Tuning-Math Archive Section 6: 5250

Message: 5250 - Contents - Hide Contents Date: Wed, 25 Sep 2002 09:25:25 Subject: Re: a reference pitch From: monz hi George,

> From: "gdsecor" <gdsecor@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, September 25, 2002 7:18 AM > Subject: [tuning-math] Re: a reference pitch (was: A common notation for

JI and ETs)

> > > --- In tuning-math@y..., "monz" <monz@a...> wrote: >>

>> umm ... well ... it only means that "middle-C" is 256 Hz. >> this would make the 12edo "A" = ~430.5 Hz. >> >> it was just my thinking that since this is not too far >> off from most of the pitch-standards already in use today, >> it makes more sense as a basis from a logical point of view. >> we commonly use "C" as the reference anyway instead of "A", >> so why not simply equate it with 1 Hz? >> >> -monz >> "all roads lead to n^0" >

> The two main obstacles are 1) getting wind instruments to > play in tune

but that's the case regardless of what intonational or notational paradigm is in use.

> and 2) getting others to accept this. I have a feeling > that the second one is the more formidable obstacle. > > --George

yeah, well ... unfortunately, that's my feeling too. but it's encouraging to me that (as i've already said) several people have written to me or the tuning list saying that they were adopting my C = 1 Hz standard. my hope is that the logic of my choice will ultimately win out. if it doesn't happen, then i guess it doesn't really matter all that much after all. -monz "all roads lead to n^0"

Message: 5251 - Contents - Hide Contents Date: Fri, 27 Sep 2002 01:20:36 Subject: interesting property concerning meantone intervals From: monz i've discovered an interesting property about 1/4-comma meantone which i haven't seen mentioned before. this concerns specifically a 12-tone chain of 1/4-comma meantone, which may be described as the "8ve"-invariant set of pitches determined by generators 5^(_p_/4), where _p_ = -3...+8. i've found that the entire set of intervals that can be found in this scale may be described as the "8ve"-invariant set of intervals determined by generators 5^(_i_/4), where _i_ = -11...+11. i was wondering if this could be generalized to the set of pitches 5^(_p_/4), where _p_ = a...b, and the set of intervals 5^(_i_/4), where i = (a-b)...(b-a). and how about generalization to other forms of meantone? to other types of scales in general? my guess is that it has been written about before, but my math-challenged brain missed it. ... ? -monz "all roads lead to n^0"

Message: 5252 - Contents - Hide Contents Date: Fri, 27 Sep 2002 02:42:47 Subject: mathematical model of torsion-block symmetry? From: monz is there some way to mathematically model the symmetry in a torsion-block? see the graphic and its related text in my Tuning Dictionary definition of "torsion" -- i've uploaded it to here: Yahoo groups: /monz/files/dict/torsion.htm * [with cont.] -monz "all roads lead to n^0"

Message: 5253 - Contents - Hide Contents Date: Fri, 27 Sep 2002 16:12:03 Subject: Re: interesting property concerning meantone intervals From: manuel.op.de.coul@xxxxxxxxxxx.xxx Ok, I'm not going to say that you shouldn't be surprised, otherwise Johnny Reinhard will start laughing. It is true for all meantone or Pythagorean generated scales. Suppose _g_ is the size of the generator, and _a_ the size of the octave. Then you can express each pitch as x g + y a, where x in your case is in -3..8, and y such that the pitch is in the range of one octave. The intervals are two pitches subtracted, and the result has the same form, say x'g + y'a. So if the range of x is -3..8 then the range of the differences of two x's is -3 - 8 .. 8 - -3 = -11 .. 11. Manuel

Message: 5254 - Contents - Hide Contents Date: Mon, 30 Sep 2002 21:46:09 Subject: Re: interesting property concerning meantone intervals From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:

> i've discovered an interesting property about > 1/4-comma meantone which i haven't seen mentioned > before. > > this concerns specifically a 12-tone chain of > 1/4-comma meantone, which may be described as > the "8ve"-invariant set of pitches determined by > generators 5^(_p_/4), where _p_ = -3...+8. > > i've found that the entire set of intervals that > can be found in this scale may be described as > the "8ve"-invariant set of intervals determined > by generators 5^(_i_/4), where _i_ = -11...+11.

i'm sorry -- isn't this completely obvious?

> i was wondering if this could be generalized to > the set of pitches 5^(_p_/4), where _p_ = a...b, > and the set of intervals 5^(_i_/4), where > i = (a-b)...(b-a). > > and how about generalization to other forms of > meantone? to other types of scales in general? > my guess is that it has been written about before, > but my math-challenged brain missed it. > > ... ?

if a scale has a generator g, and its pitches are described by g^p, where p=a...b, then its interval classes will of course be g^i, where i = 0...(b-a), and if you don't use inversional equivalence, you must include the inversions of all of those intervals, hence g^k where k = (a-b)...(b-a). it follows immediately from the definition of interval.

Message: 5255 - Contents - Hide Contents Date: Mon, 30 Sep 2002 21:47:22 Subject: Re: mathematical model of torsion-block symmetry? From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:

> is there some way to mathematically model > the symmetry in a torsion-block? > > see the graphic and its related text in my > Tuning Dictionary definition of "torsion" > -- i've uploaded it to here: > Yahoo groups: /monz/files/dict/torsion.htm * [with cont.] > > > > -monz > "all roads lead to n^0"

i see the green and red lines, but . . . which symmetry exactly are you referring to?

Message: 5258 - Contents - Hide Contents Date: Tue, 01 Oct 2002 10:18:42 Subject: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> Yes, I'm looking for equal-fretted music that's at least > intonationally close to what could be produced by the > instruments I'm working with now. > I'm less interested in "experimental" music than in > examples of some established cultural utility.

why would you think that any established culture uses equally-spaced frets? equal temperaments, yes . . . equally-spaced frets, i've heard of plenty of "experimental" examples but that's it. good luck!

Message: 5259 - Contents - Hide Contents Date: Tue, 1 Oct 2002 10:32:03 Subject: Re: mathematical model of torsion-block symmetry? From: monz hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, September 30, 2002 2:47 PM > Subject: [tuning-math] Re: mathematical model of torsion-block symmetry? > > > --- In tuning-math@y..., "monz" <monz@a...> wrote:

>> is there some way to mathematically model >> the symmetry in a torsion-block? >> >> see the graphic and its related text in my >> Tuning Dictionary definition of "torsion" >> -- i've uploaded it to here: >> Yahoo groups: /monz/files/dict/torsion.htm * [with cont.] >> >> >> >> -monz >> "all roads lead to n^0" >

> i see the green and red lines, but . . . which symmetry exactly are > you referring to?

do you see "The thin black line which divides the block in half diagonally is the torsional interval, 6561:6400 = [-8 2] = (81/80)^2 = (648/625) (2048/2025)^(-1).]" on the diagram? -monz

Message: 5261 - Contents - Hide Contents Date: Tue, 01 Oct 2002 00:25:29 Subject: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> I finally started building my mini-ensemble of > idiot-proof guitars. > > I'm rather satisfied with the results of transforming > a $9 flea market mini guitar to 3 sets of double strings > ("root/fifth/octave") and a fretboard with an octave > divided into six parts each at 1.25 inches. > > My girlfriend, who is trained as a vocalist, > finds the combination of timbre and interval > structure erotically stimulating. > > That's not a problem. > > My issue is with the first fret. > > Am I not liking those tones as much as the others > because it's basically an 11:12 kinda thing for which > my brain isn't wired, or is it just that imperfect > intonation is harder to resolve neurologically than > the other intervals? > I appreciate that, all else being equal, variations > in fret placement will have greater proportional > implications for this interval than for the others. > > I find that melodies involving the first fret > go smoother if I don't approach or leave these > tones via fret zero (open string). > Is it a common strategy in the intonation system > I'm using to use fret zero and fret 1 as > mutually exlusive, obe "substituting" for the > other or what not? > > Some reference literature might help, if anyone > can suggest it. > > I'm thinking "Morrocco". Am I on the right track?

yup. north african and arabic music is rich in these three-quarters- of-a-tone melodic intervals. if you can get a hold of as much of this music as you can to listen to for a while, i'd recommend it -- though you probably won't be able to emulate all the other notes of these "ethnic" scales on your guitar. not sure what kind of literature you're looking for . . . on equally- spaced-fret instruments?

Message: 5262 - Contents - Hide Contents Date: Tue, 01 Oct 2002 22:46:06 Subject: Re: mathematical model of torsion-block symmetry? From: wallyesterpaulrus --- In tuning-math@y..., "monz" <monz@a...> wrote:

> hi paul, > >

>> From: "wallyesterpaulrus" <wallyesterpaulrus@y...> >> To: <tuning-math@y...> >> Sent: Monday, September 30, 2002 2:47 PM >> Subject: [tuning-math] Re: mathematical model of torsion-block symmetry? >> >> >> --- In tuning-math@y..., "monz" <monz@a...> wrote:

>>> is there some way to mathematically model >>> the symmetry in a torsion-block? >>> >>> see the graphic and its related text in my >>> Tuning Dictionary definition of "torsion" >>> -- i've uploaded it to here: >>> Yahoo groups: /monz/files/dict/torsion.htm * [with cont.] >>> >>> >>> >>> -monz >>> "all roads lead to n^0" >>

>> i see the green and red lines, but . . . which symmetry exactly are >> you referring to? > >

> do you see "The thin black line which divides the block in half > diagonally is the torsional interval, 6561:6400 = [-8 2] = (81/80)^2 > = (648/625) (2048/2025)^(-1).]" on the diagram? > > > > -monz

sure -- but i don't see any symmetries unique to torsional blocks associated with reflection about that line or about the center point - - just the usual symmetry you get when you center a symmetrical shape about a point of symmetry in the lattice. however, it seems to be that if you cut the block exactly in half along the direction of one of its edges, you'd end up with two instances of a geniune (non-torsional) periodicity block. every note on the "left" half of the block is connected, with either a green or a red line, to one and only one note on the "right" half of the block, and also every note on the "top" half of the block is connected, with either a green or a red line, to one and only one note on the "bottom" half of the block. this results from seeing the black line you referred to as two syntonic commas end-to-end; using only one of these "half-black-lines" to construct the periodicity block removes the torsion problem and leaves you with a block half as big as the original (you have to do a bit of the "wedge-shifting" as described in my _excursion_ to get these blocks to fit within the borders of the original torsional block). making sense? p.s. are you reading my posts on the tuning list, monzieur? i was going to post a (fairly serious) critique of your new 12-edo page, but i'm afraid no one, not even you, would read it, since you haven't really replied to, or incorporated into your webpages, my last two lengthy tuning lists posts to you. if i e-mailed this critique to you privately, would you have a better chance of reading it?

Message: 5263 - Contents - Hide Contents Date: Tue, 01 Oct 2002 22:49:17 Subject: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> I'm just not as enthusiastic as was Harry Partch > about the number 11, and I'm not sure what > (if anything) I should do about it.

have you listened to many chords like 4:5:6:7:9:11 for example? this is where 11 really shines. melodically, i don't think ratios have much to do with melodic suitability, at least not any ratios more complex than 4:3 -- different cultures will use different melodic intervals, and melodically speaking, anything unfamiliar will sound "foreign" or "out-of-tune" whether it's more "rational", less "rational", or whatever.

Message: 5266 - Contents - Hide Contents Date: Tue, 01 Oct 2002 23:34:34 Subject: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> I'm finding that my ear tolerates the better melodic > interactions between the 1st fret and other frets > than between the first fret and an open string > or a 2:1 fret or harmonic. > > I guess I'm just slow in learning to accomodate > the weakest part of the scale, eh?

or maybe the scale just isn't so well balanced, particularly if you're favoring the open string. but yeah, i guess 11:12 would be one of the most harmonically dissonant intervals you can possibly produce on that beast.

Message: 5267 - Contents - Hide Contents Date: Tue, 01 Oct 2002 23:36:35 Subject: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> There's not a lot of really good "early music" > intonation out there,

are you making a sweeping statement or is this supposed to be limited to your record collection or whatever?

> As someone who knows tuning much better than I do, > are you seeing any kind of a pattern?

hmm . . . not at the moment . . .

Message: 5268 - Contents - Hide Contents Date: Tue, 1 Oct 2002 23:23:19 Subject: delays in responding to paul (was: mathematical model...) From: monz

> From: "wallyesterpaulrus" <wallyesterpaulrus@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, October 01, 2002 3:46 PM > Subject: [tuning-math] Re: mathematical model of torsion-block symmetry? > > > p.s. are you reading my posts on the tuning list, monzieur? i was > going to post a (fairly serious) critique of your new 12-edo page, > but i'm afraid no one, not even you, would read it, since you haven't > really replied to, or incorporated into your webpages, my last two > lengthy tuning lists posts to you. if i e-mailed this critique to you > privately, would you have a better chance of reading it?

i've been reading everything, but have been insanely busy lately. since the Sonic Arts website went under last Thursday, i spent the entire weekend putting up a mirror of my website in that new Yahoo group i created. i was a very tedious process, because Yahoo groups only allow uploading of files one at a time. then when Monday rolled around, it was back to a busy work week. please send your critique of the 12edo page, either to me or to the list. i'm always interested in your commentary on my work, and generally include it in the webpages. web stuff is simply going at a slower pace now because i'm very busy with work. -monz

Message: 5270 - Contents - Hide Contents Date: Wed, 02 Oct 2002 19:56:51 Subject: a long-overdue correction From: wallyesterpaulrus On Sat Dec 22, 2001, "genewardsmith" <genewardsmith@j...> wrote:

> 1600000/1594323 = 2^9 3^-13 5^-2 Acute Minor Third system

that should be 2^9 3^-13 5^5, right?

Message: 5271 - Contents - Hide Contents Date: Wed, 02 Oct 2002 19:58:47 Subject: Fwd: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> OK, that's fair. > > I was speaking mostly of bad recordings from the 70's

yeah, authentic intonation has improved since then.

> and university ensembles where people gravitate > toward 12tet even when they're singing "medieval" music.

actually, medieval music isn't bad in 12-equal -- it's renaissance music especially that's way off in 12-equal.

Message: 5273 - Contents - Hide Contents Date: Wed, 02 Oct 2002 20:43:07 Subject: Fwd: Re: lame-ass question From: wallyesterpaulrus --- In tuning-math@y..., <Josh@o...> wrote:

> Today I'm picking over free scrap metal to build > some metallophones. Naturally, I'm interested in > minimizing inharmonicity in lower partials > (or at least in avoiding partials that badly > approach those present on other instruments). > If you have any hints on shape or dimensions, > I'd be most appreciative.

i think there are some knowledgable people in this area on the tuning list.

Message: 5274 - Contents - Hide Contents Date: Thu, 3 Oct 2002 04:30:18 Subject: Re: Combinatorics and Tuning Systems? From: monz back around September 10,

> --- In tuning-math@y..., <Josh@o...> wrote: > > Somehow, even the great serialists failed to much > exploit combinatoriality between sets of 5 and 7. > the 5-12/7-12 aggregate is particularly interesting > in that 7-12 does not actually include any forms of 5-12. > It's such an obvious candidate for serialist treatment... > ...ok, I'll drop that.

i wasn't following this thread, and only remembered seeing the word "combinatorics" in the subject line. but i just stumbled across this: "Some Combinational Resources of Equal-Tempered Systems" by Carlton Gamer _Journal of Music Theory_ 11:1, Spring 1967 in which the opening paragraph gives the following abstract:

>> "The purpose of this article is to reveal and discuss >> certain resources available to the composer who wishes to >> employ equal-tempered systems containing either more or >> less than twelve tones per octave, with particular emphasis >> upon the former, the so-called "microtonal" systems."

i'm going to completely skirt the issue of Gamer's specific definition of "microtonal": see the Tuning Dictionary and the list archives for those arguments. anyway, i wasn't following the thread, but those who were would find this paper relevant. -monz "all roads lead to n^0"