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Message: 11200 - Contents - Hide Contents
Date: Sat, 03 Jul 2004 13:41:38
Subject: A map of starling space
From: Herman Miller
I made a chart of ET's with the comma 126;125, and drew lines on it to
illustrate some of the landmark starling temperaments. I couldn't find a
name for <<1, -8, -26, -15, -44, -38]] ("Number 114"), so I decided to
call it "grackle" (a name that originally referred to a kind of
starling, and still does in some parts of the world).
http://www.io.com/~hmiller/png/starling-map.png - Type Ok * [with cont.] (Wayb.)
Note that I haven't bothered to limit the list to 7-limit consistent
ET's, or to include ET's that might be compatible with starling in some
mapping other than their closest approximation to 3, 5, and 7.
The temperaments illustrated are:
#114 grackle <<1, -8, -26, -15, -44, -38]]
#2 meantone <<1, 4, 10, 4, 13, 12]]
#49 diaschismic <<2, -4, -16, -11, -31, -26]]
#31 tripletone <<3, 0, -6, -7, -18, -14]]
#17 diminished <<4, 4, 4, -3, -5, -2]]
#112 muggles <<5, 1, -7, -10, -25, -19]]
#27 kleismic <<6, 5, 3, -6, -12, -7]]
#30 valentine (formerly quartaminorthirds) <<9, 5, -3, -13, -30, -21]]
#29 myna (formerly nonkleismic) <<10, 9, 7, -9, -17, -9]]
Any major starling-related temperaments that I'm missing?
The original starling map, like most of the others we've seen, is based
on graphing the deviations from 3/1 and 5/1. Possibly a more useful map
for a 7-limit temperament family is one that shows the deviations from
7/1. So I created a second map.
http://www.io.com/~hmiller/png/starling-map2.png - Type Ok * [with cont.] (Wayb.)
You can see that valentine and myna look good on this chart, while
kleismic (which is a good 5-limit temperament) isn't so good in the
7-limit. Another benefit of this kind of chart is that you can use it to
look for new temperaments, like the more familiar 5-limit charts
e.g. http://www.io.com/~hmiller/png/et-scales.png - Type Ok * [with cont.] (Wayb.)
Message: 11201 - Contents - Hide Contents Date: Sat, 03 Jul 2004 19:03:12 Subject: Re: Gene's mail server From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> It says that the bra is a covariant 1-vector, and the ket is a > contravariant one-form. It also says that the combination of the two, > as in <v|w>, is an inner product.I know people talk that way. I also know it is very confusing to people trying to learn this stuff. But probably not worth worrying about in connection with your paper. I'd also flush all of that stuff about 1-forms from your brain immediately.
Message: 11202 - Contents - Hide Contents Date: Sat, 03 Jul 2004 19:06:06 Subject: Re: Gene's mail server From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Something "implicit" isn't a concept that I introduced in the paper. > Is it possible for you to think like a non-mathematician for a > moment? Look, the next version of the paper will show, a little more > explicitly, what you call the "JIP val" -- and what I would > call "pitch altitude" or something -- in the 3-limit lattice. At that > point, maybe you could meet me halfway and try to give specific > suggestions that I can implement readily, rather than overarching > criticisms which would require, essentially, a new paper . . .You've lost me. I've made no overarching criticisms, only suggestions for minute changes (such as "angle bracket" for "inner product") which would be trivial to implement.
Message: 11203 - Contents - Hide Contents Date: Sat, 03 Jul 2004 03:50:51 Subject: Re: my paper nears completion From: Paul Erlich Thanks Herman. I like the way the second one is formatted -- just the chord progression. Tying common tones would be nice too, especially where enharmonic equivalence comes into play, if it's not too difficult. What do you say? -Paul
Message: 11204 - Contents - Hide Contents Date: Sat, 03 Jul 2004 19:19:20 Subject: from linear to equal From: Kalle Aho Hi, Linear temperaments (or 2-dimensional tunings) are infinitely extendable. Once you extend a linear temperament eonugh you'll start getting different pitches that nevertheless are more or less indistinguishable from each other. Even before that you'll get approximations that are better than those the linear temperament is supposed to give. So what would be a good place to close the circle and go from linear to equal? For TOP tempered linear temperaments I suggest closing the circle when you start getting better approximations to the primes for which the tuning is optimized. What are your thoughts about this? Kalle
Message: 11205 - Contents - Hide Contents Date: Sat, 03 Jul 2004 04:58:01 Subject: Re: Gene's mail server From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Not when you've already introduced the "monzos" concept but haven't > introduced the "vals" concept.You introduce both, more or less, in your paper, at least to the extent of an implictly defined vector space togher with its dual. That is how you ended up with a bracket product despite the fact that no inner product space is being discussed, or would make any sense in the context.> Yup -- and I just said why, too.And your reason is not correct.> Anyway, since these names are so ugly, does *anyone* have suggestions > for renaming them (Dimipent, Dimisept, Negripent, Negrisept, > Sensipent, Sensisept) that preserves their approximate alphabetical > location?Just shortening them would help--dimip, dimis, negrip, negris, sensip, sensis. Possibly needs work to keep people from reading negris as Negress (a word many people find offensive) and confusing sensis with census.
Message: 11206 - Contents - Hide Contents Date: Sat, 03 Jul 2004 19:24:18 Subject: Re: dual, and inner product space (was: Gene's mail server) From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> what is the dual? if you could explain it along > the lines of my "prime-space" definition, that > would help me.Paul and I have been tossing this back and forth at each other: Dual Vector Space -- from MathWorld * [with cont.] Linear Function -- from MathWorld * [with cont.]>> That is how you ended up with a bracket product >> despite the fact that no inner product space is >> being discussed, or would make any sense in the context. > >> what's an "inner product space"? Vector Space -- from MathWorld * [with cont.] Vector space - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) Inner Product Space -- from MathWorld * [with cont.] Inner product space - Wikipedia, the free ency... * [with cont.] (Wayb.)However, since we aren't using inner product spaces it is only vector spaces which need concern us. Other relevant encyclopedia pages are for abelian group Abelian group - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) You could also look up bra-ket notation, but because it assumes we are in an inner product space and we are not, it may not be that great. This is my problem with Paul wanting to use "inner product" for the bracket--we don't actually have an inner product, whereas in quantum mechanics we do. This isn't QM, thank heavens.> is "vector space" here synonymous with my > definition of "prime-space"?I'm afraid your definition was too far from the way mathematicians define things to let me have any assurance I understand what you are saying. I define Tenney space here /root/tentop.htm * [with cont.] (Wayb.) Inside of Tenney space, monzos or musical intervals live as a lattice, which in my way of thinking must include a means of measuring the distance between the lattice points.
Message: 11207 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:16:01 Subject: Re: Gene's mail server From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> And your reason is not correct.It *isn't* the periodic unit in the tempered lattice? Where did I go wrong?> Just shortening them would help--dimip, dimis, negrip, negris, sensip, > sensis.Those don't strike me as better names, but thanks.
Message: 11208 - Contents - Hide Contents Date: Sat, 03 Jul 2004 20:45:16 Subject: Re: from linear to equal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho" <kalleaho@m...> wrote:> So what would be a good place to close the circle and go from linear > to equal? > > For TOP tempered linear temperaments I suggest closing the circle > when you start getting better approximations to the primes for which > the tuning is optimized. > > What are your thoughts about this?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 convenient for Scala score files, since the notes are now represented by (reasonably small) integers. I also sometimes use it when cooking up a Scala scl file (just did, in fact, over on the tuning list) though in that case it makes little difference. If you follow this system, 5-limit meantone closes for 81, 7-limit meantone for 31, and 11-limit meantone for 31. 5 and 7 taken together are 1/4-comma exactly, which doesn't close; 5 and 11 taken together closes at 112, and 7 and 11 of course also at 31. One rarely encounters problems; even a microtemperament like ennealimmal closes at 1053, which is perfectly reasonable for Scala applications; one does, however, need to ensure the division is divisible by 9. A different naming convention than using TOP tuning would be to give the same name iff the poptimal ranges intersect. This isn't very convenient in practice, due to the difficulty of computing the poptimal range, but clearly it leads to quite different results. Miracle, for instance, has the same TOP tuning in the 5, 7 and 11 limits, but while the 5 and 7 limit poptimal ranges intersect, the 5 and 11 or 7 and 11 ranges apparently do not, though as I say computing these is a pain, so I may have the range too small. In any case, miracle closes at 175 in the 5 and 7 limits, and at 401 in the 11-limit.
Message: 11209 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:26:57 Subject: origin of mork tuning (was: Gene's mail server) From: monz hi Paul and Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > <snip> >>> Graham, can you explain (maybe on tuning-math) what >> the melodic approach is? >> Take a look at Graham's webpages on meantone, schismic, > diaschismic . . . It seems to be closely wedded to the > val approach, which is why I suggested we use the term > "breeds" to refer to vals (as long as we're using "monzos" > to refer to lattice vectors).i like that !! ;-) i recall some posts a while back which complained of using "val", instead of a new term.>>> Why orson? >>> Son of Orwell, or ORson WELLs, or whatever . . . >>Why not just orwell? > > Different tuning, different horagrams. "Mork to Orson!" Hello? > > :)... hmm ... monz predicts that Gene's next temperament family will be named "Mork" ... that will be one zany tuning !!! ... wonder what properties it will have? i'm sure Gene will give a full description of it, including all the monzos and breeds. :) (seems like this cyber-community is becoming a nice cozy family) what tuning concept will we ever call a "smith"? ;-) ;-) -monz
Message: 11210 - Contents - Hide Contents Date: Sat, 03 Jul 2004 16:30:22 Subject: Re: from linear to equal From: Herman Miller Kalle Aho wrote:> For TOP tempered linear temperaments I suggest closing the circle > when you start getting better approximations to the primes for which > the tuning is optimized. > > What are your thoughts about this?That's pretty much my opinion, although you can go some ways beyond that point if you try to avoid the better approximations, and there may be other reasons to stick with the temperaments in particular cases. TOP father (g = 447.3863410, p = 1185.869125) has a better approximation of 5 after only 7 iterations of the generator, so you might want to switch to 8-ET. This works out fine for the 5-limit, since TOP 5-limit 8-ET <1185.032536, 1925.677871, 2814.452272] is pretty close to TOP 5-limit father <1185.869125, 1924.351908, 2819.124590]. But 8-ET isn't 7-limit consistent, so if you're using 7-limit father temperament <1185.869125, 1924.351908, 2819.124589, 3401.317477], you're probably better off sticking with the 8-note father MOS rather than going to one of the versions of TOP 8-ET.
Message: 11211 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:28:31 Subject: Re: origin of mork tuning (was: Gene's mail server) From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> what tuning concept will we ever call a "smith"? > > ;-) ;-)Aaron J. suggested that the "wedgie" be renamed the "genie" . . .
Message: 11212 - Contents - Hide Contents Date: Sat, 03 Jul 2004 22:06:41 Subject: Some naming examples From: Gene Ward Smith Augmented The 5-limit version of this, for comma 128/125, has only one poptimal generator, sqrt(10/9). Hence if we use poptimal generator range to define names nothing else will get the name. The 12&21 7-limit version, "augie" or "august", has a top tuning which is quite different, and closes at 33-equal for a poptimal version. Much closer to 5-limit augmented in tuning is "tripletone" or "augene", which has a TOP tuning not too different, and which closes poptimally at 27. 2^(2/27) is 88.889 cents, sqrt(10/9) is 91.202 cents. Is this close enough for the same name, or not? 2^(2/33) is 72.272, which is far afield. Paul's solution of three distinct names seems reasonable Orwell The 5-limit version of this, "orson" according to Paul, has a poptimal range which overlaps the 11-limit version, and in fact is almost identical with it; 19/84 being a poptimal generator for both. The 7-limit version does not seem to overlap, but it isn't so far off that 19/84 is not a decent generator for it also, though it isn't apparently poptimal, like 26/115. There are of course other versions of orwell, but none which seem good enough to bother with. To me it makes sense to name the 5, 7, and 11 limit versions all "orwell", but I don't have a systematic way of making that call. Mavila 135/128 closes poptimally at everyone's favorite equal temperament, 23. The 7-limit temperament with comma basis 15/14 and 64/63 shares a common poptimal generator in (10/3)^(1/4), but the 7-limit approximation involved is pretty hairy. Other temperaments with a common poptimal generator would add 729/700 or 126/125 to 135/128, but now the temperament has a high badness score. 27/62 or 37/85 give a common poptimal generator for the 135/128 and 126/125 temperament, which has wedgie <<1 -3 -11 -7 -20 -17||. Maybe Herman will be inspired enough by the 16 note MOS for this to tune it up, and see if it sounds like mavila to him.
Message: 11213 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:29:05 Subject: Re: Wedgies and generators From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> [to Gene] > If you were teaching a class, I would have failed out a > long time ago.imagine the trouble i'm having keeping up with Gene's work. :( -monz
Message: 11214 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:36:43 Subject: temperament names (was: Gene's mail server) From: monz hi Gene and Paul --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> <snip> > > Ah! The light dawns--you are proposing, in effect, > a naming system where two temperaments at different limits > carry the same name if, and only if, they have the same > TOP generators. This would indeed be simple and logical, > and the only problem would arise when they have *almost* > the same TOP generators, where you will be drawing a fine > distinction which may not mean much in practice.can you explain why TOP generators are so important?> I we adopt this plan, I suggest that we do link the names > of related systems, as in augmented-august-augene and > orson-orwell. I'd be interested in what other people think > of a sweeping revision of temperament nomenclature along > these lines; I've been trying to work it in this direction, > but not systematically.i support that idea whole-heartedly !!! -monz
Message: 11215 - Contents - Hide Contents Date: Sat, 03 Jul 2004 00:39:44 Subject: Re: my paper nears completion From: Herman Miller Paul Erlich wrote:> Thanks Herman. > > I like the way the second one is formatted -- just the chord > progression. Tying common tones would be nice too, especially where > enharmonic equivalence comes into play, if it's not too difficult. > What do you say?Well, I'll have to draw the arcs in by hand, since I don't have proper notation software. Hmm, Paint Shop Pro does Bezier curves, so I guess that'll have to do. http://www.io.com/~hmiller/music/porcupine-b.gif - Type Ok * [with cont.] (Wayb.)
Message: 11216 - Contents - Hide Contents Date: Sat, 03 Jul 2004 05:45:09 Subject: bimonzos, and naming tunings (was: Gene's mail server)) From: monz hi again, Gene and Paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > <snip> >>> Are you expecting people to read the comma values >> off of the bimonzo? >> No. But as long as we're on the subject here, it might > be worth reviewing here for list memmbers how you do that. > Not in the paper.yes, please do review it!> > Anyway, since these names are so ugly, does *anyone* > have suggestions for renaming them (Dimipent, Dimisept, > Negripent, Negrisept, Sensipent, Sensisept) that preserves > their approximate alphabetical location? my suggestion:i think that the past decade has seen a great expansion of the study of tuning, thanks largely to the internet. now that we have so much broader a view of large numbers of tunings, we should subject the whole "tuning universe" to deep review, and come up with a really good and logical system of classification and naming. -monz
