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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

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

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.

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

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?

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.

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.

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.

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

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

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

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).

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.

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

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!

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

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

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

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

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

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

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.

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).

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.

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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