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Message: 7975 Date: Tue, 04 Nov 2003 22:52:05 Subject: Re: hey Paul From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Is there a way for people to look them up?
> > > >Should I put up a web page? Dave, do you have an objection?
> > Graham's catalog is neither complete or up to date, last I > checked. > > The existence of a single resource is a lot to ask, I know...
I'm perfectly willing to put up web pages of named temperaments.
Message: 7976 Date: Tue, 4 Nov 2003 13:36:51 Subject: Re: ennealimmal From: Manuel Op de Coul Carl wrote:
>Manuel, is there a convenient way to get MOS-like scales with >non-octave periods in Scala?
Sure, create them with PYTHAGOREAN specifying the period as formal octave, and then use EXTEND to change the number of tones. The SHOW DATA command now also shows whether repeating blocks have Myhill's property.
>Am I correct that the first ennealimmal scale with an octave >is simply 9-equal, and the next is this 17-tone one...
You probably mean 18-tone. The generator doesn't need to be exactly 50 cents, but if I understand your question correctly, yes. Manuel
Message: 7977 Date: Tue, 04 Nov 2003 15:24:14 Subject: Re: hey Paul From: Carl Lumma
>> >Should I put up a web page? Dave, do you have an objection?
>> >> Graham's catalog is neither complete or up to date, last I >> checked. >> >> The existence of a single resource is a lot to ask, I know...
> >I'm perfectly willing to put up web pages of named temperaments.
Then don't stop 'til the break of dawn! -Carl
Message: 7978 Date: Wed, 05 Nov 2003 15:44:25 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...>
> wrote: >
> > Now if Dave says, "Should I use a 23-comma or 23-small-diesis
(23S)
> > symbol?", I immediately know what he's talking about: these are > > commas that alter tones in a 1/1 pythagorean chain to arrive at
tones
> > in a 23/16 and/or a 32/23 pythagorean chain.
> > This is fine for your very specialized purposes, but what about the > rest of us? Do you have a name for every superparticular ratio up
to
> the 23-limit?
I find it very amusing that you would place me, the *composer/theorist*, as the one with "very specialized purposes" while you, the *mathematician*, would group yourself with the "rest of us". It would seem to me that only those with very specialized purposes would actually *need* to have a name for *every* superparticular ratio up to the 23-limit. A systematic naming system should not be something that would make things more complicated for the rest of us. I could hardly imagine a professor in a microtonal music course using "minus three, zero, zero, zero, one" as a name for 26:27 when "13L-diesis" (which can even be shortened to "13L") is so much simpler and clearer. Monzos have their place as a specialized *notation* (which would also be of benefit in explaining the names), but not as *names* themselves. As for the answer to your second question: I don't know if there will be unique names for *all* of them (probably not), but I believe that there are enough names to cover all of those that would be of use to most theorists. Those with very specialized purposes (such as yourself) could devise a modification to Dave's naming system (such as appending letters a, b, c, etc. in order of ratio complexity) to distinguish equivocal names -- I think you're creative enough to come up with something that would be meaningful (or perhaps Dave might have some ideas). In summary, let's keep the simpler things simple. --George
Message: 7979 Date: Wed, 05 Nov 2003 18:08:51 Subject: Re: Eponyms From: monz hi George (and Gene) --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> A systematic naming system should not be something that > would make things more complicated for the rest of us. > I could hardly imagine a professor in a microtonal music > course using "minus three, zero, zero, zero, one" as a name > for 26:27 when "13L-diesis" (which can even be shortened > to "13L") is so much simpler and clearer. Monzos have > their place as a specialized *notation* (which would also > be of benefit in explaining the names), but not as *names* > themselves.
i see your point, and can agree with that. ... even tho *i* will always think of any rational interval as its monzo. (i guess that's self-evident, given the name of the term?) ;-) -monz
Message: 7980 Date: Wed, 05 Nov 2003 18:32:59 Subject: Re: Eponyms From: monz hi George, --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> The monzo comma-naming system is so cumbersome > (i.e., unfriendly), that I can't imagine how anyone > could follow a name above the 11-limit unless it's > written down. Imagine trying to mention to someone > in spoken conversation that you're trying to decide > whether to use the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1> > or <6, 0, 0, 0, 0, 0, 0, 1>" in a composition -- are > you expecting me to be mentally prepared to count all > those zeros so I know what prime number you mean when > you finally get to the "1" that matters?
this is an interesting and good point. in fact, over the years of working in extended-JI, i've pretty much come to the conclusion that there are so many notes available in even a rather compact 11-limit euler-genus, that via xenharmonic bridging, that system can imply many higher-prime ratios. in particular, i've noticed that lots of 11-limit ratios sound very similar to nearby 13-limit ratios. if one accepts this aspect of my theory, then monzos can easily be used to name all the relevant 11-limit kommas. on the other hand, the main reason i came up with the idea of using monzos to represent prime-factored ratios was that i wanted to avoid both having to always specify the primes, and also to avoid superscripts. at the time i originally thought of the monzo idea, i was working in 19-limit, and it seemed a easier to me to specify, to pick a totally random example, 133:72 as [-3, -2, 0, 1, 0, 0, 0, 1] (with the prime-factors 2, 3, 5, 7, 11, 13, 17, 19 understood) than to write is out as 2^-3 * 3^-2 * 7^1 * 19^1.
> The problem is, the "name" (if you can call it that) > emphasizes *powers* rather than *primes*, so it tends > to get rather cryptic.
when one works with the same set of prime-factors over and over again, one very easily gets used to remembering the primes which underly the monzo. and as i've pointed out before, the monzo allows direct visualization of the lattice, which in turn helps in comprehension of the structure of the entire tuning system. i came up with the monzo idea based on analogy with our regular numbering system. it doesn't take too long for one to learn, whether in school or in everyday life, that, for example, the number 133 is a monzo-like representation of (10^2)*1 + (10^1)*3 + (10^0)*3. the regular arabic numeral is a nice compact way of expressing what would look like a rather complicated mathematical expression if it were written out in full. but once one learns how it works, the long version is never need anymore. anyway, i'll get off my soapbox now. i've already agreed that for purposes of naming beyond 11-limit, the prime system is better than the monzo system. but i will maintain that for 3-, 5-, 7-, and 11-limit, the monzo system works just fine. -monz
Message: 7981 Date: Wed, 05 Nov 2003 18:34:19 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > ... even tho *i* will always think of any rational > interval as its monzo. > > (i guess that's self-evident, given the name of the term?) > > ;-)
Oh, dear! Now I fear that if someone brings up the word "secor" in a tuning context, they'll too readily associate that with the term "irrational". ;-) --George
Message: 7982 Date: Wed, 05 Nov 2003 20:18:15 Subject: Re: Eponyms From: monz hi George, --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > > ... even tho *i* will always think of any rational > > interval as its monzo. > > > > (i guess that's self-evident, given the name of the term?) > > > > ;-)
> > Oh, dear! Now I fear that if someone brings up the > word "secor" in a tuning context, they'll too readily > associate that with the term "irrational". ;-) > > --George
oh, not necessarily! in a series of old posts, most informatively this one: Yahoo groups: /tuning/message/23195 * [with cont.] i presented a "rational canasta" tuning. the express purpose of this was to be able to map the canasta scale to the computer keyboard in the old (JustMusic) version of my software, which was not able to accept irrational pitches. in my "rational canasta" tuning, a secor is: <3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1> ratio = 5632:5265 = ~116.657 cents this is only ~0.0101 cent less than 2^(7/72), the "standard" secor. and of course many other rational secors could be found. i mentioned in one of those old posts the irony of having to find a rational tuning which approximated the subset of the irrational 72edo MIRACLE, which in turn provides manifold approximations of rational JI intervals ... and even mentioned how it conjured up Escher images in my mind. and in fact, with regard to my original comment you quoted, i also think of many irrational intervals in terms of their monzos ... even tho many folks here find that to be pointless since irrational intervals can be prime-factored in an infinite number of ways. but i find it useful, for example, to see the generator "5th" of 1/4-comma meantone as the [2,3,5]-monzo [0, 0, 1/4], or that of 2/7-comma meantone as [1/7, -1/7, 2/7]. i just wrote "rational" to avoid having to go into details like this ... but now you've gone and forced to do it anyway! :) -monz
Message: 7983 Date: Wed, 05 Nov 2003 20:52:43 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> As for the answer to your second question: I don't know if there
will
> be unique names for *all* of them (probably not), but I believe
that
> there are enough names to cover all of those that would be of use
to
> most theorists.
Could you list what you think of as the important commas?
Message: 7984 Date: Wed, 05 Nov 2003 21:01:14 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> in my "rational canasta" tuning, a secor is: > > <3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1> > ratio = 5632:5265 > = ~116.657 cents > > this is only ~0.0101 cent less than 2^(7/72), the > "standard" secor.
though i know you wanted to use the 72-equal secor, the "standard" secor is (as you correctly state on The Proxomitron Reveals... * [with cont.] (Wayb.) arts.org/dict/secor.htm) (18/5)^(1/19) = ~116.7156 cents.
Message: 7985 Date: Wed, 05 Nov 2003 21:22:20 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi George, > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...> wrote:
>
> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > > > > ... even tho *i* will always think of any rational > > > interval as its monzo. > > > > > > (i guess that's self-evident, given the name of the term?) > > > > > > ;-)
> > > > Oh, dear! Now I fear that if someone brings up the > > word "secor" in a tuning context, they'll too readily > > associate that with the term "irrational". ;-) > > > > --George
> > oh, not necessarily! > > in a series of old posts, most informatively this one: > > Yahoo groups: /tuning/message/23195 * [with cont.] > > i presented a "rational canasta" tuning. ... > > and of course many other rational secors could be found.
Probably distant relatives of mine. ;-)
> ... > i just wrote "rational" to avoid having to go into > details like this ... but now you've gone and forced > to do it anyway! :)
Hey, you're taking my reply much too seriously. --George
Message: 7986 Date: Wed, 05 Nov 2003 21:38:35 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...>
> wrote: >
> > As for the answer to your second question: I don't know if there
will
> > be unique names for *all* of them (probably not), but I believe
that
> > there are enough names to cover all of those that would be of use
to
> > most theorists.
> > Could you list what you think of as the important commas?
I don't have the resources to do that very readily. Wouldn't putting them in order of product complexity (discarding any with factors above some particular prime limit) accomplish this? One would only need to determine at what point two commas were forced to share the same name and then decide if they were both "important". Dave already discussed this: Yahoo groups: /tuning-math/messages/7320 * [with cont.] --George
Message: 7987 Date: Wed, 05 Nov 2003 21:46:05 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi George, > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...> wrote:
>
> > The monzo comma-naming system is so cumbersome > > (i.e., unfriendly), that I can't imagine how anyone > > could follow a name above the 11-limit unless it's > > written down. Imagine trying to mention to someone > > in spoken conversation that you're trying to decide > > whether to use the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1> > > or <6, 0, 0, 0, 0, 0, 0, 1>" in a composition -- are > > you expecting me to be mentally prepared to count all > > those zeros so I know what prime number you mean when > > you finally get to the "1" that matters?
> > this is an interesting and good point. > > in fact, over the years of working in extended-JI, > i've pretty much come to the conclusion that there are > so many notes available in even a rather compact 11-limit > euler-genus, that via xenharmonic bridging, that system > can imply many higher-prime ratios. > > in particular, i've noticed that lots of 11-limit ratios > sound very similar to nearby 13-limit ratios.
No question about it! Many years ago I couldn't help noticing that the ratio 351:352 vanishes in most of the good edos below 100. More recently I also found that complex 7-limit ratios tend to *very closely* approximate ratios of 13. The 13-schismina, 4095:4096 or <- 2, -1, -1, 0, -1>, vanishes in most of the best edos between 200 and 800. This was the first step in establishing an economy of symbol- elements (flags) in Sagittal notation development, since it allows one to notate a 13M-diesis (1024:1053) as the (approximate) sum (product) of a 5-comma (80:81) and 7-comma (63:64) in both medium and high-precision JI. (Only extreme-precision Sagittal JI has separate symbols for 13-limit consonances.)
> if one accepts this aspect of my theory, then monzos > can easily be used to name all the relevant 11-limit > kommas.
Loocs like you kaught something from Dave Ceenan. ;-)
> on the other hand, the main reason i came up with the > idea of using monzos to represent prime-factored ratios > was that i wanted to avoid both having to always specify > the primes, and also to avoid superscripts. > > at the time i originally thought of the monzo idea, > i was working in 19-limit, and it seemed a easier to me > to specify, to pick a totally random example, 133:72 as > [-3, -2, 0, 1, 0, 0, 0, 1] (with the prime-factors > 2, 3, 5, 7, 11, 13, 17, 19 understood) than to write > is out as 2^-3 * 3^-2 * 7^1 * 19^1.
Sure.
> > The problem is, the "name" (if you can call it that) > > emphasizes *powers* rather than *primes*, so it tends > > to get rather cryptic.
> > when one works with the same set of prime-factors > over and over again, one very easily gets used to > remembering the primes which underly the monzo. > > and as i've pointed out before, the monzo allows direct > visualization of the lattice, which in turn helps in > comprehension of the structure of the entire tuning system.
It looks as if Dave's and your method of designating ratios are complementary. Dave's (which is at the moment restricted to ratios no larger than ~69 cents) helps more in comprehending the size of the comma and especially in identifying the Sagittal symbol used to notate ratios having the same combination of primes >3. BTW, have you ever tried collapsing an 11-limit lattice into 2 dimensions by mapping 11/8 to <10, 5>?
> i came up with the monzo idea based on analogy with > our regular numbering system. > > it doesn't take too long for one to learn, whether in school > or in everyday life, that, for example, the number 133 is a > monzo-like representation of (10^2)*1 + (10^1)*3 + (10^0)*3. > > the regular arabic numeral is a nice compact way of expressing > what would look like a rather complicated mathematical expression > if it were written out in full. but once one learns how > it works, the long version is never need anymore.
Yes, I can readily appreciate this sort of shorthand. However, its weakness as a naming system lies in the fact that you need a good way to verbalize what you're seeing. Decimal numbers work as names because, in saying a number such as 2,010,030 (as "two million ten thousand thirty"), we include words to indicate the placement of the numerals. But this is not as easy for monzos, since primes are not related to one another as are powers of ten, since exponents may be negative numbers, and since numbers may not be rounded off by dropping the lower prime exponents (as with 2.01 million). Something that you might want to consider is replacement of the comma following the powers of 3, 11, and 19 (and every 3rd prime thereafter) by a semicolon (to serve as a place marker, similar in function to a decimal point and commas in decimal numbers), so that 133:72 could be written as either [-3, -2; 0, 1, 0; 0, 0, 1] or [-2; 0, 1, 0; 0, 0, 1]. --George
Message: 7988 Date: Wed, 05 Nov 2003 22:24:37 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> BTW, have you ever tried collapsing an 11-limit lattice into 2 > dimensions by mapping 11/8 to <10, 5>?
you're probably referring to the 3-5-11 lattice? the full 11-limit lattice is at least 3 dimensional if you use one "xenharmonic bridge" as above. for the 3-5-11 case, this choice (184528125:184549376) is probably a very good one. for the full 11- limit case, 9800:9801 is probably better for most purposes, since it's both a little smaller (in cents) and much simpler (i.e., shorter in the lattice). sorry if this duplicates a previous message; that one didn't seem to show up . . .
Message: 7989 Date: Wed, 05 Nov 2003 22:02:04 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> BTW, have you ever tried collapsing an 11-limit lattice into 2 > dimensions by mapping 11/8 to <10, 5>?
a single "xenharmonic bridge" like this would only collapse the 4- dimensional 11-limit lattice into 3 dimensions, or a 5-dimensional version (with factors of 2 shown) into 4 dimensions, wouldn't it? the unison vector in this case would be 184528125:184549376, which is a good one, but 9800:9801 is both smaller and simpler . . . maybe you're specifically talking about the lattice with no 7 axis?
Message: 7990 Date: Wed, 05 Nov 2003 22:37:28 Subject: Re: Eponyms From: monz hi paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >
> > in my "rational canasta" tuning, a secor is: > > > > <3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1> > > ratio = 5632:5265 > > = ~116.657 cents > > > > this is only ~0.0101 cent less than 2^(7/72), the > > "standard" secor.
> > though i know you wanted to use the 72-equal secor, > the "standard" secor is (as you correctly state on > Definitions of tuning terms: secor, (c) 2001 b... * [with cont.] (Wayb.)) > (18/5)^(1/19) = ~116.7156 cents.
OK, thanks for pointing that out. so my "rational secor" is only a little more than half a cent smaller than that. -monz
Message: 7991 Date: Wed, 05 Nov 2003 22:52:35 Subject: Re: Eponyms From: monz hi George, --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> Loocs like you kaught something from Dave Ceenan. ;-)
i usually try to follow suggestions for standardization ... unless i very strongly disagree, as i did with Sims 72edo notation.
> BTW, have you ever tried collapsing an 11-limit lattice into 2 > dimensions by mapping 11/8 to <10, 5>?
no, i never did that. but i did do this: Yahoo groups: /tuning/message/1372 * [with cont.] Yahoo groups: /tuning/message/1380 * [with cont.] (if you're viewing on the stupid Yahoo web interface, you'll have to forward them to your email account to view the lattices properly.)
> Yes, I can readily appreciate this sort of shorthand. > However, its weakness as a naming system lies in the > fact that you need a good way to verbalize what you're > seeing.
which is the main reason why i agree with you about names in general. but i do think that for 11-limit, using only 4 exponents, it's not so bad.
> Something that you might want to consider is replacement > of the comma following the powers of 3, 11, and 19 (and > every 3rd prime thereafter) by a semicolon (to serve as > a place marker, similar in function to a decimal point > and commas in decimal numbers), so that 133:72 could be > written as either [-3, -2; 0, 1, 0; 0, 0, 1] or > [-2; 0, 1, 0; 0, 0, 1].
well, since that last monzo doesn't use 2, it should be written (following the convention proposed by Gene and accepted by me) with angle brackets instead of square: <-2; 0, 1, 0; 0, 0, 1>. anyway, yes, that's an excellent idea ... except that i was never crazy about adding the commas in the first place. i still think i prefer the nice clean look of a single space and nothing else, separating the numbers in the monzo. -monz
Message: 7992 Date: Wed, 05 Nov 2003 22:39:33 Subject: Re: Eponyms From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> > i just wrote "rational" to avoid having to go into > > details like this ... but now you've gone and forced > > to do it anyway! :)
> > Hey, you're taking my reply much too seriously.
i know ... but still, i thought of it before i wrote my original post, so i figured that since you brought it up (albeit as a joke) i might as well give a nice fat response. :) ... isn't it so much nicer when communication here is this pleasant? -monz
Message: 7993 Date: Wed, 05 Nov 2003 02:36:29 Subject: Some 13-limit TM et bases From: Gene Ward Smith 9: [21/20, 26/25, 27/25, 33/32, 45/44] 10: [25/24, 28/27, 35/33, 40/39, 49/48] 26: [45/44, 50/49, 65/64, 78/77, 81/80] 29: [49/48, 55/54, 65/64, 91/90, 100/99] 46: [91/90, 121/120, 126/125, 169/168, 176/175] 58: [126/125, 144/143, 176/175, 196/195, 364/363] 111: [176/175, 351/350, 540/539, 676/675, 1331/1323] 130: [243/242, 351/350, 364/363, 441/440, 3136/3125]
Message: 7994 Date: Thu, 06 Nov 2003 15:48:41 Subject: Re: Eponyms From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi George, > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...> wrote:
>
> > Loocs like you kaught something from Dave Ceenan. ;-)
> > i usually try to follow suggestions for standardization > ... unless i very strongly disagree, as i did with Sims > 72edo notation.
I agree with Gene that using the conventional spelling for the new (defined-range) meaning of comma and the unconventional spelling for the commonly accepted (i.e., generic) meaning for a comma would not be a good idea, so I advise that the new spelling be dropped. (It's not required for a comma-naming system to work, anyway.)
> > ... > > Something that you might want to consider is replacement > > of the comma following the powers of 3, 11, and 19 (and > > every 3rd prime thereafter) by a semicolon (to serve as > > a place marker, similar in function to a decimal point > > and commas in decimal numbers), so that 133:72 could be > > written as either [-3, -2; 0, 1, 0; 0, 0, 1] or > > [-2; 0, 1, 0; 0, 0, 1].
> > well, since that last monzo doesn't use 2, it should be > written (following the convention proposed by Gene and > accepted by me) with angle brackets instead of square: > <-2; 0, 1, 0; 0, 0, 1>.
I used the square brackets to make the point that you would no longer need the angle brackets in order to show that the exponent of 2 is omitted. But perhaps the semicolon looks too much like a comma, so that the angle brackets would be more distinctive in indicating this (however, see the following).
> anyway, yes, that's an excellent idea ... except that > i was never crazy about adding the commas in the first place. > i still think i prefer the nice clean look of a single space > and nothing else, separating the numbers in the monzo.
It's very nice that you brought this up, because that's the next thing I was going to suggest. Why not modify my suggestion above by dropping the commas entirely, then changing the semicolons that remain back to commas, so that the above example (133:72) would be done this way: [-3 -2, 0 1 0, 0 0 1] or [-2, 0 1 0, 0 0 1]. This makes the grouping by threes more obvious (and the higher primes much easier to locate), and angle brackets would no longer be necessary. --George
Message: 7995 Date: Thu, 06 Nov 2003 16:21:40 Subject: Re: Eponyms From: monz hi George, --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> I agree with Gene that using the conventional spelling for > the new (defined-range) meaning of comma and the unconventional > spelling for the commonly accepted (i.e., generic) meaning for > a comma would not be a good idea, so I advise that the new > spelling be dropped. (It's not required for a comma-naming > system to work, anyway.)
OK.
> > <snip> > > [regarding the format of monzos:] > > ... Why not modify my suggestion above by dropping the > commas entirely, then changing the semicolons that > remain back to commas, so that the above example (133:72) > would be done this way: > [-3 -2, 0 1 0, 0 0 1] or > [-2, 0 1 0, 0 0 1]. > This makes the grouping by threes more obvious (and the > higher primes much easier to locate), and angle brackets > would no longer be necessary.
i like that a lot! in fact, i find it very interesting that group the primes by threes like this also keeps them in bunches that make sense to me in terms of how i've used them and theorized about them myself! i.e., 3 is obviously extremely important both historically and theoretically, and thus deserves to be isolated by itself (or grouped with 2, if 2 is included). the next comma appears after 11, and earlier in this thread i discussed the idea that 11-limit can be a kind of boundary. Partch thought so too. (but please, don't anyone make too much of this comment.) the next comma appears after 19, which i myself used as a limit from about 1988-98. the next comma appears after 31, which is the highest limit Ben Johnston has used in his music. interesting. -monz
Message: 7996 Date: Thu, 06 Nov 2003 17:19:24 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> i.e., 3 is obviously extremely important both historically > and theoretically, and thus deserves to be isolated by itself > (or grouped with 2, if 2 is included). > > the next comma appears after 11, and earlier in this thread > i discussed the idea that 11-limit can be a kind of boundary. > Partch thought so too. (but please, don't anyone make too > much of this comment.) > > the next comma appears after 19, which i myself used as > a limit from about 1988-98. > > the next comma appears after 31, which is the highest limit > Ben Johnston has used in his music. > > interesting.
so the primes are arranged as 2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 , 61 67 71 looks like the next comma after 31 makes sense too -- isn't 43 the highest limit used by george secor at least in some context?
Message: 7997 Date: Thu, 06 Nov 2003 17:25:44 Subject: Re: Eponyms From: Paul Erlich no reply, george? --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...>
> wrote: >
> > BTW, have you ever tried collapsing an 11-limit lattice into 2 > > dimensions by mapping 11/8 to <10, 5>?
> > you're probably referring to the 3-5-11 lattice? > > the full 11-limit lattice is at least 3 dimensional if you use > one "xenharmonic bridge" as above. for the 3-5-11 case, this choice > (184528125:184549376) is probably a very good one. for the full 11- > limit case, 9800:9801 is probably better for most purposes, since > it's both a little smaller (in cents) and much simpler (i.e.,
shorter
> in the lattice). > > sorry if this duplicates a previous message; that one didn't seem
to
> show up . . .
Message: 7998 Date: Thu, 06 Nov 2003 19:25:22 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:
> I don't have the resources to do that very readily. Wouldn't
putting
> them in order of product complexity (discarding any with factors > above some particular prime limit) accomplish this?
Not really; however we can simply take everything below a certain prime limit and below a limit for what I call "epipermicity", which is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown this gives a finite list of commas if the epimermicity limit is less than one. One would only
> need to determine at what point two commas were forced to share the > same name and then decide if they were both "important".
NEVER! Two commas with the same name makes no sense at all.
Message: 7999 Date: Thu, 06 Nov 2003 19:54:55 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor"
<gdsecor@y...>
> wrote: >
> > I don't have the resources to do that very readily. Wouldn't
> putting
> > them in order of product complexity (discarding any with factors > > above some particular prime limit) accomplish this?
> > Not really; however we can simply take everything below a certain > prime limit and below a limit for what I call "epipermicity",
i think you mean "epimericity"
> which > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
this
> gives a finite list of commas if the epimermicity limit is less
than
> one.
i'll try 7-limit some other time, but since i still have my 5-limit
list in matlab's memory, here's the top rankings (for intervals < 600
cents) by epimericity -- 1/1 shows up as best but actually its
epimericity is 0/0 so is undefined:
numerator denominator
1 1
16 15
6 5
81 80
4 3
9 8
10 9
5 4
25 24
27 25
128 125
32805 32768
250 243
135 128
2048 2025
15625 15552
256 243
648 625
32 27
3125 3072
75 64
78732 78125
6561 6400
20000 19683
125 108
27 20
32 25
25 18
625 576
1600000 1594323
144 125
393216 390625
256 225
16875 16384
2187 2048
81 64
2109375 2097152
800 729
6561 6250
1125 1024
3125 2916
100 81
531441 524288
45 32
20480 19683
2187 2000
729 640
2048 1875
243 200
16384 15625
125 96
729 625
1076168025 1073741824
3456 3125
6115295232 6103515625
1224440064 1220703125
1594323 1562500
1990656 1953125
274877906944 274658203125
262144 253125
625 512
10485760000 10460353203
1215 1024
62500 59049
7629394531250 7625597484987
78125 73728
273375 262144
4096 3645
67108864 66430125
129140163 128000000
2500 2187
162 125
1638400 1594323
531441 512000
4194304 4100625
82944 78125
32768 30375
390625000 387420489
244140625 241864704
390625 373248
9765625 9565938
1953125 1889568
4294967296 4271484375
18225 16384
625 486
34171875 33554432
2560 2187
768 625
140625 131072
31381059609 31250000000
9375 8192
etc.
considering that i had numerators and denominators well in excess of
10^50 in the list, i'm inclined to believe gene that a given
epimericity cutoff will yield a finite list. and it's a good list
too -- i'm kind of pleased with this as a temperament ranking, with
meantone very near the top, augmented, schismic, pelogic,
diaschismic, blackwood, kleismic, and diminished forming a
consecutive block of interesting and eminently useful systems (given
their characteristic DE scales), while more unlikely choices for
human music making, like semisuper, parakleismic, and ennealimmal, as
well as many simpler systems with high error, fall further down --
and of course monstrosities like atomic don't appear at all. i wonder
if even dave could stomach such a ranking -- the very simple
temperaments with high error are easy enough to mentally toss out for
the user seeking a certain goodness of approximation . . .
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