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Message: 7850

Date: Fri, 31 Oct 2003 17:49:46

Subject: Re: Eponyms

From: Carl Lumma

>Another vague idea: The order of mention of primes could be different
>depending whether they are being multiplied (dot) or divided (colon).

Again a cool idea, and I find these sort of inquiries fascinating, but
I try to avoid them when I can't see them being very useful.  YMMV.

>Tanaka's kleisma (_the_ kleisma) has the systematic name of
>5^6-kleisma (five-to-the-six-kleisma)

I've so far tried my best not to mention the term "anal retentive".
:)

>> 385:383 is in that range.
>
>Yes, but the system says that the only factors omitted from the first
>part of the name are factors of 2 and 3. 383 contains other primes (in
>fact _is_ a rather large prime) which would therefore have to be
>upfront in the name.

Yeah, sorry I didn't catch that until later.

>To convert a comma ratio to its systematic name:
>
>1. Remove all factors of 2 and 3.
>2. Replace slash with colon.
>3. Swap the two sides of the ratio if necessary to put the smallest
>number first.
>4. If it now starts with "1:", eliminate the "1:".
>5. If any side of the (2,3-reduced) ratio is bigger than 125 (or maybe
>385) then give its prime factorisation in some form (details yet to be
>decided). 
>6. Calculate the comma size in cents and use it to look up and append
>the category name, preceded by a hyphen.
>
>This is not guaranteed to give a unique name (although clashes will be
>exceedingly rare). To be certain that your comma actually deserves the
>name, you have to run the process in reverse (as I've described
>already) trying 3-exponents in the series 0, 1, -1, 2, -2, 3, -3, ...
>and octave reducing, until you get a hit on the correct size-category.
>Then see if you've got your original comma ratio back again.

Again, nice touch but...

-Carl


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Message: 7851

Date: Fri, 31 Oct 2003 10:16:14

Subject: Re: Eponyms

From: Carl Lumma

>>5.7.11-kleisma has no advantages over 385/384 that I can see.

The latter must be factored to see what it's good for, and
log'ed to give an exact size.  The former gives a size range,
and with the addition of the 3 exponent tells you what it's
good for (otherwise how'reyou going to say what pythagorean
commas are good for?).  But with the addition of the 3 exponent,
we loose the ability to draft size ranges.  What say you to
this, Dave? 

-Carl


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Message: 7852

Date: Fri, 31 Oct 2003 20:51:46

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> Gene wrote:
> >5.7.11-kleisma has no advantages over 385/384 that I can see.
> 
> I think so too. It looks like it's the simplest undecimal
> kleisma and there isn't another one called that so I'll change
> the name in "undecimal kleisma".

A fine name. Go for it.


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Message: 7853

Date: Fri, 31 Oct 2003 21:09:22

Subject: Nameable 11-limit commas

From: Gene Ward Smith

Here are reasonable 11-limit commas which either don't have a name or 
(45/44) not a very good one. It seems to me that at minimum one might 
tack a name on all the superparticulars.

77/75, 45/44 (1/5 tone??), 55/54, 56/55, 245/242, 121/120, 1331/1323, 
176/175, 3136/3125, 441/440, 1375/1372, 6250/6237, 540/539, 
4000/3993, 5632/5625, 43923/43904, 3025/3024, 151263/151250, 
3294225/3294172


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Message: 7854

Date: Fri, 31 Oct 2003 21:11:43

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>5.7.11-kleisma has no advantages over 385/384 that I can see.
> 
> The latter must be factored to see what it's good for, and
> log'ed to give an exact size.  The former gives a size range,
> and with the addition of the 3 exponent tells you what it's
> good for (otherwise how'reyou going to say what pythagorean
> commas are good for?).  But with the addition of the 3 exponent,
> we loose the ability to draft size ranges.  What say you to
> this, Dave? 

If you want to make this systematic, why not simply monzo-size range?


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Message: 7855

Date: Fri, 31 Oct 2003 21:20:46

Subject: Re: Linear temperament names?

From: Paul Erlich

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

> > > You can't assume that a lack of response means that everyone 
assents
> > > to a name forever. I think you've gone overboard in naming so 
many
> > > things long before anyone actually _needed_ a name for them.
> > 
> > If you want to blame someone for that, why not pick on Paul, who 
has 
> > named 5-limit nanotemperaments no one could possibly ever use in 
> > practice.
> 
> OK. Sorry if I've unfairly "picked on" you. I wasn't aware these had
> come from Paul. Paul, please consider yourself "picked on".

Despite his opinion above, Gene first gave the data for all of these, 
as well as much more complex ones, and i just named the ones simpler 
than the atom of Kirnberger so that one would have something other 
than numbers, numbers, numbers, to refer to on Monz's ET page.

> Well I dunno about anyone else, but saying "two chains of 183 c
> generators" is something I can associate with a helluvalot better,

But it doesn't uniquely signify a temperament. I believe Graham has 
discussed 13-limit generators which are fifths within a fraction of a 
cent in optimal size. In order to actually temper, a temperament must 
have a mapping associated with it.

> IMHO the 5-limit temperament you describe above is so complex it
> doesn't need a name at all.

It describes 46, 125, 171, 217 equal temperaments, barely used so far 
but why not? Look, if it'll make you happy i'll replace all the names 
on Monz's page with SINGLE LETTERS except for meantone and schismic 
(or whatever you say) . . . that way the *signification purpose* is 
not lost . . .

> Whether we have sharp cutoffs or gradual
> rolloffs on error and complexity, there has to be _some_ such for
> naming purposes. Doesn't there? 

The atom of kirnberger is of historical interest as a result of the 
means of setting one of its associated temperaments, namely 12-equal. 
It's a bit outside the usual direct application of temperament, which 
is why i put 12 in paretheses in the table.


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Message: 7856

Date: Fri, 31 Oct 2003 21:23:52

Subject: Re: UVs for 46-ET 11-limit PB

From: Paul Erlich

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

> OK, good!  so then that means that the method being used
> in our software is producing the same results as those used
> by you and Gene, correct?

right, but you'd get better matches to your 'closest-to-origin' 
periodicity blocks if you used coordinate ranges of -.5 to .5 instead 
of 0 to 1; i.e., putting 1/1 in the center of the block instead of a 
corner.


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Message: 7857

Date: Fri, 31 Oct 2003 21:30:45

Subject: Re: UVs for 46-ET 11-limit PB

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi paul,
> 
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > when i get rid of that bad line in the program, this 896:891, 
> > 385:384, 125:126, and 176:175 block becomes:
> > 
> >     ratio         3^           5^           7^          11^
> >       1            0            0            0            0      
> >    100/99         -2            2            0           -1      
> >     33/32          1            0            0            1      
> >     25/24         -1            2            0            0      
> >     16/15         -1           -1            0            0      
> >     15/14          1            1           -1            0      
> >     35/32          0            1            1            0      
> >     10/9          -2            1            0            0      
> >      9/8           2            0            0            0      
> >      8/7           0            0           -1            0      
> >      7/6          -1            0            1            0      
> >     33/28          1            0           -1            1      
> >      6/5           1           -1            0            0      
> >     40/33         -1            1            0           -1      
> >     99/80          2           -1            0            1      
> >      5/4           0            1            0            0      
> >     32/25          0           -2            0            0      
> >      9/7           2            0           -1            0      
> >     21/16          1            0            1            0      
> >      4/3          -1            0            0            0      
> >     27/20          3           -1            0            0      
> >     48/35          1           -1           -1            0      
> >      7/5           0           -1            1            0      
> >     64/45         -2           -1            0            0      
> >     10/7           0            1           -1            0      
> >     35/24         -1            1            1            0      
> >     40/27         -3            1            0            0      
> >      3/2           1            0            0            0      
> >     32/21         -1            0           -1            0      
> >     14/9          -2            0            1            0      
> >     25/16          0            2            0            0      
> >      8/5           0           -1            0            0      
> >    160/99         -2            1            0           -1      
> >     33/20          1           -1            0            1      
> >      5/3          -1            1            0            0      
> >     56/33         -1            0            1           -1      
> >     12/7           1            0           -1            0      
> >      7/4           0            0            1            0      
> >     16/9          -2            0            0            0      
> >      9/5           2           -1            0            0      
> >     64/35          0           -1           -1            0      
> >     28/15         -1           -1            1            0      
> >     15/8           1            1            0            0      
> >     48/25          1           -2            0            0      
> >     64/33         -1            0            0           -1      
> >     99/50          2           -2            0            1     
> > 
> > i think this is the closest yet to fulfilling monz's original 
> > requirements . . .
> 
> 
> 
> yes, indeed! -- this is *very* close to the original
> pseudo-PB that i devised by eye, trying to keep all notes
> as close as possible (according to the rectangular metric)
> to the 1/1.
> 
> 
> of course, the biggest difference is that your PB contains
> only one instance of each note, whereas my pseudo-PB had
> several duplicates and triplicates which were the same
> number of steps from 1/1.  i realize that i could use your
> unison-vectors to find similar duplicates/triplicates in
> your PB.
> 
> 
> but brushing that aside (since i knew about it and expected
> it from the beginning), the two big differences between 
> yours and mine are: 
> 
> 1) your first (after 1/1) and last notes contain both 7 
>    and 11 as factors, whereas all notes in mine had either
>    7 *or* 11 *or* neither; and

the first note after 1/1 is

> >    100/99         -2            2            0           -1    

and the last is

> >     99/50          2           -2            0            1    

so i don't know whay you mean. the only notes i have with both 7s and 
11s are

> >     33/28          1            0           -1            1   

and

> >     56/33         -1            0            1           -1  

but these have *opposite* signs on 7 and 11, so are no more complex 
than having 11 by itself.

> now, to continue the puzzle:  can you or Gene (or another
> adventurous tuning-math-er) find a PB which corrects those
> two conditions?

it might be impossible, but i'll keep trying.


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Message: 7858

Date: Fri, 31 Oct 2003 13:40:29

Subject: Re: Eponyms

From: Carl Lumma

>> The latter must be factored to see what it's good for, and
>> log'ed to give an exact size.  The former gives a size range,
>> and with the addition of the 3 exponent tells you what it's
>> good for (otherwise how'reyou going to say what pythagorean
>> commas are good for?).  But with the addition of the 3 exponent,
>> we loose the ability to draft size ranges.  What say you to
>> this, Dave? 
>
>If you want to make this systematic, why not simply monzo-size range?

Example?

-Carl


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Message: 7859

Date: Fri, 31 Oct 2003 13:41:56

Subject: Re: Linear temperament names?

From: Carl Lumma

>Look, if it'll make you happy i'll replace all the names 
>on Monz's page with SINGLE LETTERS

Nooooooooooooooooooooooo!

-Carl


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Message: 7860

Date: Fri, 31 Oct 2003 22:08:30

Subject: Re: UVs for 46-ET 11-limit PB

From: monz

hi paul,

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > the two big differences between yours [46-tone 11-limit PB]
> > and mine are: 
> > 
> > 1) your first (after 1/1) and last notes contain both 7 
> >    and 11 as factors, whereas all notes in mine had either
> >    7 *or* 11 *or* neither; and
> 
> the first note after 1/1 is
> 
> > >    100/99         -2            2            0           -1    
> 
> and the last is
> 
> > >     99/50          2           -2            0            1    
> 
> so i don't know what you mean. the only notes i have with
> both 7s and 11s are
> 
> > >     33/28          1            0           -1            1   
> 
> and
> 
> > >     56/33         -1            0            1           -1  
> 
> but these have *opposite* signs on 7 and 11, so are no more complex 
> than having 11 by itself.



oops ... my bad!  i should have taken another look before
i typed that.  yes, those are the two notes i was referring to.
in the graphic i posted last night, you can see them in grey
at the bottom of the left side of your PB.



 
> > now, to continue the puzzle:  can you or Gene (or another
> > adventurous tuning-math-er) find a PB which corrects those
> > two conditions?
> 
> it might be impossible, but i'll keep trying.


thanks!

... but at this point, with the advances we've made in the
software over the last day, i can have fun myself just
trying out different unison-vectors.

i'm beginning to gain an awful lot of respect for 46-ET.



-monz


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Message: 7861

Date: Fri, 31 Oct 2003 00:08:56

Subject: Re: Linear temperament names?

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:
> 
> > Yes. I saw that one. The only posts containing the word "Unidec" 
> > come from you. 
> 
> Until now, I seem to have been the only person to take an interest in 
> the temperament. Is that my fault?

Of course not. That wasn't my point. I remind you: 

You wrote: "It may not be a very good name, but if you think that
you've waited a long time to say so."

I wrote: "... But it isn't as if lots of people are already using the
name, so I don't see a problem with changing it."

You wrote: "I've already have up a posting devoted *solely* to unidec."

I wrote: Yes. I saw that one. The only posts containing the word
"Unidec" come from you.

My point is that it wasn't actually necessary to name it back then
because there was only one person interested in it.

When in 1998 I wrote a whole article with extensive diagrams about
what is now called kleismic, it had already been discussed on the
tuning list by at least four people, but we apparently saw no need to
name it at that stage, unless you count "chain of minor thirds" as a
name. The word "kleismic" does not appear in the article, although of
course the kleisma gets a mention.

It took another 3 years before the term "kleismic" was applied to it.

> > You can't assume that a lack of response means that everyone assents
> > to a name forever. I think you've gone overboard in naming so many
> > things long before anyone actually _needed_ a name for them.
> 
> If you want to blame someone for that, why not pick on Paul, who has 
> named 5-limit nanotemperaments no one could possibly ever use in 
> practice.

OK. Sorry if I've unfairly "picked on" you. I wasn't aware these had
come from Paul. Paul, please consider yourself "picked on".

Whoever started it, it looks like everyone has gotten a bit carried
away with the fun of giving cute names to every temperament or comma
in sight. 

I think that may be unfair to those who may come after us. They may
not have a clue what we were talking about. Such names are far more in
need of a "secret decoder ring" than systematic names are.

I'm thinking maybe we've had our fun now, and we should do like in
chemistry. First you use only the systematic name (or no name at all,
just a description in terms of period and generators, in cents if
necessary), and only if a particular temperament becomes a hot topic
of conversation, particularly if someone actually uses it for music or
notation or instrument building, _then_ those people can look at
giving it a less boring common name.

> I have NOT named things needlessly; I want to talk about 
> temperaments, 

As above.

> and while the wedgie or TM comma basis might supply a 
> name, they really don't work with human beings. You can't tell someone
> "Oh yes, that's the  [12, 22, -4, -6, 7, -40, -51, -71, -90, -3]
> temperament";

I totally agree.

> but saying "Oh yes, that's unidec" gives something you 
> might be able to associate with.

Well I dunno about anyone else, but saying "two chains of 183 c
generators" is something I can associate with a helluvalot better, and
if that can be condensed systematically into a name then it's a better
name in my book. I'm sorry if that's boring, but excitement isn't
always the most important thing to me. You seem to rarely give optimum
generator sizes in cents in your posted lists of temperaments.

> > All I'm asking is - if you have a system for the more descriptive
> > names (in particular those based on the generator and period) what 
> is
> > it? And if you don't, can we make some improvements in that 
> direction?
> 
> If you want to propose a completely systematic naming proceedure, 
> then have at it. Name everything the Keenan way, and make every name 
> tell you just exactly what the temperament is. Maybe people will like 
> it, and adopt your scheme. However, I see little point in half-
> measures.

As you should realise by now, I'm not proposing "half-measures", but
double measures. Systematic names _and_ common names. Although most
temperaments would automatically _have_ systematic names, there would
of course be no point in using them for really common things like
meantone or schismic. But for temperaments that have been discovered,
but not extensively discussed or used, the systematic name should
probably be the only name. Other temperaments would be at an in
between stage where we might need to use both their systematic and
common names for a while. So we'd need two "name" columns in any
temperament database. 

The same situation occurs not only in chemistry, but also biology
(taxonomy). Athough in taxonomy there is an unfortunate fondness for
eponyms (naming after the discoverer) among the more descriptive
terminology. What's more, common names tend to differ from place to
place. People not on the tuning lists may have discovered and named
some of these temperaments, and I suppose they would be entitled to
keep using their common names, but systematic names would be, well,
systematic.

Is this a dumb idea?

Now of course I would rather such a system was obtained by consensus
(at least of those on this list who have an interest), and so would
you. You're just saying go ahead and do it the "Keenan" way because
you're still sore at me.

> Minortone is the 5-limit 50031545098999707/50000000000000000 
> comma system, extendible to 7-limit. 17 10/9's make up a 6, and 35 a 
> 40.

This brings up a good point. Log-flat badness isn't much good for
deciding which temperament gets a particular name, because you'll
always be able to go far enough out in complexity that you will find a
"less-bad" mapping with a generator that's very close to the one
you're trying to name.

IMHO the 5-limit temperament you describe above is so complex it
doesn't need a name at all. Whether we have sharp cutoffs or gradual
rolloffs on error and complexity, there has to be _some_ such for
naming purposes. Doesn't there? 

My views on that are already on record. This temperament has more than
twice the weighted rms complexity of 10 that I must have imagined some
of us had agreed on as a reasonable cutoff for 5-limit (_if_ you must
have a sharp cutoff).


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Message: 7862

Date: Fri, 31 Oct 2003 22:09:09

Subject: Re: 'neutral' intervals

From: Paul Erlich

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

> Pertaining to the discussion of interval names, I believe that the 
> term 'neutral', although it seems sensible enough, is 
> problematic in practice and should be reconsidered.
> 
> The concern arises because a so-called neutral third sounds 
> 'major' in one context and 'minor' in another.

That's usually only a result of insufficient exposure to the music in 
question. Arabic musicians don't hear their neutral thirds as 'major' 
in one context and 'minor' in another, same for neutral seconds.

> Use of these 
> 'neutral' intervals in a melodic context almost never results in a 
> musical functionality which could be described as 'neutral'.

Almost never?

> In my 
> experience, the same ambiguity presents itself when these 
> intervals are used harmonically.

So you mean almost never *in your experience*?

> This has led me to call these intervals 'narrow major' or 'wide 
> minor' depending on context. This dual perspective is 
> advantageous for descriptions of musical function.

i beg to differ.


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Message: 7863

Date: Fri, 31 Oct 2003 00:56:52

Subject: Re: UVs for 46-ET 11-limit PB

From: monz

hi paul,

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

> d'oh! i was using format rat, which approximates ratios
> that are too complex with simpler ratios! 


yeah, you have to watch out for that kind of thing!

i used to have lots of problems with that when using Excel,
until i finally just did everything using monzos instead
of ratios.



> the real ratios are . . . well who cares, they agree 
> perfectly with monz's, and of course the monzos 
> below agree . . .


OK, good!  so then that means that the method being used
in our software is producing the same results as those used
by you and Gene, correct?  that was the reason why i started
this thread in the first place, to check that we're going
about things the right way.



-monz


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Message: 7864

Date: Fri, 31 Oct 2003 22:14:26

Subject: Re: UVs for 46-ET 11-limit PB

From: Paul Erlich

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

> ... but at this point, with the advances we've made in the
> software over the last day, i can have fun myself just
> trying out different unison-vectors.

great!


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Message: 7865

Date: Fri, 31 Oct 2003 01:17:46

Subject: Eponyms

From: Dave Keenan

On the subject of eponyms:

Manuel, I'd prefer it if Scala did not refer to 384:385 as Keenan's
kleisma, although I thank Paul for his sentiments in proposing it.

Now that I've found what I think is a good system for naming kommas,
I'd prefer it to be called "385-kleisma" or "5.7.11-kleisma". I think
I prefer the latter, and would pronounce it "five seven eleven kleisma".

Does anyone have any objection to this, or want to propose another name?

Regards,
-- Dave Keenan


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Message: 7866

Date: Fri, 31 Oct 2003 22:44:12

Subject: Re: 'neutral' intervals

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "gooseplex" <cfaah@e...> wrote:
> 
> > Pertaining to the discussion of interval names, I believe that 
the 
> > term 'neutral', although it seems sensible enough, is 
> > problematic in practice and should be reconsidered.
> > 
> > The concern arises because a so-called neutral third sounds 
> > 'major' in one context and 'minor' in another.
> 
> That's usually only a result of insufficient exposure to the music 
in 
> question. Arabic musicians don't hear their neutral thirds 
as 'major' 
> in one context and 'minor' in another, same for neutral seconds.
> 
> > Use of these 
> > 'neutral' intervals in a melodic context almost never results in 
a 
> > musical functionality which could be described as 'neutral'.
> 
> Almost never?
> 
> > In my 
> > experience, the same ambiguity presents itself when these 
> > intervals are used harmonically.
> 
> So you mean almost never *in your experience*?
> 
> > This has led me to call these intervals 'narrow major' or 'wide 
> > minor' depending on context. This dual perspective is 
> > advantageous for descriptions of musical function.
> 
> i beg to differ.

I can't imagine how anyone (such as Margo Schulter or myself) who has 
spent any significant amount of time using a 17-tone temperament, 
either equal or well-tempered (to improve non-5 ratios of 7), could 
avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly 
neutral.  A prime example of this is the two-voice progression 
consisting of the interval of an augmented 6th, Eb3-C#4, resolved to 
an octave, D3-D4, by melodic intervals of 2 degrees of 17, which are 
unmistakably neutral 2nds.

--George


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Message: 7867

Date: Fri, 31 Oct 2003 22:50:27

Subject: Re: 'neutral' intervals

From: Paul Erlich

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

> I can't imagine how anyone (such as Margo Schulter or myself) who 
has 
> spent any significant amount of time using a 17-tone temperament, 
> either equal or well-tempered (to improve non-5 ratios of 7), could 
> avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly 
> neutral.  A prime example of this is the two-voice progression 
> consisting of the interval of an augmented 6th, Eb3-C#4, resolved 
to 
> an octave, D3-D4, by melodic intervals of 2 degrees of 17, which 
are 
> unmistakably neutral 2nds.
> 
> --George

George,

I'm confused. Wouldn't the resolving intervals be 1 degree of 17 each?

-Paul


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Message: 7868

Date: Fri, 31 Oct 2003 01:46:31

Subject: Re: UVs for 46-ET 11-limit PB

From: monz

hi paul,


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

> when i get rid of that bad line in the program, this 896:891, 
> 385:384, 125:126, and 176:175 block becomes:
> 
>     ratio         3^           5^           7^          11^
>       1            0            0            0            0      
>    100/99         -2            2            0           -1      
>     33/32          1            0            0            1      
>     25/24         -1            2            0            0      
>     16/15         -1           -1            0            0      
>     15/14          1            1           -1            0      
>     35/32          0            1            1            0      
>     10/9          -2            1            0            0      
>      9/8           2            0            0            0      
>      8/7           0            0           -1            0      
>      7/6          -1            0            1            0      
>     33/28          1            0           -1            1      
>      6/5           1           -1            0            0      
>     40/33         -1            1            0           -1      
>     99/80          2           -1            0            1      
>      5/4           0            1            0            0      
>     32/25          0           -2            0            0      
>      9/7           2            0           -1            0      
>     21/16          1            0            1            0      
>      4/3          -1            0            0            0      
>     27/20          3           -1            0            0      
>     48/35          1           -1           -1            0      
>      7/5           0           -1            1            0      
>     64/45         -2           -1            0            0      
>     10/7           0            1           -1            0      
>     35/24         -1            1            1            0      
>     40/27         -3            1            0            0      
>      3/2           1            0            0            0      
>     32/21         -1            0           -1            0      
>     14/9          -2            0            1            0      
>     25/16          0            2            0            0      
>      8/5           0           -1            0            0      
>    160/99         -2            1            0           -1      
>     33/20          1           -1            0            1      
>      5/3          -1            1            0            0      
>     56/33         -1            0            1           -1      
>     12/7           1            0           -1            0      
>      7/4           0            0            1            0      
>     16/9          -2            0            0            0      
>      9/5           2           -1            0            0      
>     64/35          0           -1           -1            0      
>     28/15         -1           -1            1            0      
>     15/8           1            1            0            0      
>     48/25          1           -2            0            0      
>     64/33         -1            0            0           -1      
>     99/50          2           -2            0            1     
> 
> i think this is the closest yet to fulfilling monz's original 
> requirements . . .



yes, indeed! -- this is *very* close to the original
pseudo-PB that i devised by eye, trying to keep all notes
as close as possible (according to the rectangular metric)
to the 1/1.


of course, the biggest difference is that your PB contains
only one instance of each note, whereas my pseudo-PB had
several duplicates and triplicates which were the same
number of steps from 1/1.  i realize that i could use your
unison-vectors to find similar duplicates/triplicates in
your PB.


but brushing that aside (since i knew about it and expected
it from the beginning), the two big differences between 
yours and mine are: 

1) your first (after 1/1) and last notes contain both 7 
   and 11 as factors, whereas all notes in mine had either
   7 *or* 11 *or* neither; and

2) your PB does *not* include 11/8 or 16/11, which i felt
   should be included.



now, to continue the puzzle:  can you or Gene (or another
adventurous tuning-math-er) find a PB which corrects those
two conditions?



-monz


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Message: 7869

Date: Fri, 31 Oct 2003 22:59:23

Subject: Re: 'neutral' intervals

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:
> 
> > I can't imagine how anyone (such as Margo Schulter or myself) who 
> has 
> > spent any significant amount of time using a 17-tone temperament, 
> > either equal or well-tempered (to improve non-5 ratios of 7), 
could 
> > avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly 
> > neutral.  A prime example of this is the two-voice progression 
> > consisting of the interval of an augmented 6th, Eb3-C#4, resolved 
> to 
> > an octave, D3-D4, by melodic intervals of 2 degrees of 17, which 
> are 
> > unmistakably neutral 2nds.
> > 
> > --George
> 
> George,
> 
> I'm confused. Wouldn't the resolving intervals be 1 degree of 17 
each?
> 
> -Paul

Oops, sorry -- I was in too much of a hurry!  The first interval 
should be a major 6th with tones E-semiflat and C-semisharp.

--George


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Message: 7870

Date: Fri, 31 Oct 2003 01:56:55

Subject: Re: UVs for 46-ET 11-limit PB

From: monz

i've uploaded a graphic to tuning_files, showing both my
original pseudo-PB and paul's latest PB, for 46-tone 11-limit:


Yahoo groups: /tuning_files/files/monz/compact... * [with cont.] 
et_pb.gif

or

Sign In - * [with cont.]  (Wayb.)


i know that it's too small for the numbers and letters to
be legible, but the point is simply to see by the colors
which notes are in the PB and which are not.

in both diagrams, grey shading indicates notes which occur
only one time in the PB.

my original pseudo-PB, blue indicates duplicate notes and
green indicates triplicate, which are the same number of
(rectangular metric) steps away from 1/1.  ... the brown 
shading was only used to keep track of notes and can be
ignored.



-monz







--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> yes, indeed! -- this is *very* close to the original
> pseudo-PB that i devised by eye, trying to keep all notes
> as close as possible (according to the rectangular metric)
> to the 1/1.
> 
> 
> of course, the biggest difference is that your PB contains
> only one instance of each note, whereas my pseudo-PB had
> several duplicates and triplicates which were the same
> number of steps from 1/1.  i realize that i could use your
> unison-vectors to find similar duplicates/triplicates in
> your PB.
> 
> 
> but brushing that aside (since i knew about it and expected
> it from the beginning), the two big differences between 
> yours and mine are: 
> 
> 1) your first (after 1/1) and last notes contain both 7 
>    and 11 as factors, whereas all notes in mine had either
>    7 *or* 11 *or* neither; and
> 
> 2) your PB does *not* include 11/8 or 16/11, which i felt
>    should be included.
> 
> 
> 
> now, to continue the puzzle:  can you or Gene (or another
> adventurous tuning-math-er) find a PB which corrects those
> two conditions?
> 
> 
> 
> -monz


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Message: 7871

Date: Fri, 31 Oct 2003 23:06:44

Subject: Nameable 13-limit

From: Gene Ward Smith

13-exact-limit intervals with numerator-denominator < 3 are:

13/12, 14/13, 26/25, 27/26, 40/39, 65/64, 66/65, 78/77, 91/90, 
105/104, 275/273, 144/143, 169/168, 196/195, 325/324, 351/350,
352/351, 364/363, 847/845, 625/624, 676/675, 729/728, 1575/1573,
1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655,
10648/10647, 123201/123200

The remarkable 123201/123200 might be named the chalmersia, since John
Chalmers is presumably the first to see it.


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Message: 7872

Date: Fri, 31 Oct 2003 02:30:51

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Now that I've found what I think is a good system for naming kommas,
> >I'd prefer it to be called "385-kleisma"
> 
> In your scheme, the term kleisma tells us that the denominator must
> be 384, and not 383?
> 
> >or "5.7.11-kleisma".
> 
> ...tells us how to combine factors of 5, 7, and 11 to get the
> right ratio?

I already did. Sorry I didn't give examples.

The 5, 7 and 11 are all on the same side of the ratio, or there would
have been a colon ":" in there. They are all only to the power given,
namely 1. So we can immediately fill in the monzo for all the primes
greater than two [? ? 1 1 1].

It's a kleisma so it's in the range 4.5 c (a bit arbitrary at present)
to 11.7 c (actually, exactly half a pythagorean comma).

Try successive exponents of 3 in this sequence 0, 1, -1, 2, -2, 3, -3
... and with each of those, whatever power of 2 that octave-reduces it
to lowest terms, i.e. puts it in the range -600 to +600 cents. As soon
as you hit one whose absolute value in cents is actually in the
kleisma range, you've found it. If it is a negative number of cents,
negate all the exponents in the monzo.

That's how a dumb algorithm would have to do it, but you or I
(assuming we knew something about the system) would say: Its got 385
as a factor along with some powers of 2 and 3. I know roughly how big
it is so I wonder if it's 386/385 or 385/384. Oh 384 has prime factors
of only 2's and 3's. Calculate size in cents. Yep that's it, 385/384.

The careful choice of the range boundaries for schismina, schisma,
kleisma, comma, small diesis, medium diesis, large diesis, etc. (at
square roots of various 3-kommas) is what makes the powers of 3 and 2
unambiguous.

But even before you've done any such processing, you immediately know
roughly how big it is and what its good for, namely turning 7-limit
into an approximation of 11-limit).


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Message: 7873

Date: Fri, 31 Oct 2003 23:11:52

Subject: Re: Eponyms

From: Gene Ward Smith

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

> >If you want to make this systematic, why not simply monzo-size range?
> 
> Example?

225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
[-7,-1,1,1,1]-kleisma.


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Message: 7874

Date: Fri, 31 Oct 2003 03:42:50

Subject: Komma category boundaries (was: Eponyms)

From: Dave Keenan

If you want to see potential values for komma-size-category
boundaries, put the integers from 1 to 53 in spreadsheet column A.
These represent exponents of 3. In column B calculate the associated
exponent of 2 as
=ROUND(A1 * LN(3)/LN(2), 0)
and in column C calculate the square root of the implied 3-komma in
cents as
=ABS(A1 * LN(3)/LN(2) - B1)*600

But we have to reject any where both the 2 and the 3 exponents are
even. This is because the square root would then be rational, and we
would have a 3-comma with an ambiguous category because it would be
right on the boundary. So change that to
=IF(ISEVEN(A1)*ISEVEN(B1), 0, ABS(A1 * LN(3)/LN(2) - B1)*600)

Then sort the list on column C, the size in cents.

It is more important to have category boundaries at square roots of
kommas with smaller exponents of 3, but not if they are too close to a
boundary for one with an even-lower 3-exponent.

The result follows, showing my proposal.

I note that we have to go out to 3^200 before we find a good place for
the schisma/kleisma boundary, at 4.499913461 cents = sqrt(3^200/2^317).

53	84	1.807522933  schismina/schisma
41	65	9.92248226
12	19	11.73000519  kleisma/comma
29	46	21.65248745
17	27	33.38249264  comma/small diesis
36	57	35.19001558
5	8	45.11249784  small diesis/medium diesis
46	73	55.0349801
7	11	56.84250303  medium diesis/large diesis
19	30	68.57250822  large diesis/?
22	35	78.49499048  ?
31	49	80.30251341
43	68	92.03251861
51	81	100.1474779
2	3	101.9550009  ?
39	62	111.8774831
27	43	123.6074883
26	41	125.4150113
15	24	135.3374935
3	5	147.0674987 ?
50	79	148.8750216
9	14	158.7975039
32	51	168.7199862
21	33	170.5275091
33	52	182.2575143
8	13	192.1799965
45	71	193.9875195
49	78	202.1024788
37	59	213.832484
16	25	215.6400069
25	40	225.5624892
13	21	237.2924944
40	63	239.1000173
1	2	249.0224996 ?
42	67	258.9449818
11	17	260.7525048
23	36	272.48251
18	29	282.4049922
35	55	284.2125151
47	74	295.9425203

Again, the point of these boundaries is that they let you extract the
powers of 2 and 3 from the comma name, if it includes enough
information about the exponents of the primes higher than 3.

-- Dave Keenan


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