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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 - Type Ok * [with cont.] (Wayb.) A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.) -paul

Message: 7301 Date: Wed, 13 Aug 2003 17:50:51 Subject: Re: interval vector From: monz@xxxxxxxxx.xxx hi Gene,

> From: Gene Ward Smith [mailto:gwsmith@xxxxx.xxxx > Sent: Wednesday, August 13, 2003 5:42 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Re: interval vector > > > --- In tuning-math@xxxxxxxxxxx.xxxx pitchcolor@a... wrote:

> >

> > > What's an interval vector? > > > > > > A pitch difference expressed as a list of integers, > > > which is much the same as a free albelian group. > > >

> > > > 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.

> > 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.

i already responded to this, saying the same thing.

> So far as I know, academia has not taken this as > standard. :)

i don't think other scholars would be too accepting of *me* being the first to promote the use of my own name for a term, so ... can *you* publish something with it? that would be a start in gaining acceptance. -monz

Message: 7302 Date: Wed, 13 Aug 2003 17:52:17 Subject: Re: Comments about Fokker's misfit metric From: monz@xxxxxxxxx.xxx hi Graham (and paul and Carlos),

> From: Graham Breed [mailto:graham@xxxxxxxxxx.xx.xxx > Sent: Wednesday, August 13, 2003 5:07 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Re: Comments about Fokker's misfit metric > > > 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.

well, there you go ... i wasn't following the thread and spoke up when i should have kept my mouth shut. forget it -- i'm out of this thread until and unless i go back and read from the beginning. -monz

Message: 7303 Date: Wed, 13 Aug 2003 23:10:39 Subject: Re: interval vector From: monz@xxxxxxxxx.xxx

> From: Gene Ward Smith [mailto:gwsmith@xxxxx.xxxx > Sent: Wednesday, August 13, 2003 5:52 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Re: interval vector > > > --- In tuning-math@xxxxxxxxxxx.xxxx <monz@a...> wrote: >

> > > From: Gene Ward Smith [mailto:gwsmith@s...] > > > Sent: Wednesday, August 13, 2003 5:42 PM

>

> > i don't think other scholars would be too accepting > > of *me* being the first to promote the use of my own > > name for a term, so ... can *you* publish something > > with it? that would be a start in gaining acceptance.

> > I've decided that this is the 21st century and paper is obsolecent. > It's up on www.xenharmony.org.

OK ... works for me. :) you've probably noticed by now ... almost all of my work is published in HTML format. -monz

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

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

Message: 7307 Date: Thu, 14 Aug 2003 17:53:45 Subject: Re: interval vector From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > From: Gene Ward Smith [mailto:gwsmith@s...] > > > Sent: Wednesday, August 13, 2003 5:42 PM

>

> > i don't think other scholars would be too accepting > > of *me* being the first to promote the use of my own > > name for a term, so ... can *you* publish something > > with it? that would be a start in gaining acceptance.

> > I've decided that this is the 21st century and paper is obsolecent.

Hmmm, if so, then we will be phasing out the use of music stands. And if so, then musicians must memorize their parts before coming to rehearsals. Or are live musicians also obsolescent? ;-) --George

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.

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.

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.

Message: 7312 Date: Thu, 14 Aug 2003 20:40:21 Subject: Re: interval vector From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor" <gdsecor@y...> wrote:

> > I've decided that this is the 21st century and paper is

obsolecent.

> > Hmmm, if so, then we will be phasing out the use of music stands.

You could invent an electronic music stand for us. Of course we haven't really gotten book readers right yet, so it may be premature.

Message: 7313 Date: Thu, 14 Aug 2003 20:42:40 Subject: Re: Comments about Fokker's misfit metric From: Gene Ward Smith --- 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.

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?

Message: 7315 Date: Thu, 14 Aug 2003 20:58:58 Subject: Re: Comments about Fokker's misfit metric From: Gene Ward Smith --- 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; however it's a good one to put up a web page on xenharmony for.

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

Message: 7317 Date: Thu, 14 Aug 2003 21:17:54 Subject: Re: interval vector From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx pitchcolor@a... wrote:

> 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. Clough and Myerson use "chord" to mean a set of interval classes, and "line" to mean an ordered tuple of interval classes. Has the latter been taken up in academia?

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

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?

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

Message: 7321 Date: Thu, 14 Aug 2003 21:39:55 Subject: Re: interval vector From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > I've decided that this is the 21st century and paper is

> obsolecent.

> > > > Hmmm, if so, then we will be phasing out the use of music

stands.

> > You could invent an electronic music stand for us. Of course we > haven't really gotten book readers right yet, so it may be

premature. But it may be sooner than we think. Earlier this year I wrote a piece in 19-WT using Cakewalk, and I was curious to see if I could play the solo part on the trumpet (with a couple of slides adjusted). Performance tempo was a bit beyond my ability, but I found that I was able to practice it if I set the tempo slower and set the staff view so that only a few staves displayed (with the solo part at the top). As the file played, the measures of music displayed so that I could easily read the part, which automatically scrolled, so my electronic music stand even turned the pages for me. Once I mastered the piece at a given tempo, I set it a little faster and tried again -- something like a computer game in which you complete each level, then move on to the next (which actually made practicing fun). --George

Message: 7322 Date: Thu, 14 Aug 2003 00:42:09 Subject: Re: interval vector From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx pitchcolor@a... wrote:

> <<

> > What's an interval vector? > > > > A pitch difference expressed as a list of integers, which is much

the

> > same as a free albelian group.

> > >>

> > 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. 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. :)

Message: 7323 Date: Thu, 14 Aug 2003 23:34:25 Subject: Re: Comments about Fokker's misfit metric From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> 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.

All it does is supply a different, and normally preferable, answer to the question of how to define a standard val. Choosing any val enforces consistency.

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

> > how about some graphs?

Why not.

Message: 7324 Date: Thu, 14 Aug 2003 00:45:32 Subject: Re: Comments about Fokker's misfit metric From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> 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. > > > >

> You only need 2 intervals to cover the 5-limit. The two Fokker

used,

> for example.

Eh? You need three. 2, 3, and 5 is the obvious choice. Octave reduction leads to two classes, of course.

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