Tuning-Math Digests messages 7300 - 7324

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Message: 7300

Date: Wed, 13 Aug 2003 00:25:17

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

hi carlos,

one problem i have with fokker's metric is that it doesn't consider 
the major sixth or minor third. major triads with a given absolute 
error of the fifth and a given absolute error of the third can have 
two possible errors for the minor third. when one of these is near 
zero, the triad sounds considerably purer. the effect seems at least 
as important for the minor triad.

around here we usually compute misfits using all the intervals within 
a certain odd limit (following partch), rather than all the harmonics 
or all the primes up to some limit. many weighting schemes have been 
considered, but equal weighting doesn't seem too objectionable to 
anyone, i think . . .

you might be interested to read my papers,
http://www-math.cudenver.edu/~jstarret/22ALL.pdf - Ok *
A gentle introduction to Fokker periodicity blocks, part 1, *

-paul


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Message: 7304

Date: Wed, 13 Aug 2003 23:24:50

Subject: Shenkerian analysis

From: Carl Lumma

>>Schenker's motto, appearing on the title page of Der Freie Satz,
>>was: Semper idem sed non eodem modo.
>>
>>"Always the same, but not in the same way."
>
>I just learned what Shenkerian analysis was back in January.
>Totally cool.

On these lists I've often hinted at something like "parathesis
checking" (an algorithm that checks if there is a close-bracket
for every open bracket) for studying symmetrical melodies as
found in Mozart, Sousa, Joplin, etc.  Kinda reminiscent of
Shenkerian analysis.  Also, Boomsliter & Creel's "extended
reference".

The way I was shown, two different people might produce two
very different Shenkerian breakdowns of the same piece.  I
wonder if anyone's tried to automate it in a deterministic
fashion?

-Carl


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Message: 7306

Date: Thu, 14 Aug 2003 12:45:55

Subject: Re: interval vector

From: Graham Breed

pitchcolor:

>>In case this has not been clarified, the above use of the 
>>    
>>
>term 'interval 
>  
>
>>vector' is non-standard. According to Forte / Rahn pitch-set 
>>    
>>
>theory, which is 
>  
>
>>still standard in academia, an interval vector is an ordered 
>>    
>>
>enumerated list of 
>  
>
>>six (twelve-tone) interval classes (ic) which are present in a 
>>    
>>
>pitch set.
>  
>
Oh.  Richmond Browne (1981) simply calls that a "vector", which is 
obviously far too general and not even correct in the general case.

Gene:

>I've taken to calling the row vector of exponents a "monzo", since I 
>needed a name to use when documenting my Maple programs for my own 
>purposes. So far as I know, academia has not taken this as 
>standard. :)
>  
>
But that only a subset of what I called an "interval vector" because it 
only covers JI defined using consecutive prime numbers.


                                  Graham


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Message: 7309

Date: Thu, 14 Aug 2003 19:52:31

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

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

> i was just pointing out that whatever the metric is
> measuring, if it's meant to be used over the whole
> tuning system, it only needs those 3 intervals in the
> 5-limit, or those 6 in the 7-limit, etc., to cover
> the metric for any interval in the system.

that still doesn't make any sense to me. reminds me of a mysterious 
part of your woolhouse page that i criticized way back, though.


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Message: 7310

Date: Thu, 14 Aug 2003 19:55:38

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> >actually, 53-equal suffers from an additional problem with this 
type 
> >of calculation, whether fokker's original or modified as you have 
> >above. the problem is "inconsistency"... 
> >
> Fokker only looked at prime numbers, so consistency wasn't an issue.
> 
> 
>                                   Graham

sorry, you're right -- so it's only when minor thirds and non-prime-
harmonic primary ratios are included, for example in the analyses of 
wendy carlos (!) and yunik & swift, that inconsistency calls the 
results into question. stoney had his own way of enforcing 
consistency, which was a bit ad hoc, but generally yields better 
results than just using the best approximation to each prime.


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Message: 7311

Date: Thu, 14 Aug 2003 19:58:07

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> 
wrote:
> > The intervals I have considered then are
> > 
> > Just fifth	3/2
> > Just mayor third	5/4
> > Just minor third	6/5
> > Harmonic seventh	7/4
> > Subminor fifth	7/5
> > Subminor third	7/6
> > Supersecond	8/7
> > Major tone	9/8
> > Super major third	9/7
> > Acute minor seventh	9/5
> > Trumpet interval	11/10
> > The 11th harmonic	11/8
> > Meshaqah quartertones	11/6
> > Unamed_1	11/9
> > Unamed_2	11/7
> 
> Having both 7/4 and 8/7 is redundent. If you eliminate that, you 
have 
> a table of representatives for the 14 11-limit consonances, in the 
> terminology of this group.

oops! thanks for catching that, gene!

personally, i would also include "9/6", since a complete 11-limit 
hexad contains both a 3:2 and a "9:6", so the perfect fifth should be 
weighted twice.


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Message: 7314

Date: Thu, 14 Aug 2003 20:56:37

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> stoney had his own way of enforcing 
> > consistency, which was a bit ad hoc, but generally yields better 
> > results than just using the best approximation to each prime.
> 
> There's always my suggestion of using the Zeta tuning.

ok, i just read _prime obsession_, but i think i know even less about 
the zetafunction than after reading manfred schroeder's _number 
theory in science and communication_ . . .

where can i read about zeta tuning again?


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Message: 7316

Date: Thu, 14 Aug 2003 01:04:06

Subject: Re: Comments about Fokker's misfit metric

From: Graham Breed

Carlos wrote:

>Also I have enforced that the octave should be absolutely just.  Fokker 
>actually discusses this aspects and he indicate that he sees no reason to 
>treat the octave any different. This is not my case I enforce the octave 
>to be exact and all the other intervals to be approximate.
>  
>
If the octaves weren't just, not only would you have to consider 
complements, like both 3:2 and 4:3, but also larger intervals like 3:1, 
5:2, 7:3 and whatever.  So it's a lot easier not to bother.

>Unamed_1	11/9
>Unamed_2	11/7
>
>The names are those of Ellis. Interestingly I could not find a name for the 
>two last ones.
>  
>
11/9 is a neutral third.

>The results still show that the 31 equally tempered scales is a very good 
>choice in all cases.  However, when one looks to more and more ratios 
>(like in the fourth case above) the option of 41 and 53 become more 
>attractive.
>
So it's 26.5 for 31 against 20.07 for 41.  Fokker deficiency is this 
number multiplied 2**((n-12)/12) for n notes to to the octave.  Even by 
his method, the 70.8 for 31 is so close to 71.0 for 41 as to be 
negligible.  With your figures I get a deficiency of 79.5 for 31 
compared to 109.7 for 41.  So 31 is now much better, as I suspected!

You can also see, as has been mentioned, that 72 is very good in all 
cases, and its 11-limit error is less than half the size of anything 
else.  Although, because it's so big, its deficiency is still 238.1.

>I undertstand that you like to include in the metric function then 
>intervals like 9/8 an others. Is that right?.
>  
>
Yes, for the 9 and higher limits.  But you can also look at the 7-limit, 
where 9 isn't included.

>Paul points out that the would rather include the minor thirds. I agree, I 
>thought about that myself. In fact, wouldn't  it be making more sense to 
>"a priori" select those intervals for which you want a good fit because 
>one plans to use them the most and the create a metric, maybe through a 
>weighting vector or so, that prioritizes them.
>  
>
Indeed, there's no one temperament that works in all situations.  So 
decide what intervals you want, and find a tuning that approximates them 
efficiently.


                                    Graham


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Message: 7318

Date: Thu, 14 Aug 2003 01:06:42

Subject: Re: Comments about Fokker's misfit metric

From: Graham Breed

monz@xxxxxxxxx.xxx wrote:

>i was just pointing out that whatever the metric is
>measuring, if it's meant to be used over the whole
>tuning system, it only needs those 3 intervals in the
>5-limit, or those 6 in the 7-limit, etc., to cover
>the metric for any interval in the system.
>  
>
You only need 2 intervals to cover the 5-limit.  The two Fokker used, 
for example.


                           Graham


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Message: 7319

Date: Thu, 14 Aug 2003 21:21:30

Subject: Re: Comments about Fokker's misfit metric

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > ok, i just read _prime obsession_, but i think i know even less 
> about 
> > the zetafunction than after reading manfred schroeder's _number 
> > theory in science and communication_ . . .
> > 
> > where can i read about zeta tuning again?
> 
> Somewhere lost in the archives of this list;

not lost; i just read three. i'm not sure i understand how 
consistency is enforced differently than just using the best 
approximations to the primes, though.

> however it's a good one 
> to put up a web page on xenharmony for.

how about some graphs?


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Message: 7320

Date: Thu, 14 Aug 2003 01:11:42

Subject: Re: Comments about Fokker's misfit metric

From: Graham Breed

Paul Erlich wrote:

>actually, 53-equal suffers from an additional problem with this type 
>of calculation, whether fokker's original or modified as you have 
>above. the problem is "inconsistency"... 
>
Fokker only looked at prime numbers, so consistency wasn't an issue.


                                  Graham


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