Date: Mon, 05 Jul 2004 09:49:03

Subject: Re: from linear to equal

From: monz

hi Gene and Paul,


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

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
>  
> > > You'd maybe prefer Fokker?
> > 
> > Did Fokker have a particular route along the circle of fifths that he 
> > preferred to get 11?
> 
> I doubt it. These two temperaments should both probably be melted down
> into 31 equal, however, which of course makes them the same; hence
> huygens or fokker might be good names.


huygens or fokker are indeed the two most appropriate names
for 31edo.

but if your main criteria in naming is to honor someone
who advocated 11-limit, a good choice might be Ptolemy.
his "smooth (or 'equable' as quoted by Partch) diatonic" 
and "syntonic chromatic" genera both used ratios of 11.

here are the tetrachord structures of those two tunings:


Ptolemy - _genos homalon diatonon_ = "even diatonic genus"
http://tonalsoft.com/enc/diatonic.htm#equable *

>> string-length proportions:    9 : 10 : 11 : 12
>> 
>> 
>> note ...... ratio ... ~ cents
>> 
>> mese ...... 1/1 ....... 0
>> .................................... > . 9:10 . ~ 182.4037121 cents
>> lichanos .. 9/10 .. - 182.4037121
>> .................................... >. 10:11 . ~ 165.0042285 cents
>> parhypate . 9/11 .. - 347.4079406
>> .................................... >. 11:12 . ~ 150.6370585 cents
>> hypate .... 3/4 ... - 498.0449991
>> 
>> The string-length proportions of Ptolemy's "even diatonic" 
>> have the smallest-number consecutive ratios which can describe 
>> a four-fold division of the 4:3 "perfect-4th". The top interval
>> is thus the 5-limit 10:9 "lesser tone", the middle interval is
>> the 11:10 "undecimal tone", and the bottom interval is the 
>> 12:11 "neutral 2nd" functioning as a very wide semitone.


Ptolemy - _genos syntonon chromatikon_ = "tense chromatic genus"
http://tonalsoft.com/enc/chromatic.htm#ptolemy-tense *

>> string-length proportions:    66 : 77 : 84 : 88
>> 
>> 
>> note ...... ratio ... ~ cents
>> 
>> mese ....... 1/1 ...... 0
>> .................................... >. 6:7 .. ~ 266.8709056 cents
>> lichanos ... 6/7 .. - 266.8709056
>> .................................... > 11:12 . ~ 150.6370585 cents
>> parhypate . 11/14 . - 417.5079641
>> .................................... > 21:22 . ~  80.53703503 cents
>> hypate ..... 3/4 .. - 498.0449991
>> 
>> In his "tense" version of the chromatic genus, Ptolemy used
>> the narrow 7:6 "septimal minor-3rd" as his "characteristic
>> interval" at the top of the tetrachord, thus simultaneously
>> placing his lichanos a wide 8:7 "septimal tone" above the
>> bottom note hypate. The middle interval is the 12:11 
>>"undecimal neutral 2nd", functioning here as a very wide semitone.



in a later era, Partch might be the obvious choice
... but then again, he stressed JI so much that it
would never be a good idea to honor him by naming a
temperament after him.  he was philosophically opposed
to temperament, you could almost say on moral grounds.
Ben Johnston speaks of temperament in much the same way.
and of course, here on the list Jon Szanto, Kraig Grady,
and David Beardsley and Pat Pagano would hold similar views.


-monz






 





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