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Message: 7875 Date: Fri, 31 Oct 2003 23:33:52 Subject: Re: Eponyms From: Dave Keenan --- 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
So these are two advantages of "5.7.11-kleisma" over "385/384".
> (otherwise how'reyou going to say what pythagorean > commas are good for?).
There aren't too many of them that have come to my attention so far. These are well known and have common names: Pythagorean-limma, apotome, Pythagorean-comma. If the previously described naming system was simply applied they would all be 1-<whatevers> where "<whatever>" stands for the correct size category. But I would change this to 3-<whatevers> or even better, Pythagorean-<whatevers>. So the limma and comma would be the same as their common names and the apotome would probably have the systematic name: Pythagorean-semitone. The 3-exponent can be extracted from the size category if necessary. I think that in any list of common commas you will find that more than 90% of them have a 3-exponent other than zero. So it is fairly unnecessary to include, up front, the fact that they have 3's in them; and it would be technically redundant since the 3-exponent can be extracted from the size category in conjunction with the numbers that are supplied.
> But with the addition of the 3 exponent, > we loose the ability to draft size ranges. What say you to > this, Dave?
I'm not sure what you mean by "to draft size ranges"? If you mean "to decide the boundaries of size ranges", I don't understand why you would lose that ability. You could keep the same ranges and the up front information about the 3-exponent would simply be redundant. I don't want to call 81/80 the 5:3^4-comma, but just the 5-comma. And 64/63 would be simply the 7-comma, not the 7.9-comma. That's the whole point of setting the size-category boundaries so carefully, to eliminate the need to have the power of 3 explicit in the systematic name.
Message: 7876 Date: Fri, 31 Oct 2003 05:03:32 Subject: Re: UVs for 46-ET 11-limit PB From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 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.
sorry ... i was in a hurry when i posted that. i should have added: the lattice only show 2 dimensions at a time, those of prime-factors 3 and 5. the horizontal axis is 3, the vertical is 5. the big diagrams on the left show only 3 and 5. the smaller ones on the right show, from the top down respectively, 7^1, 7^-1, 11^1, and 11^-1. -monz
Message: 7877 Date: Fri, 31 Oct 2003 23:40:57 Subject: A remarkable property of 270 From: Gene Ward Smith If we take the 12 smallest 13-limit superparticular commas, namely 1001/1000, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, 4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647, 123201/123200 we find that they all have the property of being 270-et commas. Moreover, if we take this comma list five at a time, we get the 270 et in all cases where the five commas are linearly independent. Sometimes we get what we might regard as 540, but this is just a doubling of 270.
Message: 7878 Date: Fri, 31 Oct 2003 05:11:06 Subject: Re: Eponyms From: Dave Keenan I realise I missed this the first time: --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> In your scheme, the term kleisma tells us that the denominator must > be 384, and not 383?
Yes! But not directly of course. All it tells us directly is the size range. From that, and the 385, we can get the factors of 2 and 3, as I explained in the previous message. It's such a neat trick, I guess it's hard to believe it works. But note that the number given in the name is not necessarily the numerator or demominator of the comma ratio, it's the comma ratio with the 2's and 3's removed, (and inverted if it's less than one).
> >or "5.7.11-kleisma".
> > ...tells us how to combine factors of 5, 7, and 11 to get the > right ratio?
Yes. When the dots are read as multiplication, this contains no more and no less information than "385-kleisma" since prime factorisations are unique.
Message: 7879 Date: Fri, 31 Oct 2003 07:00:34 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 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? There isn't really much point in naming commas like 385/384 in the first place, but if you do, the name should be easier than the thing it is naming. 5.7.11-kleisma has no advantages over 385/384 that I can see.
Message: 7880 Date: Fri, 31 Oct 2003 07:48:58 Subject: Re: Eponyms From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: >
> > 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?
> > There isn't really much point in naming commas like 385/384 > in the first place, but if you do, the name should be easier > than the thing it is naming. 5.7.11-kleisma has no advantages > over 385/384 that I can see.
i disagree quite strongly. to me, the only thing the ratio shows is that it's superparticular (or epimoric, if you prefer Greek over Latin). i think 5.7.11-kleisma is a much better name ... altho i think my first choice would be to use the monzo and call it the [-7 -1 1 1 1]-kleisma, or if you can do without the 2 (which i also always prefer if possible), [-1 1 1 1]-kleisma. :) -monz
Message: 7881 Date: Fri, 31 Oct 2003 17:36:58 Subject: Re: Eponyms From: Carl Lumma
>> >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.
That works. -Carl
Message: 7882 Date: Sat, 01 Nov 2003 12:10:44 Subject: Re: I get the message! From: monz hi Dave (and Gene and paul), --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> Gene, > > I apologise again for the personal insult. > > You said you did not call me names, and that's technically true. > > However I pride myself on my teaching and writing ability (perhaps > mistakenly) as well as my system design ability, and so I'm afraid I > do take it as a personal insult when someone who hasn't even read my > article makes assumptions about what my method of exposition will be > and declares them "sloppy", using the word three times, no less. > > You also said I was "going about it in a very, very bad way", and > implied that my readers would need a "secret decoder ring" to > understand my article. All without having read any of it. > > Yahoo groups: /tuning-math/message/7236 * [with cont.] > > I hope you can now understand why I found all this far more hurtful > even than being called an anal retentive.
i must say that i found Gene's comment about the "secret decoder ring" very funny and amusing. i realize it was at your expense, Dave ... sorry, but i have to be honest above all else, and i did get a laugh from it.
> But even if I felt insulted, I should not have responded in kind. > I'm sorry.
that's beautiful. i really admire you for posting it publicly here.
> Paul, > > I certainly don't want you to use letters instead of names > in those wonderful diagrams. I'm just saying I think it has > gone far enough, and besides there is at least _some_ kind > of logic to most of those names.
one side of me really enjoys the whimsy of the names paul has already coined, and the other side of me agrees with you, Dave. and i guess i also sense "some kind of logic", because i know paul well enough to know that he wouldn't simply vent his imagination on something like this without exercising his powerful logical abilites too. but i'm also anal-retentive enough to desire a nice systematic naming for everything. (remember? ... i'm the guy who wanted a tuning dictionary so badly that he simply created it.) -monz
Message: 7883 Date: Sat, 01 Nov 2003 01:45:50 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- 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.
I agree with Monz. There's definitely no need to include the 2-exponents here. They're musically irrelevant in most cases, and if you do need them, the mere fact that these are commas of some kind, and hence smaller than 600 cents, is enough to give you their 2-exponents. I wouldn't want to start using monzos in this role until things got really complex, like if you would otherwise have more than say 12 characters in the numeric part. There's no need for systematic names to be so unfriendly as to call 81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name "5-comma" can be generated and decoded systematically, as I've shown.
Message: 7884 Date: Sat, 01 Nov 2003 14:37:52 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 81/80 is the 5-comma.
using the convention Gene just proposed which i accept (OK, i'll keep the comma punctuation) : <3, 5>-monzo: <4, -1>-comma.
> 32805/32768 is the 5-schisma.
<3, 5>-monzo: <8, 1>-schisma.
> 64/63 is the 7-comma
<3, 5, 7>-monzo: <-2, 0, -1>-comma.
> 59049/57344 is the 7-medium-diesis or 7-M-diesis
<3, 5, 7>-monzo: <10, 0, 1>-diesis.
> 28/27 is the 7-large-diesis or 7-L-diesis
<-3, 0, 1>-diesis.
> 2048/2035 is the 25-comma
Dave, here i'm not sure if your ratio is correct, because 2035 = 5 * 11 * 37 .
> 6561/6400 is the 25-small-diesis or 25-S-diesis
<8, -2>-diesis.
> 128/125 is the 125-small-diesis
<0, -3>-diesis
> 250/243 is the 125-medium-diesis
<-5, 3>-diesis.
> 531441/512000 is the 125-large-diesis
<12, -3>-diesis.
> 5120/5103 is the 5:7-kleisma
<-6, 1, -1>-kleisma.
> 3645/3584 is the 5:7-comma
<6, 1, -1>-comma. -monz
Message: 7885 Date: Sat, 01 Nov 2003 16:52:50 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> > wrote: > > <snip> >
> > 59049/57344 is the 7-medium-diesis or 7-M-diesis
> > <3, 5, 7>-monzo: <10, 0, 1>-diesis.
oops ... my bad. missed a sign. that should be <10, 0, -1>-diesis.
> > 2048/2035 is the 25-comma
> > Dave, here i'm not sure if your ratio is correct, > because 2035 = 5 * 11 * 37 .
Dave made a typo here and the ratio should be 2048/2025, in monzo form: the <-4, -2>-comma. here's the whole list of Dave's "kommas", from largest to smallest, with the [2, 3, 5, 7]-monzos and cents: (if viewing on the Yahoo web interface, you'll have to forward it to your email account to see it properly.) 2 3 5 7 cents ratio [-12, 12, -3, 0]-diesis 64.51886879 531441 : 512000 [ 2, -3, 0, 1]-diesis 62.96090387 28 : 27 [-13, 10, 0, -1]-diesis 50.72410218 59049 : 57344 [ 1, -5, 3, 0]-diesis 49.16613727 250 : 243 [ -8, 8, -2, 0]-diesis 43.01257919 6561 : 6400 [ 7, 0, -3, 0]-diesis 41.05885841 128 : 125 [ -9, 6, 1, -1]-comma 29.21781259 3645 : 3584 [ 6, -2, 0, -1]-comma 27.2640918 64 : 63 [ -4, 4, -1, 0]-comma 21.5062896 81 : 80 [ 11, -4, -2, 0]-comma 19.55256881 2048 : 2025 [ 10, -6, 1, -1]-kleisma 5.757802203 5120 : 5103 [-15, 8, 1, 0]-schisma 1.953720788 32805 : 32768 or if you prefer <3, 5, 7>-monzos: 3 5 7 cents ratio < 12, -3, 0>-diesis 64.51886879 531441 : 512000 < -3, 0, 1>-diesis 62.96090387 28 : 27 < 10, 0, -1>-diesis 50.72410218 59049 : 57344 < -5, 3, 0>-diesis 49.16613727 250 : 243 < 8, -2, 0>-diesis 43.01257919 6561 : 6400 < 0, -3, 0>-diesis 41.05885841 128 : 125 < 6, 1, -1>-comma 29.21781259 3645 : 3584 < -2, 0, -1>-comma 27.2640918 64 : 63 < 4, -1, 0>-comma 21.5062896 81 : 80 < -4, -2, 0>-comma 19.55256881 2048 : 2025 < -6, 1, -1>-kleisma 5.757802203 5120 : 5103 < 8, 1, 0>-schisma 1.953720788 32805 : 32768 -monz
Message: 7886 Date: Sat, 01 Nov 2003 01:57:57 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> The idea of leaving out the 3's is clever but not beneficial, in > my opinion.
I'm a little confused here. On the one hand you'd apparently be quite happy to call something the "fartisma", because all you need is a "hook" to hang the meaning on, and names with numbers in them are boring, but when I show you a way to eliminate some of the numbers while still remaining systematic and unambiguous, you want to keep all the numbers, even redundant ones. So what would your systematic names for 81/80 and 64/63 look like?
> With the wrong ranges, you wouldn't be able to extract the 3 exponent, > I assumed.
That's true. But I don't understand what point you're making here.
Message: 7887 Date: Sat, 01 Nov 2003 17:17:32 Subject: Re: The supertemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >For any odd prime p, there is a finite list of superparticular
ratios
> >which belong to the p-limit.
> > Here's something I can believe but which isn't immediately obvious. > Can you prove it?
Been there, done that. It follows from Baker's Theorem.
> Cool. Howabout moving a fixed n down the list (or n's which, for
each
> starting point in the list, uniquely define a val)?
There's a thought, but first I've got to write up the capstone temperament (same thing but for linear temperaments.)
Message: 7888 Date: Sat, 01 Nov 2003 02:34:24 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> I think that in any list of common commas you will find > that more than 90% of them have a 3-exponent other than > zero. So it is fairly unnecessary to include, up front, > the fact that they have 3's in them;
i can pretty much agree with that, with one very important comma residing in that other 10%: the enharmonic diesis, ratio 128/125, [3,5]-monzo version: the [ 0 -3]-diesis.
> I don't want to call 81/80 the 5:3^4-comma, but just >the 5-comma.
[3,5]-monzo version: the [4 -1]-comma.
> And 64/63 would be simply the 7-comma, not the 7.9-comma.
[3,5,7]-monzo version: the [-2 0 -1]-comma.
> That's the whole point of setting the size-category > boundaries so carefully, to eliminate the need to have > the power of 3 explicit in the systematic name.
so if they're being described as monzos, just leave out the first exponent of the vector and the first prime-factor of the label. ... looks like Gene and i support each other on this method of description. -monz
Message: 7889 Date: Sat, 01 Nov 2003 17:20:35 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> >
> > > Have you actually read any of the several descriptions I've
given of
> > > the proposed komma naming algorithm and its inverse? Are they
all
> > > really that unclear?
> > > > I don't buy kommas.
> > Do you mean you don't like spelling it with a "k" when it's being
used
> as a generic term. That's fine. That's not part of the naming > algorithm. That's just me.
(1) I don't like having two words which sound the same and with related meanings (2) I think the goofy spelling, if you do this, should be for your new meaning, and not imposed on an established one. "Comma in a particular range of cents"="komma", in other words.
Message: 7890 Date: Sat, 01 Nov 2003 02:34:55 Subject: Re: Eponyms From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >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.
Seems useful to me. I think we're now past the point where all the common comma's and temperaments have been named. My purpose in proposing systematic naming methods for both linear temperaments and commas is to avoid drowning in lots of meaningless names where, if you haven't been following the tuning-math list religiously for the past x years the only way you have of figuring out what someone's talking about is to look the names up in a database somewhere.
> >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". > :)
It takes all kinds. That's funny. Gene says I'm sloppy, and you say I'm anal-retentive. A contradiction, wot? Actually, I think I'm just your classic engineer/architect type. I design systems. I'm good at it. I make my living designing systems of several different kinds. I'm apparently genetically predisposed to it. You, presumably, have other wonderful and complementary qualities. Would you care to explain what your objection's are to the proposal, as opposed to your objections to my online personality?
Message: 7891 Date: Sat, 01 Nov 2003 19:12:08 Subject: Capstone temperament From: Gene Ward Smith Before we get to the range which defines the supertemperament (where the nullspace of the matrix of monzos has dimension one, defining a val, and thus an equal temperament) we have a range where the nullspace is dimension two, and defines a linear temperament--which I've mentioned before, under the name of the capstone temperament. Here are mappings for capstones up to the 19-limit, and generators in terms of the corresponding supertemperament. (This last is a little crude, but does get us into the ballpark.) 5 limit: 81/80 meantone 7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal generators: [1/9, 1/24] 11 limit: [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] hemiennealimmal generators: [1/18, 1/72] 13 limit: [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]] generators: [1/2, 47/270] 17 limit: [[1, 35, 221, 161, -5, 197, 367], [0, -79, -517, -374, 20, -457, -858]] generators: [1, 637/1506] 19 limit: [[1, 250, 324, 62, -178, -481, 1579, 258], [0, -512, -663, -122, 374, 999, -3246, -523]] generators: [1, 4143/8539] The 13-limit capstone has reduced basis [1716/1715, 2080/2079, 3025/3024, 4096/4095] and wedgie [22, 48, -38, -34, -54, 25, -122, -130, -167, -223, -245, -303, 36, -11, -61]. It is well-covered by 494-et, and can be thought of as the 270&494 13-limit linear temperament. I don't want anyone to pitch a fit, but I suppose Cap13 is one possible name.
Message: 7892 Date: Sat, 01 Nov 2003 02:46:03 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> There's no need for systematic names to be so unfriendly > as to call 81/80 the [-4, 4, -1]-comma, or even the > [4, -1]-comma. The name "5-comma" can be generated and > decoded systematically, as I've shown.
i'm sorry to respectfully disagree with you, Dave, but i don't see anything unfriendly about "[4 -1]-comma". (note that i don't consider the comma punctuation necessary.) admittedly, "5-comma" is a whole lot easier and, yes, i'll admit, friendlier. but for me, so used to visualizing tunings on a lattice, "[4 -1]-comma" tells me exactly what i need to know. if i'm picturing the whole prime-space on a lattice, [4 -1] helps me to *immediately* set a boundary in my mind which filters out a large number of redundant lattice-points. in fact, when i hear or read the word "syntonic" the first thing i think about is the vector on the lattice which would describe it in a prime-space ... and then the second thing i think about very quickly after that is 3^4 / 5^1 . and as i've already been arguing with the ratios, forget it. there's almost always nothing valuable about retaining the data for prime-factor 2, unless it need be considered for (to cite two examples i can think of quickly): - actual orchestral scoring where the 8ve-register must be considered, or - analyzing ancient Greek and Roman theory, which was based on 4:3 "perfect-4ths" and always specified 8ves, and gave different names to notes an 8ve apart. -monz
Message: 7893 Date: Sat, 01 Nov 2003 19:34:36 Subject: Re: The supertemperament From: Carl Lumma
>>>For any odd prime p, there is a finite list of superparticular >>>ratios which belong to the p-limit.
> > > > Here's something I can believe but which isn't immediately > > obvious. Can you prove it?
> > Been there, done that. It follows from Baker's Theorem.
There are no results for "Baker's theorem" or "bakers theorem" at mathworld, but the 2nd google result for "baker's theorem" is this post of yours... Yahoo groups: /tuning-math/message/1108 * [with cont.] Hooray again for google. I wonder how much of these lists are google-searchable? -Carl
>
> > Cool. Howabout moving a fixed n down the list (or n's which, for
> each
> > starting point in the list, uniquely define a val)?
> > There's a thought, but first I've got to write up the capstone > temperament (same thing but for linear temperaments.)
Message: 7895 Date: Sat, 01 Nov 2003 11:48:40 Subject: Re: I get the message! From: Carl Lumma
>However I pride myself on my teaching and writing ability (perhaps >mistakenly) as well as my system design ability,
I think you're a great teacher, writer, and system designer. -Carl
Message: 7897 Date: Sat, 01 Nov 2003 22:09:29 Subject: Re: A remarkable property of 270 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> If we take the 12 smallest 13-limit superparticular commas, namely > > 1001/1000, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, > 4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647,
123201/123200
> > we find that they all have the property of being 270-et commas.
much like 72-equal in the 11-limit.
> Moreover, if we take this comma list five at a time, we get the 270
et
> in all cases where the five commas are linearly independent.
Sometimes
> we get what we might regard as 540, but this is just a doubling of
270. in other words, torsion?
Message: 7898 Date: Sat, 01 Nov 2003 03:18:00 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> in fact, when i hear or read the word "syntonic" the > first thing i think about is the vector on the lattice > which would describe it in a prime-space ... and then > the second thing i think about very quickly after that > is 3^4 / 5^1 .
actually that's not true. i don't visualize the numbers as 3^4 / 5^1 , but rather as 3^4 * 5^-1 , since that's exactly how the lattice works. and that visualization agrees exactly with the monzo of the syntonic comma.
> and as i've already been arguing with the ratios, forget it. > > there's almost always nothing valuable about retaining the > data for prime-factor 2, unless it need be considered for > (to cite two examples i can think of quickly): > > - actual orchestral scoring where the 8ve-register must > be considered, or > > - analyzing ancient Greek and Roman theory, which was > based on 4:3 "perfect-4ths" and always specified 8ves, > and gave different names to notes an 8ve apart.
but what i forget to emphasize here again is: even in these cases where 8ves must be considered, it's easier to use the monzo including 2's exponent, instead of the actual ratio. -monz
Message: 7899 Date: Sat, 01 Nov 2003 14:13:06 Subject: ennealimmal From: Carl Lumma Am I correct that the first ennealimmal scale with an octave is simply 9-equal, and the next is this 17-tone one... 50. 133.3 183.3 266.7 316.7 400. 450. 533.3 583.3 666.7 716.7 800. 850. 933.3 983.3 1066.7 1116.7 ...? Manuel, is there a convenient way to get MOS-like scales with non-octave periods in Scala? -Carl
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